High-quality constraints on the glacial isostatic ... · ICE-7G_NA (VM7) model Keven Roy Doctor of Philosophy Graduate Department of Physics University of Toronto 2017 The Glacial

High-quality constraints on the glacial isostatic adjustment processover North America: The ICE-7G_NA (VM7) model

by

Keven Roy

A thesis submitted in conformity with the requirementsfor the degree of Doctor of Philosophy

Graduate Department of PhysicsUniversity of Toronto

© Copyright 2017 by Keven Roy

Abstract

High-quality constraints on the glacial isostatic adjustment process over North America: The

ICE-7G_NA (VM7) model

Keven Roy

Doctor of Philosophy

Graduate Department of Physics

University of Toronto

2017

The Glacial Isostatic Adjustment (GIA) process describes the response of the Earth’s surface to variations

in land ice cover. Models of the phenomenon, which is dominated by the influence of the Late Pleistocene

cycle of glaciation and deglaciation, depend on two fundamental inputs: a history of ice-sheet loading

and a model of the radial variation of mantle viscosity. Various geophysical observables enable us to test

and refine these models.

In this work, the impact of the GIA process on the rotational state of the planet will be analyzed,

and new estimates of the long-term secular trend associated with the GIA process will be provided. It

will be demonstrated that it has undertaken a significant change since the mid-1990s. Other important

observables include the vast amount of geological inferences of past sea level change that exist for all the

main coasts of the world. The U.S. Atlantic coast is a region of particular interest in this regard, due to

the fact that data from the length of this coast provides a transect of the forebulge associated with the

former Laurentide ice sheet. High-quality relative sea level histories from this region will be employed to

generate a new model of the GIA process that includes for the first time data from the forebulge region

in its optimization process (the ICE-6G_C (VM6) model). Then, the series of analyses is extended to

include space-geodetic observations of present-day vertical uplift of the crust. A solution reconciling

all available data from the continent, named ICE-7G_NA (VM7), is obtained through modest further

modifications of both the viscosity structure of the model and the North American component of the

surface mass loading history. It provides an excellent fit to the constraining data related to the GIA

process, including observations of the time-dependent de-levelling of the Great Lakes region. Finally,

to test the global exportability of the new model, its predictions of relative sea level change are tested

against observations from the Western Mediterranean region.

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Pour mes parents, Chantal et Martin

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Acknowledgements

First of all, I would like to thank my supervisor, Prof. W. R. Peltier, for his support and thoughtful

mentoring throughout my project. Your guidance, advice and insight have helped me become a much

better scientist and communicator. I am also grateful for the privilege you have given me to present my

work and interact with other scientists at many conferences and workshops around the world during my

time at the University of Toronto.

Special thanks go to Prof. Dylan Jones and Prof. Qinya Liu for their insightful comments and

support while being on my Ph.D. committee. Rosemarie Drummond and Guido Vettoretti also deserve

credit for the invaluable help they have provided throughout the project. I would also like to thank

Prof. Ben Horton and Dr. Donald Argus for sharing their expertise with me as I was learning about

relative sea level data and space-geodetic observations. My external examiner, Prof. Geoffrey Blewitt,

also deserves credit for his insightful comments on the final version of this thesis. I would also like to

thank the very hard-working people of the Physics Department, including Ana Sousa, Pierre Savaria,

Krystyna Biel and Teresa Baptista, who have helped me so many times. Special thanks also go to Dr.

Michel Bourqui. Your introductory course on atmospheric and oceanic physics at McGill University and

your availability to discuss research in Earth sciences have helped me discover a passion for geophysics.

I would also like to thank the Natural Sciences and Engineering Research Council of Canada (NSERC),

the Ontario Ministry of Training, Colleges and Universities (through the Ontario Graduate Scholarship

program) and the Fonds Nature et Technologies (FRQNT) for the funding I received in support of my

project. I would also like to thank the Centre for Global Change Science (CGCS) for the funding I

received to attend beneficial and exciting conferences in Greenland and Australia.

Moreover, I am indebted to the amazing people I have met at the University of Toronto, and in

particular the present and former members of the Atmospheric Physics and Geophysics groups. I have

been lucky to be surrounded by such inspiring people and make lifelong friends in the process. In

particular, special thanks go to Niall, Zen, Phil, Joseph, Ryan, Federico, Andrew, Alain, Ellen, Maria

and Oliver. Your support and friendship have helped me carry this project through. I would also like to

thank Simon: I was quite fortunate to have my best friend around for most of my time at the University

of Toronto. Going rock climbing and spending time with you were always fun, and your friendship means

a lot to me. The long discussions I have had with you and Jen have also helped me in the tougher times.

Cheers to both of you. Also, thanks to all my friends back in Québec and Montréal, who have helped

me a great deal in staying balanced and motivated.

Finally, some very important people deserve recognition for the infallible support they have provided

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me throughout my project. Betty, you deserve special credit for this project. Your love and constant

support have helped me so much with finishing this project. J’aimerais finalement remercier ma famille,

et particulièrement mes parents (Chantal et Martin), à qui cette thèse est dédiée, ainsi que mes soeurs

Mélissa et Karen. Votre support inconditionnel m’a permis de mener ce projet à terme. Je vous en suis

extrêmement reconnaissant.

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Contents

1 Introduction 1

1.1 The orbital theory of the timing of the ice age cycle . . . . . . . . . . . . . . . . . . . . . 2

1.2 Ice age cycles in deep-sea sedimentary cores and ice cores . . . . . . . . . . . . . . . . . . 7

1.3 Other geomorphological evidence and the Glacial Isotatic Adjustment (GIA) process . . . 11

1.4 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2 The Earth’s rotational state and its recent evolution 18

2.1 Fundamental dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.1.1 A useful simplification: the axisymmetric case . . . . . . . . . . . . . . . . . . . . . 21

2.1.2 A special case: the free Eulerian wobble . . . . . . . . . . . . . . . . . . . . . . . . 22

2.1.3 The influence of the GIA process on the moment of inertia of the planet . . . . . . 23

2.2 Observations of rotational variability and reference frame considerations . . . . . . . . . . 25

2.2.1 Celestial reference systems and frames . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.2.2 Terrestrial reference systems and Earth Orientation Parameters (EOPs) . . . . . . 27

2.2.3 Rotational variability measurement techniques and the latest ITRF iteration . . . 32

2.3 The evolution of secular trends in the planetary rotational state: Methodology and results 34

2.3.1 Sources of variability in length-of-day measurements . . . . . . . . . . . . . . . . . 35

2.3.2 Sources of variability in polar wander measurements . . . . . . . . . . . . . . . . . 37

2.3.3 Polar wander and LOD evolution: Data set and methodology employed . . . . . . 37

2.3.4 Secular trend pivot point determination: results and analysis . . . . . . . . . . . . 41

2.4 Perspectives and future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3 GIA constraints from the U.S. East coast: the ICE-6G_C (VM6) model 48

3.1 General approach and strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.2 Modelling the GIA process: Theoretical background . . . . . . . . . . . . . . . . . . . . . 51

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3.3 The ICE-NG models of ice sheet loading history . . . . . . . . . . . . . . . . . . . . . . . 55

3.4 Analysis of the performance of the ICE-6G_C ice loading history as a function of viscosity

model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.4.1 Relative sea-level history reconstructions . . . . . . . . . . . . . . . . . . . . . . . . 57

3.4.2 The Engelhart and Horton (2012) data set of U.S. East coast RSL evolution . . . . 59

3.4.3 Analysis of the performance of the VM5a and VM5b viscosity structures . . . . . . 59

3.4.4 The fit to the Fennoscandian spectrum . . . . . . . . . . . . . . . . . . . . . . . . . 64

3.5 An exploration of alternative viscosity structures . . . . . . . . . . . . . . . . . . . . . . . 66

3.5.1 Basic assumptions and methodology employed . . . . . . . . . . . . . . . . . . . . 67

3.5.2 Error analysis and model performance . . . . . . . . . . . . . . . . . . . . . . . . . 68

3.5.3 Alternative viscosity models: V1 and V2 . . . . . . . . . . . . . . . . . . . . . . . . 69

3.5.4 Case study I: Mantle viscosity variations in the upper mantle . . . . . . . . . . . . 74

3.5.5 Case study II: Viscosity changes in the upper part of the lower mantle . . . . . . . 77

3.5.6 Case study III: Viscosity variations in the transition zone . . . . . . . . . . . . . . 77

3.5.7 Case study IV: Viscosity contrast variations between the upper mantle and lower

mantle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

3.5.8 Case study V: Lithosphere thickness variations . . . . . . . . . . . . . . . . . . . . 82

3.5.9 Case study VI: Lower mantle viscosity variations and the Earth’s rotational state . 85

3.5.10 Other considerations and a summary of the insights gained through the sensitivity

analyses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

3.6 A preferred viscosity structure: the VM6 profile . . . . . . . . . . . . . . . . . . . . . . . . 88

3.6.1 The predictions for the U.S. East coast for ICE-6G_C (VM6) . . . . . . . . . . . . 89

3.6.2 The fit to the Fennoscandian relaxation spectrum . . . . . . . . . . . . . . . . . . . 93

3.6.3 Testing the viscosity structure against data from the North American West coast . 94

3.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

4 Full GIA constraints over N. America: the ICE-7G_NA (VM7) model 100

4.1 Geophysical observables related to the GIA process in North America and the performance

of current models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

4.1.1 Constraints on former ice sheet extent and thickness . . . . . . . . . . . . . . . . . 102

4.1.2 The importance of RSL data from the North American region of forebulge collapse 102

4.1.3 Space-geodetic uplift measurements over North America . . . . . . . . . . . . . . . 104

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4.1.4 Time-dependent gravity measurements from the Gravity Recovery and Climate

Experiment (GRACE) satellites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

4.1.5 Water level gauges in the Great Lakes region . . . . . . . . . . . . . . . . . . . . . 110

4.2 Characterization of the model misfits to present-day space-geodetic uplift rate observations110

4.2.1 The impact of elastic lithospheric thickness and mantle viscosity variations upon

vertical uplift rate predictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

4.2.2 Impact of ice loading history variations on vertical uplift rate predictions . . . . . 114

4.2.3 The impact of ice loading history variations on relative sea level evolution . . . . . 118

4.2.4 The impact of mantle viscosity variations on the differential uplift of the Great

Lakes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

4.3 Parameter space search and model refinement . . . . . . . . . . . . . . . . . . . . . . . . . 122

4.3.1 Model misfits and error determination . . . . . . . . . . . . . . . . . . . . . . . . . 124

4.3.2 Initial viscosity model generation and parameter space exploration . . . . . . . . . 125

4.3.3 Ice loading history variations and full model optimization . . . . . . . . . . . . . . 128

4.3.4 The ICE-7G_NA (VM7) model of the GIA process . . . . . . . . . . . . . . . . . . 130

4.3.5 The Upper Campbell (Lake Agassiz) strandline tilt . . . . . . . . . . . . . . . . . . 141

4.3.6 The fit to the time-dependent gravity measurements of the Gravity Recovery and

Climate Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

4.3.7 Model refinement considerations: margin evolution considerations of the south-

western edge of the Laurentide Ice Sheet . . . . . . . . . . . . . . . . . . . . . . . . 148

4.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

5 A regional test of the ICE-7G_NA (VM7) model in the Mediterranean basin 151

5.1 Some geological considerations in the Mediterranean basin . . . . . . . . . . . . . . . . . . 152

5.2 The Vacchi et al. (2016) data set of Holocene RSL change . . . . . . . . . . . . . . . . . . 152

5.3 Results for the ICE-7G_NA (VM7) model in the Mediterranean basin . . . . . . . . . . . 154

5.4 Evaluation of the ICE-7G_NA (VM7) model performance . . . . . . . . . . . . . . . . . . 158

5.5 Revisiting the interpretation of Roman fish-tanks . . . . . . . . . . . . . . . . . . . . . . . 160

5.6 Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

6 Conclusion 164

6.1 Perspectives and future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

Bibliography 168

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A Copyright and contributions 191

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List of Tables

2.1 Observed values of the principal moments of inertia of the Earth . . . . . . . . . . . . . . 21

2.2 Inter-study comparison of secular trends in polar wander and in the J2 coefficient . . . . . 42

2.3 Compilation of global mass balance estimates of mountain glaciers . . . . . . . . . . . . . 45

3.1 Relaxation times in Eastern James Bay . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

3.2 Error comparison for the viscosity variations under consideration . . . . . . . . . . . . . . 79

3.3 Radial viscosity structure of the VM6 viscosity profile . . . . . . . . . . . . . . . . . . . . 89

3.4 Change in error misfit between the ICE-6G_C (VM5a/VM6) models . . . . . . . . . . . . 92

4.1 Parameter space exploration values considered . . . . . . . . . . . . . . . . . . . . . . . . 127

4.2 Radial viscosity structure for the VM5a/VM6/VM7 profiles . . . . . . . . . . . . . . . . . 130

4.3 Values of Earth rotational observables predicted by ICE-6G_C and (VM5a) and ICE-

7G_NA (VM7) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

4.4 Relaxation times determined in Eastern James Bay . . . . . . . . . . . . . . . . . . . . . . 141

5.1 Comparison of the performance of the ICE-6G_C (VM5a) and ICE-7G_NA (VM7) mod-

els in the Western Mediterranean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

5.2 Comparison of observational inference and GIA model predictions of Roman time RSL

along the Tyrrhenian coast . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

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List of Figures

1.1 Examples of erratic boulders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Orbital parameters of the Earth’s orbit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Orbital parameter evolution over the last million years . . . . . . . . . . . . . . . . . . . . 5

1.4 Examples of seasonal variations in latitudinal solar insolation . . . . . . . . . . . . . . . . 6

1.5 δ18O record from the Panama Basin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.6 Evolution of atmospheric temperature and CO2 from the EPICA Dome C ice core . . . . 10

1.7 Idealized representation of the Glacial Isostatic Adjustment (GIA) process . . . . . . . . . 13

1.8 Relative sea level change at Barbados since the Last Glacial Maximum . . . . . . . . . . . 15

2.1 Location in the sky of the 295 defining extralactic radio sources of the ICRF2 . . . . . . . 27

2.2 The five intermediate transformations linking the ICRF with the ITRF . . . . . . . . . . 30

2.3 Correspondence between nutation and polar motion in various reference frames . . . . . . 31

2.4 Polar wander values from the SPACE2008 data set . . . . . . . . . . . . . . . . . . . . . . 38

2.5 Study of secular trends in the SPACE2008 data set of polar wander motion . . . . . . . . 40

2.6 Fitted SPACE2008 data set of polar wander motion . . . . . . . . . . . . . . . . . . . . . 40

2.7 Study of secular trends in the J2 coefficient of gravitational potential . . . . . . . . . . . . 41

2.8 Total root-mean-squared error for J2 and polar wander . . . . . . . . . . . . . . . . . . . . 42

2.9 Compilation of recent mass balance estimates of the Greenland and Antarctic ice sheets . 44

2.10 Impact of recent land ice melting on the J2 Stokes coefficient . . . . . . . . . . . . . . . . 46

3.1 Topographic height anomaly at LGM in the Northern hemisphere for ICE-5G and ICE-

6G_C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.2 Schematic representation of a salt marsh environment and relative sea level indicators . . 60

3.3 Geographical location of the North American sites used in Roy and Peltier (2015) . . . . . 61

3.4 The VM5a, VM6 and VM2 viscosity profiles . . . . . . . . . . . . . . . . . . . . . . . . . . 63

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3.5 Comparison of the Engelhart and Horton (2012) data set with the ICE-6G_C (VM5a/VM5b)

model predictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

3.6 Comparison of the VM5a, VM5b, V1 (FM) and V2 (FM) viscosity profiles . . . . . . . . . 70

3.7 Comparison of the performance of selected viscosity models (VM5a, VM5b, V1, V2) along

the U.S. East coast . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

3.8 Eastern James Bay isolation basin data of Pendea et al. (2010) . . . . . . . . . . . . . . . 73

3.9 Comparison of model performance along the U.S. East coast with upper mantle viscosity

variations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

3.10 Comparison of model performance along the U.S. East coast with lower mantle (upper

part) viscosity variations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

3.11 Comparison of model performance along the U.S. East coast with transition zone viscosity

variations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

3.12 Comparison of model performance along the U.S. East coast with viscosity contrast vari-

ations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

3.13 Comparison of model performance along the U.S. East coast with lithosphere thickness

variations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

3.14 Comparison of model performance along the U.S. East coast with lowermost mantle vis-

cosity variations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

3.15 ICE-6G_C (VM6) model performance when compared to U.S. East coast relative sea

level observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

3.16 Observed and predicted rates of late Holocene sea level rise . . . . . . . . . . . . . . . . . 93

3.17 Comparison of the VM5a/VM5b/VM6 viscosity profiles . . . . . . . . . . . . . . . . . . . 94

3.18 Performance of the ICE-6G_C (VM6) model with respect to RSL observations along the

U.S. West coast . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

4.1 Deglaciation isochrones for North America . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

4.2 Space-geodetic measurements of vertical uplift rates over North America . . . . . . . . . . 106

4.3 ICE-6G_C (VM5a/VM6) model predictions of vertical uplift rates over North America

and comparison with observational data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

4.4 GRACE observations of the time dependence of the gravitional field of the Earth with

ICE-6G_C (VM5a/VM6) model predictions . . . . . . . . . . . . . . . . . . . . . . . . . . 109

4.5 Observations of vertical uplift rate around the Great Lakes from water gauge observations

and ICE-6G_C (VM5a/VM6) model predictions . . . . . . . . . . . . . . . . . . . . . . . 111

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4.6 Comparison of the modern vertical uplift rate (mm/yr) over the North American continent

using model variations based upon the ICE-6G_C (VM5a) model . . . . . . . . . . . . . . 113

4.7 Coral-based record of relative sea level evolution at Barbados with ICE-6G_C (VM5a)

model predictions (and a model variation based upon it) . . . . . . . . . . . . . . . . . . . 116

4.8 Model predictions of vertical uplift over North America for the ICE-6G_C (VM5a/VM6)

models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

4.9 Prediction of the vertical uplift rate field over North American continent for modified ice

loading histories combined with the VM5a viscosity profile . . . . . . . . . . . . . . . . . . 119

4.10 Comparison of observations of relative sea level evolution along the U.S. East coast with

modified ice loading histories based upon the ICE-6G_C (VM5a) model . . . . . . . . . . 121

4.11 Comparison of Great Lakes vertical uplift rate observations with model predictions based

upon ICE-6G_C (VM5a) with different elastic lithosphere thickness . . . . . . . . . . . . 123

4.12 Regions where modifications to the ICE-6G_C ice loading history were initially introduced128

4.13 Radial viscosity variations for the VM5a/VM6/VM7 profiles . . . . . . . . . . . . . . . . . 131

4.14 Snapshots of the thickness of the Laurentide and Greenland ice sheets for ICE-7G_NA

since LGM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

4.15 Difference in ice thickness between the ICE-7G_NA and ICE-6G_C ice loading histories

over North America . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

4.16 Predicted topography over North America at LGM with respect to sea level . . . . . . . . 133

4.17 Comparison between relative sea level observations along the U.S. East coast and ICE-

6G_NA (VM7) model predictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

4.18 Background map of predicted present-day rate of vertical motion of the crust for the

ICE-7G_NA (VM7) model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

4.19 Comparison of observations of vertical uplift rates traverses of the North American con-

tinent with ICE-6G_C (VM5a/VM6) and ICE-7G_NA (VM7) model predictions . . . . . 137

4.20 Quality-of-fit determination between space-geodetic measurements of present-day vertical

motion rates and model predictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

4.21 Comparison of observed late Holocene relative sea level rise along U.S. East coast with

ICE-6G_C (VM5a/VM6) and ICE-7G_NA (VM7) model predictions . . . . . . . . . . . 140

4.22 Comparison of water gauge observations of present-day vertical uplift rates along the

Great Lakes with ICE-6G_NA (VM7) model observations . . . . . . . . . . . . . . . . . . 140

4.23 Comparison of the modeled tilt of the Upper Campbell strandline associated with former

Lake Agassiz with the tilt measured at present . . . . . . . . . . . . . . . . . . . . . . . . 142

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4.24 Comparison of relative sea level evolution observations along the U.S. East coast with

ICE-7G_NA (VM7) model predictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

4.25 GRACE observations of the time dependence of the Earth’s gravitational field over North

America and ICE-7G_NA (VM7) model predictions . . . . . . . . . . . . . . . . . . . . . 146

4.26 Comparison between GLDAS-NOAH and WGHM hydrological contributions to the time

dependence of the Earth’s gravitational field over North America . . . . . . . . . . . . . . 147

5.1 Tectonic setting of the Western Mediterranean Sea and location of relative sea level ob-

servations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

5.2 Comparison of observations of relative sea level evolution in the Mediterranean Sea with

ICE-7G_NA (VM7) model predictions (part 1) . . . . . . . . . . . . . . . . . . . . . . . . 155

5.3 Comparison of observations of relative sea level evolution in the Mediterranean Sea with

ICE-7G_NA (VM7) model predictions (part 2) . . . . . . . . . . . . . . . . . . . . . . . . 157

5.4 Contour plot of relative sea level at 2 ka BP across the Mediterranean region predicted

by the ICE-6G_C (VM5a) and ICE-7G_NA (VM7) models . . . . . . . . . . . . . . . . . 161

6.1 Inferred GIA vertical uplift rates over Greenland and contributions of each drainage basin

to post-LGM eustatic sea level change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

xiv

Chapter 1

Introduction

One of the first geological observations connected to past variability in climatic conditions came from

the Alps, where the presence of erratic boulders, large rocks of a different rock type from their surround-

ings, was seen as a major scientific puzzle (Figure 1.1). Initially portrayed as evidence of some of the

catastrophic flooding events described in the Bible or in other ancient texts, it took until the end of the

18th century for works linking them to the action of ice, such as that of the famous early geologist James

Hutton (Krüger, 2013), to emerge, often on the account of folkloric tales of mountain glacier advance

and retreat. It was only later, in the 19th century, that the notion was extended to the presence of large

ice sheets over most of Europe, notably in the works of Jean de Charpentier (de Charpentier, 1841) or

Louis Agassiz (Agassiz, 1840), who observed most eloquently:

"The perched boulders which are found in the Alpine valleys [...] occupy at times positionsso extraordinary that they excite in a high degree the curiosity of those who see them. Forinstance, when one sees an angular stone perched upon the top of an isolated pyramid, orresting in some way in a very steep locality, the first inquiry of the mind is, When and howhave these stones been placed in such positions, where the least shock would seem to turnthem over?" (Agassiz, 1840).

These accounts sought to explain not only the presence of erratic boulders throughout the mountain-

ous regions of central Europe, but also in numerous plains or marshes to the north. The obvious impact

on oceanic depth of such large ice sheet growth was suggested a few years later (Maclaren, 1842), and

the following decades saw an increasing number of further observations from many northern regions.

However, it was not until the middle of the 20th century that plausible mechanisms to explain their

growth and decay were put forward, however limited these were by the lack of precisely dated geological

evidence.

Amidst various propositions to explain the ice age phenomenon, the Serbian mathematician Milutin

1

Chapter 1. Introduction 2

Figure 1.1: (a) A drawing of an erratic boulder, called Pierre des Marmettes, in the Rhône valley. The10-metre rock served as an inspiration to Jean de Charpentier, as its granitic composition indicatedthat it originated more than 30 kilometres up the valley. From de Charpentier (1841), and reproducedwith permission. (b) Erratic boulders perched on the top of eroded accumulations of glacier materialin Switzerland, similar to the ones that inspired Louis Agassiz in the 19th century. Picture from theGlacier Photograph Collection of the National Snow and Ice Data Center (NSIDC) (Reid, 1902), andreproduced with permission.

Milanković popularized the idea that cycles in the parameters describing the Earth’s orbit around the

Sun could induce variations in the insolation received by the Earth’s surface large enough to significantly

impact its climate and induce ice sheet growth and decay over the Northern hemisphere (Milanković,

1941). Although his original view has been modified as observations of past climate variability and models

of the astronomical processes driving such cycles became available, he put emphasis on the cyclical nature

of ice sheet formation that orbital pacing should induce, and the theory of orbitally-paced ice age cycles

(or Milanković cycles) still bears his name today.

1.1 The orbital theory of the timing of the ice age cycle

As it orbits the Sun, the Earth is subject to complex variations in the gravitational attraction it feels

towards other bodies in the solar system, and these variations induce changes in the geometry of the

Earth’s orbit. This geometry is described by three main parameters, illustrated in the left panels of

Figure 1.2. The eccentricity of the orbit describes the departure of its shape from a circle through the

ratio of the the semi-major axis of the orbit to its focus (with ratio values e ranging from 0 for a circle to

1 for an infinitely elongated ellipse), while its obliquity angle refers to the tilt of the axis of rotation with

respect to the ecliptic, or the plane with respect to which the Earth revolves around the Sun. The third

parameter is the axial and apsidal precession of the orbit. Axial precession refers to the secular variation


Figure 1.2: Presentation of the three main parameters describing the geometry of the Earth’s orbit aroundthe Sun, namely (a) its eccentricity, (b) its axial obliquity, and (c) its precessional cycle (modulated byeccentricity). The left panels show graphical illustrations of each cycle, while the right panels show thevariations in each parameter over the last 150,000 years, together with vertical lines representing thetiming of the end of the Eemian interglacial (PEGI), the Last Glacial Maximum (LGM) and the start ofthe Holocene (HO). The left axes show the range in values for each quantity. For eccentricity-precession,it indicates the position of the northern hemisphere (NH) solstices with respect to perihelion, with thedashed lines showing the eccentricity modulation. When the full curve intersects the top dashed line,the NH winter solstice (WS) occurs at perihelion (P), and when it intersects the bottom dashed line,the NH summer solstice (SS) occurs at perihelion (P). The right axis show the main impact of eachparameter on the insolation calculation. From Peltier (2007), reproduced with permission.

in the position of the rotation axis with respect to the stars caused by the differential gravitational pull

on the equatorial bulge of the planet, a phenomenon which will be discussed in detail in a later section

of this work. This effect is compounded by orbital precession of the orbit itself, which sees the apside

(the line joining the perihelion and aphelion of the orbit) drifting with respect to the fixed background

of stars because of gravitational interactions with Jupiter and Saturn, and smaller contributions from

relativistic effects and the Sun’s oblateness (Fitzpatrick, 1970).

The evolution of the three parameters can be stably computed analytically (Berger and Loutre, 1991)

or numerically (Laskar et al., 2004) for the past several million years, as an n-body problem including

the planets, their satellites and the tidal interactions between them (corrected for relativistic effects). A


reconstruction of the eccentricity, the obliquity and the precession (modulated by eccentricity) over the

past million years is presented in Figure 1.3, while a zoom over the last 150,000 years is provided in the

right panels of Figure 1.2.

Careful examination of Figure 1.3 shows the periodic or quasi-periodic nature of the variations in

orbital parameters. Obliquity is characterized by a 41,000-year cycle that brings the value of the axial

tilt from 22° to 24.5° (the present value is 23° 44 ′), while eccentricity shows variations between e = 0

and e = 0.05 (present-day value of e = 0.0167) with periods of about 405,000 and 100,000 years.

Precession/eccentricity variations exhibit cycles with periods of 19,000 years, 22,000 years and 23,000

years, due to the combined effect of the 26,000-year cycle of axial precession with the 112,000-year cycle

of orbital precession (which effectively shortens the effective cycle of the precession of the equinoxes as

seen from Earth (Fitzpatrick, 1970)).

Changes in the geometry of the Earth’s orbit induce corresponding variations in the distribution of

solar energy on the surface of the planet (in the case of obliquity and precession variations) or in the

total amount of solar energy received by the planet (in the case of eccentricity), as well as changes in

the phase of seasons as a function of the position of the Earth along its orbit. The impact of orbital

geometry variations on the distribution of solar energy can be computed as a function of latitude and

time using a geometrical expression based upon Kepler’s second law, and accounting for the various

changes in eccentricity, obliquity and precession combined in the expression:

∆Q (θ, t) = ∆RS (θ)∆ǫ (t) +m (θ)∆ [e (t) sin (ωt)] , (1.1)

where ∆Q (θ, t) represents the anomaly in solar insolation at the top of the atmosphere, measured in

W/m2, as a function of time and latitude, with respect to a chosen background value. This quantity

includes the variations in obliquity ∆ǫ (t), in eccentricity e (t) and in the precessional cycle sin (ωt), and

takes into account the latitude dependence of this quantity via the parameters ∆RS (θ) and m (θ).

Figure 1.4 displays the present-day insolation at the top of the atmosphere as a function of latitude

and time of year, together with the insolation anomaly for three other time periods. The mid-Holocene

(panel b) is the most recent manifestation of a major orbital forcing positive anomaly in the Northern

hemisphere. Around the time of the Last Glacial Maximum (LGM), the insolation anomaly was minimal

compared to present values (panel c), while there was a very strong negative summer anomaly over the

Northern hemisphere at the time of the onset of the last glaciation/deglaciation cycle, about 116,000

years ago (panel d).

The original Milankovic theory states that changes in summer insolation received by mid-latitudes in


Figure 1.3: Orbital parameter evolution over the last million years, for (A) eccentricity, (B) axial obliq-uity, and (C) precession (modulated by eccentricity). From Cronin (2010) (using data from Berger andLoutre (1991)), reproduced with permission.


Figure 1.4: Latitudinal distribution of solar insolation at the top of the Earth’s atmosphere as a functionof time of year, showing (a) present insolation values (W/m2), and anomalies with respect to presentvalues (∆W/m2) at (b) the mid-Holocene (6,000 years ago), (c) the Last Glacial Maximum (21,000 yearsago), and at (d) the onset of the last glacial cycle at the end of the Eemian interval (116,000 years ago).From Peltier (2007), reproduced with permission.


the northern hemisphere should be key to the growth or decay of ice sheets, as this parameter is directly

linked to the amount of summer melt an ice sheet will experience. He also postulated the existence

of a positive feedback loop in which the increase in snow cover induced by low insolation values would

increase the albedo and further cool these regions.

1.2 Ice age cycles in deep-sea sedimentary cores and ice cores

Evidence for the influence of orbital forcing on the climate system came to light in the 20th century with

the development of paleoceanography and deep-sea sedimentary records, which enabled the recovery of

long records of past climate conditions through the careful analysis and interpretation of isotopic proxies.

In particular, oxygen isotopic anomalies extracted from the shells of small sea-dwelling protists, called

foraminifera, found in sedimentary cores can be used as a high-quality proxy for past continental ice

sheet volume (Shackleton, 1967; Shackleton et al., 1990). This is due to the natural occurrence in sea

water of three stable isotopes of oxygen (16O, 17O and 18O, with relative abundances of 99.75%, 0.04%

and 0.2%, respectively (Bradley, 2015)), and the fact that evaporation is a kinetic mass fractionation

process that will preferentially use the lighter H162 O water molecules. As the evaporated water from

the oceans eventually gets locked-up in the growth of continental ice sheets at high latitudes during

glacial eras, the oceanic water becomes anomalously rich in the heavier H182 O molecules. This anomaly

is described by the δ18O ratio:

δ18O =

(

[18O/16O]sample

[18O/16O]standard

− 1

)

, (1.2)

where the relative concentration ratio [18O/16O] of a sample is compared to an oceanic average value. As

the foraminifera shells use oxygen from oceanic water to make their calcium carbonate (CaCO3) shells,

they will retain the oxygen isotopic composition of oceanic water during their lifetime as they eventually

get deposited at the bottom of the ocean when they die. Records of δ18O can be obtained by associating

the depth at which the foraminifera shells are found in the sedimentary core to a time of death: more

positive values indicate a glacial era while more negative values are associated with interglacial periods.

This approach was first used in a seminal paper by Hays et al. (1976) on two deep-sea sedimentary

cores in the southern Indian ocean to recover a record of δ18O anomalies over the past 450,000 years as a

proxy for continental ice cover. Beyond a clear ∼100,000-year glacial-interglacial cycle extending over the

record, a spectral analysis revealed strong quasi-periodic ∼41,000-year and ∼23,000-year periodic signals,

providing the first support to the Milankovic theory of orbital pacing of the climate. Figure 1.5 displays


Figure 1.5: (a) Oxygen isotopic δ18O record from the Ocean Drilling Program Site 677 over the past twomillion years. (b) A power spectrum of the earlier and later million years of the record, showing a strongperiodicity over the last million years in the record with a 100,000-year period, as well as smaller spectralpower peaks at other frequencies associated with the cycles in the Earth’s orbital geometry parameters(periods of ≈41,000 and ≈22-23,000 years). From Peltier (2007), reproduced with permission.

a two-million-year high-resolution record (Shackleton et al., 1990), extracted from the sedimentary core

of the Ocean Drilling Program Site 677 off the coast of Panama and Colombia (Shackleton and Hall,

1989), together with a power spectrum analysis of the signal for the 0-1 Myr BP and 1-2 Myr BP periods.

The strong 100,000-year periodicity over the last million years is evident, as well as other spectral lines

at periods of ∼41,000 and ∼22-23,000 years.

Evidence for climatic conditions appropriate for the growth and decay of such large continental ice

cover can be found in numerous other paleoclimate records, most notably in ice cores recovered from

major ice sheets. In particular, isotopic analysis of the gas trapped in small cavities found in the ice

can be used as proxies to infer past atmospheric composition, temperature and circulation. Several ice

cores have been recovered from Antarctica, including the 420-ka record from the Vostok ice core (Petit

et al., 1999) and the 800-ka record from the EPICA ice core recovered from Dome C (Augustin et al.,

2004; Jouzel et al., 2007). Ice core records from Greenland, such as the GRIP (Greenland Ice Core

Project), GISP2 (Greenland Ice Sheet Project) (Grootes et al., 1993) and NGRIP (North Greenland Ice

Core Project) (Andersen et al., 2004) records, provide an inter-hemispheric connection to the records

obtained in Antarctica, but they extend only into the last glacial cycle (or, in the case of the NGRIP


core, into the last interglacial period) due to the younger age of the deepest part of the Greenland Ice

Sheet.

δ18O anomalies can also be recovered from ice cores. However, in this case, the recovered anomaly will

provide information about the temperature of the ambient atmosphere from which the water molecule

precipitated, with the δ18O anomaly roughly falling by ∼1‰ per 1.5° Celcius, although the relationship

is also influenced by other factors, such as the source area of the precipitation or thermal diffusion during

the transition from snow to ice (Cronin, 2010).

One of the most important direct measurements which can be obtained from ice cores is related

to the past atmospheric concentration obtained from careful analysis of the gas trapped inside the ice

core. In particular, records of past greenhouse gas concentrations in the atmosphere provide a unique,

high-resolution window in the past state of the climate. Greenhouse gases are atmospheric constituents

that absorb and emit radiation at infrared wavelengths of the electromagnetic spectrum that correspond

to the thermal radiation spectrum emitted by the Earth’s surface, the atmosphere or clouds (IPCC,

2013). Although the lowermost atmosphere is dominated by latent and sensible heat transport, at upper

altitude, where radiative effects dominate, greenhouse gases, together with cloud droplets and some

aerosols, absorb the upward radiation and re-emit it in all directions, resulting in an effective warming

of the lower atmosphere. This radiative effect is the so-called greenhouse effect. The main greenhouse

gases are water vapour (H2O), carbon dioxide (CO2), methane (CH4), nitrous oxide (N2O) and ozone

(O3). An increase (decrease) in atmospheric concentration of these substances affects the energy balance

of the climate system (with the radiative forcing expressed in W/m2), provoking a warming (cooling) of

the lower atmosphere and of the surface until the radiative equilibrium of the climate system is restored.

Although water vapor accounts for a significant portion of the greenhouse effect, it is generally considered

as a positive feedback on a given episode of atmospheric warming rather than a "prime mover" of the

warming itself, since its atmospheric concentration increases with temperature. A record of atmospheric

CO2 recovered from the EPICA Dome C ice core is shown in the lower panel of Figure 1.6. Both panels

of the figure display a close correspondence and a similar 100,000-year periodicity to that found in the

sedimentary core-based δ18O record shown in Figure 1.3(a).

The link between atmospheric composition, air temperature and ice sheet growth and decay is ap-

parent from Figure 1.6, and provides the first tangible indication that the study of past continental ice

sheet extent must be considered within the broader framework of paleoclimate studies. Partly within

the framework initially put forward by Milankovic, examination of paleorecords and of insolation model

results reveals that strong positive feedbacks (from changes in greenhouse gas concentrations and surface

albedo) are required to explain the large amplitude of the late Pleistocene glacial-interglacial cycles (e.g.


Figure 1.6: Results from the EPICA Dome C ice core in Antarctica over the past 700,000 years. (a)Estimated local temperature anomaly from Dome C, inferred from oxygen isotopic δ18O anomalies, withrespect to the average of the last 1000 years, and (b) Atmospheric CO2 concentration obtained fromVostok ice core for 0-413,000 years before present (BP) (open squares), and from EPICA Dome C for413,000-650,000 years BP (open circles). From Cronin (2010) (using data from Jouzel et al. (2007) andPetit et al. (1999)), reproduced with permission.


Abe-Ouchi et al., 2007, 2015; Tarasov and Peltier, 1997, 1999). A careful analysis of the large millenial-

scale climate variability is also enabled by the high resolution of the proxy records, enabling the analysis

of Heinrich events (e.g. Hemming, 2004), Dansgaard-Oeschger (D-O) oscillations (e.g. Peltier and Vet-

toretti, 2014), the Younger Dryas cold interval (e.g. Tarasov and Peltier, 2005; Muscheler et al., 2008),

and of the existence of any time lag or inter-hemispheric phase difference in the records (e.g. Shakun

et al., 2012; Broecker, 1998).

1.3 Other geomorphological evidence and the Glacial Isotatic

Adjustment (GIA) process

Beyond the evidence of past ice sheet extent provided by erratic boulders and from proxies recovered

from sedimentary and ice cores, direct geomorphological evidence can also be found over regions that

were previously glaciated, including a large fraction of North America and Northern Europe. These

hints include the presence of:

• glacial moraines, formed by debris (till) extracted by friction from the bedrock, carried by glaciers

and accumulated on their sides or at the foot of their maximal extent;

• drumlins, hills formed by deposition of till and moraine material or the abrasion of unconsolidated

bed material, elongated in the direction of motion of the ice flow;

• grooves/striations formed by friction from ice movement over the bedrock;

• glacial valleys, long U-shaped troughs, such as fjords, carved by glaciers.

A spectacular, indirect observation of former glacial extent comes from the study of land movement

over formerly glaciated areas. Historical observations of sea retreat were already made in Scandinavia

at the end of the Middle Ages, where harbours on the Baltic Sea, such as Öregrund or Luleå, had to be

abandoned and moved forward towards the shore to compensate for its effect. For example, Stockholm

was historically founded on a chain of islands in a deep and narrow bay giving on the Baltic Sea, which

was progressively cut out as the sea retreated after the Viking age (Krüger, 2013). Today, the former

bay is a freshwater lake (Lake Vätanen) and the city straddles the two water basins.

The potential strategic and economic consequences of such movements of the sea led to concern from

the Swedish crown as early as the beginning of the 17th century, and the famous astronomer Anders

Celsius was commissioned in 1731 to investigate the possible reasons behind them. As part of his study,

he inscribed a mark at the average position of the sea near the top of a large boulder in the Gulf of


Bothnia used in his days by marine mammals as a resting spot due its near complete submergence,

and decided to track the evolution of the sea at that location (the level of the mean sea was marked

every following year on the so-called Celsius rock, and the boulder is now almost completely out of the

water). He concluded that the sea was retreating throughout Sweden, and ascribed this change to the

evaporation of water from the oceans (Ekman, 2009). After the publication of Louis Agassiz’s account

of past large ice sheet cover (Agassiz, 1840), the issue of sea variations in Northern Europe was revisited

and it was inferred that continental ice sheet formation would lead to the removal of a large volume of

water from the oceans and affect their depth (Maclaren, 1842).

The potential link between glacial cycles and land motion followed the introduction of the concept

of isotasy in the middle of the 19th century, which refers to the idealized gravitational balance of the

uppermost layer of the Earth (lithosphere) with the underlying, denser asthenosphere, developed in

response to the discovery of negative gravity anomalies in the Himalayas by Louis Bouguer, a counter-

intuitive result given the extra mass of the mountains over which the measurements were performed.

This result of apparent mountainous mass excess at the surface was explained using isostatic arguments

as either due to the existence of a low-density "root" extending deeper in the mantle that balanced

the mountainous rise (the Airy model in 1855), or due to a lateral gradient in density that resulted in

mountains where the density of the crust was lower (the Pratt model in 1854). The potential role of

flexure in the response of the crust was soon recognized by F.A. Vening Meinesz, and a hybrid solution

using components of both the Airy and Pratt hypotheses was put forward by Aleksanteri Heiskanen.

