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THE DEEP ELECTRICAL STRUCTURE OF THE GREAT GLEN FAULT, SCOTLAND
MAXWELL AZU MEJU
DOCTOR OF PHILOSOPHY UNIVERSITY OF EDINBURGH
1988
DECLARATION
This thesis has been composed by me and has not been submitted for any
other degree. Except where acknowledgement is made, the work is original.
Maxwell A. Meju
2
ABSTRACT
The deep structure of the Great Glen Fault (GGF) is an outstanding problem
of Scottish geology. The magnetotelluric (MT) technique in the frequency
range 0.016 - 640 Hz has been applied to the determination of the electrical
resistivity structure across this fault. Data from 32
stations along 3 traverses in this region have been processed using standard
tensorial techniques with attention being paid to the biasing of the impedance
tensor. Application of digital filtering techniques enhanced further the quality
of noise degraded data sets.
One-dimensional inversion algorithms which account for errors on the data
and the non-unique nature of the solutions have been developed and applied
to the processed data sets to yield an approximate geoelectric structure for the
region. Subsequent two-dimensional (2-D) finite difference modelling has
revealed some hitherto undiscovered features. To the east and west of the
fault zone, the structure is characterized by a highly resistive (2000 - 100000
S2m) upper crust which thickens from about 20Km at the traverse extremes to
30 - 42 Km near the fault and by a lower crust having a resistivity in the range
150 - 400 7m. The basic structure of the GGF is an outcropping low resistivity
(150 - 200 2m) "dyke-like" body, about 1 Km wide and connected with the
lower crust at a depth of about 30 Km. Similar concealed dykes exist to the
east and west of the fault and are interpreted as steep shear zones.
The 2-D model explains the many geophysical and geological observations
made in and around the fault and is therefore the first unequivocal model for
the region; the Great Glen is interpreted as a fossil rift or a contact zone
between accreted electrically distinct ' terrains. An integrated interpretation
of MT and gravity data sets has indicated other zones of crustal thickening in
the Scottish Caledonides. These are interpreted as ancient continental collision
(or microplate suture) zones and shown to be consistent with the known
regional geology. A tectonic model is proposed for the Scottish Caledonides
and plausible explanations are offered for the low resistivity nature of the lower
crust and the dyke-like bodies.
3
"But if any human being earnestly desire to push on to new discoveries instead
of just retaining and using the old; to win victories over Nature as a worker
rather than over hostile critics as a disputant; to attain