Isostatic considerations of the Earth structure have played a fundamental role in the much later

development of the theory of plate tectonics (Schubert et al., 2001), but it was not long before the

potential link between isostatic considerations of the crust and the fast (geologically-speaking) growth

and melt of large ice sheets was recognized by Jamieson (1882), who suggested that the melting of the

large ice cover over Northern Europe should have induced an uplift of the crust (e.g. Peltier, 1998c). It

was also an implicit realization that the rheology of the planet should have a viscous component, and

not be completely elastic, due to the continuing adjustment of the Scandinavian surface to the long-gone

ice age load.

Glacial Isostatic Adjustment (or GIA) refers to the deformation of the Earth’s surface in response

to varying land ice surface loads, which induce corresponding changes in relative sea level and in the

planetary gravitational field. A schematic view of the process is presented in Figure 1.7. When a large

ice sheet forms over a continent, global sea levels fall due to the removal of a large volume of water

from the oceans. Also, as seen in the upper panel, the surface of the Earth under the ice sheet adjusts

(in an isostatic sense) to the extra weight put upon it, whereby mantle material flows outward and the


Figure 1.7: Idealized representation of the effects of the Glacial Isostatic Adjustment (GIA) process onthe surface of the planet in a region undergoing ice sheet growth during an ice age (upper panel), andafter its removal (lower panel). Figure sourced from the Canadian Geodetic Survey (Natural ResourcesCanada).

lithosphere flexes to create a peripheral bulge (or forebulge) in front of the ice sheet. The lower panel

presents the response of the surface once the ice sheet has disintegrated. Global sea levels rise due to the

redistribution of the water that was once locked in the former ice sheet, while the crust under it rises to

recover isostatic equilibrium. The forebulge slowly collapses, which indicates that post-glacial isostatic

adjustment will result in regions of uplift under former ice sheets and subsidence in their outskirts.

The application of the isostatic adjustment theory to the problem of varying ice loading at its surface

was originally described in a series of seminal papers in the 1970s (Peltier, 1974, 1976; Peltier and

Andrews, 1976; Farrell and Clark, 1976; Clark et al., 1978; Peltier et al., 1978), and its mathematical

structure as a visco-elastic problem has been used since then to generate constraints on the effective

viscosity structure of the planetary mantle, a crucial ingredient in the design of models of the mantle

convection process, and on reconstructions of the geographical distribution of land ice cover over the

last glaciation-deglaciation cycle (e.g. Peltier, 1994, 2004; Argus et al., 2014; Peltier et al., 2015). This


knowledge has direct impacts on many scientific domains, including the determination of boundary

conditions on paleotopography and paleobathymetry that are required when studying ice-age climate

conditions using modern coupled atmosphere-ocean climate models (e.g. Vettoretti and Peltier, 2013;

Peltier and Vettoretti, 2014; Abe-Ouchi et al., 2015). These boundary conditions are also important

when studying the strong climate variability that has characterized some eras during the post-LGM

deglaciation, such as the Younger Dryas period (Tarasov and Peltier (2005); also discussed in Chapter

4), or determining the origin and characteristics of post-LGM rapid sea level rise events inferred from

tropical sea level records, such as the Meltwater pulses 1A and 1B (Peltier, 2005; Argus et al., 2014;

Peltier et al., 2015). It has also been recently suggested that they could be used to find equivalents to

present-day mass loss in Greenland (Khan, S. A. et al., 2016).

GIA models also provide a crucial way to determine coastline evolution and understand changes in

sea level around the world that accompany the glaciation-deglaciation cycle. In particular, the large

variations in ice cover that characterized the last ice age have also left an imprint on sea level worldwide.

One of the most famous examples of such changes comes from the long-term record of sea level history

based upon U-Th dating of coral terraces on the island of Barbados in the Caribbean Sea (Fairbanks,

1989; Peltier and Fairbanks, 2006). This record, once corrected for tectonic uplift of the island, is

presented in Figure 1.8 and reveals a large change in eustatic sea level of about 120 meters since the

Last Glacial Maximum, revealing the large volume of the ice sheets that once covered the Northern

hemisphere. The importance of this crucial record will be discussed in further detail in Chapters 3 and

4 of this work.

Results from GIA models are important to understand relative sea level evolution around the world

and to determine the present-day horizontal and vertical motion associated with the GIA process at

every point on the planet. This information can then be used to understand and attribute present-day

local land movements, or to interpret various archaeological sites, such as Neolithic sites in Scandinavia,

the first human settlements in the Americas and fish ponds used in former Roman settlements (Chapter

5) (Peltier, 1998c; Lambeck et al., 2004; Evelpidou et al., 2012). They can also be used to understand

mass movements within the Earth system and interpret long-term changes in the rotational state of the

planet (Chapter 2) and on the present evolution of its gravitational field (Chapter 4).

Moreover, knowledge of the GIA process is important in the context of the present-day climate change

primarily induced by anthropogenic emissions of greenhouse gases (IPCC, 2013), notably regarding

the renewed episode of continental deglaciation that is occurring due to this warming of the lower

troposphere. Our observations of the ongoing melting of both the great polar ice sheets on Antarctica

and Greenland (as well as smaller ice catchments), which is responsible for an important fraction of the


Figure 1.8: Relative sea level indicators at Barbados, indicating a very large change in relative sea levelin meters (over 120 meters) as a function of time (in thousands of years before present) since the LastGlacial Maximum (LGM) about 21-26 thousand years ago. The data points are shown in green (fromFairbanks (1989)) or in purple (from Peltier and Fairbanks (2006)), with the horizontal bar showing thetectonic uplift-corrected depth of the indicator, and the vertical bar the range with respect to sea levelin which the particular coral species from which the sample was recovered could be found. The eustaticsea level curve for a version of the precursor GIA model ICE-5G (VM2) of Peltier (2004) is shown asthe black line. From Peltier and Fairbanks (2006), reproduced with permission.


global rise of sea level occurring at the present time, are significantly impacted by the remaining isostatic

disequilibrium associated with the last ice-age cycle, the signal of which must be eliminated in order to

more clearly identify the global warming component (e.g. Peltier and Tushingham, 1989; Peltier, 2009).

In particular, the GIA correction is central to the analysis of the time dependent gravity results being

provided by the Gravity Recovery and Climate Experiment (GRACE) satellites (Chapter 4) and the

determination of the recent melting of Greenland, Antarctica, and of other smaller ice catchments such

as the Alaskan glaciers (Peltier and Luthcke, 2009; Velicogna and Wahr, 2013; Luthcke et al., 2013).

Also, the determination of global sea level rise and local relative sea level change since the mid-19th

century relies heavily on the interpretation of the data collected by tide gauges worldwide, which must

be corrected for GIA effects originating from the lasting influence of the past ice age and from the recent

increased melting of ice sheets and glaciers around the world (Hay et al., 2015).

The study of the GIA process relies on a suite of geophysical observables that provide crucial in-

formation on different parameters of the models, which will be described in detail in this work. These

include, but are not limited to:

• Observations of the rotational state of the planet (Chapter 2);

• Numerous indicators of past relative sea level (Chapters 3, 4 and 5);

• Space-geodetic observations of crustal motion (Chapters 4 and 5);

• Records of paleo-shorelines (Chapter 4);

• Space-based recovery of planetary gravitational anomalies (Chapter 4);

• Water level gauges from the North American Great Lakes (Chapter 4);

• Dated reconstructions of ice margins from geomorphological markers (Chapter 4).

In the past decade, the study of the GIA process has benefited from a large increase in the number and

quality of collected geophysical observables, most notably over North America. In fact, the advent of

gravimetric measurements from the GRACE satellites, space-based geodetic observations made using the

Global Positioning System (GPS) and concerted efforts to develop databases of relative sea level records

with standardized error treatment (Engelhart and Horton, 2012; Vacchi et al., 2014) have generated a

wide range of constraints for GIA models. Testing the most recently available GIA models against these

new data sources has largely supported their validity, but it has nonetheless raised certain issues that

might indicate systemic problems with some of the model inputs, notably regarding the mantle viscosity

structure used (Engelhart et al., 2011).


1.4 Overview

The overarching goal of this work is to construct a model of the GIA process that provides an acceptable

fit to the entire suite of geophysical observables related to the GIA phenomenon over North America and,

in particular, that reconciles the model predictions with the latest generation of observables collected

over the continent. The general procedure to be followed will be, (1) to test the ICE-6G_C (VM5a)

model of Peltier et al. (2015) against the newly available geophysical observables for the North American

continent; then, (2) to examine if the misfits that arise from these comparisons can be corrected by

appropriate changes in the parameters of the model; and finally, (3) generate a cohesive GIA model that

incorporates all of this knowledge and provides further constraints on the viscosity of the mantle and on

the history of land ice loading.

In this context, a clear hierarchical consideration of the available geophysical constraints will be

used. Chapter 2 will be based on the work of (Roy and Peltier, 2011) and focus on the rotational

state of the planet. The latest observations of the main rotational anomalies and of their secular trends

(associated with the GIA process) will provide constraints for the following steps in the work presented

here. Chapter 3 will then proceed to focus on the newly available database of high-quality relative sea

level indicators for both coasts of the United States (Engelhart et al., 2011; Engelhart and Horton, 2012;

Vacchi et al., 2014), regions that bisect the forebulge associated with the former Laurentide ice sheet.

Following the work presented in Roy and Peltier (2015), it will see the introduction a new viscosity profile

(VM6), obtained under the assumption all misfits between model predictions and the new dataset can

be removed solely through appropriate mantle viscosity variations that enable a better capture of the

behaviour of the forebulge. Chapter 4 will then relax this assumption, and following the work of Roy and

Peltier (2017), extend the methodology to the whole suite of the latest GIA-related observations over

North America to consider variations in both mantle viscosity and ice loading history. The expanded

network of space-geodetic crustal uplift rate measurements and water level gauges from the Great Lakes

will be used, in conjunction with the ice-margin reconstructions of the Laurentide ice sheet, to generate

a new model of the GIA process, ICE-7G_NA (VM7) (Roy and Peltier, 2017). Its plausibility will then

be tested (and confirmed) against spaced-based gravimetric measurements of the GRACE satellites and

paleo-shoreline records that have been used in the literature to generate alternate models of the GIA

process. Finally, Chapter 5 will see the new model being successfully tested in another region of the

world, namely in the western half of the Mediterranean basin.

Chapter 2

The Earth’s rotational state and its

recent evolution

Changes in the distribution of mass within the Earth system, whether in its interior or at its surface, result

in variations in its gravitational field (via Newton’s law of universal gravitation) and in its rotational

state (via the conservation of angular momentum). In this chapter, the impact of the redistribution of

mass associated with the previously described Late Pleistocene cycle of glaciation/deglaciation on the

rotational state of the planet will be analyzed. In particular, secular trends in observables that capture

the rotational variability of the planet have been associated with the GIA process since the 1980s (Peltier,

1982; Wu and Peltier, 1984), and the ability of GIA models to explain these trends is a major constraint

of their plausibility (Peltier, 2007).

Here, the patterns of variability which have been identified in the rotational state of the planet on a

wide range of time scales will be presented, alongside the observational data used to quantify and define

them. The most recent observational data sets will be analyzed in order to (1) determine the recent

evolution of the Earth’s rotational state, and (2) to update the rotational constraints to which GIA

models should be tuned. This chapter follows the peer-reviewed contribution of Roy and Peltier (2011)

(Roy, K., and Peltier, W. R. (2011), ’GRACE era secular trends in Earth rotation parameters: A global

scale impact of the global warming process?’, Geophysical Research Letters 38(10), L10306), with the

addition of a more substantial introduction and overview of the information currently available on the

rotational state of the planet. It will provide strong and original evidence that the Earth’s rotational

state has undergone important changes in the last few decades, and hypothesize that modern climate

change is the culprit behind this phenomenon.

18

Chapter 2. The Earth’s rotational state and its recent evolution 19

2.1 Fundamental dynamics

The Earth can be approximated as a quasi-rigid body (which can exhibit deformation and flex over time

scales much longer than the time scale associated with a full rotation), for which the three-dimensional

rotational state can be described in a fixed, inertial frame of reference by its angular velocity ~ω(t) and its

moment of inertia tensor J . The angular momentum of the planet can be separated into contributions

from the planetary moment of inertia, and from relative angular momentum ~h(t) that would result from

any motion ~u(t) with respect to the axes of rotation. Using the center of mass as the origin, and using

the Einstein summation convention, each component of the total angular momentum ~L will then take

the form:

Li = Jij(t)ωj + hi(t), (2.1)

where Jij =∫

Vρ (xkxkδij − xixj) dV are the elements of the moment of inertia tensor, ωi(t) is the i -th

component of the angular velocity vector, and the relative angular momentum hi(t) =∫

Vρ (εijkxjuk) dV

is due to relative motion ~u(t) about the principal axes of rotation. Without loss of generality, a coordinate

system coincident (momentarily) with the fixed, inertial frame of reference, and rotating at angular

velocity ~ω(t = 0) relative to it, is chosen. The Eulerian equations of motion describing the changes in

angular momentum in the presence of external torque ~τ take the form:

~τ =

(

d~L

dt

)

+ ~ω × ~L (2.2)

Equation (2.2) is called the Liouville equation of motion (Munk and MacDonald, 1960). Whereas it is

valid for any group of rigid, rotating axes, it may be useful to restrict the chosen axes to a few special

cases that take advantage of the inherent symmetry of the system, such as the so-called Tisserand’s mean-

mantle axes, which are defined such that the relative motion of the crust and of the mantle with respect

to the rotating frame of reference is null (~h(t)crust = 0 and ~h(t)mantle = 0) (e.g. Jeffreys, 1952; Tisserand,

1891). Since geodetic measurement techniques rely (at least partially) on ground-based measurements,

it is also useful to include the mean motion of the crust in the definition of the Tisseyrand axes, such

that the relative motion of both the mantle and the crust with respect to the frame of reference is null,

an extension which is called the Tisserand mean-surface axes (Wahr, 1981). Also useful are the principal

axes of rotation, with respect to which the moment of inertia tensor is diagonalized (Jij = 0 for i 6= j).

However, although these options are attractive from a mathematical perspective, it is often preferable

to determine the rotating coordinate system with respect to "fixed" celestial background objects (like a

catalogue of stars or extra-galactic objects), as will be discussed in the following section. Considering a


case in which the z-axis is approximately aligned with the rotation axis of the planet (see below), small

perturbations to the mean state of rotation about the z-axis ( ~ω0(t) = Ω0z) are expressed as:

~ω(t) = ~ω0(t) + ∆~ω(t) = Ω0z + ~m(t). (2.3)

The moment of inertia tensor will be perturbed by introducing small variations around the diagonalized

tensor J0 corresponding to the alignment of the Earth system with its principal axes of rotation:

J (t) = J0 +∆J (t) =

A 0 0

0 B 0

0 0 C

+

∆Ixx(t) ∆Ixy(t) ∆Ixz(t)

∆Ixy(t) ∆Iyy(t) ∆Iyz(t)

∆Ixz(t) ∆Iyz(t) ∆Izz(t)

(2.4)

In Equation (2.4), A, B and C refer to the principal moments of inertia of the Earth, which are ordered

such that A < B < C. Substituting equations (2.3) and (2.4), and keeping only first-order terms (Munk

and MacDonald, 1960), equations for the perturbations to the rotation of the planet become (Gross,

2007):

dmx

dt+

[

B (C −B)

A (C −A)

]1/2

αrmy(t) = −αr

[

B

A

]1/2 [1

Ω0

dβx(t)

dt− βy(t)

]

+ τx(t) = Ψx(t) (2.5a)

dmy

dt−

[

A (C −A)

B (C −B)

]1/2

αrmx(t) = −αr

[

A

B

]1/2 [1

Ω0

dβy(t)

dt+ βx(t)

]

+ τy(t) = Ψy(t) (2.5b)

dmz

dt= −

dβz(t)

dt+ τz(t) = Ψz(t) (2.5c)

The above equations are the governing equations for the evolution of the angular velocity vector for the

Earth. Physically, changes recorded along the reference rotation axis (z-axis) in Equation (2.5c) are

perceived as variations in the length of the day, while the coupled differential equations (2.5a) and (2.5b)

are perceived as a motion of the instantaneous axis of rotation of the Earth with respect to its reference

rotation axis, a phenomenon which is also called polar wander. Above, the frequency of oscillation αr is:

αr =

[

(C −A)(C −B)

AB

]1/2

Ω0, (2.6)

and the Ψi(t) elements are the so-called excitation functions, in which βi(t) corresponds to the changes

in moment of inertia:

βx(t) =1

√

(C −A)(C −B)

[

∆xz(t) +1

Ω0hx(t)

]

, (2.7a)


βy(t) =1

√

(C −A)(C −B)

[

∆yz(t) +1

Ω0hy(t)

]

, (2.7b)

βz(t) =1

C

[

∆zz(t) +1

Ω0hz(t)

]

. (2.7c)

Changes in the rotational state of the planet can be expected to impact both the Earth’s moment of

inertia and relative angular momentum values. By using Tisseyrand’s mean-mantle frame, there will

be, by definition, no change to the relative angular momentum of the planet caused by the motion of

the mantle (Tisserand, 1891). This extension of Tisseyrand’s definition to include both the motion of

the mantle and motion of the crust leaves only the contributions of the core, the oceans and of the

atmosphere to contribute to the variations in relative angular momentum of the planet.

2.1.1 A useful simplification: the axisymmetric case

A useful simplification of Equations (2.5a) to (2.7c) can be obtained from the fact that the Earth is

almost axisymmetric, suggesting that the two equatorial moments of inertia will be similar (A ≈ B, see

Table (2.1)).

Parameter Value (×1037 kg·m2)A 8.0101B 8.0103C 8.0365

Table 2.1: Observed values of the principal moments of inertia of the Earth (Groten, 2004)

Using the average value of A and B (A′ = A+B2 ≈ A ≈ B) greatly simplifies Equations (2.5a) to

(2.7c) (Lambeck, 1980; Peltier, 2007), and the Liouville equations become

dmx

dt+ α′

rmy = Ψ′

x(t), (2.8a)

dmy

dt− α′

rmx = Ψ′

y(t), (2.8b)

dmz

dt= Ψ′

z(t), (2.8c)

with a natural frequency which takes the simplified form

α′

r =(C −A′)

A′Ω0 (2.9)


The excitation functions are also greatly simplified by the previous approximations and become

Ψ′

x(t) = α′

rβ′

y(t)−α′r

Ω0

dβ′x(t)

dt+ τx(t), (2.10a)

Ψ′

y(t) = −α′

rβ′

x(t)−α′r

Ω0

dβ′y(t)

dt+ τy(t), (2.10b)

Ψ′

z(t) = Ψz(t) = −dβz(t)

dt+ τz(t), (2.10c)

where the changes in moment of inertia are written:

β′

x(t) =1

C −A′

[

∆xz(t) +1

Ω0hx(t)

]

, (2.11a)

β′

y(t) =1

C −A′

[

∆yz(t) +1

Ω0hy(t)

]

, (2.11b)

βz(t) =1

C

[

∆zz(t) +1

Ω0hz(t)

]

. (2.11c)

2.1.2 A special case: the free Eulerian wobble

In the absence of excitation and external torques (βi(t) = 0 and τi(t) = 0, hence Ψi(t) = 0), the right-

hand side of Equations (2.5a) to (2.5c) becomes null, and the coupled equations for polar wander are

solved by:

mx(t) =M cos (αrt+ ψ) (2.12)

my(t) =M

[

A(C −A)

B(C −B)

]1/2

sin (αrt+ ψ) (2.13)

for a phase ψ determined by the initial conditions of the system, an amplitude M along the x-axis and a

frequency of oscillation αr described by Equation (2.6). This motion, first predicted by Euler (1765), is

the free periodic motion that a rigid Earth should exhibit if the rotation axis were momentarily displaced

from its principal axis, and should have a period of about 305 days using the values for the principal

moments of inertia of the Earth from Table 2.1.


2.1.3 The influence of the GIA process on the moment of inertia of the

planet

Mathematical models of the GIA process, which have been developed since the 1970s, have been success-

fully used to predict the impact of ice-age influence on the rotational state of the planet (e.g. Peltier,

1982; Wu and Peltier, 1984; Peltier, 2007). Although the detailed mathematical structure of these mod-

els will be discussed in the following sections, it is useful to present here the way in which moment of

inertia perturbations that lead to polar wander and length-of-day variations can be generated by the

large mass redistribution characteristic of the GIA process. As this relationship has been thoroughly

reviewed in the literature (e.g. Peltier, 2007), only the key results will be shown here.

First, it should be noted that the length-of-day variations that are induced by changes in the axial

component of the moment of inertia of the Earth can be equivalently expressed over long time scales as a

change in the gravitational potential of the Earth. This effect is measured in terms of the evolution of the

parameter J2, the dimensionless Stokes coefficient of degree 2 and order zero of the Earth’s gravitational

field, which is directly related to the oblateness of figure of the Earth. This time evolution, denoted as

J2, is linked to the angular velocity of the Earth by a simple expression (Lambeck, 1980; Peltier, 2007):

J2 =3C

2a2me

ω3

Ω0, (2.14)

where a represents the average radius of the Earth, me represents the mass of the Earth, ω3 is the

angular velocity of the Earth with respect to its rotation axis, and Ω0 is the modern average angular

velocity of the Earth.

With regards to the polar wander components of rotational variability, two sources of perturbation

from the GIA process have been identified. The first relies on the direct effect of the viscoelastic nature

of the modelled Earth undergoing isostatic adjustment. This is expressed through the so-called Love

numbers (Love, 1911), which are dimensionless parameters that capture the elastic (or viscoelastic)

response of a planetary object to tidal potential or surface mass loads (to be described in further detail

in a following section of this work). In particular, Peltier (1982) has shown that the direct, loading-related

impact from the GIA process on the moment of inertia of the Earth, ∆ILOADij , takes the form:

∆ILOADij =

(

1 + kL2 (t))

⋆ Irigidij (t), (2.15)

where kL2 (t) is the degree 2 surface mass loading Love number, ⋆ represents a temporal convolution, and

∆Irigidij (t) is the perturbations to the moment of inertia that the same surface load would have induced


on a totally rigid equivalent of the Earth.

The second contribution to the moment of inertia perturbations originates from the fact that the

rotation itself is changing, and is described using the tidal potential loading Love number of degree 2,

kT2 (t), of Farrell (1972) in the so-called MacCullagh’s formula (Munk and MacDonald, 1960; Peltier,

2007):

∆IROTiz =

kT2kf

⋆ mi (C −A′) , (2.16)

where kf =(

3Ga5Ω2

0

)

(C −A′), i can represent either the x or y component, the axisymmetric approxi-

mation of section 2.1.1 has been used, and G is the universal gravitational constant.

Substitution of these moment of inertia perturbations in the Laplace-transformed versions of the

polar wander components of Equations (2.8) and (2.9), while dropping the subscripts for clarity’s sake,

leads to:

smx + αr

(

1−kT2 (s)

kf

)

my = Ψx(s), (2.17a)

smy + αr

(

1−kT2 (s)

kf

)

mx = Ψy(s), (2.17b)

where s is the transform variable.

Assuming that externally applied torques vanish, the excitation functions expressed in Equation

(2.10) take the form, in the Laplace domain:

Ψx(s) =Ω0

A∆Iyz −

s

A∆Ixz, (2.18a)

Ψy(s) =Ω0

A∆Ixz −

s

A∆Iyz, (2.18b)

where, based on Equation (2.15), ∆Iij(s) =(

1 + kL2 (s))

∆Irigidij (s).

Thus, the impact of the GIA process on the polar wander component of rotational variability is

obtained by solving for its two components mx and my:

mx(s) =Ω0

Aα′r

1 + kL2 (s)

1−kT2(s)

kf

∆Irigidxz (s), (2.19a)

my(s) =Ω0

Aα′r

1 + kL2 (s)

1−kT2(s)

kf

∆Irigidyz (s), (2.19b)

In Equations (2.19), the rigid equivalent to the moment of inertia perturbations induced by the

surface load can be easily determined from the history of ice sheet loading used in the GIA model


realization (to be described in a following chapter), using the surface integral:

∆Irigidij (t) =

∫ ∫

v (θ, φ, t)(

a2δij − xij)

dS, (2.20)

where v (θ, φ, t) is the surface mass loading at time t for latitude θ and longitude φ.

Further details for these calculations are provided in Peltier (2007), together with an in-depth dis-

cussion of the assumptions that are integral to their expression.

2.2 Observations of rotational variability and reference frame

considerations

Determining the evolution of the Earth’s rotation takes advantage of a wide range of terrestrially-based

and space-based measurement techniques, which have considerably gained in precision over the last few

decades. To achieve the precision required to understand the minute variations in the Earth’s rotation,

all of the techniques discussed below require both a precise, uniform time scale and a suitable reference

frame against which measurements can be established.

To determine the position of the Earth in space and the properties of its rotation, two rotation frames

need to be defined: an Earth-fixed reference frame which follows the Earth’s mean rotation and is fixed

to the Earth’s surface as best as possible, and a celestial, inertial reference frame that is based upon

stellar, galactic or extra-galactic sources assumed to be fixed in space. In astronomy and geodesy, they

are based upon a reference system, which specifies the rules concerning the origin and the axes to be

used, which is then realized specifically for a given reference frame by giving the coordinates of a series

of reference observations that follow the rules prescribed by the reference system. For celestial frames

of reference, a list of specific astronomical sources to be located is prescribed, while the survey control

stations, also called geodetic reference points, are the equivalent prescribed locations when defining a

terrestrial reference frame (Boucher, 2001).

2.2.1 Celestial reference systems and frames

An inertial reference system should ideally be based upon a precise knowledge of all forces acting on the

main bodies of a given system and their exact position, such that the equations of motion describing its

evolution could be re-written in such a way that all rotational terms are removed. However, it is easier

in practice to rely upon a kinematic definition for the reference system, which states that far objects


exhibiting no proper motion should remain fixed and show no residual rotational or translational motion.

Locations of astronomical objects have traditionally been given using two angles: the declination

of an object being its angular distance measured perpendicularly to the extension to infinity of the

Earth’s equatorial plane, and its right ascension being the eastward angular distance measured along

the equator between the object and a given reference point (that defines a zero meridian). This reference

point is traditionally given by the vernal equinox, defined as the south to north (or upward) intersection

of the Sun’s apparent yearly motion in the celestial sphere (the ecliptic) and of the equatorial plane

of the Earth. Since both the ecliptic and the equatorial plane are not fixed, the coordinate systems

that use them need to be associated with a specific time at which the origins were determined. Recent

measurements are given with respect to the equinox at 12h (on the Terrestrial Time scale, measured by

atomic clocks and uncorrected for irregularities in the rotation of the Earth) of January 1, 2000 (Julian),

represented as J2000.0.

In order to develop the first celestial reference frames, catalogues of star positions and proper motions

were developed from the 19th century onwards using the timing of their lunar occultations, the precision

of which was limited before the advent of atomic clocks. The ground-based catalogues of stars have

been supplanted in the past decades by wide-ranging space-based surveys. In particular, the Hipparcos

catalogue of star positions and proper motion (ESA, 1997) provides over 118,000 astrometric observations

collected over the 1989-1993 period by the eponymous satellite of the European Space Agency (ESA).

The GAIA satellite, its ESA successor, has been in orbit since the end of 2013 and should provide a

substantial update on the star catalogue, as it aims to make over a billion astrometric measurements in

the galaxy (de Bruijne, 2012).

Since a given reference frame should be usable for any future observation, the accuracy of the proper

motion of the reference objects used in its realization is a crucial limiting factor of its accuracy. This

limitation is compounded by the fact that a large fraction of the astrometric observations used in the

recent star catalogues are based upon relatively short time series, or ones for which the earliest mea-

surements are unreliable (Feissel and Mignard, 1998). This implies that the quality of a given reference

frame realization will deteriorate quickly with time if unaccounted systematic errors in the determina-

tion of proper motions exist. The issue was largely addressed with the advent of radio-astronomy and

the discovery of powerful extragalactic compact radio sources such as quasars, highly energetic galactic

nuclei located billions of light-years away, at such a distance that their proper motion is undetectable

even when using the most precise techniques available today. The development of radio interferometry

(Very Long Baseline Interferometry (VLBI), described below) enabled the International Astronomical

Union (IAU) to locate these sources with a sub-milliarcsecond (mas) accuracy (Feissel and Mignard,


Figure 2.1: The location in the sky of the 295 defining extragalactic radio sources of the Second Realiza-tion of the International Celestial Reference Frame (ICRF2). The celestial sphere is represented usingan Aitoff equal area projection. From Schuh and Behrend (2012), reproduced with permission.

1998).

The first celestial reference system that used the newly gained precision of extragalactic radio sources

is the International Celestial Reference System (ICRS) (Arias et al., 1995), adopted in 1998 (Feissel and

Mignard, 1998). Its origin is defined at the barycenter of the solar system in order to facilitate the

introduction of general relativistic corrections (Kovalevsky, 2001). The pole of the ICRS is given, for

the sake of continuity from previous systems, in the J2000.0 direction, while the right ascension origin

is aligned with a specific source from previous systems. The latest (second) version of the International

Celestial Reference Frame (ICRF) is an implementation of the ICRS that relies on a catalogue of 3414

extragalactic radio source positions determined by VLBI observations over two observational bands

(wavelengths of 3.6 cm and 13 cm), in which a subset of 295 sources are used as defining sources that

precisely align the frame with the J2000.0 equator and equinox (Feissel and Mignard, 1998).

2.2.2 Terrestrial reference systems and Earth Orientation Parameters (EOPs)

A terrestrial reference system is a spatial reference system that rotates together with the Earth’s surface

in its diurnal cycle, in such a way that the ensemble of all defining points on the surface displays no

residual rotation (Kovalevsky, 2001; Petit and Luzum, 2010). Coordinates of points on the surface do

not display variations beyond tidal and tectonic effects.

After Euler’s suggestion that the Earth should exhibit wobbling around its rotation axis, the first

observational hints corroborating its existence came from observations of periodic variations in latitude

by German astronomers in 1881, followed a few years later by the detection of their anti-phased counter-

part in Hawaii (at roughly 180° of longitude away from Germany). In a seminal paper, H.C. Chandler

discovered a 14-month phase in the signal (Chandler, 1891), which convinced the International Asso-


ciation of Geodesy (IAG) to set up the International Latitude Service (ILS) to study this motion and

monitor its evolution. It oversaw the installation of six observatories placed at roughly the same latitude

(39° 8 ′) but at different longitudes (three in the United States, one in Italy, one in Japan, and one in

Turkmenistan, later replaced by a station in Uzbekistan), which were tasked with the careful monitoring

of the position of twelve groups of six pairs of stars (Yokoyama et al., 2000). The comparison of the

collected measurements at each site, once corrected for refraction corrections, provided a means to record

the evolution of the latitude measurements of the stars, while the spread of the observatories along the

same latitude simplified the analysis.

The first global terrestrial reference frame developed from the combination of site position measure-

ments was performed by the Bureau International de l’Heure (BIH) in 1984. The rules defining how

realizations of terrestrial reference frames should be performed were laid down in 1988, with the creation

of the International Terrestrial Reference System (ITRS):

• The frame should be geocentric (the origin of the system being at the center of mass of the entire

Earth system, which includes the atmosphere and the oceans);

• The time coordinate is determined by a geocentric local frame (accounting for general relativistic

effects)

• The orientation of the poles is coincident with the BIH (1984) orientation;

• The time evolution of the frame is determined by the condition that the frame should undergo no

net rotation under the horizontal tectonic motion at the Earth’s surface.

Then, once the terrestrial reference system is realized into a particular frame of reference using a set

of space-geodetic and/or astronomical observations, time-dependent coordinates of interest in the body-

fixed, rotating terrestrial frame ~xT (t) can be transformed into the inertial celestial frame ( ~xC(t)), using

a coordinate transformation of the type (e.g. Gross, 2007; Petit and Luzum, 2010):

~xC(t) = QRW ~xT (t) (2.21)

= PNRXY ~xT (t) (2.22)

where Q = PN is associated with the celestial part of the motion of the pole in the celestial reference

system caused by precession (P) and nutation (N), R represents the rotation of the Earth about this pole,

and W = XY accounts for the remaining polar motion along two specified x and y directions. While

only three time-dependent transformations should be necessary to perform a coordinate transformation


between the two frames, the International Astronomical Union (IAU) traditionally uses an intermediate

frame of reference in order to separate celestial and terrestrial components of the rotation variability. In

more detail, the celestial parameters under consideration in the transformation matrices linking the two

frames are:

Axial precession (P) The precession of the Earth’s rotation axis was the first observed source of

variability in the rotational state of the planet in the 2nd century BC by the Greek astronomer

Hipparcus (however, anecdotal evidence suggests it might have been discovered earlier in India and

Ancient Egypt), who observed that the position of the Sun in the celestial sphere at the vernal

equinox was slowly precessing along the zodiac constellations. This slow motion, of about 50 ′′ per

year, is mostly due to the gravitational torque induced on the equatorial bulge of the planet by the

Moon and the Sun (the so-called lunisolar component of the precession, or precession of the equa-

tor). It causes the rotational axis of the planet to trace an imaginary cone around the ecliptic pole

axes with a period of about 28,500 years. Another precessional component comes from the move-

ment of the ecliptic itself due to the gravitational interactions between the planets (the so-called

planetary component of precession), although it occurs about 500 times slower than the lunisolar

component. Both the lunisolar and planetary components are included in the determination of

precession constants and transformation matrices, as well as dynamical processes that influence

the shape of the Earth and of its equatorial bulge (to be discussed later). The axial precession

transformation matrix links the geocentric version of the International Celestial Reference Frame

(ITRF) to the mean celestial equator system.

Nutation (N) Nutation originates from the same gravitational interaction of the Sun, of the Moon and

of the other planets (to a lesser extent) with the non-spherical shape of the Earth that gives rise to

precession, but refers to the slight wobbles of the precessional motion caused by variations in the

orbits of these objects. Much smaller in amplitude than those caused by the precessional cycle of

the rotation axis, they also operate on shorter timescales. The Earth’s nutation is dominated by

the relative positions of the Moon and of the Sun (lunisolar nutation), which is characterized by

cycles ranging from a few days all the way to 18.6 years. The most important of these oscillations is

the 18.6-year cycle in the position of the lunar orbital node (intersection between the lunar orbital

plane and the ecliptic) caused by the 5° inclination of the orbital plane of the Moon with respect to

the ecliptic. An accounting of the various nutation effects that impact the Earth’s precession are

generated using time-dependent harmonic series in precession-nutation models (such as the IAU

2006/2000A model of Mathews et al. (2002)), which include the direct gravitational effect of other


Figure 2.2: The five intermediate transformations linking the space-fixed International Celestial Refer-ence Frame (ICRF) and the Earth-fixed International Terrestrial Frame (the W includes both x and ycomponents). From Seitz and Schuh (2010), reproduced with permission.

planets on the Earth and their indirect effect on the Sun, ocean tidal effects, mantle anelasticity,

as well as core/mantle and inner/outer core electromagnetic coupling (Mathews et al., 2002). By

definition, these models are limited to motions with periods greater than two days as measured

from the celestial frame of reference. These models are used to determine the elements of the

precession and nutation matrices P and N and generate the true celestial equator system and the

Celestial Intermediate Pole (CIP) with respect to which the rotation rate of the planet and the

transient motion of the pole can be determined.

Diurnal spin (S) This transformation matrix represents the diurnal rotation of the planet around the

Celestial Intermediate Pole.

Polar motion (X and Y) The transformation matrices record the position of the Celestial Interme-

diate Pole in the frame obtained from the International Terrestrial Reference System in two di-

mensions, where the x direction is in the direction of the Greenwich meridian and the positive y

direction is along the 90° West meridian, following IERS conventions.

Figure (2.2) presents the link between the transformation matrices and the interim reference sys-

tems between the quasar position-based celestial frame (translated to its geocentric form instead of its

barycentric form) and the crust and mantle-based terrestrial reference frame. Thus, for a particular

frame of reference based on a set of space-geodetic and/or astronomical observations, only five variables

are required to define the coordinate transformation between the space-fixed and Earth-fixed reference

frames. These quantities, called Earth Orientation Parameters (EOPs), capture the variability of the

Earth’s rotation, and are defined as:

Universal time (UT1) Length of day variations (∆LOD) are given as the difference between the

measured rotation speed (UT1) and the uniform, reference atomic time (UTC). From it, a measure

of the difference in instantaneous rotation speed with respect to a nominal value (taken to be 86,400

seconds) can be inferred.


Figure 2.3: Conventions relating the frequency (in cycles per sidereal day) between nutation and polarmotion as measured in the celestial (top) and terrestrial reference frames (bottom). Low-frequencymotion is defined as nutation in the celestial reference frame but as polar motion in the terrestrialreference frame. From Gross (2007), reproduced with permission.

Celestial pole offsets (dΨ, dǫ) (2) The celestial pole offsets represent the difference in longitude and

obliquity between the measured celestial pole and the Celestial Intermediate Pole (CIP), with a

temporal resolution of 2 days as seen in the celestial frame of reference. They are the residuals

between the observed cycles in axis motion and the modelled predictions from precession-nutation

models for motions that are characterized by a frequency (prograde or retrograde) of less than

two days (as seen in Figure 2.3). The measured offsets will include phenomena that are still not

modelled, such as the free nutation of the core (originating from the difference in axis direction

between the core and the rest of the planet).

Polar motion Complementary to nutation being defined as motion with a frequency (prograde or

retrograde) of less than two days as seen in the celestial reference frame, any motion of the Celestial

Intermediate Pole that falls out of this range is called ’polar motion’. To determine how the

frequency of any motion in the celestial frame (ωcel) will be observed in the rotating terrestrial

frame, the diurnal rotational frequency of the planet needs to be removed from the celestial-based

frequency, as seen in the lower half of Figure 2.3. This implies that nutation as seen on Earth

encloses all retrograde motions that have a close-to-diurnal frequency, while polar motion includes

all other terms, including secular trends and high-frequency motions.

Another key consideration in determining rotational variability is to set up a time reference stan-

dard from which anomalies can be inferred. In this respect, Universal Time (UT1) is based upon the

instantaneous rotation rate of the planet, determined with respect the International Celestial Reference

Frame (ICRF). However, since the advent of atomic clocks, an atomic time scale has been provided

(International Atomic Time, or TAI). Coordinated Universal Time (UTC), has been introduced to be

an intermediary between TAI and UT1.


2.2.3 Rotational variability measurement techniques and the latest ITRF

iteration

To infer a suitable terrestrial frame of reference, the variability of the Earth’s rotational state must

be precisely monitored. Although optical methodologies (lunar occultations or meridian transit times)

extend back well before the space era (into the 19th century), recent iterations of such terrestrial reference

frames have increasingly relied on the much increased precision reached by space-geodetic observing

systems. These methodologies include Very Long Baseline Interferometry (VLBI), Global Navigation

Satellite System (GNSS) measurements, satellite and lunar laser ranging (SLR and LLR, respectively),

and Doppler-based orbitography (DORIS). Each methodology is briefly summarized here:

Lunar occultation A lunar occultation refers to the passage of the Moon between an object and an

observer on Earth. Before the atomic era, lunar occultations of stellar objects were the most

precise way by which to measure variations in the Earth’s rotational rate (Gross, 2007), and a

substantial number of such events have been collected since the beginning of the 19th century (for

instance, the Jordi et al. (1994) reduction contains over 53,000 events from 1830 to 1955). For

each collected occultation event, the stellar position is determined from star catalogues, a chosen

lunar ephemeris (giving the evolution of the position of the Moon in the sky over time) and a

correction for the uneven limb profile of the Moon from photographic charts. Reductions then

compare the terrestrial time measurements (TT), based on astronomical measurements, to the

Universal Time measurements (UT1), based on the Earth’s rotation, to produce time series for

the length of day, albeit with significant uncertainties originating from the ephemeris and limb

models used. Although the term can also be used to describe the passage of the Moon in front

of the Sun itself, otherwise known as a solar eclipse, the data sets of such events are not used in

the generation of the ITRF, due to the timing uncertainty associated to the large relative size of

both the source and screen. Solar (and lunar) eclipses provide valuable information, however, on

millenial variations in rotation speed.

Local meridian transit times With the advent of atomic clocks, it became more advantageous to

measure star transits directly with respect to a given reference meridian, rather than with respect

to an intermediary like the Moon. Measurements of this type include the previously discussed

International Latitude Service (ILS), from 1899 to 1978 (Yumi and Yokoyama, 1980), with its

six stations located at about the same latitude in the Northern hemisphere, that recorded any

variations in the orientation of the planet’s rotation axis by monitoring the position of stellar

pairs. Other series include the later Bureau International de l’Heure (BIH) data set, taken from


136 stations, over the period 1962-1981 (Li and Feissel, 1986) and the more recent and precise

astrometric measurements from the Hipparcos system, which have been combined with the ILS

data to create a long uniform data set from 1899 to 1992 (Vondrák, 1999).

Very long baseline interferometry (VLBI) Very long baseline interferometry (VLBI) is based on

the measurement of the differential arrival time of radio signals emitted by distant extragalactic

radio sources at different radiotelescopes located around the globe. The methodology is sensitive

to any change in the relative position of the radiotelescopes with respect to the radio source,

which would be caused by a change in orientation of the planet or by the tectonic displacement of

the radiotelescopes on the surface. Measurements using more than two radiotelescopes (so-called

"multibaseline" measurements) enable the independent determination of all Earth orientation pa-

rameters (the only space-geodetic technique able to do so, as the other satellite-based methods

only record the rate of change of the length of day but not its instantaneous value) (Gross, 2007).

Global navigation satellite system (GNSS) Global navigation satellite systems consist of a net-

work of satellites with on-board atomic clocks broadcasting orbital data and precise emission time,

with a ground-based multi-channel receiver combining satellite signals to orient itself precisely with

respect to the satellite network. The American Global Positioning System (GPS) is the most ex-

tensive (as of October 2016, 32 satellites (U.S. Coast Guard Navigation Center, 2016)) and oldest

truly global system (worldwide coverage since 1995), while the Russian Global Navigation Satellite

System (GLONASS) network has also reached full global operational capacity in 2011 (after a

brief stint in the mid-1990s). The Galileo (European Union) and Compass (China) systems are

planned to go in fully global operational mode around 2020. Trilateration is used to generate the

position of the ground receiver (using four or more signals), taking into account a wide range of

error sources (relativistic effects, ionospheric effects, etc.) (Gross, 2007; Leick, 2015).

Satellite laser ranging and lunar laser ranging Laser ranging relies on the timing of a laser pulse

trip from an observing station towards and back from a reflector installed on a satellite orbiting the

Earth (satellite laser ranging) or on the surface of the Moon during the Apollo and Luna missions

(lunar laser ranging). Although many artificial satellites carry reflectors to aid with their tracking,

the geodetic community mostly relies on the specialized LAGEOS-1 and LAGEOS-2 satellites (from

LAser GEOdynamics Satellite) for geodetic measurements, because of their high orbital stability

(spherical shape) and highly reflective surface (ease of tracking) (Tapley et al., 1985). Range

measurements are realized at many tracking stations around the globe, and the orbit of each

satellite fitted to them, while taking into account the geophysical parameters which may influence


the position of each tracking station (tectonics, tidal motion, Earth orientation changes). From this

fitting process, most Earth orientation parameters, except absolute length-of-day measurements,

can be obtained (Tapley et al., 1985). Lunar ranging is more challenging version of the technique

because of the much weaker signal to be detected, and only two observatories actively perform the

measurements. Earth orientation parameters are obtained from analysis of the range measurement

residuals, once the Moon’s orbit and the other ground effects have been taken into account (Gross,

2007).

Doppler orbitography The Doppler Orbitography and Radio positioning Integrated by Satellite (DORIS)

system is a French network based on the Doppler shift analysis of dual-frequency signals emitted

from ground stations towards an orbiting receiver. Once corrected for ionospheric effects, the orbit

of the satellite can be inferred from the measurements, as well as changes in the orientation of the

planet (Willis et al., 2006).

Each methodology described above has its strengths and limitations, which are the topic of much liter-

ature and analysis by the geodesy community (Gross, 2007; Petit and Luzum, 2010). In particular, the

information gathered obtained from each of these methodologies can then be combined together, given

appropriate weight, to minimize the individual uncertainty associated with each methodology. Thus,

the International Earth Rotation Service (IERS) has aimed to provide periodic realizations of the most

accurate reference frame. There have been thirteen iterations of the International Terrestrial Reference

Frame (ITRF), each of them providing a different long-term solution to the Earth’s orientation axes and

their motion based on different observational data sets. They are denoted by the ITRF acronym followed

a number representing the last year for which observational data was used to constrain the computed

solution (starting from ITRF88 to ITRF2014), with each iteration provided with full variance-covariance

information. Since ITRF2000, the relative motion of all geodetic sites on the surface of the planet are

accounted for by aligning the orientation rates of each site with a plate tectonic motion model (the

NNR-NUVEL-1A model of Argus and Gordon (1991)) (Altamimi et al., 2001).

2.3 The evolution of secular trends in the planetary rotational

state: Methodology and results

The large gains in precision obtained in the space era using geodetic measurement techniques, combined

with the development of suitable reference frames to which measurements can be compared, has enabled

us to build upon the previous optical methods and study precisely the variability characteristic of the


Earth’s rotational state. The impact of various physical processes has now been characterized and

compared to model realizations (Gross, 2007). The study of planetary rotation follows the evolution of

two particular observables, namely the wander of the pole with respect to the geographic background

(polar wander) and changes in the length of the day (LOD) (or equivalently, in the Earth’s dynamic

oblateness of figure, also known as the J2 coefficient).

Predictions of the impact of the GIA process on these observables have been realized and tested

against observations since the 1980s (Peltier, 1982; Wu and Peltier, 1984), and have explained secular

trends in these observables as being mostly due to the GIA process (Peltier, 2007). However, recent

changes in these observables have been noted. For polar motion, a possible kink in its secular trend

was suggested to have begun in the mid-1990s (Gross and Poutanen, 2009), while studies of the Earth’s

dynamic oblateness (J2) have also pointed towards a recent change in its secular trend (Cox and Chao,

2002). However, no simultaneous study of those two observables had ever been attempted before the

study of (Roy and Peltier, 2011), which is presented here. We first start by briefly reviewing known

sources of variability in both observables, and then show in what follows that they both began to depart

from their previously established, Late Quaternary ice-age related, GIA-controlled trends, at a somewhat

earlier time. The simultaneity of the onset of this new regime of behaviour is suggestive of a common

cause, which is discussed thereafter.

2.3.1 Sources of variability in length-of-day measurements

The existence of anomalies in the Earth’s rate of rotation was first suggested in the context of tidal

dissipation and angular momentum conservation in the Earth-Moon system, resulting in the transfer of

rotational angular momentum to its orbital counterpart (Halley, 1695; Spencer Jones, 1939). The study

of this phenomenon over historical periods can be performed through the careful analysis of numerous

records of the timing of past lunar and solar eclipses, as performed by ancient Babylonian, Chinese, Arab

and Greek astronomers (Stephenson and Morrison, 1995; Morrison and Stephenson, 2001). These studies

have inferred, from the difference between historical records the "expected" timing of eclipses based on

constant, modern rotational rates and historical records, the average acceleration in the Earth’s rate of

rotation since then. However, the rate of planetary rotation can be impacted through both variations in

the relative angular momentum of the Earth-Moon system and by variations in the moment of inertia

of the planet, as demonstrated in Equation (2.7). This becomes apparent as the component of the total

acceleration resulting from the tidal braking of the Earth’s spin can be precisely known from laser ranging

observations of the recession of the Moon. Using these values, Stephenson and Morrison (1995) inferred


the existence of a non-tidal component acceleration of 1.6(±0.4)·10−22 rad·s−2, primarily associated with

the change in oblateness of the Earth resulting from the ongoing glacial isostatic recovery of the planet.

Following the direct correspondence between changes in the J2 zonal component of the gravitational

field of the planet (the oblateness of figure of the Earth) and length-of-day variations, this non-tidal

acceleration corresponds to a rate of change in the oblateness of the Earth of −3.5(±0.8) · 10−11 yr−1.

The advent of the space age also enabled the study of the Earth’s gravitational field from the precise

monitoring of satellite orbital parameters, which are in turn connected to the rotational rate of the planet

(Yoder et al., 1983). From orbital tracking of the orbits of geodetic satellites using the satellite laser

ranging (SLR) technique, a measurement of the time rate of change of the oblateness of the Earth can

be obtained. From a constellation of seven satellites, Cheng and Tapley (2004) studied the variation of

the J2 component over a 28-year period, and found a secular trend of −2.75 ·10−11 yr−1, consistent with

the Stephenson and Morrison (1995) inference based upon the analysis of the timing of ancient eclipses.

The existence of strong variability in the J2 component of the Earth’s gravitational field and in

length-of-day measurements has been established at various time scales. On the daily scale, the periodic

displacement of fluid at the surface of the planet associated with tides is responsible for large cyclical

signals in LOD measurements at tidal frequencies, which are easily distinguishable in high-accuracy VLBI

measurements (Gross, 2007). Effects from longer-period ocean tidal effects can be accounted for using

assimilated models of tide influence from radar altimetry data sets, such as the TOPEX-POSEIDON

(T/P) sea surface height record (Kantha et al., 1998; Gross, 2007). Long-period solid-Earth tidal effects

have also been identified at the biannual, annual and multi-year (9.3 years and 18.6 years) time scales

(Yoder et al., 1981). On the annual and seasonal scales, measurable cycles in LOD measurements have

been associated with meteorological processes, such as annual and semiannual cycles in the angular

momentum carried by zonal winds (Munk and MacDonald, 1960; Lambeck, 1980; Gross et al., 2004),

the Madden-Julian oscillation (Gross et al., 2004) or by ocean currents (Gross et al., 2004). Strong

ENSO-induced interannual variations over timescales of 4 to 6 years have also been identified, and have

been associated with the transfer of angular momentum between the atmosphere and the solid Earth

induced by the collapse of tropical easterlies during an ENSO event. Finally, the existence of strong

decadal variations in the length of the day has also been revealed by the study of the 400-year lunar

occultation record (Munk and MacDonald, 1960; Lambeck, 1980; Gross, 2007).


2.3.2 Sources of variability in polar wander measurements

The study of polar wander variability benefits from the existence of a long time series extending back

to the end of the 19th century with the previously described record of the International Latitude Ser-

vice (ILS), which has collected data until 1980 (Yumi and Yokoyama, 1980). The development of

precise space-era star catalogues, such as Hipparcos, has greatly increased the precision of transit time

techniques, and space-geodetic techniques have provided supplementary means to infer changes in the

position of the poles.

Studies of the long-term ILS record have revealed the dominating presence of the free Chandler

wobble with a period of about 433 days (amplitude of 100-200 milliarcseconds, or mas), caused by a

combination of atmospheric, oceanic and hydrological processes (Gross, 2007). An annual cycle in polar

motion has also been detected. At the decadal time scale, the Markowitz wobble, with a period of 24

years and an amplitude of about 30 mas, dominates the variability seen in the ILS data (Markowitz, 1961;

Gross and Vondrák, 1999; Gross, 2007). Beyond the existence of these periodic signals in polar motion,

secular trends in the ILS and Hipparcos polar motion series have been identified (Gross and Vondrák,

1999), with an estimate of 3.51(±0.01) mas/year towards the meridian 79.2(±0.20)°W. If adjusted for

a different terrestrial frame of reference (in the so-called hotspot frame), the rate is increased to 4.03

mas/year towards the meridian 68.4°W (Argus and Gross, 2004).

2.3.3 Polar wander and LOD evolution: Data set and methodology employed

The recent evolution of polar wander is determined through a study of the previously described Earth

Orientation Parameters (EOPs), which provide the transformation between the International Terrestrial

Reference System (ITRS) and its celestial counterpart. They are determined from a set of independent

space-geodetic techniques, which includes very long baseline interferometry (VLBI), global positioning

system (GPS) measurements, as well as lunar and satellite laser ranging (SLR) techniques. Once a

specific terrestrial frame of reference is determined, keeping track of any anomalies recovered from

these techniques provides useful supplementary information on the evolution of the rotational state of

the planet. Moreover, combining the information gained from the strengths and weaknesses of each

methodology has resulted in the generation of an unbiased measure of polar motion variations since the

mid-1970s (Ratcliff and Gross, 2010).

The analysis of the recent evolution of rotational parameters presented here takes advantage of the

recent SPACE2008 data set of Ratcliff and Gross (2010), which is part of a family of Earth orientation

parameters generated annually from a combination of space-geodetic measurements by the Geodynamics


Figure 2.4: Polar wander values from the SPACE2008 data set (Ratcliff and Gross, 2010), covering the1976-2009 time period, with the raw x-component (a) and y-component of polar wander shown in theupper panels, with the corresponding 1-σ uncertainty presented in the lower panels, for the x-component(c) and y-component (d), respectively. From Ratcliff and Gross (2010), reproduced with permission.

and Space Geodesy Group of the Jet Propulsion Laboratory (JPL), and obtained via anonymous ftp

at <ftp://euler.jpl.nasa.gov/keof/combinations/2008>. Once the time components of the measurement

series were corrected for solid-Earth tide effects (Yoder et al., 1981), and for the impact of the roughly

fortnightly (Mf ) and monthly (Mm) tides, the individual data sets are combined by iterative comparison

to a combination of all the other data sets, using a Kalman filter (Gross et al., 1998). Bias-rate corrections

and uncertainty scale factors are provided for each data set, and a final combined time series is generated

with its corresponding uncertainty range. The particular set used in this study, denoted SPACE2008,

spans the period from 28 September 1976 to 2 July 2009. The values of the polar motion components are

presented in Figure 2.6 at daily intervals, with components extending from the Conventional International

Origin (CIO) along an x-axis aligned with the Greenwich meridian (panel (a) of Figure 2.6), and along

a y-axis aligned with the 90° W meridian (panel (c) of Figure 2.6) (Ratcliff and Gross, 2010).

Linear trends in the polar motion series are studied by removing the large high-frequency variations

that dominate the signal. In this study, a zero-phase Butterworth-type low-pass filter is applied to the

polar motion series (Butterworth, 1930), with a period cut-off set at 6 years, a value that is consistent


with previous studies of polar motion trends (Gross and Vondrák, 1999) and chosen to eliminate the signal

associated with the beat period between the annual wobble and the Chandler wobble. The Butterworth

filter is preferred to simpler, idealized digital filters such as the boxcar filter, as it has a maximally

flat frequency response and phase shift effects that are simple to compensate for. The smoothed series,

shown in panels (a) and (c) of Figure 2.5 (for the x- and y-components, respectively), serve as the

basis for the analysis of quasi-periodic low-frequency residual variability, using a methodology similar to

that employed in Gross and Vondrák (1999). Low-frequency and quasi-periodic terms are determined

by performing a simultaneous weighted least-squares fit for a mean, a linear trend and low-frequency

periodic terms that correspond to the most prominent peaks in the amplitude spectrum of the time series,

including terms with periods of 8.3 years, 10.7 years and 18.8 years. Changes in the linear trend are then

found by dividing the time series in two segments. The position of the "pivot-time" that separates the

two segments is varied systematically, and the linear trend and mean are determined for each segment in

order to find the pivot-time that minimizes the overall root-mean-square error of the fit. The quality of

the best two-segment fit is then compared to the best single-segment fit. To be preferred, the best two-

segment fit must be characterized by a substantially lower root-mean-square error. This simple analysis

provides an approximation to any broad changes having occurred in the linear trend, but it should be

remembered that such changes may not be as abrupt as approximated by the methodology used here.

The resulting two-segment fits to the low pass filtered polar wander observations are presented in panels

(b) and (d) of Figure 2.5 for the x- and y-components, respectively. Figure 2.6 demonstrates the quality

of the fit provided by the final fitted function for both components.

Recent variations in the linear trends for the non-tidal acceleration of planetary rotation rate are

analysed using a single data set, namely the satellite laser ranging (SLR) data set of Cheng and Tapley

(2004), which uses a subset of seven geodetic satellites (Starlette, Ajisai, Stella, LAGEOS 1 and 2,

Etalon-1 and -2, and BE-C, from 1999 onwards). The data used for this analysis is derived from the

acceleration of the precession rate of the orbital node of each satellite. It was obtained via anonymous

ftp from the Center for Space Research at the University of Texas at Austin, spans January 1976 to the

end of 2009, and is presented in panel (a) of Figure 2.7. In order to study variations in the linear trend

over this period, low-frequency terms are also eliminated using a low-pass Butterworth filter. The cut-off

for the analysis of this data is fixed at 20 years, which is sufficient to remove the suggested 18.6-year

tidal and decadal variations, and is similar to the cut-off frequency employed by Cheng and Tapley

(2004). The smoothed series, plotted in panel (a) of Figure 2.7, reveals a marked change approximately

in the mid-1990s. Linear trends are then determined in the smoothed J2 series, by separating the time

series into two parts and fitting the trend separately for each part, as for the polar motion. The pivot-


1975 1980 1985 1990 1995 2000 2005 2010−400

−300

−200

−100

0

100

200

300

400

Year

Polar Motion (mas)

(x−component)

1975 1980 1985 1990 1995 2000 2005 20100

10

20

30

40

50

60

Year

Polar Motion (mas)

(x−component)

1975 1980 1985 1990 1995 2000 2005 20100

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200

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Year

Polar Motion (mas)

(y−component)

1975 1980 1985 1990 1995 2000 2005 2010240

260

280

300

320

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Year

Polar Motion (mas)

(y−component)

a)

c) d)

b)

Figure 2.5: Study of the polar wander from the SPACE2008 data set (Ratcliff and Gross, 2010). (a)-(b)Raw x-component of polar wander (dark grey), its subjection to a 6-yr, low-pass, Butterworth filter(red), and the two linear fits for the time periods 1976-1992 (rate: 1.7 mas/yr), and 1992-2009 (rate:0.9 mas/yr). (c)-(d) Raw y-component of polar wander (dark grey), its subjection to a 6-yr, low-pass,Butterworth filter (red), and the two linear fits for the time periods 1976-1992 (rate: 4.1 mas/yr), and1992-2009 (rate: 1.5 mas/yr). From Roy and Peltier (2011).

Figure 2.6: Representation of the SPACE2008 polar wander data set (Ratcliff and Gross, 2010) subjectedto a 6-yr, low-pass, Butterworth filter (red), together with the fitted signal used in the determination ofthe secular trends (black). This fit function includes a constant term, a linear trend and the periodicsignals described in the text (black), for the x-component (a) and the y-component (b) of the signal.


1975 1980 1985 1990 1995 2000 2005 2010−8

−6

−4

−2

0

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4

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8

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Variation in J2 Component (x 10−10)

1975 1980 1985 1990 1995 2000 2005 2010−5

−4

−3

−2

−1

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1

2

3

4

Year

Variation in J2 Component (x10−10)

b)a)

Figure 2.7: Changes in the J2 Stokes coefficient (×10−10), for the period 1976-2009, from the orbitalparameters of seven geodetic satellites (Cheng and Tapley, 2004). (a) Raw variations of J2 (×1010)(dark grey), and subjected to a 20-year, low-pass filter (green). (b) Two linear fits for the time period1976-1992 [rate: −0.37(±0.01) ·10−10 yr−1], and 1992-2009 [rate: −0.09(±0.02) ·10−10 yr−1]. From Royand Peltier (2011).

time is again shifted along the series to find the position of the knot in time that minimizes the overall

root-mean-square error of fit. The final result of this process is presented in panel (b) of Figure 2.7.

2.3.4 Secular trend pivot point determination: results and analysis

Changes in the linear trends of the secular drift of the pole relative to the surface geography and of the

non-tidal acceleration of the rate of rotation have been clearly identified. It is most revealing, however,

to compare the variation of the total root-mean-square error for the two rotation-related time series as

the position of the transition in the linear trend is varied systematically. These results are presented

in Figure 2.8 and demonstrate that the transition in the linear trends for both anomalies occurs in

approximately 1992.

Our results are compared to historical inferences in Table 2.2. Our analysis of the polar motion time

series demonstrates that its secular rate of drift remained approximately fixed, prior to 1992, at 4.5(±0.1)

mas/yr (or 1.3° Myr−1), along the 68(±8) °W meridian, while it slowed to 1.8(±0.4) mas/yr (or 0.5°

Myr−1), along the 58(±9) °W meridian subsequently. Although not corrected for the hotspot frame of

reference, as done by Argus and Gross (2004), the current analysis demonstrates that dividing the time

series into two distinct epochs increases the magnitude of the established secular rates of change prior

to 1992, but decreases them for modern times subsequent to this. As no uncertainties were provided in

previous inferences of these quantities, it is hard to state if the observed differences between our inferred

values and previous ones are significant or not, but different data processing methodologies and data

set time spans might explain part of the differences. Concerning the corresponding changes in J2, our


1980 1985 1990 1995 2000 20058

10

12

14

Root−mean Squared Error (rms)

Year

1980 1985 1990 1995 2000 20050

0.2

0.4

0.6

Root−mean Squared Error (rms)

Figure 2.8: The total root-mean-squared error for the J2 Stokes coefficient fit (dark green) and for thepolar wander fit (red), as a function of the position of the pivot-time, with a common minimum observedaround 1992. From Roy and Peltier (2011).

Table 2.2: Inter-study comparison of secular trends in polar wander and in the J2 coefficient. From Royand Peltier (2011)

analyses estimate a rate of change of −3.7(±0.1) · 10−11 yr−1 before 1992, and −0.9(±0.2) · 10−11 yr−1

after 1992. Splitting the time series fully reconciles the observed rate of change for J2 presented in

Cheng and Tapley (2004) with the estimate from historical records of Stephenson and Morrison (1995)

(−3.5(±0.8) · 10−11 yr−1). No uncertainty is provided for the Cheng and Tapley (2004) inference, which

complicates any comparison to our values, it would be expected to lie between our pre-1992 and post-1992

inferences for J2 as it straddles both time periods.

The identification of significant shifts in the secular rates of change of the two primary Earth rotation

observables, which become evident in the same epoch of time, is highly significant. This is especially

the case as the variation in J2 and polar wander are dependent upon completely independent elements

of the Earth’s moment of inertia tensor (e.g. Munk and MacDonald, 1960; Peltier and Luthcke, 2009).


In models of the GIA process, which are able to simultaneously fit those two rotational anomalies,

variations in the elements of the moment of inertia tensor of the planet are determined by a specific

model of time-dependent continental ice sheet loading and of the depth-dependence of mantle viscosity.

It would therefore be very surprising if the observed recent changes in the secular trends inferred to have

occurred since 1992 were not caused by recent changes in surface ice sheet loading.

Our hypothesis is that these simultaneous shifts in Earth’s rotational state are caused by additional

melting of continental ice cover induced by recent global climate change. The existence of a link between

the timing of the changes in the linear trends for the rotational state anomalies and recent changes in

the ice sheet loading on the planet is supported by studies of the year-to-year melting rates of major

continental ice sheets and of small ice sheets and glaciers distributed globally. For instance, Thomas et al.

(2006) compared various estimates of the time dependence of elevation changes and mass balance for the

Greenland Ice Sheet and found that snow-accumulation rates and airborne laser altimetry measurements

support the idea of a sharp decrease in ice sheet mass beginning in the mid-1990s. The mass decrease is

inferred to come predominantly from the coastal areas of the ice sheet, as seen in Figure 2.9(a). Figure

2.9(b) shows mass balance estimates for the Greenland Ice Sheet for the 1958-2007 period resulting from

aircraft radar campaigns, estimates of ice discharge, and positive-degree-day estimates (Box et al., 2006),

which indicate that the ice sheet was close to mass equilibrium during the 1970’s and 1980’s, with a rapid

melting onset during the 1990’s and onwards (Rignot et al., 2008). Other studies of uplift rates measured

from GPS ground stations located around the Greenland ice sheet (Jiang et al., 2010; Khan, S. A. et al.,

2015a, 2016), have found regions peripheral to the Greenland ice sheet to be uplifting at an accelerating

rate, also indicating an accelerated melting of land ice having also occurred since the mid-1990s. These

results, combined with radar altimetry, laser altimetry and gravimetric measurements (for the post-2002

GRACE era) are seen in Figure 2.9(c) (Khan, S. A. et al., 2015a). Other studies also indicate a similar

behaviour for the West Antarctic Ice Sheet, albeit with a later onset of the melting phase (Velicogna and

Wahr, 2013; IPCC, 2013). Mass balance results for the ice sheets covering Antarctica and Greenland,

expressed in Sea Level Equivalent (SLE), are summarized in Figure 2.9(d).

The results indicating an accelerated melting of large ice sheets are robust, but extending these results

to the smaller mountain glaciers that cover high-altitude areas is not straightforward. These smaller ice

catchments are widely distributed, and can be found notably in northern and Arctic regions (Canada,

Russia, Svalbard, Iceland, Scandinavia and Alaska), the western United States and Canada, the southern

Andes (including Patagonia), Central Europe, New Zealand and the Central Asian mountains. Estimates

of the mass balance of these mountainous regions over the 20th century indicate that there are strong

regional differences between the response of mountain glaciers, with some showing an accelerated negative


Figure 2.9: Representation of the best estimates of ice sheet mass balance values or indicators from theGreenland and Antarctic ice sheets. (a) Rates of surface elevation change (dS/dt) for the central (red)and coastal (blue) regions of the Greenland Ice Sheet, coming from snow-accumulation estimates (box a),solely airborne radar measurements (boxes b and c), or from a combination of airborne and satellite radarmeasurements (box d). Figure from Thomas et al. (2006), reproduced with permission. (b) Total massbalance (TMB) and surface mass balance (SMB) estimates for the Greenland ice sheet over 1958-2007,using interpolated (light blue diamonds) and observed (dark blue circles) values (uncertainty of ± 60Gt/yr). The other lines show the anomalies comprising the total values (SMB anomalies in green circles,interpolated ice discharge (D) in red squares and observed ice discharge (D) in pink triangles). Figurefrom Rignot et al. (2008), reproduced with permission. (c) Total mass balance estimates of the Greenlandice sheet over 1992-2012, using GRACE observations (red), laser altimetry (green), radar altimetry(blue), so-called "input-output" methods (precipitation and ice discharge estimates, in black), and theIMBIE (Ice-sheet Mass Balance Inter-Comparison Exercise) combination of the previous observations(light blue). Figure from (Khan, S. A. et al., 2015a), reproduced with permission. (d) Rate of ice-sheetloss equivalent averaged over 5-year periods over Antarctica (red), Greenland (blue), and a combinationof the two (green), with corresponding uncertainties, coming a large combination of estimates over thetwo regions collected by IPCC (2013). Figure from IPCC (2013), reproduced with permission.


mass balance since the mid-century (in particular in Patagonia and Western North America), and other

regions showing an inconclusive signal. Overall, the studies of Cogley (2009) and Marzeion et al. (2012)

suggest a likely acceleration in the melting rates of the ice cover in most of these mountainous regions

(although not conclusive, given the uncertainty level), but it should be noted that Gardner et al. (2013),

who focused solely on the 2003-2009 time period, have found no significant acceleration in mountain

glacier mass loss. Whether this inconsistency is indicative of any long-term mass balance signal that

applies to the earlier time periods not covered by the study or simply a short-term discrepancy is not

clear (IPCC, 2013; Gardner et al., 2013; Cogley, 2009; Dyurgerov, 2010). In the context of the work

presented in this chapter, the regional variability in the mass balance estimate would need to be taken

into account, as ice catchments at lower latitudes would be expected to contribute more significantly to

changes in polar wander. The summarized results of these analyses were presented in IPCC (2013) for

different time periods are shown in Table 2.3.

Table 2.3: Average annual rates of global mass change in Gt yr−1 for different time periods for all glaciersglobally, except those peripheral to the Antarctica and Greenland ice sheets, with 90% uncertainty range,with data sources. The data was originally compiled in IPCC (2013).

Time period Reference Mass balance for all glaciers (Gt yr−1)

1901-1990 Marzeion et al. (2012) -197 ± 241971-2009 Cogley (2009); Marzeion et al. (2012) -226 ± 1351993-2009 Cogley (2009); Marzeion et al. (2012) -275 ± 1352005-2009 Cogley (2009); Marzeion et al. (2012) -301 ± 1352003-2009 Gardner et al. (2013) -215 ± 26

2.4 Perspectives and future work

We infer that these melting events would induce, depending on their origin, changes in either or both of

the rotational observables under consideration, with the melting events closer to the rotation axis (i.e.

the poles) inducing changes in the oblateness of figure of the planet, and hence of the LOD, while melting

events occurring further from the axis of rotation affecting predominantly polar wander observations.

It is also probable that these melting events have an impact on any comparison between observations

of the time derivatives of the degree two and order one Stokes coefficients observed by the Gravity

Recovery and Climate Experiment (GRACE) satellites and the values predicted by GIA models, such

as the earlier ICE-5G (VM2) model of the GIA process (Peltier and Luthcke, 2009). In their analysis,

the authors suggested that the misfit could be due to the fact that the ICE-5G (VM2) model of the

GIA process is designed to fit only data that is unambiguously associated with the Late Quaternary

ice-age. The model specifically does not include the influence of the additional changes in continental


Figure 2.10: Simple test of the hypothesis of the impact of recent land ice melting on inferences of therate of change of the J2 coefficient of the Earth’s gravitational field, prior and after the break in theJ2 time series identified in Roy and Peltier (2011). The post-1992 change is roughly explained by theprogressive addition of additional land ice melting from the polar regions of Greenland (GRN), WestAntarctica (WAN) and Alaska (AK) in the model. The rates of melting for these regions (in global sea-level rise equivalent, in mm/yr) were inferred by Peltier (2009) from the Gravity Recovery and ClimateExperiment (GRACE) satellite observations of the time dependence of the Earth’s gravitational field.Reproduced from Peltier et al. (2012), with permission from the American Geophysical Union.


ice-cover associated with the modern global warming process. The validation of the Peltier and Luthcke

(2009) hypothesis concerning the origins of the inferred misfit between the ICE-5G (VM2) prediction of

the time derivatives of the degree two and order one Stokes coefficients and the GRACE inferences of

the same properties of Earth’s gravitational field requires the development of a forward model that is in

accord with the known characteristics of modern continental ice-sheet disintegration and global sea-level

observations.

Initial work on this front has indicated that a relatively good fit to the post-1992 value of J2 inferred

in Roy and Peltier (2011) can be recovered using a simple linear melting rate over the Alaskan mountain

ranges, Greenland and Antarctica (Peltier et al., 2012), presented in Figure 2.10. Recently, Adhikari

and Ivins (2016) have shown that, using a simple model considering only the elastic response of the solid

Earth to recent changes in terrestrial water storage and land ice cover can explain a good fraction of

the change in polar wander and J2. However, to fully reconcile recent observations with the full theory

describing the dynamic response of the Earth system to the recent melting of ice sheets and mountain

glaciers induced by modern climate change, future work will require the augmentation of the latest GIA

model structure (presented in the following chapters of this work) to include this additional source of

rotational forcing. In the absence of the demonstration herein of the occurrence of a marked change in

rotational state prior to the launch of the GRACE satellite, work of this kind would not be warranted.

Chapter 3

GIA constraints from the U.S. East

coast: the ICE-6G_C (VM6) model

The study of the Glacial Isostatic Adjustment (GIA) process, in which the solid Earth responds visco-

elastically to varying ice and water loads at its surface which are associated with the Late Quaternary ice-

age cycle, has significantly contributed to our understanding of both paleoclimatological phenomenology

and solid Earth geophysics. A contribution in the latter area has involved the provision of robust

constraints on the effective viscosity of the planetary mantle, a crucial ingredient in the design of models

of the mantle convection process. Furthermore, comparisons of the relative sea-level histories predicted by

GIA models to the large and globally distributed database of geologically-derived records of such histories,

such as that originally assembled at the University of Toronto and described in Tushingham and Peltier

(1992), has led to significant advances in our understanding of the most recent cycle of Late Quaternary

glaciation and deglaciation. As previously mentioned, understanding the geographical distribution and

temporal variability of land ice cover over the approximately 100,000-year period of this cycle provides

the detailed boundary conditions of paleotopography and paleobathymetry that are required as a basis

for the reconstruction of ice-age climate conditions using modern coupled atmosphere-ocean climate

models (Peltier, 1994, 2004; Vettoretti and Peltier, 2013; Abe-Ouchi et al., 2015; Ivanovic et al., 2016).

Because the quality of the Relative Sea Level (RSL) database is especially high in the period since LGM

(which occurred between 26,000 and 21,000 years ago), the focus upon this period, both geophysically

and climatologically, has been especially intense. This has led to the development of regional RSL

databases with exceptional quality control such as that for the United Kingdom (Shennan et al., 2002),

for the Canadian land mass that was once covered by the vast Laurentide/Cordilleran/Innuitian ice

48

Chapter 3. GIA constraints from the U.S. East coast: the ICE-6G_C (VM6) model 49

sheet complex (Dyke et al., 2002; Dyke, 2004), the Caribbean region (Khan, N. S. et al., 2015b), the

Mediterranean region (Vacchi et al., 2014, 2016) and for both the East and West coasts of the continental

United States (Engelhart et al., 2011; Engelhart and Horton, 2012; Engelhart et al., 2015). These two

last data sets will be of special interest for the purpose of the present section.

It is also important to appreciate the role played by GIA modelling in the context of the current

renewed episode of continental deglaciation caused by the rapidly increasing atmospheric concentration

of greenhouse gases. Our observations of the ongoing melting of the great polar ice sheets on Antarctica

and Greenland, and of many small ice catchments and glaciers (responsible for an important fraction

of the global rise of sea level occurring today) is significantly impacted by the remaining isostatic dis-

equilibrium associated with the last ice-age cycle, the signal of which must be accounted for in order

to more clearly identify the global warming component (e.g. Peltier and Tushingham, 1989; Peltier,

2009; Velicogna and Wahr, 2013). In particular, the GIA correction is central to the analysis of the

time-dependent gravitational measurements provided by the Gravity Recovery and Climate Experiment

(GRACE) satellites, which provide constraints on the the recent melting rate of ice sheets and other

smaller ice catchments (e.g. Peltier, 2009; Peltier and Luthcke, 2009; Velicogna and Wahr, 2013; IPCC,

2013), but which can show some degree of variability depending on the GIA correction (Ivins et al., 2013;

Argus et al., 2014). Given the large potential economic and social impacts of future sea level change, it

becomes crucial to evaluate models of the GIA process against all available geophysical observables, at

all stages of the post-LGM deglaciation, and to consider them from a global perspective.

As will be discussed, the Atlantic coast of the continental United States is a region of special interest

for the study of the GIA process, given that data from this coast provides a transect of RSL history as-

sociated with the collapse of the forebulge induced outboard of the former Laurentide ice sheet that once

covered all of the Canadian landmass to the north. Given the availability of a newly compiled database

of 14C-dated records of sea-level evolution of very high quality for the Holocene period (Engelhart and

Horton, 2012), it has become possible to more stringently test the accuracy of recently constructed

GIA models than had previously been possible given the modest quality control to which previous such

compilations were subject. Testing the performance of these models and determining whether further

adjustments are necessary to their current iteration to recover a best fit to this new high-quality data

for the forebulge region are primary goals of the work presented in this chapter. The results presented

here follow the peer-reviewed contribution of Roy and Peltier (2015) (Roy, K., and Peltier, W. R. (2015),

’Glacial isostatic adjustment, relative sea level history and mantle viscosity: reconciling relative sea level

model predictions for the U.S. East coast with geological constraints’, Geophysical Journal International

201(2), 1156-1181), with a slightly expanded theoretical background section providing more information


about the mathematical structure of the model (section 3.2), as well an additional section to introduce

the relative sea level records used in the study (section 3.4.1).

3.1 General approach and strategy

Models of the GIA process depend on two fundamental inputs: a history of ice-sheet loading and a

model of the radial variation of mantle viscosity. The methodology traditionally employed to address

the complex interaction between the depth-dependent viscosity of the mantle and ice loading history

inputs to GIA models has taken advantage of the fact that certain characteristics of some of the geological

observables are relatively insensitive to either the exact nature of the ice loading history or to the viscosity

in certain ranges of depth, thereby enabling the development of an iterative approach that maximizes the

information content gathered from all available data (an overview of these considerations is presented in

Peltier (1998b)). This has involved the determination of a spherically-symmetric viscosity structure for

the Earth’s mantle from certain specific constraints that are understood to be sufficiently independent

of the deglaciation history, such as the relaxation times that are characteristic of the "rebound" of the

surface of the solid Earth in regions that were previously covered by thick continental ice sheets at the

Last Glacial Maximum in central parts of North America and Fennoscandia (Mitrovica and Peltier, 1991,

1993a,b, 1995; Wieczerkowski et al., 1999), or from rotational state constraints that provide information

about the viscosity of the lower mantle (Wu and Peltier, 1984; Peltier and Jiang, 1996).

GIA models may be tested and refined by comparing their local predictions of relative sea-level history

to geological inferences based upon appropriate sea level indicators. The U.S. Atlantic coast is a region

of particular interest in this regard, due to the fact that data from the length of this coast provides a

transect of the forebulge associated with the former Laurentide ice sheet. High-quality relative sea level

histories from this region are employed herein to explore the ability of current models of mantle viscosity

to explain the inferred evolution of relative sea level that have accompanied forebulge collapse following

deglaciation. Existing misfits are characterized, and alternatives are explored for their reconciliation.

Then, it will be demonstrated that a new model of mantle viscosity (referred to herein as VM6) is able

to eliminate the majority of these misfits when coupled with the latest model of deglaciation history

ICE-6G_C, while continuing to reconcile a wide range of other important geophysical observables, as

well as additional relative sea-level data from the U.S. West coast which also record the collapse of the

forebulge but which have not been employed in tuning the viscosity profile to enable ICE-6G_C (VM6)

to fit the East coast data set.

In this work, the global ice-sheet loading history is fixed to that of the most recently constructed


model, known as ICE-6G_C (Peltier et al., 2015), while the radial profile of mantle viscosity is initially

fixed to the VM5a profile of Peltier and Drummond (2008) or to the variant upon it labelled VM5b

(Engelhart et al., 2011). Characterizing these misfits and attempting to remove them solely through

appropriate variations in mantle viscosity will lead to a further refined model of mantle viscosity referred

to as VM6.

The systematic series of sensitivity analyses that have led to this refined model, which involves a

search through the space of plausible parametrization of viscosity depth dependence, will in many ways

be ad hoc rather than based upon the application of the formal Bayesian methodology that led to the

original VM2 model of Peltier (1996b, 1998b,c). The reason this approach has been followed is that

there does not appear to be an equally simple parametrization of the RSL histories from the forebulge

region as that possible for histories from the regions previously covered by thick accumulations of land

ice, where sea level histories have a simple exponential form described entirely by an amplitude and

a characteristic relaxation time. Prior to implementing a brute force statistical search based upon

Monte Carlo methods, a restricted search will be pursued, during which a physical understanding of the

sensitivity of RSL histories from sites along the U.S. East coast to variations of viscosity over various

depth intervals will be discussed.

3.2 Modelling the GIA process: Theoretical background

The mathematical structure of the global GIA process was first developed in the 1970s in a series of

seminal papers (Peltier, 1974, 1976; Farrell and Clark, 1976; Peltier and Andrews, 1976; Clark et al.,

1978; Peltier et al., 1978). It has since been extended and further refined from this earliest form. The

current and most elaborate version of the theory (and the one that is employed herein) takes the form

of a Fredholm equation of the second kind, an integral equation whose solution provides a prediction of

the time varying level of the sea with respect to the continuously deforming (visco-elastically) surface of

the solid Earth (Peltier et al., 2015). The explicit form of this Sea Level Equation (SLE) is as follows:

S(θ, λ, t) = C(θ, λ, t)

t∫

−∞

∫∫

Ω

[

L(θ′, λ′, t′)GL(γ, t− t′) + ΨR(θ′, λ′, t′)GTR(γ, t− t′)

]

dΩ′dt′

+

C(θ, λ, t)

[

Φ(t)

g

]

,

(3.1)

where S(θ, λ, t) represents the local level of the sea relative to the local surface of the solid Earth at

time t and at the geographical location with co-latitude θ and longitude λ. The function L(θ′, λ′, t′) is


the surface mass loading history per unit area which consists of both grounded land ice and ocean water

components. It may be written in the composite form:

L(θ, λ, t) = ρII(θ, λ, t) + ρWS(θ, λ, t), (3.2)

where ρI and I(θ, λ, t) represent ice density and ice thickness history, whereas ρW and S(θ, λ, t) respec-

tively represent water density and relative sea level. The integral nature of Equation (3.1) is thus clear

on the basis that the unknown field S(θ, λ, t) appears both in the integrand of Equation (3.1) and on its

left-hand side. The function ΨR(θ′, λ′, t′) represents the change in centrifugal potential associated with

the GIA process itself. These rotational changes are induced by the large mass movements associated

with the glaciation and deglaciation processes. The solution of the Sea-Level Equation to be employed

herein includes the renormalized representation of the rotational feedback terms discussed explicitly in

Peltier et al. (2012) and Peltier et al. (2015). This renormalization (see equation (3b) of Peltier et al.

(2015)) is necessary to ensure that changes in the centrifugal potential at any single location do not

impact the centrifugal potential at any other location (such that they only exert "local" influence).

The function C(θ, λ, t) in Equation (3.1) is the so-called ocean function (Munk and MacDonald,

1960), which has a value of unity over oceans and zero elsewhere. The time-dependent function Φ(t)g

is constructed so as to ensure that there is a precise match between the mass of water generated by

melting land ice and that which appears in the ocean basins (see Peltier (1998c); Peltier et al. (2012) for

thorough discussions of this term). The complete Sea-Level Equation (3.1) is solved in such a way as

to simultaneously compute the time dependence of the ocean function using the iterative methodology

introduced in Peltier (1994, 1998a) and Peltier and Fairbanks (2006).

ΨR(θ, λ, t), the time and space dependent change in the centrifugal potential associated with the

GIA process, can be expressed as a first-order perturbation, following Dahlen (1976):

ΨR(θ, λ) = Ψ00Y00(θ, λ) +

1∑

m=−1

Ψ2mY2m(θ, λ), (3.3)

where Ψ00 = 23mz(t)Ω

20a

2,

Ψ20 = − 23mz(t)Ω

20

√

1/5,

Ψ2,+1 = (ω1 − iω2)(

Ω2

0a2

2

)

√

2/15,

Ψ2,−1 = −(ω1 + iω2)(

Ω2

0a2

2

)

√

2/15,

mx,y,z =ωx,y,z

Ω .


In Equation (3.1), the functions GL(γ, t− t′) and GTR(γ, t− t′) are Green functions representing the

impulse response of the surface of the Earth which, when convolved with the surface loading history and

the centrifugal potential loading, predict the evolving separation between the surface of the solid Earth

and the surface of the sea. The angle γ represents the separation between the source point and the field

point at which the response is to be determined. The impulse function GL was first described in Peltier

(1974), subsequently refined in Peltier and Andrews (1976) and Peltier (1985), and takes the form:

GL(γ, t) =a

Me

∞∑

l=0

[

1 + kLl (t)− hLl (t)]

Pl(cos γ), (3.4)

where a and Me refer to the Earth’s mean radius and mass, while the Pl(cos γ) terms are the standard

Legendre polynomials evaluated at angle γ. A similar formulation exists for the Green function GTR(γ, t)

associated with the change in centrifugal potential ΨR(θ′, λ′, t′), the form of which has recently been

reviewed in Peltier et al. (2012):

GTR(γ, t) =

1

g

∞∑

l=0

2l + 1

4π

[

1 + kTl (t)− hTl (t)]

Pl(cos γ). (3.5)

In Equation (3.5), kLl (t) and hLl (t) are so-called surface load Love numbers (Love, 1911), dimensionless

quantities that capture the nature of the visco-elastic response of the Earth’s surface to surface loading.

They are time-dependent, visco-elastic extensions of the equivalent elastic surface load Love numbers of

Farrell (1972), and are related to the radial displacement of the surface induced by the load (h) and the

induced gravitational potential perturbation (k). The exact form of the impulse functions is determined

using a linear Maxwell rheology for the Earth’s interior and can be expressed, following Peltier (1976,

1985), at spherical harmonic expansion degree l, as:

hLl (t) = hL,El δ(t) +

m∑

j=1

rlje−sljt, (3.6a)

kLl (t) = kL,El δ(t) +

m∑

j=1

qlje−sljt, (3.6b)

where hL,El and kL,E

l are the elastic surface load Love numbers of Farrell (1972), while rlj , qlj and slj

are the amplitudes and relaxation times of the m exponential decay responses that are required to

define the time-dependent behaviour of the Love numbers, determined as the zeros of an appropriate

secular function (Peltier, 1985) or using a collocation method (Peltier, 1974, 1976) while using a specific

model of the elastic properties of the Earth’s interior (using here the Preliminary Reference Earth Model


(PREM) of Dziewonski and Anderson (1981)) and of the radial variations in viscosity. Another suite of

Love numbers, lLl (t), describes the tangential displacement resulting from an applied surface load, and

takes the form:

lLl (t) = lL,El δ(t) +

m∑

j=1

tlje−sljt, (3.7)

where lL,El is the equivalent elastic surface load Love number and the tlj coefficients are the amplitudes

of the normal mode expansion. The Love numbers at expansion degree l (hl, kl and ll) are linked to

the induced radial displacement Ul, tangential displacement Vl and perturbation in the gravitational

potential Φ3,l:

Ul

Vl

Φ3,l

= Φ2,l

hl

g

llg

kl

, (3.8)

where Φ2,l refers to the gravitational potential perturbation of Legendre degree l due to a point mass

load applied to the surface of the Earth to determine its impulse response, the exact expression of which

is derived in Peltier (1974).

In Equation (3.6), corresponding time-dependent visco-elastic tidal loading Love numbers kTl (t) and

hTl (t) are included, and can be expressed as:

hTl (t) = hT,El δ(t) +

m∑

j=1

r′jle−sljt, (3.9a)

kTl (t) = kT,El δ(t) +

m∑

j=1

q′jle−sljt, (3.9b)

lTl (t) = lT,El δ(t) +

m∑

j=1

t′jle−sljt. (3.9c)

The required inputs to Equation (3.1) consist of two geophysical fields, namely a global ice loading

history I(θ, λ, t), and a spherically-symmetric radial viscosity structure (on the basis of which the exact

forms of the time-dependent visco-elastic Love numbers included in the equation, kLl (t), hLl (t), k

Tl (t)

and hTl (t), are determined). Using this information, Equation (3.1) is solved iteratively to determine

the rotational feedback component and the time dependence of the ocean function C(θ, λ, t) (the land-

ocean interface) induced by the changing ice cover, with the first step neglecting the rotational feedback

component and fixing the ocean function C(θ, λ, t) to its present value. The time dependence of the ocean

function is determined following the steps of Peltier (1994), in which topographical self-consistency is

obtained by recognizing the arbitrariness of the point with respect to which S(θ, λ, t) is defined. In other


words, the difference between the predicted present sea level, S(θ, λ, tpresent), and modern topography

with respect to sea level, (Tpresent), is calculated (using Tdiff (θ, λ) = T (θ, λ, tpresent)−S(θ, λ, tpresent)).

Past topography at any time t is then determined by adjusting the sea level history with respect to the

observed topographic difference at present (Tdiff ), such that T (θ, λ, t) = S(θ, λ, t) + Tdiff (θ, λ). The

ocean function is then updated taking into account this new topographic field T (θ, λ, t), the ice sheet

loading history I(θ, λ, t), and effects related to grounded ice and the inundation that results when regions

initially covered by ice become inundated when it melts (also known as "implicit ice") (Peltier, 1998a).

For the rotational feedback component (Equation 3.3), an initial estimate of its value is determined from

ice and ocean loading histories obtained from solving the Sea Level Equation (3.1) without the rotational

feedback. Then, Equation (3.1) is solved again, but this time incorporating the influence of this first

estimate of the rotational feedback ΨR(θ, λ, t). The iterative process is repeated until convergence is

achieved. For both the rotational feedback component and the time dependence of the ocean function,

this convergence occurs after only a few iterations, because of the relative low amplitude of the rotational

and topographic corrections (Peltier, 1998c). It results in a solution for the sea level history that is both

gravitationally and topographically self-consistent.

3.3 The ICE-NG models of ice sheet loading history

The first input required to infer a global evolution of relative sea level is a representation of the history

of ice sheet loading associated with the last glaciation/deglaciation process. Such models are largely

constrained on the basis of surface geomorphological evidence, such as carbon-datable material from

terminal moraines, which record the location of the margins of a retreating ice sheet. In the recent past,

cosmogenic exposure age dating, which may be invoked to determine the time since a surface outcrop

or glacial erratic boulder has been exposed following the retreat of an ice sheet that once covered the

location, has also become a widely employed tool (e.g. Dyke et al., 2002; Dyke, 2004; Whitehouse et al.

(2012); Argus et al. (2014)). These methods are particularly suited to infer the evolution of the margins

of large concentrations of ice, but most often only weakly constrain their former thickness. Significant

progress in the estimation of the evolving thickness of ice was primarily realized through the application

of GIA analyses. Because of the simple exponential relaxation form of relative sea level records from

regions that were previously covered by thick ice sheets (e.g. Northern Canada, Fennoscandia), one may

employ the amplitude of the exponential rebound curves to "weigh" the thickness of the ice that must

have been removed to induce the amplitude of the rebound observed. This requires, however, that the

viscosity structure of the Earth’s deep interior be known. It is possible to separate both problems by first


determining the viscosity of the Earth on the basis of observations of the relaxation times characteristic

of the RSL time series in previously glaciated regions, before using the amplitude of these rebound time

series to focus on ice sheet thickness (of course, granted that such additional constraints from the time

of load removal exist) (e.g. Peltier, 1982, 1998a, 2007).

However, the early attempts to determine the history of ice sheet loading during the last ice age cycle

(e.g. Peltier and Andrews, 1976; Wu and Peltier, 1983) benefited subsequently from the development of

large databases of relative sea level histories spanning the whole globe, which led to the development of

the ICE-3G structure, presented in Tushingham and Peltier (1991). Subsequent changes, which included

refinements in the analysis methodology and the use of further observational constraints (such as the

high-quality Barbados RSL history based upon U-Th dating of coral terraces (Fairbanks, 1989)) resulted

in the improved ICE-4G model (Peltier, 1994). Later, the ICE-5G model (Peltier, 2004) was obtained

from a careful analysis of geodetic and gravity-based data, most notably over North America (Argus

et al., 1999), and extended the ice loading history to the entirety of the most recent glacial-interglacial

cycle. Other improvements to the model included the application of refined GIA data over the British

Isles (Peltier et al., 2002; Shennan et al., 2002) and new observations of far-field sea-level histories

(Peltier, 2004). The ICE-5G model has been widely used in conjunction with the VM2 viscosity profile

of the mantle to provide a global theory of relative sea-level adjustment due to the glaciation/deglaciation

process.

Further refinements to the ICE-5G loading history over North America have more recently been

introduced, in which the misfits to Global Positioning System (GPS) observations of vertical motion of

the crust documented in Argus and Peltier (2010) have been eliminated by appropriate modifications of

the ice thickness history over North America, Northwestern Eurasia and Antarctica. This process has

led to the ICE-6G_C model (Argus et al., 2014; Peltier et al., 2015). Figure 3.1 shows a comparison

of the LGM topographic height anomaly generated by this most recent model with respect to LGM

sea level over North America with that of the precursor ICE-5G model. Inspection will show that the

primary modifications of the ice thickness distribution consist of a thinning of the Laurentide ice sheet in

central Canada and a thickening over northern Quebec and Labrador, as well as over the northern border

region between the Canadian provinces of Alberta and British Columbia. The Antarctic component of

the global model is discussed in Argus et al. (2014). The ICE-6G_C model will be employed for the

purpose of the analyses presented in this chapter.


Figure 3.1: (a) Geographical extent and topographic height anomaly at LGM in the Northern hemi-sphere for the ICE-6G_C ice loading history. (b) Difference in ice thickness between the ICE-6G_Cloading history and its ICE-5G predecessor (Peltier, 2004). From (Roy and Peltier, 2015), adapted fromVettoretti and Peltier (2013) and used with permission.

3.4 Analysis of the performance of the ICE-6G_C ice loading

history as a function of viscosity model

Among the regions for which high quality records of past sea level based on the recovery of biological

material from past coastal environments now exist, the eastern seaboard of the United States is of

particular interest, as it straddles the glacial forebulge consisting of an upwarping of the crust induced

by the viscous flow of material from the Earth’s interior beneath the previously ice sheet covered region

into its periphery. The regional extent of this forebulge covers the entirety of the coast, from just south of

the LGM margin of Laurentian ice in New England, through the peak of present-day subsidence around

New Jersey and Chesapeake Bay, and into the Caribbean Sea, with its trailing edge being located in the

near vicinity of the critical island of Barbados.

3.4.1 Relative sea-level history reconstructions

Reconstructions of past sea level rely on different types of information sources, which include, in order

of increasing reach in the past, (1) instrumental records of sea level (tide gauges, satellite altimetry), (2)

archaeological evidence, (3) biological indicators and (4) geological evidence of past shoreline features

(Kearney, 2009).


One of the primary observational data sets providing constraining information on GIA model re-

constructions of past sea level evolution is the isotopic dating of relevant biological markers interpreted

in the context of their possible elevation range with respect to relative sea level at the time of their

growth (e.g. van de Plassche, 1986). Examples of such indicators include shells of intertidal or marine

mollusc species (providing a sea level range or a marine limit, respectively), high/low marsh plants and

salt marsh micro-fossils indicating a freshwater limit, or coral samples that indicate a possible range of

relative sea level based on the living depth range of specific species recovered in situ. Today, an extensive

number of such records exist at a wide range of locations around the globe, and their use has greatly

benefited from the availability of precise geolocation techniques based on the Global Positioning System

(GPS). These measurements also take advantage of precise dating techniques, and in particular of the

14C and U/Th methods of radioactive dating. 14C dating (Arnold and Libby, 1949; Libby, 1952) relies

on the natural presence of 14C radionuclides in the atmosphere, which are produced by the constant

bombardment of the Earth’s atmosphere by natural cosmic rays and eventually get absorbed by living

organisms after mixing in the atmosphere. After their death, 14C atoms progressively undergo β-decay

to become 14N, allowing the determination of the moment of death of a biological sample if 14C atoms

can be properly counted (using accelerator mass spectrometry) and if a proper transfer function from

14C age to sidereal age is used (e.g. Shennan and Horton, 2002). This mapping is not linear due to

changes in cosmic ray bombardment rates, and depends on cross-calibration with other methodologies,

such as dendrochronology or U/Th methods (van de Plassche, 1986; Kearney, 2009). Information on the

indicative range of sea level of each species from which biological samples are recovered is crucial in the

inference of past sea level (e.g. van de Plassche, 1986). Various organisms have been used to determine

past sea level extent, including numerous shell species living in shallow waters close to tidal mean sea

level, tree stumps or whale bones.

One of the most promising means of recovering past sea level ranges relies on the careful study of

intertidal deposits in past salt marsh environments. Sediments from these environments, common in most

coastal areas around the world, provide information about the separation between high (freshwater) and

low (intertidal) marshes, when correctly interpreted in light of reference water levels, mean tide levels

and highest astronomical tides and the cross-referencing of indicative altitude ranges for the available

vegetation and intertidal microfossil assemblages (van de Plassche, 1986; Engelhart and Horton, 2012).

Figure 3.2 shows a schematic representation of a coastal salt marsh environment and how the information

recovered from sediments can be interpreted and turned into a relative sea level curve. Beyond providing

a direct inference of past sea level height with respect to present (so-called index points), biostratigraphic

data can also indicate deposition in a purely terrestrial or marine environment. In this case, it will only


provide limiting information (either marine or terrestrial) about past sea level. Once an elevation and an

age are determined, a careful error analysis must be performed in order to correctly account for altitude

errors, extrusion errors and past tidal range inferences (Engelhart and Horton, 2012). Another potential

source of error is related to the potential natural compaction of the sediment column. In this respect,

so-called base of basal index points are of particular interest, as they are recovered from the sediments

that are located within a few centimeters of an incompressible substrate, such that they are held to be

unaffected by compaction (Engelhart and Horton, 2012; Horton et al., 2013).

3.4.2 The Engelhart and Horton (2012) data set of U.S. East coast RSL

evolution

In this analysis, the quality-controlled database presented by (Engelhart and Horton, 2012) (with a

preliminary iteration first used in Engelhart et al. (2011)) will be employed. The data set consists of

686 individual sea-level indicators from locations along the coast ranging from northern Maine to South

Carolina. These data may be assigned to 16 distinct locations, which are indicated on Figure 3.3.

A substantial advantage of this particular data set over the previously employed tools used for

the same region (e.g. Tushingham and Peltier, 1992; Peltier, 1996a) derives from the fact that the

methodology employed for its construction was consistent throughout the coast and significant efforts

have been invested in assigning meaningful, consistent error estimates to each of the sea level indicators

employed, especially with regards to the indicative range of habitat of each sample, altitude, extrusion

errors and tidal level errors Engelhart and Horton (2012). It should be noted, however, that compaction

errors were not included in the original data set, but were subsequently added for the New Jersey site

(Horton et al., 2013). The data used here for the New Jersey site includes the effect of compaction.

3.4.3 Analysis of the performance of the VM5a and VM5b viscosity struc-

tures

The availability of this new, high-quality data set of relative sea level index points and limiting indicators

provides an opportunity to test the performance of current GIA models and their inference of the viscosity

structure of the Earth’s interior. Early models of its radial structure, such as the VM1 model (Peltier,

1986), were simple two-layer models that attempted to distinguish the viscosity between the average

values appropriate for the upper and lower mantle with the boundary between these regions defined

by the phase transition interface at 660 km depth (Peltier et al., 1986; Tushingham and Peltier, 1992).

Using a theoretical framework based on the formal Bayesian methodology (Mitrovica and Peltier, 1991,


Figure 3.2: Schematic representation of a salt marsh environment and how local RSL evolution canbe eventually reconstructed from radiocarbon-dated biological markers in sediment cores. (a) Typicalvegetation zones in the typical tidal frame for a mid-latitude coastal environment, such as the U.S. Eastcoast. (b) Depth and altitude of samples in a typical core showing a transition from freshwater to marineenvironment. (c) Conversion of the recovered information into a collection of dated sea level index points(estimate of past sea level height), or limiting data. The figure notes Reference Water Levels (RWL),Indicative Range (IR), Altitude (A), Mean Tide Level (MTL), Mean High Water (MHW), Mean HigherHigh Water (MHHW) and Highest Astronomical Tide (HAT). Figure from Engelhart and Horton (2012),reproduced with permission.


Figure 3.3: (a) Geographical location of the 16 composite sites identified in Engelhart and Horton (2012)along the U.S. East coast and used in the current work. (b) Geographical extent of the 14 compositesites used in the current work for the U.S. West coast (Engelhart et al., 2015). From Roy and Peltier(2015).


1993a, 1995), various observables were considered in an effort to better understand their resolving power

for the inference of viscosity depth dependence. Thereafter, Peltier (1996a) used the totality of the

data, each type having its own resolving power insofar as viscosity depth dependence is concerned, to

demonstrate that when all such data were employed one could obtain estimates of mantle viscosity from

the surface to the core-mantle boundary. These data included the Fennoscandian relaxation spectrum of

relaxation times (described below), a suite of site-specific relaxation times in regions heavily influenced

by the former Laurentide and Fennoscandian ice sheets, as well as two anomalies of the Earth’s rotational

state: the speed and direction of the true polar wander phenomenon and the rate of nontidal acceleration

of the Earth’s axial rotation rate (Peltier and Jiang, 1996). The resulting profile of radial variations

in viscosity, capped by a 90-kilometre elastic lithosphere, is referred to in the literature as VM2 and

is presented in Figure 3.4(a). Despite the high-quality fits that the VM2 viscosity profile combined

to the ICE-5G history of ice sheet loading provided when compared to a large number of globally

distributed 14C-dated relative sea level histories (Peltier, 2004), discrepancies were subsequently found

to exist between predicted and observed modern-day horizontal motion of the crust of the solid Earth

over North America (Argus et al., 1999; Sella et al., 2007). These results were obtained from an extensive

network of GPS observations over the continent. Peltier and Drummond (2008) demonstrated that the

horizontal velocity misfits could be eliminated by a simple modification of the shallow structure that was

characteristic of the VM2 viscosity profile by introducing radial viscosity stratification of the near-surface

lithosphere, a feature that is expected based on the exponential temperature dependence of the creep

resistance of a solid. Their preferred viscosity profile, referred to as VM5a, is a 5-layer approximation to

the VM2 viscosity profile. It incorporates at the surface a 60-km elastic lithosphere overlaying a 40-km

region of high viscosity (1022 Pa·s), and becomes a multi-layer average of the VM2 model at greater

depths.

Engelhart et al. (2011) compared predictions of relative sea-level history at locations along the U.S.

East coast for the VM5a viscosity structure when combined with either the ICE-5G or an early version

of the ICE-6G ice loading history with their observational database and found persistent misfits between

them, irrespective of which of these loading histories was employed. In an initial simple attempt to

eliminate these misfits, they introduced the VM5b viscosity profile, a modified version of the VM5a

profile for which the viscosity was halved in the upper mantle and transition zone. Such a modification

was found to remove some of the identified discrepancies in the mid-Atlantic region, although significant

misfits remained for the southernmost part of the region covered by the data set. The VM5a and VM5b

profiles are presented in Figure 3.4(a), together with the parent VM2 profile.

The model predictions of relative sea level histories based upon the use of the VM5a and VM5b


Figure 3.4: (a) Radial variations of the viscosity in depth for the viscosity profiles VM5a (green), VM5b(blue) (note that VM5a and VM5b are identical except in the upper mantle), and VM2 (blue/green),for which VM5a is a five-layer approximation with added stratification (high viscosity layer between 60and 100 km depths) just below the elastic lithosphere (0-60 km depth). (b) Inverse relaxation time asa function of spherical harmonic degree obtained from observations of the glacial isostatic adjustmentof Fennoscandia, with the relaxation spectrum (black dashed line) and corresponding 1-σ uncertainties(dark gray area) calculated by Wieczerkowski et al. (1999), with the inferred 2-σ uncertainties extendingto the light gray area. The predicted spectra for the VM5a model (green) and for the VM5b model (blue)are superimposed. (c) Fit of the ICE-6G_C model predictions to the Barbados record of coral-basedsea level indicators when combined to the VM5a viscosity structure. From Roy and Peltier (2015).


viscosity profiles and of the ICE-6G_C ice loading history are first revisited and compared to the

Engelhart and Horton (2012) data set, using the most recent version of the University of Toronto model.

All model predictions include the impact of rotational feedback. The resulting comparisons are shown

in Figure 3.5 for the 16 distinct locations for which data is included in the compilation.

For the northernmost locations (Southern Maine, Massachusetts), inspection of the fits to the obser-

vational data shows that the VM5a profile performs better than VM5b, as the softer model of the upper

mantle and transition zone predicts the existence of a highstand, which is not observed in the observa-

tional records, and significantly under-predicts relative sea level changes over the past few millennia. It

is worthwhile to note that for Eastern Maine, neither VM5a nor VM5b succeeds in reconciling model

predictions and observations. In fact, the predictions of the two models bracket the observations. Pro-

gressing further south along the coast, the performance of the VM5a model degrades noticeably, while

the VM5b model is either the equal of VM5a or an improvement, depending on the region of interest. For

coastal Connecticut and New York, neither of the models adequately reproduces the observed relative

sea level evolution, but they rather bracket the observations. Continuing further south, the VM5b profile

begins to perform better than VM5a for coastal locations (in Long Island, New Jersey, Outer Delaware

and East Virginia), although misfits remain for observations older than 6.5 ka. VM5a and VM5b bracket

the available observations for the regions further inland (Inner Delaware, Inner Chesapeake). However,

progressing into North Carolina, the misfits between the observations and the model predictions for

both VM5a and VM5b become larger for the earliest part of the records. Model predictions become very

similar for both VM5a and VM5b, and a clear misfit can be identified prior to 5 ka for the locations in

North Carolina and South Carolina. Thus, although the VM5b profile does indeed provide an improve-

ment in performance for mid-Atlantic locations (Engelhart et al., 2011), important misfits remain for

the southernmost sites.

3.4.4 The fit to the Fennoscandian spectrum

An important caveat needs to be addressed concerning the marginal increase in performance of the

VM5b viscosity profile, which concerns its fit to one of the most important observational data sets

relating to the GIA process, namely the fit to the so-called Fennoscandian spectrum of relaxation times,

first introduced by McConnell (1968).

Fennoscandia is a region of considerable interest with regards to the study of the GIA process, due

to the large ongoing surface adjustments due to the influence of the former ice sheet which covered it

during the last glaciation-deglaciation cycle (e.g. Steffen and Wu, 2011 for a comprehensive review).


Figure 3.5: Comparison of the Engelhart and Horton (2012) data set of relative sea-level histories alongthe U.S. Atlantic coast for the 16 composite regions with the predicted relative sea-level history atthose locations for the ICE-6G_C model of ice-loading history combined to the VM5a (blue) and VM5b(black) radial viscosity profiles. Green data points represent sea-level index points, whereas blue crossesrepresent marine-limiting data and orange crosses represent terrestrial-limiting data. From Roy andPeltier (2015).


The Fennoscandian spectrum provides, as a function of deformation wavelength, a series of constraints

on the relaxation time associated with the GIA process, which have been inferred using a database of

six former strandlines from different locations along the coast of the Gulf of Bothnia and of the Gulf of

Finland. The analysis of McConnell (1968) has been shown to provide a constraint on Earth rheology

to a depth of approximately 1,500 km, with the considerable advantage of being relatively insensitive to

the temporal and geographical evolution of the thickness of the Fennoscandian ice sheet. This important

property has led to considerable interest being devoted to the area and to the data set employed (e.g.

Mitrovica and Peltier, 1993a; Peltier, 1998a; Wieczerkowski et al., 1999; Steffen and Wu, 2011). In

particular, Wieczerkowski et al. (1999) applied a damped least-squares solution methodology to the

strandline data of Donner (1995) to revisit the McConnell (1968) constraints. Their result provided a

somewhat modified inference of the relaxation time dependence on deformation wavelength, complete

with a 1-σ uncertainty range, which is reproduced in Figure 3.4(b) and extended to 2-σ, together with

the prediction of the spectrum for the VM5a and VM5b viscosity profiles.

While the VM5a profile succeeds in producing a relaxation time spectrum consistent with the infer-

ences of McConnell (1968) and Wieczerkowski et al. (1999) (unsurprisingly so, one might say, given that

it is a 5-layer approximation to the VM2 profile, which is heavily constrained by an earlier analysis of the

spectrum (Peltier, 1996a)), the VM5b profile fails to reproduce the Fennoscandian spectrum, especially

at longer wavelengths (lower spherical harmonic degree). The inability of the VM5b model to reproduce

this constraint indicates that the gain in performance from the VM5b profile in the mid-Atlantic might

not be acceptable from a global GIA perspective, as accepting it would require one to entertain models

with significant lateral heterogeneity of viscosity and our goal in the first instance is to attempt to define

the best possible spherically symmetric model of the Earth’s viscosity structure.

The focus of the further analyses to follow is therefore to attempt to optimally reconcile observations

of relative sea level evolution on the U.S. East coast with predictions of a spherically symmetric viscosity

model, while using other key global constraints related to the GIA process (such as the Fennoscandian

relaxation spectrum), as further independent checks of the final solution.

3.5 An exploration of alternative viscosity structures

The nature of the misfits identified between observational data and model predictions of relative sea

level evolution on the U.S. East coast is investigated in what follows. Whether they can be eliminated

using suitable variations of the radial mantle viscosity structure, while maintaining a good fit to other

geophysical constraints, such as the Fennoscandian spectrum of relaxation times, will be investigated.


3.5.1 Basic assumptions and methodology employed

In this analysis, a two-pronged approach will be employed, in which the reasonableness of some alternative

viscosity structures that have been suggested in the literature, and which differ significantly from the

VM2/VM5a profiles, will be explored. In particular, the V1 and V2 structures, based on the work of

Forte and Mitrovica (1996), will be analyzed. Following this initial analysis, and having demonstrated

that these alternatives are not in fact acceptable substitutes for the Toronto models, our second step

will be to perform a direct analysis of the sensitivity of model predictions of sea-level evolution on U.S.

East coast to radial viscosity variations at different depths. These tests are performed using the ICE-

6G_C ice loading history (Argus et al., 2014; Peltier et al., 2015), since it is a self-consistent global

model constrained, among other observables, by the total eustatic sea level change that has occurred

since the Last Glacial Maximum at the far-field Barbados location Peltier and Fairbanks (2006) (the

fit of the precursor ICE-6G_C (VM5a) model is provided in Figure 3.4(c)). The final step involves

merging the knowledge gained from all of the sensitivity analyses into a single model of radial viscosity

structure that minimizes the misfits between observations of past RSL evolution along the U.S. East

coast and model inferences of the same quantities, which will lead to a new viscosity structure named

VM6. The high quality of the fit of this model to the totality of the geophysical observables will then be

demonstrated, including not only the relative sea level evolution along the U.S. East coast to which the

model is tuned (as the current work is especially focused on recovering a fit to this data set), but also

other independent tests that are not included in the optimization process, including the Fennoscandian

spectrum of relaxation times, a comparison to relative sea level evolution data along the U.S. West coast

and the overall shape of the late Holocene forebulge.

In this analysis, it is important to note that the focus will be solely on radially symmetric models

of mantle viscosity. Even though lateral heterogeneity in mantle viscosity is fully expected to exist

based on our knowledge of the temperature dependence of the creep resistance of mantle material, our

goal is to develop a model of minimal complexity. It is therefore seen as important to explain as many

geophysical observables as possible using a simple one-dimensional viscosity model, as this is intended to

provide a suitable background structure onto which accurately inferred lateral viscosity variations may

be superimposed.

Another caveat is related to the ice model being fixed in all cases explored to ICE-6G_C. As models

of the GIA process have as inputs both the viscosity structure of the planet’s interior and a suitable

history of ice sheet and ocean loading on its surface, it should be noted that this work is thus part of an

iterative process during which those two input fields should be allowed to vary in turn. The goal of such


an approach is to reduce the overall misfit between GIA model predictions and all available relevant

geophysical observables, using the knowledge that the misfits to different observables are sensitive to

different model inputs in order to eventually reach convergence. The work presented in this section

focuses solely on the impact of the viscosity structure on these misfits, while follow-up work will also

focus on the ice loading history, and will enable us to address the issue of potential non-uniqueness in a

more thorough manner.

3.5.2 Error analysis and model performance

The performance of the viscosity structure variations considered in this study is first determined by a

visual inspection of the difference in relative sea level evolution at all locations along the U.S. East coast,

which provides an invaluable source of information about the complex variations in the temporal and

geographical responses of the forebulge. This analysis is complemented by a quantitative analysis of the

misfit between the relative sea level index and limiting data points and the model predictions, which is

described by a χ2-like relationship defined as:

χ2 =1

N

N∑

i=1

(

∆i

σi

)2

. (3.10)

In this relationship, N represents the number of historical data points (index points and limiting data

points) and σi is the uncertainty in the sea level height of the i-th data point (95% confidence level).

Also, ∆i represents the difference between an observed sea level indicator and the model prediction. For

sea level index points, it takes the simple form:

∆i = Si,obs − Si,pred, (3.11)

where Si,obs represents the observed sea level height and Si,pred is the model prediction. Limiting data

points are treated differently, as they are given as an upper or lower restriction on relative sea level. Their

contribution is considered to be non-zero only if the predicted sea level falls above a maximal terrestrial-

limiting data height or below a minimum marine-limiting data height. In the terrestrial-limiting case,

the expression for the difference ∆i is written as:

∆i =

Si,pred − hi,obs + σi, if Si,pred ≥ (hi,obs − σi)

0, otherwise

, (3.12)

where hi,obs is the height of the observed limiting observational data point. In the case of a marine-


limiting data point, ∆i takes the form:

∆i =

hi,obs − Si,pred + σi, if Si,pred ≥ (hi,obs − σi)

0, otherwise

. (3.13)

It should be noted that while this treatment of the limiting data induces some limited departure from

a pure χ2 methodology, it accurately captures the physical meaning of each observation. The uncertainty

in the age of the observational data points is taken into account in a simple manner: if the predicted

sea level falls below the 95% confidence height interval of an observation, the difference ∆i is calculated

for the time at the upper edge of the age uncertainty range, while it is calculated at the lower edge of

the age uncertainty range if the predicted sea level is above the 95% confidence height interval of an

observation.

In all cases, the χ2-like values are defined for each site along the U.S. East coast as well as for the

entire coast and a sub-ensemble of the four southernmost locations, in order to focus specifically on the

regional model response to depth-dependent mantle viscosity perturbations.

3.5.3 Alternative viscosity models: V1 and V2

An alternative family of viscosity models under consideration in this study is derived from a methodology

that involved a joint inversion of data that are related both to the mantle convection process and to glacial

isostatic adjustment (Forte and Mitrovica, 1996; Mitrovica and Forte, 1997, 2004). The convection data

sets used in those two studies include a map of the free-air gravity field, the non-hydrostatic ellipticity of

the core-mantle boundary from measurements of the Earth’s free-core nutation, the horizontal divergence

of tectonic plates (e.g. forte.etal.1991) and a map of dynamic topography (Forte et al. (1993); see also

Pari and Peltier (1996, 2000)). The GIA data includes postglacial relaxation times for the Fennoscandian

region and for Hudson Bay (two separate locations in James Bay and at Richmond Gulf). Their analysis

was performed using a non-linear, iterative procedure, and was further refined in the upper mantle using

site-specific decay times (Forte and Mitrovica, 1996; Mitrovica and Forte, 2004). In their analysis, a soft

layer was included just above the 660 km discontinuity (as had been earlier suggested by Forte et al.

(1991) and Pari and Peltier (1995)), based on the idea that it should be a consequence of transformational

superplasticity due to grain size reduction during the transformation of mineral phase that occurs as

material flows across the transition interface because of mantle convection. The resulting preferred

model of Mitrovica and Forte (2004), referenced herein as V1 (FM), is presented in Figure 3.6. The

second model shown in the same figure, referenced herein as V2 (FM), was presented in Moucha et al.


Figure 3.6: Comparison of the viscosity profiles VM5a (black) and VM5b (red) with the V1 (FM) profileof Mitrovica and Forte (2004) (green), and the V2 (FM) variation profile of Forte et al. (2009) (blue).From Roy and Peltier (2015).

(2008) and Forte et al. (2009), and is a variation based upon the same methodology, but where the thin

low-viscosity layer is practically absent and where the lower mantle is more viscous.

The relative sea-level history predictions for these two viscosity structures when combined with the

ICE-6G_C ice loading history are shown in Figure 3.7, where they are compared to the Engelhart and

Horton (2012) data set for a subset of four locations (Southern Maine, New York, Inner Delaware and

Southern South Carolina). For the northernmost part of the Atlantic coast of the United States, the two

FM profiles display predictions that are very similar to those of VM5a. However, for locations further

south along the East coast, into regions located within the remains of the proglacial forebulge associated

with the maximal extent of the former Laurentide Ice Sheet, the two FM profiles become less and less

able to explain the observations of past relative sea level, with a systematic decrease of the quality of

the fit as a function of latitude, which becomes particularly acute in the Carolinas. In particular, in

South Carolina, the misfit characterizing the use of the VM5a and VM5b profiles with ICE-6G_C is

notably worsened by a switch to the alternative FM models (V1 and V2). This could be explained by

their overall higher viscosity in the lower mantle, which would further extend the region of proglacial

forebulge collapse associated with deglaciation following the Last Glacial Maximum.

One possible limitation of this simple comparison might come from the fact the ICE-6G_C ice

loading history is optimized to be used with the VM5a model, and might thus be expected to fail when

combined with viscosity structures that are very different. One way to explore this possibility is to look

at predictions that do not depend significantly on the ice loading history, such as the relaxation time


Figure 3.7: Comparison of a subset of regions of the Engelhart and Horton (2012) data set of relativesea-level histories along the U.S. Atlantic coast with the predicted relative sea-level history at thoselocations for the ICE-6G_C model of ice-loading history combined to the V1 (FM) (green) and V2(FM) (blue) radial viscosity profiles. Green data points represent sea-level index points, whereas bluecrosses represent marine-limiting data and orange crosses represent terrestrial-limiting data. From Royand Peltier (2015).


experienced by the solid Earth under formerly heavily glaciated areas like Hudson Bay (at the center of

the former Laurentide Ice Sheet) or Fennoscandia. As these regions were under considerable ice cover at

Last Glacial Maximum (LGM), model-predicted relaxation times at locations near the centre of the ice

loads are relatively protected from load history uncertainties near the ice sheet margins. For the Hudson

Bay area, analyses performed by Peltier (1998a) on a complete set of 14C-dated RSL constraints suggested

a best estimate for the relaxation time in Hudson Bay (Richmond Gulf, Quebec) of approximately 3.4

ka, whereas the ICE-4G (VM2) model predictions suggested a relaxation time of approximately 4.1 ka.

A subsequent analysis of this data set by Mitrovica et al. (2000) led them to revisit the Peltier (1998a)

inference, on the premise that close sites in the area might have largely different relaxation times which

would be reflected on the individual curves for each site and that only a single type of RSL data should

be employed. They suggested that the relaxation time at the Richmond Gulf site is approximately

5.0 ka (with a permissible range of 4.0-6.6 ka) and that it should be significantly lower at James Bay,

at around 2.5 ka (with a permissible range of 2.0-2.8 ka) (Mitrovica et al., 2000). This difference is

important, because whereas a single relaxation time is used to represent the whole Hudson Bay region in

the formal Bayesian inversion leading to the VM2 radial viscosity model (of which VM5a is a five-layer

approximation) (Mitrovica and Peltier, 1995; Peltier, 1998a), the inversion performed by Mitrovica and

Forte (2004) that leads to the FM family of viscosity profiles uses two different relaxation times for the

area. Given that these region are in such geographical proximity this would appear to be unphysical.

A further contribution to this debate was provided by Dyke and Peltier (2000), who revisited the

exponential model RSL curve to qualitatively assess the impact of having various types of sea-level

indicators, in particular with respect to 14C-dated mollusc shells whose habitat could extend (for many

species) to a significant depth beneath mean sea level. However, the revised relaxation times produced by

this analysis, which were around 2.5 ka, further exacerbated the misfit between the observed relaxation

time and model predictions based upon the ICE-4G loading history coupled to the VM2 viscosity profile.

An important recent additional perspective is provided by the work of Pendea et al. (2010), who

have published the first marine-to-freshwater transition shoreline displacement model for the James Bay

area using Accelerator Mass Spectrometry (AMS) dating of organic material in paleo-tidal wetlands as

emergence indicators. As pointed out by Pendea et al. (2010), the isolation basin methodology had

been used in the past for individual lakes in the Hudson Bay region (Saulnier-Talbot and Pienitz, 2001;

Miousse et al., 2003), but not using a sequence of distinct basins, and never reaching as far south as

James Bay. A similar methodology had previously been applied to great effect by Shennan et al. (1994)

in constructing the RSL history record for the Arisaig location and other sites in western Scotland. The

Pendea et al. (2010) data set provides an independent take on the debate concerning relaxation times


Figure 3.8: High quality sea-level index points from the Pendea et al. (2010) analysis in the easternpart of James Bay, and the exponential decay fit that best represents the relaxation of the area after itsdeglaciation. The locations used in the Pendea et al. (2010) analysis are shown in the inset. From Royand Peltier (2015).

in the James Bay area, and is significant given the high accuracy of the methodology. The location

of the individual sites included in their analysis is shown in Figure 3.8. Following Peltier (1998a), a

simple exponential decay curve was fitted to the Pendea et al. (2010) data, while constraining it to zero

present-day change in relative sea level S(t), following Equation (3.14):

S(t) = A[

expt/τ −1]

, (3.14)

where τ is the inferred relaxation time and A is the amplitude characterizing the post-glacial RSL history.

This best fit, along with the Pendea et al. (2010) data, is presented in Figure 3.9. A relaxation time of

3.9(±0.8) ka is obtained for this site. This result is noteworthy, as it calls into question the validity of

the lower estimates for relaxation times for the area, such as the estimates of Dyke and Peltier (2000),

as well as that of Mitrovica et al. (2000).

A comparison of the various a posteriori predictions of relaxation time for the various radial viscosity

profiles under discussion in this analysis is presented in Table 3.1. The relaxation time for the Hudson

Bay region predicted by VM2 and that of its related 5-layer approximation, VM5a, is approximately

4.1 ka. Understandably, the VM5b profile, which has an upper mantle half as viscous as VM5a does,

exhibits a much lower relaxation time than the other stiffer models. All previous Toronto models have

relaxation times that are within the new observational inference obtained from the Pendea et al. (2010)

data. On the other hand, the FM family of models exhibits relaxation times that are much higher than


the other models, with values of approximately 5.7 ka for V1 (FM) (a similar value to Mitrovica and

Forte (2004)), and 6.4 ka for V2 (FM). These high values are due to the excessively high lower mantle

viscosity that is delivered by the inversion procedure when a high relaxation time is assumed to govern

the rebound process in the region near the centre of the Laurentide ice sheet. The FM family of models

fails to predict relaxation times that fall within the error on the relaxation time estimated from the

highly accurate isolation basin record of Pendea et al. (2010) as well as the previous estimates in Peltier

(1998a).

Table 3.1: Relaxation times determined in Eastern James Bay for various viscosity profiles coupled withthe ICE-6G_C ice loading history

Initial observations strongly suggest that neither of the FM models (V1 and V2) are plausible insofar

as the understanding of the GIA process is concerned, as comparisons to the relative sea level data along

the U.S. East coast (when combined with the realistic ICE-6G_C loading history) has revealed important

misfits between the model predictions and observations, especially along the southernmost portion of

this coast. This view is now reinforced by the fact that these viscosity models deliver excessively high

relaxation times in the James Bay area when compared to the latest data set of high-quality information

from this region. This observation puts into question the assumptions made regarding relaxation times

in the Hudson Bay and James Bay areas in the design of these models. Although the claim was that the

V1 and V2 models were constrained to fit both GIA and convection related constraints, it seems clear

that they are not in fact able to explain GIA observables.

3.5.4 Case study I: Mantle viscosity variations in the upper mantle

Rather than further mining the existing literature for additional alternative models of mantle viscosity

that might be employed to reconcile observations of relative sea level history with GIA model predictions

along the U.S. East coast, a systematic exploration of the sensitivity of model predictions to mantle

viscosity variations at different depths is preferred. It is well-known that, for regions previously located

beneath a former ice sheet and therefore undergoing uplift today, the sensitivity of the characteristic


relaxation time for uplift of the crust depends strongly on the horizontal scale of the previously ice-

covered region (e.g. Peltier, 1998a), but the sensitivity to mantle viscosity variations is more complex

for regions located in the surrounding glacial forebulge region, as the variation of viscosity with depth

strongly impacts both the extent and amplitude of the forebulge itself (Tushingham and Peltier, 1991).

The viscosity of the upper mantle and transition zone is known to have a profound impact on RSL

predictions associated with ice sheets of moderate horizontal extent, such as the Fennoscandian ice

sheet (Shennan et al., 2002). In the context of the larger Laurentide ice sheet, the impact is expected

to be focused on the region in the ice sheet periphery. A wide range of upper mantle and transition

zone viscosity structures has been investigated, and a representative subset of this family of models is

presented here in Figure 3.9, where upper mantle and transition zone viscosity values are varied from

0.25 · 1021 Pa·s (the VM5b model of Engelhart et al. (2011)) to 0.75 · 1021 Pa·s, in a layer between the

660-km discontinuity and the base of the rheologically-stratified lithosphere (100 km depth), while the

remainder of the radial viscosity structure is assumed to be identical to that of the VM5a profile. The

range of viscosities considered here is representative of various viscosity values for this depth range in the

literature. The RSL evolution predictions resulting from these models when combined to the ICE-6G_C

loading history are presented in Figure 3.9 for representative locations on the U.S. East coast, while a

quantitative estimate of the error in the fit to the observational data is provided in Table 3.2.

Varying progressively the value of the viscosity in the upper mantle and transition zone from 0.25 ·

1021 Pa·s to 0.75 · 1021 Pa·s results in a broad but consistent range of predictions. Focusing first on

the northernmost locations, the viscosity of the target region strongly impacts the predictions of RSL

evolution. Models characterized by a more viscous upper mantle have reduced forebulge amplitudes

which develop later in time, while models with less viscous upper mantles recover more quickly towards

equilibrium. Progressing further south into the forebulge, softer models such as VM5b perform better at

reconciling GIA predictions and observations, especially for the most recent 4 ka. However, beginning in

North Carolina, such variations in mantle viscosity fail to explain the older data, and by South Carolina,

relative sea level history predictions become insensitive to variations of viscosity in the target region.

However, it should be noted that the spectrum of Fennoscandian relaxation times is highly sensitive to

the viscosity in the region above the 660 km seismic discontinuity (Peltier, 1998a), so much so that the

fit to this spectrum constitutes an important test of any spherically symmetric viscosity model.


Figure 3.9: Comparison of a subset of regions of the Engelhart and Horton (2012) data set of relative sea-level histories along the U.S. Atlantic coast with the predicted relative sea-level history at those locationsfor the ICE-6G_C model of ice-loading history combined to viscosity profiles where the viscosity of theupper mantle (UM) is allowed to vary (as discussed in the text). (a) Viscosity variations for the uppermantle (values are shown in the legend). The region of the mantle in which the variations occur ishighlighted in the inset. (b) For the models shown in (a), comparison of RSL history predictions withobservational data at four locations of the Engelhart and Horton (2012) data set, namely Southern Maine(2), New York (6), Inner Delaware (9), and Southern South Carolina (16). Green data points representsea-level index points, whereas blue crosses represent marine-limiting data and orange crosses representterrestrial-limiting data. From Roy and Peltier (2015).


3.5.5 Case study II: Viscosity changes in the upper part of the lower mantle

Given the large horizontal scale of the Laurentide Ice sheet, it is known that the predictions of relative

sea level evolution under the former Laurentide ice sheet are sensitive to mantle viscosity variations in

the upper part of the lower mantle (as determined from the site-specific Fréchet kernels, or sensitivity

kernels, which link arbitrary perturbations in the radial profile of mantle viscosity to resulting variations

in the post-glacial recovery of the surface (e.g. Peltier and Andrews, 1976; Peltier and Jiang, 1996;

Peltier, 1998a; Wu et al., 2010)). This sensitivity can be expected to extend into the peripheral bulge

region associated with the former Laurentide ice sheet, and to impact not only the geographical extent

of the forebulge, but also its temporal evolution. The sensitivity of RSL predictions along the U.S.

East coast to a wide range of variations in the viscosity of the upper part of the lower mantle has been

investigated, and in a representative subset of those models, the viscosity of the upper part of the lower

mantle has been taken to vary over the range from 0.518 ·1021 Pa·s (one third of the value of the viscosity

of the VM5a model at this depth), through the sequence 0.78 · 1021 Pa·s, 1.57 · 1021 Pa·s (the VM5a

profile), and 3.14 · 1021 Pa·s, while the viscosity structure of the rest of the mantle is kept fixed to that

of the VM5a profile. In this initial treatment, the range of values is selected only to provide a broad

overview of the dependence of RSL predictions on viscosity variations at this range of depths.

Figure 3.10 compares the RSL predictions resulting from these variations of the VM5a structure

when combined to the ICE-6G_C ice loading history for the same subset of locations from Engelhart

and Horton (2012) that were previously employed, with a quantitative estimate of the error in the fit

to the observational data being provided in Table 3.2. In the northernmost regions of the coast, the

softer models produce a peripheral bulge of larger amplitude that reaches its maximum earlier following

initiation of the deglaciation process. Moving further south along the coast reveals that this difference

increases in magnitude, and leads to the presence of a mid-Holocene highstand south of Virginia for the

softest of the models analysed, a feature that is incompatible with the RSL data from the region. From

these observations, when compared to the predictions for the VM5a viscosity structure, a lower viscosity

in the upper part of the lower mantle would be expected to partly eliminate the misfits observed along

the southern part of the U.S. East Coast, although the optimal viscosity must be high enough to prevent

the formation of highstands in the regions south of Virginia and at Barbados.

3.5.6 Case study III: Viscosity variations in the transition zone

When characterizing the sensitivity of RSL history predictions to mantle viscosity, a further avenue that

warrants explicit study concerns the possibility of the existence of a harder transition zone between


Figure 3.10: Comparison of a subset of regions of the Engelhart and Horton (2012) data set of relative sea-level histories along the U.S. Atlantic coast with the predicted relative sea-level history at those locationsfor the ICE-6G_C model of ice-loading history combined to viscosity profiles where the viscosity of theupper part of the lower mantle (ULM) is allowed to vary (as discussed in the text). (a) Viscosity variationsfor the upper part of the lower mantle (values are shown in the legend). The region of the mantle inwhich the variations occur is highlighted in the inset. (b) For the models shown in (a), comparison ofRSL history predictions with observational data at four locations of the Engelhart and Horton (2012)data set, namely Southern Maine (2), New York (6), Inner Delaware (9), and Southern South Carolina(16). Green data points represent sea-level index points, whereas blue crosses represent marine-limitingdata and orange crosses represent terrestrial-limiting data. From Roy and Peltier (2015).


depths of 410 kilometers and 660 kilometers, as has been previously shown to trade off perfectly with

a softening of viscosity in the vicinity of the deeper endothermic phase transition itself (e.g. Peltier,

1998a). Because the mineral assemblage present in the transition zone is especially rich in garnet, there

are a priori reasons to believe that a feature of this kind in the depth dependence of mantle viscosity

could exist. However, the recent discovery of ringwoodite, a high-pressure polymorph of olivine thought

to originate in the transition zone, embedded in a diamond from a Brazilian diamond pipe (Pearson

et al., 2014), suggests that viscosity of the transition zone might instead be lower than the viscosity of

the upper mantle, since ringwoodite is a hydrous phase and this might be interpreted to suggest that

the transition zone is rich in water. In the context of the GIA response to the melting of the Laurentide

ice sheet over the East coast of North America, modifying the viscosity structure in this region alone

and not in the upper mantle could result in a change in the characteristics of the forebulge, while also

modifying somewhat the ability of the model to fit the Fennoscandian relaxation spectrum.

Table 3.2: Error comparison for the viscosity variation cases considered

To investigate the impact of viscosity variations in the transition zone on sea level dynamics in the

glacial forebulge region, a sequence of modifications of the VM5a profile was once again looked at. These

new models are identical to the VM5a profile, except in the channel between the base of the transition


zone at 660 km depth and the top of the transition zone at 410 km depth. The RSL predictions at

a few selected sites along the U.S. East coast are presented in Figure 3.11 for a subset of the tests,

covering a viscosity range from 0.30 · 1021 Pa·s to 1.50 · 1021 Pa·s. A quantitative estimate of the error

in the fit to the observational data is provided in Table 3.2. In the northern and central regions of the

coast, increasing the viscosity in the channel between the depths of 410 and 660 kilometers creates a

less pronounced forebulge that achieves its maximum displacement at a later time. In these regions,

a softer transition zone seems to be favoured when comparing to sea level index point data, although

stiffer models enable a slightly better fit to the marine-limiting observational data at some locations (e.g.

Massachusetts). The difference between the predictions resulting from the stiffer viscosity profiles is quite

limited for sites in the mid-Atlantic regions (such as Connecticut and New York), but the soft transition

zone models perform notably better in those regions. Indeed, for New Jersey, Delaware, Virginia and the

index points of the Chesapeake Bay record, the softer transition zone models provide a better fit to the

observational data provided by the Engelhart and Horton (2012) database. For regions further south,

the inter-model comparison differences become less important. One explanation for this behaviour could

be that, as the viscosity under the lithosphere is decreased, the geographical extent of the forebulge

decreases, a feature that is accompanied by a larger amplitude of response at locations closer to the

former Laurentide ice sheet. Correspondingly, for models with a stiffer transition zone, the impact of the

forebulge is felt further south along the eastern U.S. seaboard, and the predicted RSL curves for those

models are shifted to shallower depth compared to the original VM5a model and with less curvature in

the recovery phase. Hence, increasing the viscosity in the transition zone provides a poorer fit to the RSL

observations for most locations along the U.S. East coast. Thus, large changes in mantle viscosity in the

transition zone would have to be incorporated simultaneously with large viscosity changes in either the

upper mantle or the upper part of the lower mantle to account for the RSL evolution misfits introduced

in the northernmost and southernmost parts of the U.S. East coast.

3.5.7 Case study IV: Viscosity contrast variations between the upper mantle

and lower mantle

Most simple models of mantle viscosity include a discontinuity in mantle viscosity associated with the

presence of the 660-km seismic discontinuity (e.g. Peltier, 1989), and the issue of determining an ap-

propriate contrast between the upper and lower mantle is explored here in the context of relative sea

level evolution predictions for the Eastern seaboard of the United States. Results for a representative

subset of viscosity structures are shown here, where two different viscosity values are explored for both


Figure 3.11: Comparison of a subset of regions of the Engelhart and Horton (2012) data set of relative sea-level histories along the U.S. Atlantic coast with the predicted relative sea-level history at those locationsfor the ICE-6G_C model of ice-loading history combined to viscosity profiles where the viscosity of thetransition zone (TZ) is allowed to vary (as discussed in the text). (a) Viscosity variations for thetransition zone (values are shown in the legend). The region of the mantle in which the variations occuris highlighted in the inset. (b) For the models shown in (a), comparison of RSL history predictionswith observational data at four locations of the Engelhart and Horton (2012) data set, namely SouthernMaine (2), New York (6), Inner Delaware (9), and Southern South Carolina (16). Green data pointsrepresent sea-level index points, whereas blue crosses represent marine-limiting data and orange crossesrepresent terrestrial-limiting data. From Roy and Peltier (2015).


the upper part of the lower mantle (1.57 · 1021 Pa·s and 0.785 · 1021 Pa·s) and for the upper mantle and

transition zone (0.75 · 1021 Pa·s and 0.37 · 1021 Pa·s), which results in four different viscosity structure

combinations.

Figure 3.12 presents the relative sea-level history predictions resulting from these structures for the

subset of representative locations on the U.S. East coast, while a quantitative estimate of the error in

the fit to the observational data is provided in Table 3.2. Beginning with the northernmost locations, it

is noticeable that for the lowest upper mantle and transition zone viscosity (0.37 · 1021 Pa·s), increasing

the contrast between the upper and lower mantle (by increasing the value of the lower mantle viscosity)

leads to a less prominent forebulge that reaches its maximum amplitude at a slightly later time, a

direct consequence of the slower mantle material flow caused by its higher overall viscosity. In terms

of relative sea level evolution at the locations of interest on the U.S. East coast, this increase of the

lower mantle viscosity results in flatter sea level curves and slower recovery to today’s observed sea

level. Repeating the analysis for a higher upper mantle viscosity (0.75 · 1021 Pa·s), a similar behaviour

is observed for the northernmost sites. However, for mid-Atlantic sites (Massachusetts to Maryland),

the higher upper mantle viscosity masks any impact arising from a higher lower mantle viscosity. In

Virginia, North Carolina and South Carolina, as was the case for the models with the lower upper mantle

and transition zone viscosity, relative sea level history predictions are highly dependent on the value of

lower mantle viscosity employed. Models with softer lower mantles tend to reproduce the observational

data of Engelhart and Horton (2012) more accurately, although in the two sets of experiments shown

here, the overall sea level evolution curvature is not captured accurately.

3.5.8 Case study V: Lithosphere thickness variations

Another important sensitivity analysis concerns the thickness of the elastic lithosphere. Here, lithospheric

thickness is varied from 60 kilometers to 195 kilometers (through the sequence of 60, 90, 130 and 195

kilometers), while the rest of the mantle viscosity structure is fixed to the VM5a profile. The rheological

stratification of the lithosphere present in the VM5a and VM5b viscosity profiles is ignored in this test.

Relative sea level evolution predictions for the U.S. East coast are compared to the sea level indicators

in four regions of the Engelhart and Horton (2012) database in Figure 3.13. A quantitative estimate of

the error in the fit to the observational data is provided in Table 3.2.

The model predictions for the northernmost sites (in particular Eastern and Southern Maine) are

particularly interesting. While the forebulge is almost absent when using models with a very thin

lithosphere, as lithospheric thickness is increased from 60 to 130 kilometers, the forebulge gains in


Figure 3.12: Comparison of a subset of regions of the Engelhart and Horton (2012) data set of relative sea-level histories along the U.S. Atlantic coast with the predicted relative sea-level history at those locationsfor the ICE-6G_C model of ice-loading history combined to viscosity profiles where the viscosity contrastbetween the upper and lower mantles is allowed to vary (as discussed in the text). (a) Upper/lowermantle viscosity contrast variations (values are shown in the legend). The region of the mantle in whichthe variations occur is highlighted in the inset. (b) For the models shown in (a), comparison of RSLhistory predictions with observational data at four locations of the Engelhart and Horton (2012) dataset, namely Southern Maine (2), New York (6), Inner Delaware (9), and Southern South Carolina (16).Green data points represent sea-level index points, whereas blue crosses represent marine-limiting dataand orange crosses represent terrestrial-limiting data. From Roy and Peltier (2015).


Figure 3.13: Comparison of a subset of regions of the Engelhart and Horton (2012) data set of relativesea-level histories along the U.S. Atlantic coast (Southern Maine (2), New York (6), Inner Delaware(9), and Southern South Carolina (16)) with the predicted relative sea-level history at those locationsfor the ICE-6G_C model of ice-loading history combined to viscosity profiles where the thickness ofthe elastic lithosphere is allowed to vary (as discussed in the text). Green data points represent sea-level index points, whereas blue crosses represent marine-limiting data and orange crosses representterrestrial-limiting data. From Roy and Peltier (2015).


maximal amplitude and reaches it at earlier times. However, the behaviour becomes less marked for

a thicker lithosphere (or even slightly reverses for the highest values considered), which implies that

there might be a lithospheric thickness value that maximizes the amplitude of the local forebulge for

the northernmost sites. Similarly, progressing further south along the coast, models with a thinner

lithosphere demonstrate a wide range of predictions, but models with very thick elastic lithospheres (130

km and above) display a similar RSL evolution. In the mid-Atlantic region (e.g. New York and Inner

Delaware), this situation manifests itself in the form of a forebulge of lower amplitude for the thickest

lithosphere under consideration. Conversely, in the southernmost regions, models with a thin lithosphere

exhibit a similar behaviour, while the viscosity structures with a thicker lithosphere display a broader

range of predictions with a more prominent forebulge, a feature which is not desirable given the small

change in RSL history over the past 6 ka captured by the sea level indicators in the region.

This series of features can be explained in terms of the physical nature of the movement of mass in the

mantle associated with the former Laurentide ice sheet and the corresponding flexure of the lithosphere

it induces. For a thicker lithosphere, the greater stiffness of the Earth’s solid surface tends to spread the

forebulge over a larger geographical area, while diminishing its maximal amplitude. Viscosity models

with thinner lithospheres result in a more pronounced but localized shape. A simple variation of this

parameter does not provide a satisfactory improvement in the ability of the VM5a viscosity profile to

explain the RSL observational data over all regions along the coast.

3.5.9 Case study VI: Lower mantle viscosity variations and the Earth’s ro-

tational state

It has been long established that the Earth’s rotational state continues to be influenced by the isostatic

adjustment of the planet resulting from the deglaciation process that followed the Last Glacial Maximum

(e.g. Peltier, 1982; Wu and Peltier, 1984; Peltier, 1996a), in particular concerning the two main rotational

anomalies observed today and previously discussed, namely the non-tidal acceleration of the rate of

planetary rotation and the secular drift of the pole of rotation with respect to the surface of the planet

(true polar wander). Modern observations of these anomalies are numerous, and recent observations

of changes in the non-tidal acceleration of the planetary rotation rate and in the secular drift of the

poles have been described earlier in this work (based upon measurements of satellite orbital parameters,

historical records of lunar and solar eclipses and space-geodetic techniques (Yoder et al., 1983; Stephenson

and Morrison, 1995; Gross and Vondrák, 1999; Cheng and Tapley, 2004; Roy and Peltier, 2011; Cheng

et al., 2013)).


Predictions of these GIA-induced anomalies are mostly sensitive to the viscosity of the lowermost

mantle (Peltier, 1982; Wu and Peltier, 1984). Although predictions of RSL evolution following the

deglaciation were found to be sensitive to the viscosity of the upper part of the lower mantle but only

weakly sensitive, if at all, to changes in the viscosity of the deepest portion of the lower mantle (Peltier,

1998a), it is important to investigate the impact of such viscosity changes in the deepest mantle on RSL

predictions along the U.S. East coast. Such modifications might be required to maintain the quality

of fit to Earth rotation constraints when an appropriately softened shallower structure is introduced

to improve the quality of GIA predicted RSL histories along its southernmost parts. In Figure 3.14,

predictions of RSL evolution for four key sites along the U.S. East coast are presented when the viscosity

of the lowermost lower mantle is allowed to vary between three different values (chosen for illustrative

purposes) in panel (a), while a quantitative estimate of the error in the fit to the observational data is

provided in Table 3.2. The impact of constraining a potential viscosity increase to the lowermost section

of the lower mantle was also investigated. The panels shown in Figure 3.14(b) demonstrate the minimal

impact induced on RSL evolution predictions by slight changes in lower mantle viscosity, with the only

notable difference seen at the very edge of the former Laurentide ice sheet in Maine. Furthermore, by

concentrating the increase in lower mantle viscosity to the very deepest part of the mantle and keeping

the rest of the viscosity structure fixed to VM5a, it should be noted that the RSL predictions are identical

to the ones made using the VM5a structure.

3.5.10 Other considerations and a summary of the insights gained through

the sensitivity analyses

Synthesizing the results of these calculations, it appears that, for the ICE-6G_C ice loading model,

combinations of viscosity changes in different layers of the mantle could potentially lead to a good fit

to RSL observations in the northernmost locations along the U.S. East coast. Indeed, for these regions,

the thickness of the elastic lithosphere and the viscosity of the upper mantle and transition zone have

an important influence on the RSL predictions, while the viscosity of the upper part of the lower mantle

has a significant impact on the shape of the forebulge and its decay. However, the RSL data from the

southern part of the U.S. East coast is only better represented with a viscosity in the upper part of

the lower mantle that is slightly lower than that characteristic of the VM2 and VM5a profiles, which

reduces the ambiguity as to which of the possible combinations one should first apply to better fit the

data from the northernmost points along the coast. For instance, an increase in lithospheric thickness

decreases the quality of the fit in the southernmost locations while a reduction of its thickness does not


Figure 3.14: Comparison of a subset of regions of the Engelhart and Horton (2012) data set of relative sea-level histories along the U.S. Atlantic coast with the predicted relative sea-level history at those locationsfor the ICE-6G_C model of ice-loading history combined to viscosity profiles where the viscosity of thelower part of the lower mantle is allowed to vary (as discussed in the text). (a) Viscosity variations inthe lower part of the lower mantle (values are shown in the legend). (b) For the models shown in (a),comparison of RSL history predictions with observational data at four locations of the Engelhart andHorton (2012) data set, namely Southern Maine (2), New York (6), Inner Delaware (9), and SouthernSouth Carolina (16). Green data points represent sea-level index points, whereas blue crosses representmarine-limiting data and orange crosses represent terrestrial-limiting data. From Roy and Peltier (2015).


reconcile RSL observations with the model predictions, which restricts the range of possibilities in the

quest for an improved model that will enable us to fit the majority of the observational data. Changes

in the viscosity of the transition zone have some impact on RSL predictions at northern locations along

the U.S. East coast. Also, a softer transition zone fits the mid-Atlantic data points better, but as the

viscosity of the upper part of the lower mantle must be decreased to fit the observational data from

the southernmost locations along the U.S. East coast, it is interesting to note that the viscosity in the

transition zone could potentially be used as a fine-tuning parameter in balancing the quality of the fit to

the geologically derived RSL data from the southernmost locations of the Engelhart and Horton (2012)

data set and to other global geophysical constraints (such as the Fennoscandian spectrum of spherical

harmonic degree-dependent relaxation times).

3.6 A preferred viscosity structure: the VM6 profile

Here, using the previous analyses and the most recent ICE-6G_C globally consistent ice loading history

(Argus et al., 2014; Peltier et al., 2015), a new radially-symmetric viscosity profile for the Earth’s mantle

that addresses the shortcomings of the VM5a and VM5b profiles along the U.S. East coast is found. In

each layer listed in the previous section, the viscosity was perturbed to minimize the misfit between the

GIA prediction of RSL along the U.S. East cost and the Engelhart and Horton (2012) data set, starting

with the thickness of the lithosphere, progressing with the viscosity of the upper mantle and of the

layers beneath it until reaching the upper part of the lower mantle (RSL predictions along the U.S. East

coast have no meaningful sensitivity to viscosity variations in the lower part of the lower mantle). The

process was repeated for the whole mantle until convergence was achieved. In this process, a rheological

stratification of the lithosphere, with a 30-km thick layer with higher than upper mantle viscosity located

right under the elastic lithosphere, was used, following its introduction in the VM5a and VM5b models

in order to reconcile GIA model predictions of present-day horizontal motion of the solid Earth surface

with space-geodetic observations (Peltier and Drummond, 2008). The resulting new viscosity structure,

which is referred to as VM6, is illustrated in Figure 3.17(a) and its properties are listed in Table 3.3. It

contains a slightly thicker elastic lithosphere (90 km thickness) superimposed upon a 30-km thick layer

with higher than upper mantle viscosity representing the rheological stratification of the lithosphere. In

the VM6 profile, the viscosity of the upper part of the mantle, located between the base of the lithosphere

and the base of the transition zone at 660 km depth (0.45 ·1021 Pa·s), lies between the viscosity of VM5a

and VM5b. No additional structure in the transition zone (between the 420-km and 660-km seismic

discontinuities) was found to provide a consistent decrease in the error function values obtained from


comparisons to the Engelhart and Horton (2012) data set, but it was preferable to reduce the viscosity

in the upper part of the lower mantle, where it is set to 0.90 · 1021 Pa·s, or about half the viscosity

characteristic of the VM5a structure in this depth range. This decrease is key to addressing the misfits

identified between the observational data in the Carolinas and the model predictions for the VM5a and

VM5b viscosity profiles. The lower part of the lower mantle requires a slight decrease in viscosity. These

modifications do enable the revised model to address decisively the shortcomings of the VM5a viscosity

profile regarding the reconstruction of past relative sea level from the Engelhart and Horton (2012) data

set.

Table 3.3: Radial viscosity structure of the VM6 viscosity profile down to the core-mantle boundary

The new ICE-6G_C (VM6) model is tested here not only against the RSL data from the U.S. East

coast of Engelhart and Horton (2012), but also against geophysical observables that were not used in

the optimization process that led to it. This enables us to test the plausibility of the new VM6 viscosity

structure and its potential global exportability, but it should be remembered that these extra constraints

have not been used in the tuning process itself.

3.6.1 The predictions for the U.S. East coast for ICE-6G_C (VM6)

Comparisons between the complete Engelhart and Horton (2012) data set of relative sea level indicators

and the ICE-6G_C (VM6) predictions of RSL history are shown in Figure 3.15, while a summary of

the changes in reduced error function performance introduced by the new model when compared to

the fit provided by the ICE-6G_C (VM5a) model is shown in Table 3.4. For indicative purposes, the

change in performance resulting from a transition from the VM5a viscosity profile to the VM5b viscosity

profile of Engelhart et al. (2011) is also included in Table 3.4. However, as demonstrated earlier, it

should be noted that VM5b is not as favoured as a spherically-symmetric model of mantle viscosity, as

it provides a worse fit to the Fennoscandian spectrum of relaxation times than that provided by VM6.

For the northernmost sections of the U.S. East coast, VM6 provides some improvement to the fit to

the observations over that of the VM5a profile over the latest 4 ka of the observational record, but a


decrease in the quality of the fit at earlier times. Since the region was co-located with the margin of the

Laurentide ice sheet during the initial phases of the last deglaciation and is relatively close to a region of

complex margin evolution (e.g. Borns et al., 2004), slight modifications to the ice loading history would

be able to resolve this localized misfit. It should be noted, however, that VM6 provides an improved fit

compared to VM5a for most of the sea-level index points south of Connecticut. In particular, in New

York (site 6), VM6 provides a fit to the observational data much superior to both VM5a and VM5b,

especially for the more recent sea-level index points. For Long Island and New Jersey (sites 7 and 8), the

change in viscosity structure from VM5a to VM6 improves the fit to the more recent index points, but

does not provide a substantial improvement for indicators older than 4 ka. The performance difference

observed between VM5a and VM6 (or between VM5b and VM6) in Long Island (site 7) is strongly

affected by a few outlying data points (mostly marine-limiting) that none of the models can explain,

and which dominate the percentage change shown in Table 3.4 for these locations. Progressing further

south, the VM6 model is found to be much superior to VM5a, increasing substantially the quality of the

fit of the GIA model to the Engelhart and Horton (2012) data set in Maryland, Virginia and Delaware

(sites 9 to 12). In the Carolinas (sites 13 to 16), particularly in South Carolina, VM6 provides a noted

improvement over both VM5a and VM5b. As the misfits associated with the use of these models for the

southern part of the Atlantic coast of the United States was an outstanding problem (Engelhart et al.,

2011), the introduction of this new viscosity structure is important. However, it should be noted that

the error function is strongly impacted by a limited number of relatively old marine-limiting data points

in the Chesapeake Bay region (site 11) and in the northern part of North Carolina (site 13), where some

misfits remain.

Another key result is related to the age dependence of the performance of the fit with latitude.

As indicated by Table 3.4, for sites south of Delaware, the increase in performance provided by VM6

is particularly notable for older RSL geological data points (older than 4 ka). In particular, for the

southern part of the coast, using the new VM6 viscosity structure removes practically all the misfits

with respect to older geological RSL data that were identified when using the VM5a model, with some

misfit reductions of more than 95% at some locations. This result is important, as the ability to model

older geological data points is crucial if the behaviour of the forebulge collapse observed along the U.S.

East coast is to be explained correctly, a feature which we now turn our attention to.

Looking in detail at the form of the ongoing collapse of the forebulge predicted to extend over much of

the U.S. East coast is an important test of the new VM6 viscosity structure, as the accurate description

of the properties of the forebulge is a key component of the study of relative sea-level changes observed

over the later part of the Holocene and during the 20th century along the U.S. East coast (e.g. Engelhart


Figure 3.15: Comparison of the Engelhart and Horton (2012) data set of relative sea-level histories alongthe U.S. Atlantic coast for the 16 composite regions with the predicted relative sea-level history at thoselocations for the ICE-6G_C model of ice-loading history combined to the VM5a (green), VM5b (blue)and VM6 (black) radial viscosity profiles. Green data points represent sea-level index points, whereasblue crosses represent marine-limiting data and orange crosses represent terrestrial-limiting data. FromRoy and Peltier (2015).


Table 3.4: Percentage change in the reduced error misfit for the transition from ICE-6G_C (VM5a) toICE-6G_C (VM6) along the U.S. East coast (negative changes indicate a better fit to the observationaldata, with -100% representing a complete removal of the misfits observed with ICE-6G_C (VM5a)).The change in error misfit performance resulting from the transition from the VM5a viscosity profile tothe VM5b viscosity profile of Engelhart et al. (2011) is also shown for indicative purposes. From Royand Peltier (2015).

et al., 2009). The shape of the forebulge inferred from the geological records described in Engelhart and

Horton (2012) is presented in Figure 3.16 by considering the rates of relative sea-level rise over the

late Holocene. Following Engelhart et al. (2009), these late Holocene rates of relative sea-level rise are

shown in Figure 16 for a series of locations along the U.S. East coast as a function of distance from one

of the main centers of glaciation of the Laurentide ice sheet (Keewatin dome), approximated to be in

modern-day Churchill, Manitoba, on the western shore of Hudson Bay. These values were determined

in Engelhart et al. (2009) by running a linear regression over the geological data covering the past 4,000

years at each site under consideration (with 2-σ error bars). The results obtained for the ICE-6G_C

(VM5a) and ICE-6G_C (VM6) models at the same locations are superimposed on Figure 3.16.

As shown in Figure 3.16, using the VM5a mantle viscosity profile over-estimates the late Holocene

rates of relative sea-level rise for most locations along the U.S. East coast, most notably for locations

in the mid-Atlantic states of New Jersey and Delaware (around 2,500 kilometers away from the former

centre of glaciation), where the most significant forebulge collapse occurs. This results in a forebulge

collapse shape (illustrated in Figure 3.16 by the slope of uplift rate change as a function of distance


Figure 3.16: Comparison of observed Late Holocene relative sea level rise (mm/a) for locations alongthe U.S. East coast (dark grey diamonds) (with 2-σ uncertainty ranges in light gray) (Engelhart et al.,2009), with predicted values for the ICE-6G_C (VM5a) (green dots) and ICE-6G_C (VM6) models(red dots). The data points are plotted as a function of distance from the city of Churchill, Manitoba,on the western shore of Hudson Bay (km). From Roy and Peltier (2015).

from the center of former glaciation) that compares unfavourably with the geological inference. The

VM6 profile captures much better the amplitude of forebulge collapse in most locations, in particular its

maximal range, which results in a forebulge collapse shape that is very close to that geologically inferred.

The ability of the ICE-6G_C (VM6) model to capture the geographical extent and the amplitude

of forebulge collapse is very significant, since this data was not used in the development of the model.

It should also enable an improvement in the characterization of the relative sea level rise observed

during the late Holocene and during the 20th century (see Engelhart et al. (2009); Kemp et al. (2013)).

Discrepancies remain for the northernmost locations (around Maine), but as these locations are very

close to the margin of the former Laurentide ice sheet, these misfits could be eliminated by slight

changes in the timing of margin retreat or in the thickness of ice cover in the margin regions. The slight

remaining misfit observed in the southernmost locations (around South Carolina) at the trailing edge of

the forebulge can be eliminated by a minor change in the stratification of the elastic lithosphere, which

will be demonstrated elsewhere.

3.6.2 The fit to the Fennoscandian relaxation spectrum

A global constraint to which the new model may be tested, and which is directly linked to the properties

of the viscosity structure itself, is the fit to the Fennoscandian spectrum, provided in Figure 3.17(b)


Figure 3.17: (a) Comparison of the radial variations of the viscosity in depth for the new VM6 viscosityprofile (black) with the VM5a (green) and VM5b (blue) profiles. (b) Inverse relaxation time as afunction of spherical harmonic degree obtained from observations of the glacial isostatic adjustmentof Fennoscandia, with the relaxation spectrum of Wieczerkowski et al. (1999) (black dashed line) andcorresponding 1-σ and 2-σ uncertainties (dark gray and light gray areas, respectively). The predictedspectra for the new VM6 viscosity structure is shown in red, while the predictions for the VM5a (green)and VM5b (blue) models are superimposed. Adapted from Roy and Peltier (2015).

for the VM6 viscosity profile. The new model provides a generally adequate fit to this constraint. The

relaxation times inferred for the largest deformation scales are somewhat lower than for the VM5a profile

(also shown), largely owing to the slight decrease in effective viscosity across much of the upper mantle

(including the slight reduction in the transition zone) and the upper part of the lower mantle. However,

they still remain within the 2-σ uncertainty range of the Fennoscandian spectrum suggested by the formal

analysis of Wieczerkowski et al. (1999). For high spherical harmonic degrees (e.g. smaller deformation

scales), the relaxation times for VM6 lie well within the 1-σ uncertainty range. As the optimization

process leading to VM6 was focused on the U.S. East coast, modifications to the VM6 profile which

may have improved the fit to the Fennoscandian spectrum at lower spherical harmonic degree (such as a

higher viscosity in the transition zone) were not included, as these variations would have decreased the

quality of the fit to RSL observations for a substantial number of sites along the U.S. East coast, most

notably in the mid-Atlantic region. A slight reduction in mantle viscosity in the transition zone from

the VM5a model, similar to the change applied to the rest of the upper mantle, was found to fit the

observational data at these sites optimally. This conclusion may have to be revisited in future analyses

that expand the methodology employed here to include a broader suite of global constraints.

3.6.3 Testing the viscosity structure against data from the North American

West coast

The new viscosity structure can also be tested by comparing its relative sea level evolution predictions

to other regional data sets. Such a region is the western coast of North America, where a large data set


of calibrated 14C-dated sea level indicators has also recently become available (Engelhart et al., 2015).

The locations of the sites under consideration in this analysis were shown in Figure 3.3(b) and span

the entire West coast of Canada and the United States. Since the area is another region undergoing

postglacial forebulge collapse, it will provide another important test of the quality of the new model,

given that it has not been employed in its development. It is important to note, however, that unlike

the Atlantic coast of North America, where the continental margin is relatively "passive", the Pacific

coast is much more active tectonically, as it is notably host in the south to the San Andreas strike slip

fault and in the north to the subduction zone that marks the eastern flank of the Juan De Fuca plate.

For example, the West coast is known to have been host to the great Cascadia earthquake of 1700AD

(magnitude 8-9), which was accompanied by significant vertical motion (Hawkes et al., 2010, 2011). The

area is also affected by uplift/subsidence associated with the subduction of the Juan de Fuca plate,

especially over Vancouver Island (James et al., 2000). Thus, when comparing our results for relative

sea level history predictions with the observational constraints, the existence of a potential tectonic

overprint should be considered. However, as the database of 14C-dated sea level indicators covers only

a few millennia, this impact should be relatively minor compared to the other uncertainties involved in

the study of the GIA problem (viscosity structure of the mantle, ice loading history uncertainty, large

spontaneous subsidence/uplift events, impact of sediment compaction on geological records, etc.).

The performance of the VM6 viscosity profile is evaluated along the North American West coast by

comparing its RSL history predictions at the sites shown on Figure 3.3(b) with the data presented in

Figure 3.18, starting with the northernmost locations. The model provides a good fit to the available

sea level indicators for Queen Charlotte Strait (site 1), as well as the eastern coast of Vancouver Island

(site 3). However, on the west coast of Vancouver Island (region 2), a region strongly affected by

tectonic impact associated with the subduction of the Juan De Fuca plate, the model not surprisingly

fails to reproduce the relative sea level evolution inferred from observational constraints, even though

it performs marginally better than VM5a in this regard. For the southeastern part of Georgia strait

(site 4), a region located directly on the mainland, the model fails to predict the subsidence revealed by

the geological record for all viscosity structures employed. Instead, it predicts a complex RSL history

probably impacted by the rebound associated with the melting of the adjacent Cordilleran ice sheet, the

collapse of the forebulge associated with the more remote but much larger Laurentide ice sheet to the

east, and the global rise in sea level associated with the last deglaciation. These discrepancies could be

explained by various phenomena. In fact, not only is the response in this region heavily impacted by

tectonic activity, but the site is also very close to the advancing front of Cordilleran ice during the Last Ice

Age and is thus highly sensitive to variations in the ice thickness in its immediate vicinity, most notably


Figure 3.18: Comparison of a data set of sea-level indicators along the U.S. Pacific coast (Engelhart et al.,2015) with the predicted relative sea-level history at those locations for the ICE-6G_C model of ice-loading history combined to the VM5a (green), VM5b (blue) and VM6 (black) radial viscosity profiles.Green data points represent sea-level index points, whereas blue crosses represent marine-limiting dataand orange crosses represent terrestrial-limiting data. From Roy and Peltier (2015).


in the earlier part of the sea level record. Furthermore, there might be subsidence effects associated

with sediment loading caused by glacier meltwater-fed (and thus sediment-rich) water outflows towards

Georgia Strait (such as the Fraser and Columbia rivers), which could have impacted certain locations

on the eastern shore of Georgia Strait.

For the southern coast of Vancouver Island (site 5), the new ICE-6G_C (VM6) model performs

adequately with regards to the limiting data points, although ICE-6G_C (VM5a) fits marginally better

the index points. Progressing further south to the Juan de Fuca Strait (site 6) and onwards, the quality

of the fit provided by the new model improves substantially. Most notably, along the western coast of

Washington State and Oregon, the ICE-6G_C (VM6) model is able to fit most of the observational con-

straints derived from both sea level index points and terrestrial- or marine-limiting data at all locations.

Finally, for the southernmost part of the data set, in California, the models struggle to reproduce the

flattening of relative sea level observed between 7.5 ka and 4 ka. However, it should be noted that the

northern part of the Californian coast could be affected by the complex tectonic setting of the region,

close to the Mendocino Triple Junction (e.g. Merritts and Bull, 1989), while the central coast site, being

located within the Sacramento River basin, could also be affected by sediment compaction effects.

Although the new viscosity structure VM6 provides a notably improved fit to the observational data

of relative sea evolution on both the East and West coasts of North America, small misfits persist for

some locations and time periods. For example, Eastern Maine is a region where a misfit persists, but

as mentioned earlier, the issue should not be significant given its location at the margin of the former

Laurentide ice sheet. In fact, as relative sea level evolution predictions close to the edges of a former ice

sheet are highly dependent on the exact melting history and geometry of the ice sheet, it is expected

that these misfits could be removed by modifying the ice loading history in the margins of the former

ice sheets. A similar situation applies to the two sites on the eastern shore of Georgia strait on the West

coast of the continent. Also, some misfits remain for the U.S. East coast locations situated right at the

center of the forebulge associated with the former Laurentide ice sheet, namely Long Island and New

Jersey, regions that could be perhaps affected by compaction issues (Horton et al., 2013).

3.7 Conclusion

The East coast of the continental United States is a very important region in the study of the glacial

isostatic adjustment process. The availability of a new, high-quality data set that covers the entire

coast provides an opportunity to test current GIA models, most notably concerning models of the radial

variation of mantle viscosity. Here, misfits resulting from the use of either the VM5a model (Peltier and


Drummond, 2008) or the VM5b model (Engelhart et al., 2011) have been analysed in detail and shown

to be effectively eliminated by the new VM6 model when coupled to the new ICE-6G_C model of global

planetary glaciation and deglaciation. Alternative models in the literature (the V1 and V2 models of

Mitrovica and Forte (2004) and Forte et al. (2009)) were also tested and rejected on the basis that the

misfits to the data associated with them were found to be even greater than those characteristic of the

VM5a and VM5b models.

Through a detailed series of sensitivity tests designed to investigate the response of U.S. East coast

RSL histories to variations in the radial variation of viscosity over specific ranges of mantle depth, a new

model of the depth dependence of viscosity in an assumed spherically symmetric model of the Earth’s

interior was constructed using a simple, iterative methodology. This new model is referred to as VM6,

the sixth in the series of models that continue to be refined using the iterative methodology developed to

converge upon an optimal structure of this kind. Although the predecessor model VM5a is a perfectly

acceptable model insofar as the reconciliation of relative sea level histories and geodetic data from the

ice covered regions is concerned, the model fails in the region of forebulge collapse along the U.S. East

coast, especially in its southern section. Demonstrating this has required the availability of the new high

quality data base that has been assembled in Engelhart and Horton (2012). The new viscosity model,

when combined with the ICE-6G_C loading history, has been shown to eliminate most misfits that

otherwise exist at sites along the U.S. East coast which record the process of forebulge collapse. Some

misfits still remain in isolated regions, most notably in the northernmost locations of the U.S. East coast

in the data set of Engelhart and Horton (2012), and in some locations near the crest of the forebulge,

but they are relatively minor when compared to those associated with the predictions of the previous

ICE-6G_C (VM5a,b) models. These remaining misfits appear to be resilient, however, in the sense that

they cannot be eliminated without sacrificing the fit the model provides to the Fennoscandian spectrum

of scale-dependent relaxation times (those tests were not explicitly shown here). Such persistent misfits

could be an indication of lateral heterogeneity in the viscosity structure of the Earth, which is expected

to exist based on the nature of the mantle convection process, where upwellings and downwellings are

respectively associated with hotter (and less viscous) and colder (and more viscous) material. However,

these misfits could also be related to compaction effects in some areas (Horton et al., 2013), and will

be the subject of further work. Independent additional tests of the new model have also been discussed

in this paper. The first such independent test involved the analysis of RSL data from the U.S. West

coast which is also undergoing the process of glacial forebulge collapse but the data from which were not

employed to constrain the new VM6 viscosity structure. With the exception of misfits associated with

tectonic influence at a few sites along this coast and with the close proximity of the glacial advance of


the Western Cordilleran ice sheet, the new model was shown to reconcile equally well the data from this

region as it does along the U.S. East coast.

Another independent test of the new model in the North American domain would verify the impact

upon the misfit to the voluminous data base of GPS observations of vertical motion of the crust to which

the ICE-6G_C (VM5a) model has also been tuned. An analysis of this nature will be presented in the

following section, and will identify remaining regions of misfit. This presents a potentially major issue

that arises when one proceeds using the methodology employed in the analyses presented in this chapter,

as it focuses solely on changes to the viscosity structure to eliminate any existing misfits. As the loading

history was fixed here to the ICE-6G_C model, which tuned to enable a good fit to a rich database of

GPS observations of vertical motion of the crust and to a large number of carbon-dated RSL histories

from both Northern Canada and Fennoscandia (Peltier et al., 2015) when used with the VM5a viscosity

model. Therefore, whether these critical data sets will remain well reconciled by the new ICE-6G_C

(VM6) model of the global GIA process remains an important question. If significant misfits to these

data were found to exist even when variations to the ice loading history were allowed, then the iterative

process we are employing to refine the model might not be convergent. This iterative process consists

of alternately fixing the viscosity and the loading history and then refining the other. By returning

to investigate the fit to GPS observations over the North American continent, the circle can be closed

to test convergence. Such a process, using the advocated modifications to the radial profile of mantle

viscosity presented herein, will be detailed in the following chapter.

Chapter 4

Full GIA constraints over N. America:

the ICE-7G_NA (VM7) model

The increasing availability of large global and regional databases of relative sea level history records

(Tushingham and Peltier, 1992; Engelhart and Horton, 2012; Engelhart et al., 2015; Khan, N. S. et al.,

2015b; Vacchi et al., 2016) has greatly benefited the study of the GIA process. Together with the con-

current development of space-geodetic constraints on the rotational state of the planet (Peltier, 1982;

Wu and Peltier, 1984; Peltier and Jiang, 1996), they have led to the successful development of compre-

hensive models such as the ICE-5G (VM2) model of Peltier (2004). Further significant improvements to

GIA models have been enabled by the availability of large networks of space-geodetic measurements of

crustal motion (Argus et al., 1999; Milne et al., 2001; Argus and Peltier, 2010; Argus et al., 2014), and

of satellite-based time-dependent gravity observations from the Gravity Recovery and Climate Experi-

ment (GRACE) satellites (Peltier, 2004; Peltier et al., 2015), which have been used most recently in the

development of the ICE-6G_C (VM5a) global model of Peltier et al. (2015). In the previous section,

a detailed analysis of the performance of this model in terms of its ability to reconcile data from the

region of forebulge collapse outboard of the Laurentide ice sheet has been invoked to further tighten

the constraint on mantle viscosity (beyond that provided by RSL data from the region that was previ-

ously ice covered). It led to the ICE-6G_C (VM6) model of Roy and Peltier (2015), which reconciled

observations of past RSL along the U.S. East coast with model predictions, while producing a good or

excellent fit to a suite of other observables linked to RSL evolution or the post-deglaciation relaxation of

the centers of the former ice sheets covering North America and Fennoscandia. However, modifications

to the viscosity structure of the mantle may also be expected to impact the ability of the new model to

100

Chapter 4. Full GIA constraints over N. America: the ICE-7G_NA (VM7) model 101

maintain its fit to additional constraints related to the present-day crustal uplift rate of the continent

that had led to the precursor ICE-6G_C (VM5a) model. This situation is addressed in the present

chapter, where the optimization methodology used in Roy and Peltier (2015) is expanded to encompass

not only the U.S. East coast relative sea level records, but also those space-geodetic constraints on the

uplift rate of the crust. Also, whereas the previous analysis was only concerned with mantle viscosity

variations, variations in continental ice cover over the region will also be allowed. The plausibility of the

Roy and Peltier (2015) assumption that the ICE-6G_C loading history may be considered converged

will thus be directly addressed.

It will demonstrate that updating the most recently published refinement of the global model (ICE-

6G_C (VM5a)) with the VM6 viscosity structure leads to significant misfits with the space-geodetic

uplift rate observations which the original ICE-6G_C (VM5a) model fit with high accuracy. This raises

the issue of the convergence of the iterative methodology being employed in the process of model con-

struction. Through modest further modifications of both the viscosity structure of the model and the

North American component of the surface mass loading history, a locally optimized solution which elim-

inates these misfits and maintains an excellent fit to the RSL constraints from the U.S. East coast is

obtained. The new model presented herein, ICE-7G_NA (VM7), is then tested against other indepen-

dent constraints to which it was not tuned, including the time dependent de-levelling of the Great Lakes

region, the time dependent gravity results being provided by the GRACE satellites and the shape of the

late Holocene forebulge. In this chapter, the observables used in the study will first be described. Then,

the predictions of the same observables resulting from the latest ICE-6G_C (VM5a) and ICE-6G_C

(VM6) models (Peltier et al., 2015; Roy and Peltier, 2015) will be assessed, the sensitivity of any re-

sulting misfits to appropriate variations in mantle viscosity and ice cover determined, and a new model

adjustment, ICE-7G_NA (VM7), which removes those misfits, will be presented. It follows closely the

contribution of Roy and Peltier (2017), under review for publication (Roy, K., and Peltier, W. R. (2017),

’Space-geodetic and water level gauge constraints on continental uplift and tilting over North America:

Regional convergence of the ICE-6G_C (VM5a/VM6) models’, Geophysical Journal International).

4.1 Geophysical observables related to the GIA process in North

America and the performance of current models

Models of the GIA process can be constrained, refined and tested by comparing their predictions of

various observables to geophysical and geological inferences of the same quantities. An overview of the


main data sets used in this analysis is provided here.

4.1.1 Constraints on former ice sheet extent and thickness

In the construction of the ICE-5G (VM2) and ICE-6G_C (VM5a) models of the GIA process, one of

the main parameters to be constrained is the evolution of ice sheet cover over the continents during

the last glaciation-deglaciation cycle (Peltier, 2004; Peltier et al., 2015). The determination of the time

dependence of the area covered by grounded continental ice may be constrained on the basis of radio-

carbon dated biological material from ice sheet margin locations (end moraines, former pro-glacial lakes,

etc.) and exposure dating from cosmogenic nuclide abundances (10Be, 14C,26Al, 36Cl, 3He and 21Ne)

which are based on the inhibition of their natural formation on the bedrock when the surface is covered

by a thick ice sheet. These clues have been successfully used to reconstruct past ice sheet margins over

North America for the Laurentide, Cordilleran and Innuitian ice sheets (Dyke et al., 2002; Dyke, 2004),

shown in Figure 4.1, and over Fennoscandia (Gyllencreutz et al., 2007; Hughes et al., 2016).

Estimates of ice sheet thickness variations within the evolving margins are provided by the amplitude

of the emergence of the land with respect to evolving sea level provided by radio-carbon dated relative

sea level histories or from the application of fully coupled models of ice-Earth-ocean interactions in

conjunction with them (Tarasov et al., 2012; Stuhne and Peltier, 2015). Appropriate records of sea level

evolution at far-field sites also provide strong constraints on the time dependent total volume of water

locked in grounded land ice: in particular, the high-quality record of relative sea level change at Barbados

provides an excellent approximation to the eustatic (globally averaged) changes in sea level owing to its

location at the trailing edge of the forebulge associated with the former Laurentide ice sheet (Fairbanks,

1989; Fairbanks et al., 2005; Peltier and Fairbanks, 2006). This record, shown earlier in Figure 1.8, will

play an important role in the current discussion.

4.1.2 The importance of RSL data from the North American region of fore-

bulge collapse

Regional databases of RSL history, such as that for the British Isles (Peltier et al., 2002; Shennan

et al., 2002), the U.S. East coast (Engelhart et al., 2011; Engelhart and Horton, 2012), the U.S. West

coast Engelhart et al. (2015), the Caribbean Sea (Khan, N. S. et al., 2015b) or the western half of the

Mediterranean Sea (Vacchi et al., 2016), can be used to provide further constraints on the GIA process,

but great care must be taken in the interpretation of any misfits between observations and GIA model

predictions, given the potential additional contribution of other local geophysical effects such as tectonics,


Figure 4.1: Deglaciation isochrones for North America, based on the work of Dyke et al. (2002) andDyke (2004), which are the margin constraints applied to the ICE-5G (Peltier, 2004) and ICE-6G_C(Peltier et al., 2015) ice sheet loading histories. From Peltier et al. (2015).


sediment compaction or tidal range change (e.g. Horton et al., 2013; Engelhart et al., 2015; Roy and

Peltier, 2015). The high-quality database for the U.S. Atlantic coast of Engelhart and Horton (2012)

(see also Engelhart et al. (2011)) was employed by Roy and Peltier (2015) to further constrain the depth

dependence of viscosity, and the individual time series of which the database is comprised were employed

to demonstrate that by slightly modifying the viscosity model VM5a to VM6, a marked improvement

in the fit to RSL data along the U.S. East coast could be achieved, particularly in the southernmost

part of the coast, where systematic misfits had been previously identified (Engelhart et al., 2011). This

work will use the same crucial data set to constrain the evolution of relative sea level along the forebulge

associated with the former Laurentide ice sheet.

4.1.3 Space-geodetic uplift measurements over North America

A further set of geophysical observables which is critical for present purposes is the large number of space-

geodetic measurements of vertical crustal motion currently available for the North American continent.

The data set to be employed for the purpose of the discussion herein is a refinement and updated

version of the initial database presented in Argus and Peltier (2010), and recently employed in Argus

et al. (2014) and Peltier et al. (2015). It includes GPS records from the most recent JPL solution,

SLR measurements, VLBI records, Doppler Orbitography and Integrated Radio-Positioning by Satellite

(DORIS)-based measurements, as well as GPS measurements from the Canadian Base network, with

appropriate reference frame and tectonic plate motion considerations in order to measure accurately

long-term ground motions (Argus, 2007; Argus et al., 2010). It covers the time period from 1994 to

2012. A suitable transformation from the GPS measurements to a realization of the 2008 International

Terrestrial Reference System (ITRF2008) (Altamimi et al., 2011) is realized using a subset of the GPS

data, and using appropriate tidal standards (Petit and Luzum, 2010), from which estimates of the time

evolution of each site is determined (Argus et al., 2010, 2014). For the purpose of this work, the linear

trends in vertical motion (together with its 2-σ confidence range) found in each record are used. The

total suite of measurements includes not only 509 JPL GPS sites, but also 52 VLBI, 20 SLR and 37

DORIS sites, as well as 142 GPS sites from the Canadian Base Network (see Peltier et al. (2015)). An

in-depth discussion of the methodology employed can be found in Peltier et al. (2015), Argus et al. (2014)

and Argus and Peltier (2010). In the reference frame construction (and velocity of the Earth’s center

determination), estimated parameters include the velocity (rotational and translational components)

between the reference frames of each of the four space-geodetic techniques, the angular velocities of the

major plates, and the velocities of sites on sites that are not currently undergoing significant post-glacial


recovery or ice loss.

In Peltier et al. (2015), it was shown that the ICE-6G_C (VM5a) model provides an excellent fit to

almost all such available measurements. The locations of the vertical motion measurements upon which

we will focus for present purposes are shown on Figure 4.2, together with a graphical representation

of the values of the measurements and their associated error bars. Clearly evident is the well defined

boundary between the region of present day uplift in the region that was once covered by the Laurentide

ice sheet and the forebulge region south of the U.S.-Canada border in which present day crustal sinking is

occurring. Figure 4.3 presents maps of the predicted present-day uplift rate by the ICE-6G_C (VM5a)

and ICE-6G_C (VM6) models over North America (in panels (a) and (b), respectively). Inspection

of the results in this figure demonstrates that the predictions of both models are characterized by the

existence of a clearly evident double "bull’s eye" pattern of uplift which straddles Hudson Bay. Switching

the viscosity structure from VM5a to VM6 dramatically reduces the amplitude of predicted present-day

vertical crustal motion, as the reduced viscosity in the upper part of the lower mantle leads to much

more rapid relaxation, thereby diminishing the uplift rate extrema seen over both former Laurentide ice

domes (by as much as 5 mm/yr for the Keewatin Dome located west of Hudson Bay, and by 3 mm/yr

for the dome that was located over present-day Quebec and Labrador) and over all the regions that were

once covered by the Laurentide ice sheet. The forebulge region is also strongly impacted both in extent

and amplitude from a switch to the VM6 profile, with strong extrema occurring on the oceanic floor of

the Labrador Sea and in the American Midwest. Along the U.S. East coast, a reduction in subsidence is

also observed (reaching 1.4 mm/yr in the mid-Atlantic portion of the coast). The demarcation line that

separates regions of uplift and subsidence is, however, not affected by the change in viscosity structure.

The large difference in vertical motion predicted by these models has consequences for their ability

to explain the space-geodetic measurements recording the crustal vertical uplift rate over the continent.

The lower panels of Figure 4.3 provide a closer look at the ability of both the ICE-6G_C (VM5a)

and ICE-6G_C (VM6) models (in Figs 4.3(d) and 4.3(e), respectively) to explain them within their

observational uncertainties. Clearly, the fit to the space-geodetic data is significantly degraded when

the VM5a viscosity structure is simply replaced with VM6. A primary issue addressed here is whether

there exists or not a modified version of the ICE-6G_C (VM6) GIA model that is simultaneously able

to reconcile both the RSL data from the region of forebulge collapse along the U.S. East coast and

the space-geodetic observations of crustal uplift rates, while also continuing to reconcile all of the other

geophysical observables (global and regional) successfully explained by these models (including RSL

history observations from the ice covered regions of the planet).

Finally, an examination of the quality of the fit provided by the ICE-6G_C (VM5a) and ICE-6G_C


Figure 4.2: Measurements of vertical motion at sites from which space-geodetic measurements are avail-able for the central and eastern parts of the North American continent in the Jet Propulsion Laboratory(JPL) data set, following the reduction methodology of Peltier et al. (2015). The color of each circlerepresents the vertical uplift inferred at each site, whereas its radius is inversely proportional to thestandard error of the individual measurements, following the scale provided at the bottom left of themap. From Roy and Peltier (2017).


Figure 4.3: Model predictions of vertical uplift rates over North America for the (a) ICE-6G_C (VM5a)and (b) ICE-6G_C (VM6) models in mm/yr. The difference between the predictions of those two modelsover North America is shown in panel (c) (in mm/yr). The bottom panels show a comparison of anyexisting misfits between space-geodetic measurements of present-day vertical motion rates (mm/yr) withGIA predictions of those same rates (taking into account observational uncertainties) for the ICE-6G_C(VM5a) (panel (d)) and ICE-6G_C (VM6) (panel (e)) models at the site of each uplift rate measurement.From Roy and Peltier (2017).


(VM6) models to the (mostly) GPS-based observations of modern-day vertical uplift rates reveals the

existence of two regions of persistent misfit in south-western Canada and in the vicinity of the Gulf of

St. Lawrence. A recent analysis of three-dimensional crustal velocities over North America by Snay

et al. (2016) also found residual misfits in these regions. This work will attempt to address whether any

changes in ice margin evolution in this region are required (and sufficient) to remove these misfits.

4.1.4 Time-dependent gravity measurements from the Gravity Recovery and

Climate Experiment (GRACE) satellites

Another independent test of the quality of the ICE-6G_C (VM5a) and ICE-6G_C (VM6) models is

provided by measurements of time-dependent gravity being provided by the GRACE satellites, which

have not been employed to tune the model parameters.

The GRACE system relies on two identical satellites that orbit on the same near-circular low Earth

orbit (altitude of ∼500 km), but separated by about 220 km (Tapley et al., 2004). A microwave ranging

system is used, together with GPS receivers, attitude sensors and accelerometers, to monitor variations

in the separation between the two satellites. These variations are converted into gravity anomalies using

a precise sequence of data manipulations, together with a suitable background gravity model to which the

anomalies can be determined and a reference frame. These "release products" are currently generated by

the U.S. Center for Space Research (CSR) and the GeoForschungsZentrum (GFZ) in Potsdam, Germany.

By monitoring the evolution of these anomalies over many orbits, a sequence of gravity estimates can be

obtained from the measurements, from which gravity anomaly maps and time series can be generated.

The signal observed by GRACE over North America, obtained using the Release 5 product from the

U.S. Center for Space Research, is shown in Figure 4.4(a) in terms of the rate of thickness change of

an equivalent water layer at Earth’s surface, and clearly reveals the double "bull’s eye" pattern (with

extrema existing on either side of Hudson Bay, as described in the recent analysis of Peltier et al. (2015)).

The only processing applied to the Release 5 data before analyzing the linear rate of change has involved

use of a Gaussian filter with a 300-km half-width. In Figure 4.4(b), the prediction of the time-dependent

gravitational changes over North America from the ICE-6G_C (VM5a) model is presented, while Figure

4.4(c) shows the residual when the GIA prediction is subtracted from the GRACE observation so as

to clearly reveal the remaining signal over Greenland and Alaska, which has been associated with the

modern global warming induced rate of loss of grounded ice from these regions (for example, in Luthcke

et al. (2013)). Figure 4.4(d) further corrects the GRACE inference for the gravitational influence of

changes in surface hydrology based upon data from the Global Land Data Assimilation System (GLDAS-


Figure 4.4: (a) Gravity Recovery and Climate Experiment (GRACE) observations of the time dependenceof the gravitational field over North America from the Release 5 data from the Center for Space Research(CSR) for the period covering January 2003 to October 2013, on which no correlated filter was applied buta spatial filter of Gaussian half-width of 300 km was applied; (b) ICE-6G_C (VM5a) model predictionsof this field; (c) Difference between the raw GRACE field (a) and the ICE-6G_C (VM5a) prediction(b); (d) Difference between the GRACE field (a), corrected for hydrological effects using the GLDAScorrection of Rodell et al. (2004), and the ICE-6G_C (VM5a) prediction; (bottom panels) Same figuresas (b) to (d), with using the ICE-6G_C (VM6) model predictions instead. All results are presented inequivalent water layer thickness change (cm/yr). From Roy and Peltier (2017).

NOAH) model (Rodell et al., 2004). Application of this correction amplifies the strength of the "bull’s

eye" pattern. Figs 4.4(e) to 4.4(g) show the same information as Figs 4.4(b) to 4.4(d), except using

ICE-6G_C (VM6). As described in Peltier et al. (2015), ICE-6G_C (VM5a) is able to reproduce the

double "bull’s eye" pattern extremely well while the application of the hydrology correction slightly

degrades the quality of the fit provided by the model over formerly glaciated areas. The use of the VM6

viscosity profile, however, results in a notable degradation of the quality of the model fit to the GRACE

observations, especially with respect to the strength of the bull’s eye over southern Hudson Bay, a result

that is fully consistent with the misfit to the GPS observations introduced when the VM6 viscosity model

is employed in place of VM5a. This misfit becomes even more prominent if a hydrological correction is

applied to the GRACE data. These questions, in particular with respect to the characteristics of the

hydrology correction provided by the GLDAS-NOAH model (and also by competing representations),

will be explored in a subsequent section of this chapter.


4.1.5 Water level gauges in the Great Lakes region

An additional data set that will be employed for present purposes is related to the evolving tilt of the

North American Great Lakes basin associated with the differential uplift of the crust between the regions

north and south of the lakes and which is recorded in the time dependent warping of their shorelines.

This differential uplift is accurately recorded by long-term records from water level gauges. Analyses

of such data were initially published in Peltier (1986) but higher quality and longer records have since

become available (Mainville and Craymer, 2005), based on water level measurements at 55 lake shore

locations (some of which extending back to 1860). In their methodology, the seasonal and systemic water

level variations that are common to all locations within each lake are taken into account by pair-wise

comparisons of water level change, and a least-squares fit is performed to remove any inconsistencies

between each pair of sites (Mainville and Craymer, 2005). Long-term uplift measurements are provided

with respect to the outlet of each lake. The differential uplift measurements obtained by Mainville and

Craymer (2005) for each of the Great Lakes are presented in Figure 4.5(a) (with respect to each lake

outlet, shown in white). They are then compared to the ICE-6G_C (VM5a) and ICE-6G_C (VM6)

model predictions in panels (b) and (c) of the same figure, where the observations in each lake are adjusted

uniformly by the predicted uplift at each lake outlet. Panels (d) and (e) show the difference between

the model predictions of differential uplift for each lake and the Mainville and Craymer (2005) inference

when accounting for the 2-σ uncertainty in the observations. Both models can adequately represent the

differential uplift observed in Lakes Ontario and Erie, while misfits exist for the other Great Lakes. In

particular, both ICE-6G_C (VM5a) and ICE-6G_C (VM6) do not accurately capture the differential

uplift seen across the northern and southern shores of Lake Superior and in the southern half of Lake

Michigan, while ICE-6G_C (VM5a) also predicts misfits in the vicinity of the Upper Peninsula of Lake

Michigan.

4.2 Characterization of the model misfits to present-day space-

geodetic uplift rate observations

Since a primary goal of this chapter is to demonstrate that the degradation of the fit to the space-geodetic

measurements of vertical crustal motion may be simply eliminated without sacrificing the quality of the

fit to the U.S. East coast data that constrain the time dependence of forebulge collapse, the nature of the

misfits to these data induced by the switch from VM5a to VM6 is explored. A simple sensitivity analysis

is performed, in which the impact of a systematic scan of depth-dependent changes in the viscosity


Figure 4.5: (a) Mainville and Craymer (2005) observations of vertical uplift for sites along the shores ofthe Great Lakes (mm/yr), compared to the reference point of each individual lake outlet (indicated as awhite circle); (b) Predicted vertical uplift rate field from the ICE-6G_C (VM5a) model superimposed bythe Mainville and Craymer (2005) measurements of vertical uplift (the data of each lake being correctedby the amount of uplift predicted by the model at each lake outlet); (c) Same field for the ICE-6G_C(VM6) model; (d) Difference between the modeled vertical uplift rate and the adjusted observations ofMainville and Craymer (2005) at each outlet, accounting for the 2-σ uncertainty in the inferences, forthe ICE-6G_C (VM5a) model (with its predicted uplift rate field in the background); (e) Same as (d),but for the ICE-6G_C (VM6) model. From Roy and Peltier (2017).


structure on vertical motion predictions is assessed. Following this analysis, the strong constraint of a

loading history to be fixed to that of ICE-6G_C (as in Roy and Peltier (2015)) is relaxed.

4.2.1 The impact of elastic lithospheric thickness and mantle viscosity vari-

ations upon vertical uplift rate predictions

Variations in the viscosity structure of the mantle are known to significantly affect the sea level evolution

predictions obtained from GIA models (e.g. Roy and Peltier, 2015), but the response of the solid Earth

to ice loading and the range of mantle depth over which the response to deglaciation will be sensitive to

viscosity is also highly dependent on the horizontal scale of the former ice sheets (Peltier, 1974; Wu and

Peltier, 1984; Mitrovica and Peltier, 1993b; Peltier, 1996b, 1998a). In Fennoscandia and in the British

Isles, due to the relatively small horizontal extent of the ice sheets that once covered these regions,

the relative sea level evolution predictions are mostly sensitive to the viscosity structure at shallower

depths (upper mantle and lithosphere) (Peltier et al., 2002; Shennan et al., 2002). For North America,

it was demonstrated that, due to the larger scale of the former Laurentide ice sheet, the evolution of

the forebulge associated with it is sensitive primarily to the viscosity at a greater depth in the mantle,

namely the lower part of the upper mantle and the upper part of the lower mantle (Roy and Peltier,

2015). However, the initial shape of the forebulge (namely, its maximum amplitude and geographical

extent) was found to be significantly impacted by the thickness of the elastic lithosphere and the viscosity

contrast between the upper and lower mantles (e.g. Tushingham and Peltier, 1991).

It is expected that this degree of sensitivity to variations in mantle viscosity and elastic lithosphere

thickness should extend to predictions of vertical uplift rates. To this end, the same simple approach

employed by Roy and Peltier (2015) is now applied to the case of vertical uplift rate predictions. A

series of experiments is performed in which the thickness of the elastic lithosphere and the viscosity of

the mantle are allowed to vary independently at various depth ranges in the mantle, while the rest of the

profile remains identical to the VM5a profile of Peltier and Drummond (2008). Throughout this initial

process, the ice loading is fixed to the ICE-6G_C model.

A summary of the most important results of this sensitivity study is presented in Figure 4.6. The

impact of variations in the viscosity of the upper mantle (UM) is shown in the first row. The resulting

uplift rate field for two VM5a variations is presented (upper mantle viscosities of 0.2 · 1021 Pa·s (panel

(a)) and 0.8 · 1021 Pa·s (panel (b)), together with the difference between each of these signals and the

original VM5a-based prediction. An increase in upper mantle viscosity leads to a stronger uplift signal

over the formerly glaciated area and more subsidence over the inner forebulge. On the eastern seaboard,


Figure 4.6: Comparison of the modern vertical uplift rate (mm/yr) over the North American continentusing model variations based upon the ICE-6G_C (VM5a) model, where the ICE-6G_C loading historyis kept fixed, but a single feature of the VM5a viscosity model is modified. (Upper row) Predictedvertical uplift rate field over North America, together with the difference between this field and theoriginal ICE-6G_C (VM5a) model, for the cases in which the viscosity of the upper mantle (UM) ischanged to (a) 0.2·1021 Pa·s, and (c) 0.8·1021 Pa·s; (2nd row) Same as the upper row, except for modifiedversions of the VM5a structure in which only the viscosity of the upper part of the lower mantle (ULM)is changed to (e) 0.8 · 1021 Pa·s, and (g) 2.4 · 1021 Pa·s. From Roy and Peltier (2017).


the impact of viscosity variations in the upper mantle upon present-day uplift rates is negligible south

of New Jersey. The second row of Figure 4.6 shows the results of a similar analysis, but for viscosity

variations in the upper part of the lower mantle (ULM: 420 to 1260 km depth), for two examples (one

below and one above the VM5a value), namely 0.8 · 1021 Pa·s (panel c) and 2.4 · 1021 Pa·s (panel d),

together with the difference between these fields and the initial ICE-6G_C (VM5a) result. A visual

comparison of the UM and ULM results shows that the impact of viscosity variations on the measured

uplift rate under previously glaciated regions becomes more geographically uniform with increasing depth

of the viscosity perturbation. As the geographical extent of the region of the forebulge sensitive displays

a strong dependence to variations in viscosity in the upper part of the lower mantle, whether a significant

gain in the quality of the fit could be obtained by increasing the resolution of the viscosity profile in this

region will be considered later in this analysis.

4.2.2 Impact of ice loading history variations on vertical uplift rate predic-

tions

To evaluate the sensitivity of model predictions of present-day vertical uplift rates to changes in ice

loading history, two families of scenarios will be considered. The first involves changes in ice sheet melting

rates, notably in the vicinity of the Younger Dryas cold interval (that extended from approximately

12,800 to 11,500 years Before Present (BP) (Muscheler et al., 2008)), and the second will consider

mass-conserving changes in the geographical distribution of ice thickness. The first family of sensitivity

analyses focuses upon the global eustatic change of sea level that has occurred since LGM, recorded by

the Barbados record of relative sea-level evolution (Fairbanks, 1989; Peltier and Fairbanks, 2006). The

fit provided by the ICE-6G_C (VM5a) model of Peltier et al. (2015) to this crucial record of past sea

level evolution is shown in Figure 4.7. The model provides an excellent fit through most of the period

around LGM and during the deglaciation phase, but an important misfit to this coral based record is

evident in the Younger Dryas period, which ends at approximately 11.5 ka. These errors of fit, which

were not present in the precursor ICE-5G (VM2) model (Peltier, 2004; Peltier and Fairbanks, 2006),

could conceivably be due to the slight difference in total ice melt (and its timing) in Antarctica and

North America between the two models. It may be most natural to seek an improvement of fit to the

Barbados record in the Younger Dryas interval of time in terms of a modification to the ice melting

history of the Laurentide ice sheet, since it has been argued that the return to near-glacial conditions

itself was triggered by a meltwater injection into the Arctic Ocean from proglacial lakes associated with

the Laurentide deglaciation (Tarasov and Peltier, 2005). In a simple sensitivity experiment, variations


in the history of the Laurentide Ice Sheet complex are introduced, with successively larger reductions

in ice sheet melt rates over the entire ice sheet during the Younger Dryas period, until a best fit to

the Barbados record is achieved. The reductions in melt rate are introduced abruptly at the start

of the Younger Dryas (reflecting the abrupt nature of its onset), while progressively returning to the

original ICE-6G_C ice loading model at the end of the Younger Dryas (with the difference between

the original and the perturbed model following a decaying exponential curve). The fit provided by the

best-fitting model variation based upon this simple methodology is presented in Figure 4.7. In this

version, called ICE-6G_C_BARB, the melt rate falls to zero during the Younger Dryas period. It is

clear that this new loading model eliminates the misfit to the Barbados record through the interval

from 12.8 ka to 11.5 ka, which was an evident flaw in the ICE-6G_C (VM5a) model of Peltier et al.

(2015) (who commented explicitly on the issue). This modification is an expected consequence of the

return to extreme glacial conditions during this interval of time. Finally, in this context, it should be

noted that some recent analyses of the living depth range of various coral species (Hibbert et al., 2016)

have suggested an uncertainty depth range for the Acropora Palmata species of 3 meters (1-σ error) or

9.4 meters (2-σ error), indicating the very shallow depth at which most individuals can be expected to

reside. In the analysis of Peltier and Fairbanks (2006) (and subsequent studies based upon this record,

including here), an uncertainty of 5 meters is used (Fairbanks, 1989), which effectively falls between the

two new interpretations of the biological record. It should be noted that the misfit identified around the

Younger Dryas period would remain even if one were to consider the larger depth range value of Hibbert

et al. (2016).

Figure 4.8 presents the vertical uplift rate predictions over North America provided by this modified

version of the ICE-6G_C ice loading history when it is employed in conjunction with the VM5a (upper

row) and VM6 (lower row) profiles of radial variations in viscosity. The first figure of each row presents

the predicted field of vertical uplift rate predicted for each viscosity structure using the ICE-6G_C

model (panels (a) and (d), respectively), while the central figure of each row presents the same field,

but using the ICE-6G_C_BARB test scenario (panels (b) and (e) for VM5a and VM6, respectively).

The difference between the test ICE-6G_C_BARB scenario and the ICE-6G_C model is shown on

the rightmost panels of Figure 4.8 (panels (c) and (f)). The modifications in melting rate that lead to

the ICE-6G_C_BARB model not only recover a better fit to the Barbados sea-level record, but also

substantially increase the amount of predicted present-day uplift under the former Laurentide ice sheet

that is obtained with both viscosity structures, with the maximum occurring in the core of the former

Laurentide ice sheet, and minor maxima located west of Hudson Bay and over Baffin Island. For both

viscosity structures, given the short timescale over which the modified melting scenario has been applied,


Figure 4.7: The coral-based record of relative sea level evolution at the Caribbean island of Barbados ofPeltier and Fairbanks (2006), with the vertical bars representing the depth range of the correspondingcoral species (including 5 meters for Acropora palmata and 20 meters for Monastera annularis). TheU/Th age uncertainties are small and close to the thickness of the plotted lines, so are not shown.Comparison of the fit provided by the ICE-6G_C (VM5a) (green), together with the modified ICE-6G_C_BARB (VM6) model described in the text (black). From Roy and Peltier (2017).


Figure 4.8: Model predictions of vertical uplift over North America for the (a) ICE-6G_C (VM5a) and(d) ICE-6G_C (VM6) models in mm/yr. The predictions for the models using a modified version of theICE-6G_C model that corrects the fit to the Barbados data is shown for the (b) VM5a structure, and(e) VM6 structure. The difference between the ICE-6G_C (VM5a) and ICE-6G_C_BARB (VM5a)models is shown in (c), whereas the difference between the ICE-6G_C (VM6) and ICE-6G_C_BARB(VM6) models is shown in (f) (in mm/yr). From Roy and Peltier (2017).

the impact of the modified ice loading history is not felt in the forebulge, except in the Labrador Sea

region. This slight shift in the melting history through the Younger Dryas interval significantly increases

the predicted present-day rates of vertical motion by delaying the timing of deglaciation, an effect that

could partly cancel the decrease in predicted uplift rate caused by a shift towards slightly less viscous

model (in particular for the upper part of the lower mantle, as in the transition from VM5a to VM6).

These results support the idea that there could potentially be a model of the GIA process that is able

reconcile both RSL records from the region of forebulge collapse and the present day measurements of

vertical motion of the crust.

The second series of sensitivity analyses concerning glaciation history will consider the thickness

distribution of the ice at LGM and during the deglaciation phase, and explore the impact that variations

in Laurentide ice sheet "shape" have on predictions of present-day vertical uplift. Here, only the impact

of a simple redistribution of the existing ice between the northern and the southern sectors of the ice


sheet are considered. The goal of this simple experiment is to determine whether simple variations

in the distribution of ice could also be involved in understanding the marked reduction in predicted

uplift rate delivered by the ICE-6G_C (VM6) model. The result of these experiments is shown in

Figure 4.9. Two families of variations are considered: one in which mass from the northern half of

the Laurentide ice sheet is redistributed to the southern half between 19 ka and 13 ka (the "Early"

case, upper-row panels of Figure 4.9), and one in which the same variation is introduced between 15 ka

and 9 ka (the "Late" case, lower-row panels of Figure 4.9). Two amplitudes are considered (thickness

redistributions that reach a maximal amplitude of 200 and 400 meters) and, in each case, the amplitude

of the perturbation is smoothed at the beginning and at the end of the time period under consideration

to ensure continuity in the local thickness of the ice sheet. It should be noted that the amplitudes of the

redistributions remain small compared to the overall thickness of the ice sheet at the majority of time

steps under consideration (except perhaps for the youngest time under consideration in the "Late" case,

but it should be remembered that this simple test is, at this early stage in the methodology, focused on

obtaining a physical understanding of the overall sensitivity of the predictions to such redistributions and

that any final modifications of this kind to the model would be required to be fully physically consistent).

The vertical uplift rate field predicted by the model and its difference with respect to the original ICE-

6G_C (VM5a) prediction are investigated. This simple experiment results in a reduction of the uplift

predicted over northern Canada and a corresponding increase in uplift predicted in southern Canada,

but the uplift in the forebulge is once more unaffected. Increasing the magnitude of the modification

in ice thickness from 200 to 400 meters simply increases the magnitude of the uplift change, without

impacting its geographical extent. For a given change in ice thickness, an earlier application results in a

less pronounced and smoother variation across the entire region over which the modification was applied.

4.2.3 The impact of ice loading history variations on relative sea level evo-

lution

The sensitivity of the evolution of relative sea level in the forebulge to the simple variations in ice melting

history and ice redistribution described above is now considered. In Figure 4.10, the inferences of relative

sea level evolution from the Engelhart and Horton (2012) compilation are presented for a subset of sites

along the U.S. East coast (Southern Maine, New York, Inner Chesapeake Bay and South Carolina), and

are compared to predictions provided by the variations of the ICE-6G_C model discussed above (all of

which are used in combination with the VM6 viscosity profile). The cases considered here all include

the simple modifications to the ice sheet melting rate around the Younger Dryas cold event and the


Figure 4.9: Prediction of the vertical uplift rate field over the North American continent from modifiedICE-6G_C (VM5a) models (mm/yr), in which the ice loading history is modified following the descrip-tion in the text, for the case of (a) an early application of a 200 m change; (c) an early application ofa 400 m redistribution; (e) a late application of a 200 m redistribution, and (g) a late application of a400 m redistribution. The difference between each field and the original ICE-6G_C (VM5a) predictionis shown in (b), (d), (f), and (h) for each respective case (mm/yr). From Roy and Peltier (2017).


simple "Early" (19 ka to 13 ka) and "Late" (15 ka to 9 ka) ice redistribution scenarios (amplitude of 200

meters).

The impact of ice thickness redistributions within the Laurentide ice sheet on sea level history deter-

mination along the U.S. East coast is small, except for the northernmost sites where edge effects are to

be expected. Changing the ice melting rate to recover a fit to the Barbados record of sea level history

around the Younger Dryas cold event has a small impact on the sea level history predicted along the

U.S. East coast. The effect is stronger for the northernmost sites, and fades as one moves south into

the forebulge. This seems to indicate that the impact on RSL evolution along the U.S. East coast will

be small, though not entirely negligible, at most sites if such a modification to the ice melting history is

introduced, due to the limited memory of the system to these relatively small mass loading variations.

Nevertheless, since VM6 was optimized to be used in conjunction with the ICE-6G_C loading history,

it will be necessary to determine if another viscosity profile is necessary to maintain (or improve upon)

the high quality of the fit provided by the VM6 model.

4.2.4 The impact of mantle viscosity variations on the differential uplift of

the Great Lakes

The final data set considered in this sequence of preliminary sensitivity analyses will focus upon the

differential uplift recorded by water gauge measurements in the Great Lakes that record the ongoing

post-glacial tilting of the region and the transition from the region of deglaciation-induced uplift to that

of forebulge collapse. The impact of shallow Earth structure variations on the ability of the model to

reproduce the Great Lakes coastal water gauge uplift rate compilation of Mainville and Craymer (2005)

is investigated by considering changes in elastic lithosphere thickness and upper mantle viscosity, while

the rest of the viscosity structure is kept identical to that of the VM5a profile and the ice loading history

is fixed to ICE-6G_C. In this analysis, as the observed uplift data is presented for each of the Great

Lakes with respect to its outlet, the observations need to be adjusted by an amount corresponding

to the model prediction at the outlet of the Great Lake in which it is located, in order to capture

the uplift gradient across each one of the Great Lakes. Results for a subset of two elastic lithosphere

thicknesses are considered in the upper panels of Figure 4.12 (40 km and 220 km). Inspection of this

figure demonstrates that increasing the thickness of the lithosphere results in a much smoother transition

between the uplifting regions of the continent and the forebulge collapse region. These variations do not

significantly impact the ability of the model to reproduce the uplift observed in most of the Great Lakes

area, except for Lake Erie, where the observed gradient in vertical motion is not compatible with the


Figure 4.10: Comparison of the relative sea level history predicted at four key sites along the U.S. Eastcoast from the (Engelhart and Horton, 2012) data set, namely Southern Maine, New York, the InnerChesapeake Bay and Southern South Carolina. The ICE-6G_C (VM6) prediction (black) is comparedto the ICE-6G_C_BARB (VM6) model (red), and the two cases where a 200 meter redistribution ofice is performed "early" (green) and "late" (blue) on the ICE-6G_C model (description in the text).Green data points represent sea-level index points, whereas blue crosses represent marine-limiting dataand orange crosses stand for terrestrial-limiting data. From Roy and Peltier (2017).


smooth gradient predicted when choosing models with a thick elastic lithosphere. All models are equally

able to reproduce the gradient observed in Lakes Huron and Superior.

The equivalent experiment, but performed with viscosity changes in the upper mantle, is presented in

the lower panels of Figure 4.12 for two different values (0.2 · 1021 Pa·s and 0.8 · 1021 Pa·s). In this exper-

iment, the rest of the viscosity structure is kept identical to that of VM5a. It can be seen that changing

the viscosity of the upper mantle strongly impacts the amplitude of the observed uplift/subsidence, but

does not affect the shape of the contours (and in particular of the zero uplift transition). A much stronger

gradient is observed when using a stiffer upper mantle. The stiffest model (0.8 · 1021 Pa·s) overestimates

the north/south gradient that is observed across the majority of the lakes (in particular for Lake Huron),

while the softest model considered (0.2 · 1021 Pa·s) underestimates it.

4.3 Parameter space search and model refinement

Using the knowledge gained from the analysis of the sensitivity of the fit to the suite of geophysical

observables that may be invoked to constrain the GIA process provided by variations of the ICE-6G_C

(VM5a/VM6) models (both in mantle viscosity and ice loading history over North America), we investi-

gate whether a single best-fitting model can be identified. In particular, we wish to answer the question

as to whether the iterative process employed in GIA model construction actually converges to an optimal

solution, and to address the meaning of such convergence (or otherwise). This work may also provide

insight into the issue of the uniqueness of the global solution to the GIA problem.

In this analysis, we will continue to focus on spherically-symmetric models of mantle viscosity. The

reason for this approach, even if the temperature dependence of the creep resistance of mantle material

would lead one to expect the existence of lateral viscosity heterogeneity to be characteristic of the

convecting mantle on a priori grounds, is to fully explore the explanatory capabilities of a model of

minimum complexity. The best-fitting spherically symmetric model thereby determined is expected to

provide a suitable background viscosity structure upon which lateral variability can be superimposed

and additional degrees of freedom added (if necessary). In the absence of a well-constrained background

model of this kind, it is our view that the ad hoc addition of new degrees of freedom embodied in models

characterized by significant lateral heterogeneity of viscoelastic structure would not be meaningful.

The strategy to be employed here will expand the "brute force" methods originally employed in

the iterative process, where the ice loading history is first kept intact to focus on viscosity refinements,

and then fix viscosity to refine loading history. A stage has now been reach where the refinement of

both components of the model structure will be required. In broad terms, a voluminous suite of over 380


Figure 4.11: Comparison between the Mainville and Craymer (2005) observations of vertical uplift forsites along the shores of the Great Lakes presented with respect to the value modelled at the referencepoint of each individual lake outlet (filled circles representing the value in mm/yr) with a modified ICE-6G_C (VM5a) scenario where only the thickness of the elastic lithosphere is changed to (a) 40 km, and(b) 220 km, or where only the viscosity of the upper mantle (UM) is modified to (c) 0.2 · 1021 Pa·s and(d) 0.8 · 1021 Pa·s. From Roy and Peltier (2017).


mantle viscosity profiles is first generated, covering a plausible range of parameter space in each viscosity

layer of the mantle determined by the sensitivity analyses discussed above and in Roy and Peltier (2015).

The ability of each model to recover a good fit to both the vertical uplift rate data and the U.S. East

coast relative sea level data of Engelhart and Horton (2012) is then carefully determined. Then, the 25%

top performing viscosity structures are kept, and perturbations in the ICE-6G_C ice loading history are

introduced upon them. These include both the introduction of a perturbation in the time evolution of

the Laurentide ice sheet mass to recover a fit to the Barbados record during the Younger Dryas, as well

as perturbations in the geographical distribution of the ice melt during the deglaciation phase across

various regions of the Laurentide ice sheet. These modifications are applied successively, with the idea

that the misfits should be reduced as significantly as possible with each iteration. Finally, predictions

from the best performing model resulting from the parameter space exploration will be tested against

a series of additional important geophysical observables that were not employed in the optimization

process, such as observations of modern-day differential tilt of the Great Lakes inferred on the basis of

water level gauges, and time-dependent gravity measurements from the GRACE mission.

4.3.1 Model misfits and error determination

The performance of the GIA models considered in this study will be evaluated using both quantitative and

qualitative approaches. A quantitative analysis of the fit of a given model to the data will be performed

by comparing two sets of important geophysical observables and corresponding model predictions of the

same quantity, namely for the extensive database of relative sea level history observations of Engelhart

and Horton (2012) for the U.S. East coast and for the observed present-day vertical uplift rate of the

North American continent. Although the fit of the model to the totality of the relative sea level records

available from the previously ice covered region is also, of course, of critical importance, they are not

included in this further stage of model iteration, as we are already sufficiently close to a converged

solution that the fit of the current set of model variations to these data remains of the same high quality

as that provided in the Supplementary Materials section of Stokes et al. (2015), such that no useful

purpose would be served by reproducing the fits to these data herein.

The determination of the quality of the fit provided by the predictions of the model to geological

records of relative sea level index and limiting data points is evaluated using the same error function

presented in the previous chapter (Roy and Peltier, 2015). The contribution of limiting data points to

the error function is only non-zero if the predicted sea level falls above a terrestrial-limiting data height

or below a marine-limiting height, as these data provide only an upper or lower bound on relative sea


level at a given time in the past. The uncertainty related to the accuracy of the dating of the geological

inferences of past sea level is taken into account following the methodology of Roy and Peltier (2015),

which takes into account the uncertainty range associated with the age determination of each sample.

The misfits are calculated for each site along the U.S. East coast, and also combined in a single composite

value for the entire coast.

For the misfits between present-day continental uplift rates observed from space-geodetic techniques

and corresponding model inferences, a similar methodology is employed, with the total misfit between

observational data and the prediction of a GIA model at the given location for the present-day vertical

uplift rate presented as:

χ2G =

1

NG

NG∑

i=1

(

∆G,i

σG,i

)2

, (4.1)

where NG represents the number of present-day uplift observations in the data set as compiled and

tabulated in Peltier et al. (2015), σG,i is the uncertainty in each uplift rate observation (95% confidence

level), and ∆G,i represents the difference between each measurement and the GIA model prediction for

the same location.

The misfits for each major data set are combined into a single composite value. For relative sea-level

evolution along the U.S. East coast, the performance indicator of a given GIA model is given as a ratio of

the χ2-like error it generates to that provided by the ICE-6G_C (VM6) model of Roy and Peltier (2015),

a model which will be taken as a benchmark for present purposes. To ensure that the focus of the fitting

process is on determining the best fit to the shape of the forebulge, the ratio is weighted for the density

of constraining data at each site (an average of all regional ratios is used). It should be noted that, as

opposed to Roy and Peltier (2015), where the stated goal was to remove specific misfits pertaining to the

earlier part of the relative sea level record, the performance indicator used here encompasses the entirety

of the geological record. With regards to the fit to the space-geodetic constraints on vertical uplift rates,

a similar ratio is generated with respect to the fit provided by the ICE-6G_C (VM5a) model. Then, a

final, synthesized value is generated as a simple average of the two ratios generated.

4.3.2 Initial viscosity model generation and parameter space exploration

First, a wide family of initial viscosity structures covering a large region of parameter space is generated,

using the VM6 profile as an initial model upon which modifications are introduced. However, based

on the previous sensitivity analysis, mantle viscosity variations in the upper part of the lower mantle

(or slightly below) appear to play a fundamental role in reconciling the fit to both important data sets


at hand, but in opposite directions, as a lower viscosity at this depth, favored by the RSL constraints,

exacerbates the misfits to the space-geodetic observations. To account for this observation, the upper

part of the lower mantle is further subdivided in two sections to account for potential differences in peak

sensitivity with depth between the two data sets. Also, the transition to the lower part of the lower

mantle is pushed slightly deeper in the mantle to account for the residual sensitivity of forebulge RSL

observations observed in the lower part of the lower mantle in Roy and Peltier (2017). Thus, the layers

under consideration are the elastic lithosphere (of variable thickness), a high-viscosity stratification layer

beneath the elastic lithosphere, the upper mantle and transition zone (down to a depth of 670 km), the

upper part of the lower mantle (depth range of 670-980 km), the middle part of the lower mantle (depth

range of 980-1470 km), and the lower part of the lower mantle (from 1460 km depth to the core-mantle

boundary). The thickness of the high-viscosity stratification layer beneath the elastic lithosphere is

fixed to 40 km, the value found by Peltier and Drummond (2008) to reconcile observations of horizontal

crustal displacement with model predictions (model predictions of North American RSL and vertical

uplift rates were found to be quite insensitive to changes in its thickness (not shown here)).

The range of viscosity values and elastic lithosphere thicknesses considered in each of these layers

is shown in Table 4.1. The greater number of values considered in the parameter space exploration for

lithospheric thickness, upper mantle viscosity and for the upper and middle parts of the lower mantle

reflects the high sensitivity of model predictions of RSL along the U.S. East coast to even small variations

in mantle viscosity in the upper mantle and in the upper part of the lower mantle (Roy and Peltier, 2015)

and the high sensitivity of model predictions of vertical uplift rates to changes in lithospheric thickness.

The values of mantle viscosity considered in each layer or lithospheric thickness are above and below

that of the initial VM6 viscosity profile, and their range is determined following a simple methodology.

In the lithosphere, the upper mantle and the upper and middle parts of the lower mantle, an initial test

is performed in which the value in each layer is varied independently (while the rest of the mantle is

fixed to VM6), until the error function increases by ≈50%, determining the upper and lower ranges to

be considered. To reduce computational costs, and given the lower sensitivity of the predictions to a

change in its viscosity, in the lower part of the lower mantle, only two values (VM5a-like and VM6-like)

are considered for the initial run.

A first range of 384 spherically-symmetric viscosity models is thus generated, and their misfits to the

uplift rate observations over North America and to the U.S. East coast relative sea level data computed.

Each viscosity structure is first run over the last glacial cycle in conjunction with the global ICE-

6G_C ice loading history of Peltier et al. (2015). The misfits between model predictions and the

geological record of RSL evolution of Engelhart and Horton (2012) and the space-geodetic measurements


Table 4.1: Elastic lithosphere thickness and mantle viscosity values considered in the parameter spaceexploration, with the values in each layer varied independently. From Roy and Peltier (2017).

of vertical uplift rates of Peltier et al. (2015) are generated using the methodology outlined in the previous

section.

Several important inferences follow on the basis of this initial analysis. Concerning the performance

of the viscosity structure variations with respect to relative sea level evolution along the U.S. East coast,

our results support the Roy and Peltier (2015) observation that models with a thick elastic lithosphere

are rejected, especially for the southernmost sites, where they systematically exacerbate the misfits

identified in the region when using the ICE-6G_C (VM5a) model. However, these thick-lithosphere

models tend to perform better than their thin-lithosphere counterparts with regards to the vertical

uplift predictions over North America. This apparent conundrum can be resolved only by an increase in

viscosity in the deeper mantle, a region to which the present-day uplift measured near the center of the

former Laurentide ice sheet still displays weak, though non-negligible, sensitivity.

Models based upon viscosity structures featuring the further subdivision of the lower mantle into

three sections provide a fit that is significantly better than the equivalent models that do not have this

feature, as the changes in viscosity can be focused at a greater depth in the lower mantle. This is

beneficial, since RSL predictions for the Atlantic coast are not nearly as sensitive to the viscosity at this

depth as are vertical uplift rates over the center of the former Laurentide ice sheet.

After an exploration of the errors generated by this scan of parameter space and a careful analysis

of the results, the best-fitting mantle viscosity variations that are consistent with these observations

favour a viscosity structure that is somewhat similar to the VM6 profile of Roy and Peltier (2015).

Indeed, among all the alternatives considered in this analysis, only a few are able to provide a significant

improvement over ICE-6G_C (VM5a) and ICE-6G_C (VM6). These models are all found to have a

thin elastic lithosphere (60-80 km), and an upper mantle viscosity equal to that of the VM6 viscosity

profile of Roy and Peltier (2015).


Figure 4.12: Map of the regions in which modifications to the ICE-6G_C ice loading history were initiallyintroduced. From Roy and Peltier (2017).

4.3.3 Ice loading history variations and full model optimization

The second step of the analysis involves the introduction of appropriate ice loading history pertur-

bations. The first ice loading history variations to be introduced is the modification that led to the

ICE-6G_C_BARB model introduced in the earlier sensitivity analysis, with its Laurentide ice sheet

component modified from ICE-6G_C by ice melt rate changes in the vicinity of the Younger Dryas

event (between 12.8 ka and 11.5 ka). This modification, while helping the model recover a good fit to

the extensive coral-based record of sea level evolution at the island of Barbados, also reduces the misfits

observed between observations of vertical uplift rates over the center of North America and ICE-6G_C

(VM6) predictions of the same quantity.

Then, mass-conserving redistribution of ice between different regions of the Laurentide ice sheet

during the initial phase of deglaciation are introduced, smoothed at the beginning and at the end of each

time period, which effectively acts as a change in the regional melt rate of the ice sheet. The regions of

equal area for which such redistribution scenarios are investigated are shown in Figure 4.12, and include

the western (Keewatin) dome of the Laurentide ice sheet, the south-western part of the ice sheet over

Alberta, Nunavut, the southern shore of Hudson Bay, the northern and southern halves of the Great

Lakes region, central Quebec, southern Quebec, and the Gulf of St Lawrence.

The modifications were allowed to occur in the early part of the deglaciation phase (from 26 ka to


16 ka), or in its late phase (from 16 ka to 8 ka, or until the ice sheet had retreated from the region,

whichever comes first). The location of the ice sheet margins are kept identical to those of the ICE-

6G_C model at all times, which are based upon the best estimate isochrone compilation of Dyke et al.

(2002) and Dyke (2004). Each step of the optimization procedure allows mass conserving ice load

redistribution to occur between any two of the regions described above in the early or late phase of the

deglaciation (for various amplitudes, including a smooth merger of the modification introduced for each

region with the surrounding areas, and including ice sheet smoothing inside each region if necessary),

and the redistribution resulting in the most significant reduction of the misfit function is chosen from

an extensive list of such trials.

Subsequent to this step, slight perturbations to the viscosity structure or to the elastic lithosphere

thickness are introduced, and these further modifications to the viscosity structure are kept if any such

change results in a significant further reduction of the misfit function. The process is repeated until

convergence is achieved (i.e. no further ice loading redistribution or viscosity perturbation results in a

significant reduction of the misfit function).

Then, as a final step, the viscosity in the lowermost part of the lower mantle is considered. The

viscosity layer found at great depth in the VM6 model is similarly introduced here, and the viscosity

of the lowermost mantle is adjusted to fit the constraints on the rotational state of the planet provided

by the recent analyses of Roy and Peltier (2011) (namely, on the speed and direction of the true polar

wander phenomenon and on the rate of non tidal acceleration of the Earth’s axial rate of rotation). As

the fit to the space-geodetic uplift rate constraints has only limited sensitivity to viscosity changes at

the top of the lowermost mantle, and none at greater depths, the best-fitting viscosity model is one that

shows a gradual increase in mantle viscosity throughout the layer.

To check the final result of this process, the best-fitting ice loading history resulting from the opti-

mization procedure is re-run with the entirety of the viscosity profiles considered in the initial parameter

space exploration. Also, small perturbations to the mantle viscosity structure are introduced in each

layer (with changes of ±0.1×1021 Pa ·s in the upper mantle and the upper part of the lower mantle, and

±0.5× 1021 Pa ·s in the deeper layers of the mantle), but no change in our best-fitting viscosity profile

provided notable improvements to the quality of the fit, suggesting that our methodology is converging

towards a best, locally optimal spherically symmetric viscosity solution (within the range of parameter

space considered).


4.3.4 The ICE-7G_NA (VM7) model of the GIA process

The fine-tuning process performed in this analysis results in a final model, named ICE-7G_NA (VM7),

which will be shown to satisfy all of the major geophysical constraints related to the GIA process over

the North American continent. In the model, the ice loading history for all regions outside of the North

American continent is kept fixed to that of the ICE-6G_C model of Peltier et al. (2015). The final

viscosity profile, called VM7, is shown in Figure 4.13, where it is seen to be very similar to the VM6

profile of Roy and Peltier (2015) in the upper mantle and upper part of the lower mantle. The profile

is characterized by a much more systematic increase in viscosity in the lower mantle, and it has the

same viscosity in the upper mantle as VM6, but a slightly thinner elastic lithosphere thickness (75

km). A detailed comparison of the viscosity variations with depth found in the VM5a, VM6 and VM7

viscosity profiles is provided in Table 4.2. The modification in lower mantle viscosity from VM6 to VM7

is consistent with the idea that the tests employed in the Roy and Peltier (2015) study, which focused

exclusively on reconciling the model predictions with the geological records of past sea level along the

U.S. East coast, did not provide sufficient resolution in the lower mantle to justify a further refinement

of the viscosity structure in this depth range. However, this increase of viscosity with depth in the lower

mantle is fully expected on a priori grounds based upon a homologous temperature argument, given the

greater rate of increase with depth of the solidus as compared to that of a mantle adiabat. Because the

mantle convective circulation operates at very high Rayleigh number (of O(107)), the depth dependence

of mantle viscosity is expected to be adiabatic and therefore to depend only on depth throughout the

vast majority of the mantle volume (e.g. Peltier and Solheim, 1992; Butler and Peltier, 2000; Shahnas

and Peltier, 2010, for specific examples based upon mantle convection model results).

Table 4.2: Radial viscosity structure of the VM5a, VM6 and VM7 viscosity profiles down to the core-mantle boundary. From Roy and Peltier (2017).

The ICE-7G_NA model of ice loading history includes slight redistributions of ice from the region


Figure 4.13: Comparison of the radial variations of the viscosity in depth for the VM7 viscosity profile(blue/green) with the VM5a (black) and VM6 (red) profiles. From Roy and Peltier (2017).

south of the western (Keewatin) dome to the southern and eastern shores of Hudson Bay and in southern

Quebec, and from the southern part of the Great Lakes region to the Gulf of St. Lawrence region. The

thickness of the Laurentide ice sheet in the ICE-7G_NA model is shown in Figure 4.14 for several time

steps during the deglaciation phase, while differences in Laurentide ice sheet thickness between the initial

ICE-6G_C model and the new ICE-7G_NA model are shown in Figure 4.15 for a selection of time steps.

In Figure 4.16, the predicted topography over North America at LGM for both the ICE-7G_NA (VM7)

and the ICE-6G_C (VM5a) models are shown, together with the difference between the two.

The fit provided by the new ICE-7G_NA (VM7) model to the database of relative sea-level histories

of Engelhart and Horton (2012) is shown in Figure 4.17. The model provides a better fit to the sea-

level index points in the northernmost sites than did ICE-6G_C (VM6), improving one of the minor

drawbacks of that model, and even provides a slight improvement over the ICE-6G_C (VM5a) model

predictions in this region. For the rest of the U.S. East coast, the ICE-7G_NA (VM7) model provides

an equally good fit to the observational data as did ICE-6G_C (VM6).

Figure 4.18 presents the vertical uplift rate predicted by the ICE-7G_NA (VM7) model over North

America in its upper panel, upon which a number of transects have been superimposed. Explicit point-

by-point comparisons between model predictions and observations along these transects are performed

for the new ICE-7G_NA (VM7) model and its precursors ICE-6G_C (VM5a) and ICE-6G_C (VM6).

These transects are labelled A-A’ to P-P’. The comparisons along each transect are presented in groups


Figure 4.14: Snapshots of the thickness of the Laurentide Ice Sheet (m) and of the Greenland Ice Sheetin the ICE-7G_NA ice loading history since the Last Glacial Maximum. The state of the ice sheets ineach panel is given at the time indicated at the bottom left corner, in ka. From Roy and Peltier (2017).


Figure 4.15: Difference in ice thickness between the ICE-7G_NA and ICE-6G_C ice loading histories(meters) over the Laurentide Ice Sheet (m) and of the Greenland Ice Sheet for a selection of time steps(given in the corner of each panel, in ka) in the reconstructions since the Last Glacial Maximum. FromRoy and Peltier (2017).

Figure 4.16: Predicted topography over North America (meters) with respect to sea level at LGM forthe (a) ICE-6G_C and for the (b) ICE-7G_NA ice loading histories, with (c) the difference betweenthe ICE-7G_NA and ICE-6G_C topography at LGM. From Roy and Peltier (2017).


Figure 4.17: Comparison of the Engelhart and Horton (2012) data set of relative sea-level histories alongthe U.S. Atlantic coast for the 16 composite regions with the predicted relative sea-level history at thoselocations for the ICE-6G_C model of ice-loading history combined to the VM5a (green) and VM6 (blue)radial viscosity profiles, together with the prediction of the new ICE-7G_NA (VM7) model (black).Green data points represent sea-level index points, whereas blue crosses represent marine-limiting dataand orange crosses represent terrestrial-limiting data. From Roy and Peltier (2017).


of four in Figure 4.19, together with a schematic location of each transect on the continent (note that

all data points within 200 kilometers of a given transect are considered to be "on" the transect following

a projection normal to it). The transects A-A’ to K-K’ are the same as those employed in Peltier et al.

(2015) for their evaluation of the performance of the ICE-6G_C (VM5a) model, while the other eight

transects provide new insight on additional features of the uplift field over North America: the L-L’

transect, in particular, follows the southern forebulge associated with the former Laurentide ice sheet

west to east; the M-M’ transect follows the southern shore of Hudson Bay, through Quebec and into

the Gulf of St. Lawrence; the N-N’ and O-O’ transects provide information on the uplift along the

U.S. Atlantic coast and along the St. Lawrence River, respectively, while the P-P’ transect cuts across

Hudson Bay between the former ice domes that existed on either side of it. In general, VM6 predicts

significantly lower uplift rates than VM5a in regions that were once under significant Laurentide ice

cover. VM6 also predicts lower subsidence rates in the forebulge (L-L’, N-N’ and O-O’) and fails to

reproduce the gradient in observed uplift along the eastern shore of Hudson Bay (P-P’), but performs

well in the transition between the uplifting and subsiding regions and for some western transects (I-

I’ and J-J’). However, the vertical uplift rate field provided by the new model over North America is

much closer to that of the ICE-6G_C (VM5a) model than ICE-6G_C (VM6), in fact performing better

than both models along a few transects. The maximum uplift rate value predicted in central Quebec

is 12 mm/yr, similar to the ICE-6G_C (VM5a) prediction and still within the constraints provided

by the GPS observations of uplift in the region. West of Hudson Bay, the amplitude reached by the

uplift rate is slightly lower than that predicted by ICE-6G_C (VM5a) while, in the mid-Atlantic, the

predicted subsidence by the new model is lower and much closer to that of the ICE-6G_C (VM6) model,

maintaining one of the main advantages of that model over the original model of Peltier et al. (2015).

The new model also performs slightly better in the region surrounding the Gulf of St. Lawrence, as

visual inspection of the B-B’ transect will demonstrate.

The fit provided by this model to the observations of present-day vertical uplift rate (considering

their observational uncertainty) is presented in Figure 4.20(b), where it is seen that most misfits that

existed in comparison to the predictions of the ICE-6G_C (VM6) model (shown in Figure 4.20(a)) are

now eliminated. The more sophisticated iterative process that has been employed in this further step of

analysis leading to the ICE-7G_NA (VM7) model enables a fit to the uplift rate observations that is at

least as good as the fit provided by the parent ICE-6G_C (VM5a) model (previously shown in Figure

4.3(d)). The two areas of misfit previously identified for the ICE-6G_C (VM5a) model in the Gulf

of St. Lawrence and in south-western Canada deserve special attention. Using the new ICE-7G_NA

(VM7) model reduces the misfits found in the former region, but it does not reduce the significant


Figure 4.18: Background map of predicted present-day rate of vertical motion of the crust for the ICE-7G_NA (VM7) model of the GIA process following the scale at the bottom of the panel (mm/yr),superimposed by the traverses along which comparisons of the model to the observations of crustaluplift rate are performed in Figure 4.19, and for which the end points are labelled. From Roy andPeltier (2017).


Figure 4.19: Comparison of the predicted uplift rate along the traverses identified and labelled in Figure4.18 for the new model (green), the ICE-6G_C (VM5a) model (red) and ICE-6G_C (VM6) models(blue) with the observations measured at sites located within 200 km of each traverse, and which areprojected onto it. Schematic maps for each subgroup of four traverses show their location on the map.From Roy and Peltier (2017).


misfits observed in the latter. This persistent misfit will be addressed in a subsequent section. Following

the methodology of Argus et al. (2010), Argus et al. (2014), and Peltier et al. (2015), Figure 4.20(c)

presents the normalized quality of the fit provided by the new ICE-7G_NA (VM7) model with regards

to the space-geodetically observed vertical uplift rates, together with the same quantity determined for

precursor models. It can be seen that the excellent gain in quality of fit to the North American uplift

rates that the ICE-6G_C (VM5a) model provided over the ICE-5G (VM2) model was partially lost when

switching to the VM6 viscosity model. Introducing the vertical uplift rate predictions in our methodology

(which led to the VM7 viscosity profile) substantially reduced the misfit compared to VM6. The rest of

the gain in quality of fit is provided by the slight ice loading melting rate variations introduced in the

passage from ICE-6G_C to ICE-7G_NA.

The fit provided by the ICE-7G_NA (VM7) model to the history of sea level change observed along

the U.S. East coast during the late Holocene (since 4ka) as a function of distance from western Hudson

Bay (following Engelhart et al. (2009)) is shown in Figure 4.21, together with the same model predictions

for the ICE-6G_C (VM5a) and ICE-6G_C (VM6) models. It was shown in Roy and Peltier (2015) that

one of the main strengths of the new VM6 viscosity profile was its ability to reproduce the shape of

the forebulge associated with the former Laurentide Ice Sheet much more accurately than was possible

with the VM5a viscosity structure when this was employed with the ICE-6G_C loading history. Not

only does the new model fit the data from the mid-coast region as well as ICE-6G_C (VM6) did, but

it also better captures the location of the maximum subsidence of the forebulge and its amplitude, as

well as the behaviour of the tail of the forebulge (in South Carolina), which is a strong indication of the

improvement that this further refined model represents.

The fit provided by the ICE-7G_NA (VM7) model to the uplift data based upon water gauges in the

Great Lakes is shown in Figure 4.22, where it is demonstrated that the ICE-7G_NA ice loading history

with its slightly modified deglaciation rates compared to ICE-6G_C enable the full model to recover a

much better fit to the observational data in the southern region of the Great Lakes than that provided

by ICE-6G_C (VM6) or ICE-6G_C (VM5a). The only remaining misfit is in the southern half of Lake

Michigan.

It should be also noted that ICE-7G_NA has less ice than the Laur16 model of Simon et al. (2016)

over Manitoba and more ice over the southern shores of Hudson Bay at LGM. While the two models

generally seem to be qualitatively similar in other regions at LGM (an entirely expected consequence of

the fact that both are variants of ICE-6G_C), a detailed comparison of these models will be described

elsewhere.

Moreover, the fit provided by the new ICE-7G_NA (VM7) model to the Roy and Peltier (2011)


Figure 4.20: Quality-of-fit determination between the JPL space-geodetic measurements of present-day vertical motion rates (mm/yr) and the GIA predictions of those same rates (taking into accountobservational uncertainties) for the ICE-6G_C (VM5a) (panel (a)) and the new ICE-7G_NA (VM7)(panel (b)) models at the site of each uplift rate measurement; (c) Misfit between GIA model predictionsof vertical uplift rates and observations over North America, with the vertical bars representing thenormalized quality of the fit, and the values at the top the square root of the reduced chi-square value(also called the Normalized Sample Standard Deviation, or NSSD), for the ICE-5G (VM2) (red), ICE-6G_C (VM5a) (cyan), ICE-6G_C (VM6) (green), ICE-6G_C (VM7) (purple), and ICE-7G_NA (VM7)(dark blue) models. From Roy and Peltier (2017).


Figure 4.21: Comparison of observed late Holocene relative sea-level rise (mm/yr) for locations alongU.S. East coast (dark grey diamonds) (with 2-σ uncertainty ranges in light gray) (Engelhart et al., 2009),with predicted values for the same quantity from the ICE-6G_C (VM5a) (green dots), ICE-6G_C (VM6)(blue dots), and from the new ICE-7G_NA (VM7) (red dots) models. The data points are plotted asa function of distance from the city of Churchill, Manitoba, on the western shore of Hudson Bay (km).From Roy and Peltier (2017).

Figure 4.22: (a) Mainville and Craymer (2005) inferences of vertical uplift rate for sites along the shores ofthe Great Lakes presented with respect to the reference point of each individual lake outlet (filled circlesrepresenting the value in mm/yr); (b) Map of the ICE-7G_NA (VM7) model prediction of vertical upliftrates in the Great Lakes region, together with the Mainville and Craymer (2005) observations of thesame quantity adjusted to the modeled uplift rates at each reference outlet point; (c) Difference betweenthe modeled ICE-7G_NA (VM7) vertical uplift rates and the adjusted observations of Mainville andCraymer (2005) (b), accounting for the 2-σ uncertainty in the inferences. From Roy and Peltier (2017).


constraints on true polar wander (speed and direction) and on the change of oblateness of figure of the

planet (the change in the J2 zonal harmonic of the gravitational field of the Earth) is provided in Table

4.3.

Table 4.3: Values of Earth rotational observables predicted by ICE-6G_C and (VM5a) and ICE-7G_NA(VM7)

Finally, the relaxation time at the heart of the former Laurentide ice sheet, namely along the south-

eastern coast of Hudson Bay, provided by the new ICE-7G_NA (VM7) model is provided in Table 4.4,

where its excellent fit to the relaxation time inferred from high-quality isolation basin data from the

region can be observed.

Table 4.4: Relaxation times determined in Eastern James Bay

4.3.5 The Upper Campbell (Lake Agassiz) strandline tilt

A further data set to which GIA model predictions can be compared, and which was not used in

the optimization process leading to the ICE-7G_NA (VM7) model, is the observed present-day tilt of

ancient strandlines associated with former proglacial lakes that existed during the deglaciation phase

of the Laurentide ice sheet. In particular, a recent attempt at reconstructing the ice thickness history

of the western part of the Laurentide ice sheet by Gowan et al. (2016) was explicitly tuned to the

amount of observed tilt along strandlines in the region. A central constraint of this optimization was the

Upper Campbell strandline, the longest such feature in the vicinity spanning more than 1000 kilometers

from Saskatchewan to the tri-border area between North Dakota, South Dakota and Minnesota. The

ice loading history for the western Laurentide ice sheet of Gowan et al. (2016) is characterized by a

significant reduction of ice volume in the area as compared to that of the ICE-6G_C model (a decrease

in ice volume at LGM of about 30% has been suggested to be appropriate), and the authors suggest that

the observed tilt along the Upper Campbell strandline of Glacial Lake Agassiz "provides strong support


Figure 4.23: Comparison of the modeled tilt of the Upper Campbell strandline associated with formerLake Agassiz with the tilt measured at present along the strandline with respect to the point of lowestelevation along its length (Gowan et al., 2016). The black dashed line represents the line of best fit.Results for the ICE-6G_C (VM5a) (blue) and ICE-7G_NA (VM7) (red) models are shown, togetherwith the linear fit best characterizing the modeled values (unconstrained at the origin) and the valueof its slope. The inner panel shows the location of the Upper Campbell strandline in central NorthAmerica. From Roy and Peltier (2017).

for thin ice cover" in the region (Gowan et al., 2016). We revisit this statement here in the context of the

new ICE-7G_NA model, which embodies slight modifications to the ice loading history over the region

of interest.

Figure 4.23 compares model predictions of the amount of tilting that has occurred along the Upper

Campbell strandline of former Lake Agassiz since its formation approximately 10,500±300 years BP with

present-day elevation differences observed along the strandline. The methodology used in this analysis

follows that of Gowan et al. (2016) to favour inter-model comparisons with their NAICE model, and

uses the same conversion of elevation values to orthometric height differences along the strandline as

compared to the point of present-day lowest elevation (Gowan et al., 2016) . In Figure 4.23, modeled

values of tilt at locations for which elevation values are available are plotted against the observed values

(thus, observations follow a perfect one-to-one slope in the figure). Model predictions for the ICE-6G_C


(VM5a) and ICE-7G_NA (VM7) models are shown in Figure 4.23, with the uncertainty provided by

using the range of ages allowed for the age of the strandline. Gowan et al. (2016) observed that the ICE-

6G_C (VM5a) predicted slightly too much uplift in the northern part of the strandline, thus lowering

the quality of the fit it provides to the data, but it should be noted that this inference is somewhat

dependent on the predicted uplift in the southern part of the strandline and on potential errors in

elevation measurements (suggested to be ±5 m by the authors). The fit provided by the ICE-7G_NA

(VM7) model to this observational data set is also provided in Figure 4.23, and shows that the new

model entirely corrects the misfit predicted by the ICE-6G_C (VM5a) model. This is due to the slight

adjustment of the ice loading history introduced over central Canada that was necessary to recover a

good fit to the network of space-geodetic uplift rate measurements over the continent. It is clearly

important to understand that our new model was not tuned to fit the strandline tilt observation but

rather to fit the space-geodetic observations.

This result is highly significant because it demonstrates the ability of the ICE-7G_NA (VM7) model

to fit observations to which it was not tuned. Also, it is important to note that the ICE-7G_NA model

is only marginally thinner than the original ICE-6G_C ice loading history over the region over which

the NAICE model is defined. This means that the total ice volume found in ICE-7G_NA at LGM over

this area of the western Laurentide ice sheet is very close to that of ICE-6G_C, and thus far above that

of the NAICE model of Gowan et al. (2016). The fact that a fit to the Upper Campbell strandline tilt

observations can be obtained with the "thicker" ICE-7G_NA model contradicts the assertion of Gowan

et al. (2016) that the Upper Campbell strandline tilt observations should necessarily support a "thin ice

cover" in the region. The slight misfits identified by Gowan et al. (2016) for the ICE-6G_C ice loading

history can be corrected by slight variations in ice thickness over only a small fraction of central Canada

that corresponds to the northern portion of the strandline.

As the strandline is found in a region that was completely ice-covered at LGM, it is also important

to recognize that the quality of the fit provided by a given ice sheet reconstruction to strandline tilt

observations is sensitive only to the differential ice thickness over the region, rather than to the thickness

of ice removed. Moreover, the preference expressed by Gowan et al. (2016) for thin ice cover over the

region is a consequence of their preference for a viscosity structure that is characterized by a very high

viscosity in the lower mantle (1.0 · 1022 Pa·s) below 670 km depth, which requires a significant reduction

in ice thickness of the Laurentide ice sheet to maintain an acceptable fit to the space-geodetic constraints

on present-day vertical uplift rates. In fact a viscosity model of this kind is entirely ruled out by the

characteristic exponential relaxation times that are characteristic of every relative sea level curve that

has ever been measured from coastlines of the region that was once covered by Laurentide ice (e.g.


Figure 4.24: Comparison of the relative sea level history predicted at four key sites along the U.S. Eastcoast from the Engelhart and Horton (2012) data set, namely Southern Maine, Southern Massachusetts,Northern South Carolina and Southern South Carolina, between the ICE-7G_NA (VM7) model (black)and the viscosity model used by Gowan et al. (2016), when coupled with the ICE-7G_NA ice loadinghistory (red). Green data points represent sea-level index points, whereas blue crosses represent marine-limiting data and orange crosses represent terrestrial-limiting data. From Roy and Peltier (2017).


Peltier (1998b) and the more recent analysis based upon isolation basin data in Roy and Peltier (2015)).

Furthermore, it should be noted that in the present analysis we have shown that using a viscosity

structure with a very high lower mantle viscosity (and a thick elastic lithosphere as also suggested by

these authors) has a significant impact on the ability of the model to fit constraints from the forebulge

region. Figure 4.24 compares the fit at some sites in the U.S. East coast RSL database of Engelhart

and Horton (2012) between VM7 and the viscosity model used by Gowan et al. (2016) when combined

to the ICE-7G_NA ice loading history. Significant misfits are introduced in the ability of the model

to reproduce the sea level evolution along the coast, especially at the southernmost sites. It should

also be noted that, even though Gowan et al. (2016) suggest that supplementary ice might perhaps be

required in other parts of the Laurentide ice sheet to maintain an adequate sea level budget over the

last glaciation-deglaciation cycle, the misfit to the RSL data in the southernmost parts of the U.S. East

coast remains even if a significant thinning is applied to the ICE-7G_NA model. Finally, it should be

noted that inspection of the results of their parameter space exploration does not in fact preclude the

validity, in the first place, of a viscosity structure with a lower mantle viscosity that is much closer to

that of VM7 (see figure 6 of Gowan et al. (2016)).

4.3.6 The fit to the time-dependent gravity measurements of the Gravity

Recovery and Climate Experiment

Another important data set in terms of which the quality of a given model of the GIA process may be

assessed is that provided by the time-dependent gravity measurements of the GRACE satellites. The

GRACE inference of this field, presented in terms of the time rate of thickness change of an equivalent

water layer placed upon the surface of the Earth (cm/yr) is provided in Figure 4.25(a). As before,

the Release 5 GRACE product from the U.S. Center for Space Research (CSR) is used without the

application of a correlated error filter, but with a 300-km half-width Gaussian filter applied to the raw

data. The equivalent field predicted by the new ICE-7G_NA (VM7) model is presented in panel (b) of

the same figure. The new model displays the same double "bull’s eye" pattern that GRACE observes in

the change in the gravitational field over North America, and predicts a higher maximum amplitude of

this field than did ICE-6G_C (VM6). This difference enables the model to provide a much better fit to

the time-dependent gravity signal over central Quebec than that provided by ICE-6G_C (VM6), as can

be observed in panel (c) where the residual between the new model’s predicted field and the GRACE

observations is presented.

Variations in surface hydrology impact the gravitational signal seen by the GRACE satellites, and


Figure 4.25: (a) Gravity Recovery and Climate Experiment (GRACE) observations of the time depen-dence of the gravitational field over North America from the Release 5 data from the Center for SpaceResearch (CSR) for the period covering January 2003 to October 2013, on which no correlated filterwas applied but a spatial filter of Gaussian half-width of 300 km was applied; (b) ICE-7G_NA (VM7)model predictions of this field; (c) Difference between the raw GRACE field (a) and the ICE-7G_NA(VM7) model predictions (b); (d) Difference between the GRACE field (a), corrected for hydrologicaleffects by the application of the GLDAS correction of Rodell et al. (2004) and the ICE-7G_NA (VM7)model predictions. All results are presented in equivalent water layer thickness change (cm/yr). FromRoy and Peltier (2017).

the time-dependent gravity signal provided by the mission has been successfully used to determine

continental water storage trends for many regions of the world (e.g. Ramillien et al., 2008; Landerer and

Swenson, 2012; Longuevergne et al., 2013). Due to the background signal that the GIA process imprints

upon the gravity signal, these efforts have mostly focused on regions where the GIA signal is expected to

be small, although some effort has been expended in an attempt to separate the GIA and hydrological

components over North America and Scandinavia (Wang et al., 2012; Lambert et al., 2013; Wang et al.,

2015). In spite of these difficulties and the limited nature of the surface hydrological observations,

hydrological models can be used to estimate their contribution to the gravity signal and may potentially

be employed to evaluate the quality of GIA models (for example, see Peltier et al. (2015)). As in

the case of ICE-6G_C (VM5a) and ICE-6G_C (VM6), adding the hydrological correction of GLDAS-

NOAH (panel (d)) results in a degradation of the quality of the fit provided by the model, especially in

Quebec and Ontario, but the degree of degradation that occurs is not as marked as was the case for the

ICE-6G_C (VM6) model (see Figure 4.4).

It is useful to investigate in this context the different impact that the application of different hydrolog-

ical models can have upon the portion of the GRACE signal that should be explicable as a consequence

of the ongoing GIA process. The two most prominent surface hydrology models that may be employed

for the purposes of such analyses are the Global Land Data Assimilation System model (GLDAS-NOAH)

(Rodell et al., 2004) and the WaterGAP hydrology model (WGHM) (Döll et al., 2014). The different


Figure 4.26: (a) GLDAS-NOAH model prediction of the time dependence of the gravitational fieldover North America induced from hydrological effects from Rodell et al. (2004); (b) Residual from theGRACE observation of the time dependence of the gravitational field corrected for hydrological effectsusing the GLDAS-NOAH model results and the ICE-7G_NA (VM7) prediction of this field; (c) WGHMmodel prediction of the time dependence of the gravitational field over North America induced fromhydrological effects from Döll et al. (2014); (b) Residual from the GRACE observation of the timedependence of the gravitational field corrected for hydrological effects using the WGHM model resultsand the ICE-7G_NA (VM7) prediction of this field. All results are presented in equivalent water layerthickness change (cm/yr). From Roy and Peltier (2017).


hydrology corrections for time dependent gravity provided by these models over North America, when

the same filtering is applied to them as to the GRACE signal, is shown in Figs 4.26(a) (GLDAS-NOAH)

and 4.26(c) (WGHM). Comparing the two panels demonstrates the existence of clear differences between

the two models over North America. The difference in model design and input data accounts for the most

prominent signal difference found over Greenland. The residual between the GRACE time-dependent

gravitational signal corrected for hydrology and the GIA signal predicted by the ICE-7G_NA (VM7)

model is presented in Figure 4.26(b) when using GLDAS-NOAH and Figure 4.26(d) when using WGHM.

Given the limited availability of hydrological constraints and the large differences of approach between

the two models, it is hard to determine on a priori grounds which one is more plausible, but the fact

that there are such significant differences underscores the need to develop a sufficiently robust model

of surface hydrological change that can be confidently used to facilitate the interpretation of GRACE

measurements.

4.3.7 Model refinement considerations: margin evolution considerations of

the south-western edge of the Laurentide Ice Sheet

The analysis of the ability of the ICE-6G_C (VM5a) and ICE-6G_C (VM6) models to match the

observations of present-day vertical uplift provided by the extended network of space-geodetic observa-

tions over North America (see Figure 4.3) reveals a region of persistent misfit in the southern part of

the Canadian Prairies, centered in Saskatchewan, which is also found in the comparison to the GRACE

time-dependent gravity signal (Figure 4.4), even with the inclusion of a hydrological correction (GLDAS-

NOAH) to the GRACE data (see also Peltier et al. (2015)). These misfits were also mentioned by Snay

et al. (2016), who found in their analysis of observed crustal motion that large residuals in vertical

velocities remained in the region even if GIA effects from the ICE-6G_C (VM5a) model were taken into

account. The persisting misfit is also visible along the L-L’ transect of Figure 4.19. From Figure 4.3, it is

clear that this region of misfit remains highly localized within the borders of the Canadian provinces of

Saskatchewan and Manitoba. This misfit is insensitive to large variations in elastic lithosphere thickness

and lower mantle viscosity, while it is relatively sensitive to changes in upper mantle viscosity. However,

the best-fitting models would provide a highly significant reduction in the quality of the fit provided to

RSL data along the U.S. East coast, and are thus excluded.

The possibility that this regional misfit could be influenced by the position and timing of the retreating

ice sheet margin or by the past thickness of the ice sheet covering the region is addressed by revisiting

the ICE-6G_C and ICE-7G_NA models in the area. Slight changes in the configuration of the ice sheet


margin in the southern Prairies were considered within the acceptable range permitted by the limiting

data provided by the Dyke (2004) (also Dyke et al. (2002)) reconstruction of ice sheet margin evolution,

taking into account the relative paucity of constraints at the onset of deglaciation in this region. Also,

changes in the ice thickness covering the region at LGM and during the earlier part of the deglaciation

were introduced, with variations of up to 30% in total ice thickness around LGM and during the onset

of the deglaciation (from 26 ka to 14 ka). These variations were accompanied by slight changes in ice

thickness elsewhere on the ice sheet to keep its total mass similar to that of the ICE-7G_NA model.

It was found that the misfits to the GPS observations of vertical uplift were insensitive to even large

increases in ice sheet thickness of the region, or to the slight changes in ice margin location that would

be consistent with the constraints provided in Dyke (2004) and Dyke et al. (2002).

Misfits between GIA model inferences of vertical uplift and gravity change and both GPS observations

of vertical uplift rate and GRACE observations of time-dependent gravity in the Canadian Prairies were

identified in previous studies (Wang et al., 2012, 2015). Wang et al. (2012) suggested a methodology that

might be employed to separate the signal from hydrology to that of GIA in the GRACE observations

of time-dependent gravity, in a way that should be GIA model-independent (making the assumption

that a generated uplift field from the GPS observations would, within the first 60 degrees of a spherical

harmonic decomposition except the first, show the GIA signal alone), and the subsequent analysis of

Wang et al. (2015) found that a positive water storage trend over the Prairies and the Western Great

Lakes could explain the remaining misfit. This possibility is consistent with our determination of a

positive residual in the GRACE observations of time-dependent gravity over the region, which cannot

be explained by GIA influence.

4.4 Conclusion

The current analysis has introduced the voluminous quantity of space-geodetic measurements of crustal

motion available over North America into the methodology that has led to the recent development of

the ICE-6G_C (VM6) model of Roy and Peltier (2015). The resulting parameter space exploration and

careful characterization of the misfits between the observational data sets and the GIA model predictions

has led to the ICE-7G_NA (VM7) model of the GIA process, which is able not only to reproduce the

sea level history database of Engelhart and Horton (2012), but also explain the present-day crustal uplift

and subsidence over the North American continent as observed by the GPS network. Furthermore, it

also explains the crustal tilting over the Great Lakes region that has occurred following deglaciation and

provides a good fit to the geological inference of the shape and extent of the Late Holocene forebulge


evolution along the U.S. East coast.

Our goal has been to provide a model that enables a further improvement of fit to the totality of

the available geophysical observables, while continuing to employ the same basic iterative methodology

that has been so successfully employed in the previous steps in global GIA model development. Given

the selected region of parameter space that was explored, and the strong focus put on the importance

of global constraints, we suggest the ICE-7G_NA (VM7) model to be robust and the preferred solution

within the region of parameter space that has been explored. The work performed here also establishes

a framework in terms of which an explicit measure of the uncertainty associated with the ICE-7G_NA

(VM7) model can be provided. However, given how interconnected the ice loading history and the mantle

viscosity structure are, it is important to consider carefully how this uncertainty should be quantified.

Work to quantitatively define such a measure of uncertainty will be presented elsewhere. Finally, given

the inherently global nature of the problem we are addressing, the methodology used in this analysis

will have to be expanded to other regions that have undergone large changes in ice cover since LGM to

fully understand this uncertainty within the global framework in which this analysis must be applied.

As the added constraints were heavily focusing upon the North American component of the model, it

remains important to test the global exportability of the adjustments that were made to the viscosity

structure and to the Laurentide ice sheet loading history. The next chapter of this work presents the

first such test, focusing on the Mediterranean basin, a region where a large number of past relative sea

level observations have been collected.

Chapter 5

A regional test of the ICE-7G_NA

(VM7) model in the Mediterranean

basin

The study of past relative sea level change in the Mediterranean Sea has been an active field of research,

which has greatly benefited from the collection of a large number of biological, geological and archaeo-

logical indicators (e.g. Lambeck et al., 2004; Antonioli et al., 2001, 2003, 2006, 2009; Vacchi et al., 2016,

among others). Among the regions from which such knowledge of past relative sea level is available, the

Mediterranean basin is of particular interest, given the large quantity of archaeological evidence that

exists along its coasts (owed to the long human presence there), as well as its location south of the

former ice sheets that covered Northern Europe and the British Isles at the height of the last glacial

cycle. This wealth of past sea level information can be used in the context of the development and

testing of the global exportability of the ICE-7G_NA (VM7) model, as constraining information from

the Mediterranean basin was not included in the optimization process that led to it.

Using a recent database of relative sea level indicators gathered by Vacchi et al. (2016), it will be

shown that the new model performs very well in this new regional comparison test. It provides strong

support for the global exportability of the ICE-7G_NA (VM7) model, while identifying isolated regions

of remaining misfit which would benefit from further study, and providing further clues towards the

correct interpretation of certain archaeological evidence.

This chapter follows closely a publication in preparation (Roy, K., and Peltier, W. R. (2017), ’Relative

151

Chapter 5. A regional test of the ICE-7G_NA (VM7) model in the Mediterranean basin152

sea level in the Western Mediterranean basin: A regional test of the ICE-7G_NA (VM7) model’, in

preparation).

5.1 Some geological considerations in the Mediterranean basin

The western part of the Mediterranean basin is a tectonically complex region, marked by the existence

of a large convergent margin (Nubia-Eurasia), two small oceanic basins (the Tyrrhenian basin under

the homonymous regional sea extent, and the Liguro-Provencal basin along its northern coast) and a

continental block under the islands of Sardinia and Corsica (Faccenna et al., 2014). There is evidence

suggesting the existence of an Adriatic micro-plate, extending in an arc covering the Po Valley, the

Adriatic Sea, the Ionian Sea and the southern part of Sicily, which is thought to move independently

from the main Eurasian plate and to be responsible for considerable tectonic activity along its margins

(Devoti et al., 2002; Faccenna et al., 2014). Some parts of the basin margins are undergoing compressional

tectonics (Billi et al., 2011), with significant tectonic activity exists along the southern Mediterranean and

southern Tyrrhenian margins, while contractional activity characterizes the Adriatic region. The effect of

long-term tectonic activity on geological records is also compounded by numerous co-seismic events that

can result in rapid local sea level change after volcanic activity or earthquakes of large magnitude (e.g.

Pirazzoli et al., 1994; Mastronuzzi and Sansò, 2002; Antonioli et al., 2007). The existence of numerous

inter-glacial shorelines enables a comparison of tectonic uplift rates over different time scales, and has

been used to argue for the long-term tectonic stability of certain regions in the Mediterranean basin

when interpreting relative sea level data (e.g. Lambeck et al., 2004; Antonioli et al., 2015). Figure 5.1

shows the tectonic setting of the basin, with known tectonic faults, present-day GPS uplift rates, and the

elevation of MIS 5e shorelines recovered from ancient beaches. This complex regional geological setting

is very different from the geological setting of the eastern coast of the United States, where the relative

sea level records upon which the ICE-7G_NA (VM7) model of Roy and Peltier (2017) was calibrated

were originating.

5.2 The Vacchi et al. (2016) data set of Holocene RSL change

The Vacchi et al. (2016) data set of relative sea level change is formed of 469 sea-level index points

(capturing past local mean sea level value) and 177 terrestrial-limiting and marine-limiting constraints

(which provide upper and lower limits for the past level of the sea), gathered around the western part

of the Mediterranean basin and separated in 22 individual regions (Vacchi et al., 2016). These include


Figure 5.1: Tectonic setting of the Western Mediterranean Sea and precise location of the 22 regionalsubdivisions of the Vacchi et al. (2016) data set, presenting tectonic faults as red lines (Faccenna et al.,2014), the color-coded elevation of MIS 5e shorelines as squares (Ferranti et al., 2006; Pedoja et al.,2014), and modern GPS vertical motion as color-coded circles (Serpelloni et al., 2013). The 22 regionsare shown as numbered black squares. Figure from (Vacchi et al., 2016), reproduced with permission.

a large number of re-interpreted 14C-dated biological material, and a large number of archaeological

inferences from historical coastal infrastructure that takes advantage of the long human presence around

the Mediterranean Sea. Several other data points were excluded from the analysis in regions potentially

affected by large co-seismic or tectonic uplift/subsidence. However, whereas most past sea level indi-

cators are based in other data sets on salt marsh environments (Shennan and Horton, 2002; Engelhart

and Horton, 2012), the small tidal range observed in the Mediterranean basin has limited the develop-

ment of salt-marsh areas (except near large deltas). Most sea level index points are thus based on the

interpretation of past coastal lagoonal environments, most notably from the transition from open marine-

influenced lagoons to semi-enclosed, brackish environments (Vacchi et al., 2016). Index points are also

provided through the study of beachrocks, sedimentary rock formations cemented by carbonate minerals

in coastal environments that will often imprison shell fragments (Mauz et al., 2015). Terrestrial limiting

data points are provided by biological markers deposited in freshwater marshes or alluvial plains, while

marine limiting points will be obtained from open lagoonal settings or in situ marine benthos. Coastal

archaeological evidence recovered along the coastlines include former harbour structures, coastal quarries

and Roman fish tanks and can provide limiting or index points for the past few thousand years, given


the long occupation of the Mediterranean basin (Auriemma and Solinas, 2009). Errors on each analyzed

data point are described in detail in Vacchi et al. (2016), but follow largely the methodology of Shennan

and Horton (2002). Compaction-free and basal index points comprise 28% and 14% of the data points,

while the rest are intercalated data points (Vacchi et al., 2016), which could potentially be impacted by

compaction effects.

The separation of the data set into 22 different regions by Vacchi et al. (2016) is based upon the

geographical proximity of the data points and the regional tectonic setting, which is derived from the

height of the last interglacial (MIS 5e) shorelines and from modern GPS-derived vertical uplift or subsi-

dence rates (Vacchi et al., 2016). These results, and the precise location of the regional subdivisions of

the data set, are presented in Figure 5.1. The RSL data for each of the 22 sub-divisions of the Vacchi

et al. (2016) data set is presented in Figs. 5.2 and 5.3.

5.3 Results for the ICE-7G_NA (VM7) model in the Mediter-

ranean basin

While the ICE-7G_NA (VM7) model of Roy and Peltier (2017) was able to reconcile the latest space-

geodetic observations over the North American continent and the extensive observations of RSL evolution

along the U.S. East coast of Engelhart and Horton (2012) with GIA model predictions of the same

quantities, the extensive data set of past sea level evolution in the western half of the Mediterranean

Sea of Vacchi et al. (2016) provides a new testing ground for the model and a crucial verification of its

global exportability.

The predictions of the ICE-7G_NA (VM7) model for the 22 regional subdivisions are provided in

Figs. 5.2 and 5.3, together with the predictions provided by the precursor ICE-6G_C (VM5a) model of

Peltier et al. (2015). Along the north-western coast of the Mediterranean Sea, the new model improves

on the performance of the precursor along the southern coast of France (site 3), where the data set is

comprised of a large fraction of basal index points in the mid-Holocene that are less likely to be impacted

by compaction (Vacchi et al., 2016). The new model does not perform as well in Central Spain (site 1),

however, as it misses some of the older index points in the earlier part of the Holocene.

In the Ligurian Sea (sites 4 and 5), the new ICE-7G_NA (VM7) model performs very well. It is able

to fit all index points on its western flank, while the lone basal index point on its eastern shore (at 6 ka)

is also fit by the model. There are discrepancies with the older intercalated data points on the eastern

Ligurian shore (site 5), but this might be due to compaction. The sea level index points recovered from


Figure 5.2: Comparison of the Vacchi et al. (2016) data set of relative sea-level histories in the WesternMediterranean Sea with the predicted relative sea-level history at those locations for the ICE-6G_C(VM5a) (green) and the ICE-7G_NA (VM7) (black) models of the GIA process, for the first 14 locationsin the data set (out of 22). Green data points represent sea-level index points, whereas blue crossesrepresent marine-limiting data and orange crosses represent terrestrial-limiting data. The top mapshows the location of each site in the Mediterranean basin.


Corsica (site 6) are entirely based on biological markers and archaeological inferences from Roman ruins

on Pianosa Island, located between Elba and Corsica, and only extend to a bit more than 4 ka. Both

models perform equally well in the area, given the uncertainty range associated with each relative sea

level inference. In Southern Corsica (site 7), the ICE-7G_NA (VM7) model enables a better fit to the

terrestrial limiting data points, and it generally fits the expected RSL evolution in the region. In Sardinia

(site 8), the new model provides an excellent fit to the basal index points slightly older than 9 ka, but

suggests that the younger intercalated data points might be impacted by compaction. Between 4.6 ka

and 4.0 ka, Vacchi et al. (2016) identified considerable scatter between the two intercalated data points

from Cagliari (−6.8 ± 1.0 m) and Is Mistras (−2.2 ± 1.0 m). The new model suggests an intermediate

RSL value around this time, and its suggested RSL evolution is consistent with the younger index points.

In central Latium (site 9), there is significant scatter between the various index points used to

constrain RSL evolution, which were associated by Vacchi et al. (2016) to the potential influence of

long-term tectonic uplift (also seen on the large elevation of the last interglacial beaches in the area from

Figure 5.1) and sedimentary deposition from the Tiber Delta. In this context, it is hard to evaluate the

performance of each GIA model, but it should be noted that the new model tends to overestimate past

relative sea level changes in the area. It remains to be seen whether any tectonic correction (consistent

with the MIS 5e shoreline elevation) would reduce this misfit. In the Gulf of Gaeta (site 10), the new

model presents a significant performance upgrade over the precursor ICE-6G_C (VM5a) in the late

Holocene. However, for the earlier part of the Holocene, a misfit remains between the terrestrial-limiting

data points and the predictions offered by both GIA models. In Salerno Bay (site 11), the new model

provides a slight degradation in the quality of the fit it provides to the data.

In Northwestern Sicily (site 12), a region of considerable interest because of its low local tectonic

activity (Antonioli et al., 2002), the new model provides an excellent fit to the available RSL data,

including a pair of limiting data points, marine and terrestrial, of similar age and elevation. In mid-

eastern Sicily (site 13), there is considerable scatter among the late Holocene index points (Vacchi et al.,

2016), and the earlier part of the record is limited to three intercalated and a single marine-limiting data

points. In this context, the predictions of both GIA models are consistent with the limiting data points,

but the index data points do not enable a clear evaluation of their relative performance, except that

they both seem to underestimate relative sea level change since the mid-Holocene (albeit with respect

to intercalated data points that might be affected by compaction). In Malta and Southern Sicily (site

14), the new model provides a significant improvement over the precursor ICE-6G_C (VM5a) model.

The data from Southern Tunisia (site 15 on Figure 5.3), collected around the Gulf of Gabes, is

of considerable interest as it suggests the existence of a late Holocene highstand in this part of the


Figure 5.3: Comparison of the Vacchi et al. (2016) data set of relative sea-level histories in the WesternMediterranean Sea with the predicted relative sea-level history at those locations for the ICE-6G_C(VM5a) (green) and the ICE-7G_NA (VM7) (black) models of the GIA process, for the last 8 locationsin the data set (out of 22). Green data points represent sea-level index points, whereas blue crossesrepresent marine-limiting data and orange crosses represent terrestrial-limiting data. The top mapshows the location of each site in the Mediterranean basin.


Mediterranean Sea, which was inferred to provide information about the late post-LGM melting of the

Antarctic ice sheet (Stocchi et al., 2009). Although the ICE-6G_C (VM5a) model supports the existence

of a strong highstand in the region, similar to that predicted by Stocchi and Spada (2009), the new ICE-

7G_NA (VM7) model reduces its amplitude. Further study is required to characterize and determine

the cause and significance of this difference, especially in the context of the sensitivity analysis performed

by Stocchi et al. (2009).

In the Adriatic Sea, where the observational data set is able to provide useful constraints, the new

ICE-7G_NA (VM7) model generally performs better than the precursor ICE-6G_C (VM5a) model.

At the northern end of the sea, close to Venice and its lagoon (site 16), there is significant scatter

among the mid-Holocene and late Holocene data, most likely caused by compaction (Vacchi et al.,

2016). Nevertheless, the new model provides a marked improvement in the qualitative shape of the

RSL curve, as the precursor ICE-6G_C (VM5a) model suggests the existence of a highstand, a feature

that is absent from the observational data. This observation extends to the other sites in the Adriatic

Sea (sites 17 to 20), where the new ICE-7G_NA (VM7) provides a higher rate of late Holocene RSL

change, consistent with the observational record for the region. The fit provided by the new model is of

particularly high quality in the middle part of the Adriatic Sea, including the Italian mainland and the

islands of Vis and Bisevo (sites 19 and 20, respectively). In Istria and northern Dalmatia (North-East

Adriatic, site 17), the new model provides an excellent fit to the high-quality RSL data points sourced

from marshes and lagoons.

In the north-western extent of the Adriatic Sea, in a region corresponding to the Romagna and

Commachio coastal plains (site 18), both models are unable to reproduce all characteristics of the

inferred sea level history. Although they are able to fit the older basal data points. they are unable to

reproduce the rapid rise in relative sea level suggested by the younger data. Compaction effects may be

important when studying the younger data points, however Vacchi et al. (2016).

Finally, in the southern extent of the Adriatic Sea, the new model provides a significant improvement

over its precursor. In both Northern Apulia (site 21) and Southern Apulia (site 22), the new model

is able to reproduce the entirety of the index points, although there are some inconsistencies with the

younger terrestrial limiting data points along the Salento coast (site 22).

5.4 Evaluation of the ICE-7G_NA (VM7) model performance

The gain in performance provided by the new model over its ICE-6G_C (VM5a) precursor is shown in

Table 5.1. It is evaluated following the χ2-like error function introduced in the previous chapters (see


Roy and Peltier (2015) and Roy and Peltier (2017)). The increase in performance obtained by the new

ICE-7G_NA (VM7) model for most locations is evident. This is a highly significant result, given how

the evolutionary step between each model was driven by constraints from North America.

In particular, an important gain in the ability of the model to explain the observational constraints

is seen at most sites in the Western Mediterranean, especially in the Adriatic Sea and along the Ligurian

Sea. The two GIA models perform similarly well at most locations in Sicily. A performance decrease from

the use of the ICE-7G_NA (VM7) model is observed at only two locations out of the 22 locations included

in the Vacchi et al. (2016) data set: in Central Spain (site 1) and along the Salerno Bay (site 11). The

existence of significant misfits between GIA model predictions of past sea level evolution and geological

inferences in Central Spain has been identified as an open issue (Vacchi et al., 2016), and the ICE-7G_NA

(VM7) is unable to reconcile model predictions with observations at this specific site. As the present-day

vertical uplift rates, inferred from GPS observations (Serpelloni et al., 2013), are not sufficient to easily

ascribe this difference to tectonic effects, further investigation of this situation is warranted. In Salerno

Bay (site 11), the ICE-7G_NA (VM7) displays a significant decrease in performance when compared to

the predictions of its ICE-6G_C (VM5a) precursor. However, the southern Italian coast is very complex

tectonically (Faccenna et al., 2014), and although several MIS-5e shorelines in the area have been used

to infer little to no long-term uplift (Vacchi et al., 2016), our model results suggest that it would be hard

to fit both the RSL data for the Salerno Bay area (site 11) and for the Gulf of Gaeta (site 10), which is

located only 100 kilometers to the north, without accounting for some local effects (perhaps due to the

geographical proximity of the Naples volcanic area).

However, these local issues are heavily compensated by the strength of the model at all other sites,

where important gains in model fit are obtained at most locations. The new ICE-7G_NA (VM7) model

performs very well in France, around the Ligurian Sea and in Corsica (sites 3 to 7). In Northern Corsica

(site 6), the increase in model performance is entirely due to the ability of the new model to fit the lone

terrestrial-limiting data point at about 2 ka BP, while remaining within the uncertainty range of the

other index points. In the Gulf of Gabes, in Southern Tunisia (site 15), it is important to note that both

models perform equally well, due to the uncertainties associated with the geological record. The ICE-

6G_C (VM5a) and ICE-7G_NA (VM7) models perform equally well in Sicily, with the remaining misfit

in Mid-Eastern Sicily (site 12) due to the scatter in late Holocene data points, associated by Vacchi et al.

(2016) to the probable uplift trend in the area. In the Adriatic Sea, the new model significantly reduces

the misfits observed between the ICE-6G_C (VM5a) model predictions and the geological inferences of

past relative sea level. However, while the ICE-7G_NA (VM7) model is able to eliminate almost entirely

the misfits for the mid-Adriatic Sea locations (sites 19 to 22), some misfits remain in the northern section


of the Adriatic Sea (sites 16 to 18), probably due to the general subsidence of the area (Antonioli et al.,

2009).

Table 5.1: Comparison of the performance of the ICE-6G_C (VM5a) and ICE-7G_NA (VM7) modelsin the Western Mediterranean basin in absolute and relative terms. Gains (drops) in the χ2-like errorgreater than 20% are displayed with a green (red) background, with changes of less than 20% are shownwith a yellow background.

5.5 Revisiting the interpretation of Roman fish-tanks

One of the main archaeological tools used to infer relative sea level during Antiquity is based on the

interpretation of Roman-era piscinae (fish tanks), structures built close to sea level that acted as storage

pens for live fish, since their architecture and age was thought to be well constrained (e.g. Lambeck

et al., 2004; Auriemma and Solinas, 2009; Morhange and Marriner, 2015). However, the interpretation

of the mechanism controlling the water inflow and outflow, based on field studies and study of original

descriptions in Latin, has recently been the subject of much debate in the community (e.g. Lambeck

et al., 2004; Auriemma and Solinas, 2009; Evelpidou et al., 2012; Morhange et al., 2013). The main

features from which sea level can be inferred include the sluice gate that enabled the flow of water while

keeping the fish inside the tank, water exchange channels and foot-walks that delimited the extent of the

basins and enabled workers to move between them (Lambeck et al., 2004; Antonioli et al., 2007; Vacchi

et al., 2016). While some authors (Lambeck et al., 2004; Antonioli et al., 2007; Auriemma and Solinas,


Figure 5.4: Contour plot of relative sea level at 2 ka BP across the Mediterranean region, as predictedby (a) the ICE-6G_C (VM5a) model of Peltier et al. (2015), and (b) the ICE-7G_NA (VM7) model ofRoy and Peltier (2017). A full contour line represents a relative sea level rise while a dashed contourline represents a relative sea level fall, with each contour line signifying a RSL increment of 20 cm fromthe bolder zero contour line.

2009) have suggested that the upper RSL limit was 0.2 meters below the lowest foot-walks and that

water inflow was tidally controlled, others (Evelpidou et al., 2012; Morhange et al., 2013) have suggested

that this interpretation would not have allowed a sufficient water flow and instead that the fixed gates

were built in a sub-tidal position.

These different interpretations lead to different estimates of relative sea level during Roman times.

Along the Italian coast of the Tyrrhenian Sea, Lambeck et al. (2004) place Roman RSL in the range of

-1.37 m to -1.20 m with respect to present, while Evelpidou et al. (2012) place it at -0.58 m to -0.32 m

with respect to present. This difference has strong implications in the determination of late Holocene

rates of relative sea level change, indispensable for the correct interpretation of modern tide gauge rates

of sea level change in the context of anthropogenic climate change (e.g. Engelhart et al., 2009; Church

and White, 2011; Hay et al., 2015; Vacchi et al., 2016).

In this context, it is interesting to determine whether the new ICE-7G_NA (VM7) model can shed

some light on the disagreement. Figure 5.4 displays the predicted values of relative sea level around

the time at which the fish tanks were constructed (at 2 ka BP) across the Mediterranean region for the

ICE-6G_C (VM5a) and ICE-7G_NA (VM7) models. Table 5.2 compares the observational inference


of relative sea level at 2 ka BP along the Italian coast of the Tyrrhenian Sea, obtained from the inter-

pretation of Roman fish-tanks of Lambeck et al. (2004) and Evelpidou et al. (2012), together with the

predictions from GIA models of the same quantity at the location where the main fish tanks are located

(Civitavecchia). The new ICE-7G_NA (VM7) model predicts a sea level rise of 0.54 m since Roman

times at that location, a value that is consistent with the upper margin of the Evelpidou et al. (2012)

result. The ICE-6G_C (VM5a) model of Peltier et al. (2015) predicts a sea level change since Roman

times that cannot be easily reconciled with the two interpretations of Roman fish tank indicative mean-

ing. As the available geological data points are not able to differentiate between the two interpretations

once altitude errors are accounted for (Vacchi et al., 2016), the predictions of GIA models (which were

not tuned to this data source) are an important tool by which the interpretation of archaeological infor-

mation can be assessed. This preliminary result supports, albeit at the limit of its uncertainty range,

the Evelpidou et al. (2012) analysis.

Table 5.2: Comparison of observational inference and GIA model predictions of Roman time RSL alongthe Tyrrhenian coast

5.6 Perspectives

It has been shown herein that the new ICE-7G_NA (VM7) model of Roy and Peltier (2017) is able to

reproduce the available observations of past relative sea level in the western part of the Mediterranean

basin gathered by Vacchi et al. (2016). Observations from this region were not used in the optimiza-

tion process that led to the ICE-7G_NA (VM7) model, which focused on constraints from the North

American continent. Thus, the current analysis is highly significant, as it is the first supplementary

regional test of the model. Some misfits remain in specific regions of interest, most notably in Central

Spain and Southwestern Italy. Further work will be required to analyze carefully these misfits in the

context of the complex geological setting of the Mediterranean basin. Also, some further assessment

of the model predictions in the Gulf of Gabes will be required, in particular to revisit the sensitivity


analyses performed by Stocchi et al. (2009) regarding the far-field influence of the melting history of the

Antarctic ice sheet on the small highstand observed in the geological record. Finally, it will be necessary

to extend similar tests of the global exportability of the ICE-7G_NA (VM7) model to other regions,

including the eastern half of the Mediterranean basin, for which geological inferences of past relative sea

level also exist (among others, Vacchi et al. (2014) for the Aegean Sea).

Chapter 6

Conclusion

This work has focused on the study of the Glacial Isostatic Adjustment (GIA) process, and has presented

the clear opportunities that high-quality data sets of geophysical observables related to the process can

provide to further constrain our models of the phenomenon.

First, the impact of the GIA process on the rotational state of the planet, caused by the large redis-

tribution of mass associated with the Late Pleistocene cycles of glaciation/deglaciation, was analyzed.

An inference of the impact of this cycle on the most recent observational data sets of the J2 coefficient

and of polar wander was presented, together with evidence of the impact of present-day anthropogenic

climate change on the secular trends inferred in those observables (Roy and Peltier, 2011).

Then, the importance of the Atlantic coast of the continental United States for the study of the GIA

process and the availability of a newly compiled database of 14C-dated records of sea-level evolution of

very high quality for the Holocene period in the region (Engelhart and Horton, 2012) were discussed.

It was shown that misfits identified between observations of Holocene relative sea level change for the

southern part of the coast and model inferences of the same quantities (Engelhart et al., 2011) could be

eliminated by carefully considered changes in the structure of the mantle viscosity profile used in the

ICE-6G_C (VM5a) model of Peltier et al. (2015). This work, which was the first time that constraints

from regions of forebulge collapse were actively used in the optimization process leading to a global GIA

model, has led to the new VM6 viscosity profile (Roy and Peltier, 2015), which was also tested against

data for the West coast of the United States.

The following step led to the expansion of the methodology used in Roy and Peltier (2015) to include

the voluminous quantity of space-geodetic measurements of crustal motion available over North America

(Argus et al., 2010; Peltier et al., 2015), and an exploration of the impact of ice sheet loading history

164

Chapter 6. Conclusion 165

variations on the resulting quality of the fit to the combined suite of observables that also included

the relative sea level history database of Engelhart and Horton (2012). The resulting parameter space

exploration and careful characterization of the misfits between the observational data sets and the GIA

model predictions has led to the ICE-7G_NA (VM7) model of the GIA process, which is able not

only to reproduce the sea level history database of Engelhart and Horton (2012), but also explain the

present-day crustal uplift and subsidence over the North American continent as observed by the GPS

network. The model also explains the crustal tilting over the Great Lakes region that has occurred

following deglaciation and provides a good fit to the geological inference of the shape and extent of the

Late Holocene forebulge evolution along the U.S. East coast.

The resulting ICE-7G_NA (VM7) model enables a further improvement of fit to the totality of the

available geophysical observables over the North American continent, while continuing to employ the

same basic iterative methodology that has been so successfully employed in the previous steps in global

GIA model development. It takes advantage of the high quality of the new observational data sets

related to the GIA process, particularly in regions of forebulge collapse.

Finally, the performance of the ICE-7G_NA (VM7) model in the Mediterranean basin, a region

to which the model was not tuned, was presented. This work, to be published imminently, shows the

increased quality of fit provided by the viscosity structure variations that led to the VM7 viscosity profile.

6.1 Perspectives and future work

Several avenues could be explored to further exploit and enhance the advances described in this work.

In particular, our knowledge of the post-LGM evolution of the ice sheets covering Antarctica has

benefited from the rapid development of an extended GPS observation network over the region and of an

increase in suitable geological inferences of ice sheet retreat (notably bedrock exposure age estimates),

which have recently been used in generating new models of Antarctic ice sheet evolution (Argus et al.,

2014; Whitehouse et al., 2012). In light of the slight adjustments that led to the ICE-7G_NA (VM7)

model, and furthering the exportability process initiated in the present work, predictions of geophysical

observables related to the GIA process over Antarctica will be addressed. However, as most of the

sensitivity of the GIA model predictions of geophysical observables over West Antarctica to changes in

mantle viscosity lies within the upper mantle (Argus et al., 2014), it is expected that the impact of

the switch to the new ICE-7G_NA (VM7) model on the regional GIA predictions over Antarctica will

be minor, since the new VM7 viscosity structure is very similar to its precursors VM6 and VM5a in

the upper mantle. This work would also enable a revisit of the Stocchi et al. (2009) hypothesis of the


Figure 6.1: (a) Inferred GIA vertical displacement rates (mm/yr) inferred over the GPS sites of theGreenland Global Positioning System Network (GNET) by Khan, S. A. et al. (2016), together withestimates of the contribution of each drainage basin of the ice sheet to total eustatic sea level changesince LGM. (b) GIA vertical displacement rates inferred from the simple GIA model of Khan, S. A. et al.(2016), that shows the GIA uplit rates over Greenland that would be required to explain the GNETobservations. The path of the Iceland hot spot, inferred by the authors to be a source of potential lateralheterogeneity in mantle viscosity that could explain this misfit, is shown as a dashed line. From Khan,S. A. et al. (2016). Reprinted with permission from AAAS.

impact of Antarctica post-LGM deglaciation on relative sea level evolution in some isolated sections of

the Mediterranean Sea.

Similar analyses will be performed to carefully assess the quality of the new GIA model predictions

over Fennoscandia (Steffen and Wu, 2011) and Greenland (Wake et al., 2016). Although these analyses

are also expected to lead only to minor adjustments, if any, to the model, they are nonetheless necessary

in order to turn ICE-7G_NA (VM7) into a truly global model. The model could then be used to revisit

the present-day mass balance estimate of the Greenland and Antarctic ice sheets and estimates of 21st

century sea level rise in the context of anthropogenic climate change.

A further topic of interest is related to Greenland, where space-geodetic measurements from the

Greenland Global Positioning System Network (GNET), maintained by the Danish National Space

Institute and other partners, have revealed the existence of regional misfits between observed vertical

uplift rates and GIA model predictions of the same quantities (Khan, S. A. et al., 2016). Although these


misfits have been associated by the authors to potential lateral heterogeneity in the viscosity of the

mantle associated with the Iceland hot spot (Khan, S. A. et al., 2016), it would be enlightening to see if

the methodology applied in Roy and Peltier (2015) and Roy and Peltier (2017) could introduce suitable

ice loading history variations that would explain this regional discrepancy (shown for indicative purposes

in Figure 6.1) with a simple 1D model of the viscosity of the mantle. If they cannot be reconciled using

the current model structure, an interesting research question would be to address the extent to which

the GIA-related predictions and the optimization method that led to the ICE-7G_NA (VM7) model

might be impacted by lateral heterogeneity in mantle viscosity. Various techniques, including coupled

Laplace finite-element methods (Wang and Wu, 2006), have been used in previous studies, and have

also relied on further constraints provided by modern global seismic tomography models (such as the

one of Ritsema et al. (2011)). As the Greenland Ice Sheet is already showing evidence of accelerated

decay (Khan, S. A. et al., 2015a), it is imperative to better understand how it responded to past climate

change, in order to better constrain its current response and predict its future behaviour.

Finally, as the ICE-7G_NA (VM7) model is able to explain available GIA-related geophysical ob-

servables over North America, it will be interesting to use this model in the study of recent relative sea

level change associated with the global warming process, and recorded over the past century or so in a

large number of tide gauge measurements that cover the globe. As an accurate inference of modern-day

changes in relative sea level from these instruments depends on an appropriate correction for GIA effects

(e.g. Douglas and Peltier, 2002; Church and White, 2011; Hay et al., 2015), and since a large fraction of

the longest North American tide gauge records are found along the United States East coast (e.g. Hay

et al., 2015), it might be insightful to revisit the GIA-related assumptions of these studies in the context

of the new ICE-7G_NA (VM7) model. This work looks especially interesting in light of the extension of

the Jet Propulsion Laboratory network of GPS receivers along the U.S. East coast (unpublished work).

As sea level changes are one of the most striking manifestations of past and present climate change, it

is crucial to develop our understanding of the processes that influence its characteristics. Their breadth

is very extensive, and includes processes that control the growth and disappearance of large continental

ice sheet cover and the feedbacks that relate these components. It also relies on a knowledge of the

Earth’s interior, of its orbital properties and of the evolution of its gravitational field. In the context

of understanding the present and future climate perturbations induced by anthropogenic emissions of

greenhouse gases, it is important, more than ever, to understand correctly how the past climate responded

to various forcings in order to correctly capture the potential physical and societal impacts brought by

current climate change.

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Appendix A

Copyright and contributions

This section contains the copyright agreements for the previously published material in this work.

The following figures are shown in the current work under a License Agreement between the author,

Keven Roy, and the publisher (in parentheses for each listed figure), as provided by the Copyright

Clearance Center for the reuse in a thesis. Figure 1.2: (Peltier, 2007) (Elsevier). Figure 1.3: (Cronin,

2010) (Columbia University Press). Figure 1.4: (Peltier, 2007) (Elsevier). Figure 1.5: (Peltier,

2007) (Elsevier). Figure 1.6: (Cronin, 2010) (Columbia University Press). Figure 1.8: (Peltier and

Fairbanks, 2006) (Elsevier). Figure 2.1: (Schuh and Behrend, 2012) (Elsevier). Figure 2.2: (Seitz

and Schuh, 2010) (Springer). Figure 2.3: (Gross, 2007) (Elsevier). Figure 2.9(a): (Thomas et al.,

2006) (John Wiley and Sons). Figure 2.9(b): (Rignot et al., 2008) (John Wiley and Sons). Figure

5.1: (Vacchi et al., 2016) (Elsevier). Figure 6.1: (Khan, S. A. et al., 2016) (American Association for

the Advancement of Science).

Figure 1.1(a) is from de Charpentier (1841), is in the public domain and was obtained from the

Open Internet Archive (Open Library ID - ia:essaisurlesglac02chargoog; Internet Archive ID - essaisurles-

glac02chargoog). The permission to use Figure 1.1(b) was obtained from the National Snow and Ice

Data Center, in Boulder (CO), USA. Figure 1.7 is modified from a figure provided courtesy of the

Canadian Geodetic Survey (Natural Resources Canada), covered under a Creative Commons Attribution

Share-Alike 3.0 License (CC BY-SA 3.0). The rights for Figure 6.1 are also covered under a Creative

Commons Attribution NonCommercial 4.0 License (CC BY-NC) linked to the American Association for

the Advancement of Science. Figure 2.6 (Ratcliff and Gross, 2010) was reproduced with permission,

courtesy of NASA/JPL-Caltech. Figure 2.9(c) (IPCC, 2013) was reproduced with permission from the

IPCC Secretariat (World Meteorological Organization).

191

Appendix A. Copyright and contributions 192

The work presented in Chapter 2 contains material from Roy and Peltier (2011) (Roy, K., and Peltier,

W. R. (2011), ’GRACE era secular trends in Earth rotation parameters: A global scale impact of the

global warming process?’, Geophysical Research Letters 38(10), L10306) and Figure 2.10 from Peltier

et al. (2012) (Peltier, W. R., Drummond, R., and Roy, K. (2012), Comment on "Ocean mass from

GRACE and glacial isostatic adjustment" by D.P. Chambers et al., Journal of Geophysical Research:

Solid Earth, 117, B11403). All necessary permissions were obtained from the publishers. The work in

Chapter 3 contains material from Roy and Peltier (2015) (Roy, K., and Peltier, W. R. (2015), ’Glacial

isostatic adjustment, relative sea level history and mantle viscosity: reconciling relative sea level model

predictions for the U.S. East coast with geological constraints’, Geophysical Journal International 201(2),

1156-1181). All necessary permissions were obtained from the publishers. The work in Chapter 4 is under

review at the time of printing (accepted with minor modifications) under Roy and Peltier (2017) (Roy,

K., and Peltier, W. R. (2017), ’Space-geodetic and water level gauge constraints on continental uplift and

tilting over North America: Regional convergence of the ICE-6G_C (VM5a/VM6) models’, Geophysical

Journal International). The work in Chapter 5 will be submitted to Geophysical Research Letters as

(Roy, K., and Peltier, W. R., ’Relative sea level in the Western Mediterranean basin: A regional test of

the ICE-7G_NA (VM7) model’ ).

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High-quality constraints on the glacial isostatic ... · ICE-7G_NA (VM7) model Keven Roy Doctor of Philosophy Graduate Department of Physics University of Toronto 2017 The Glacial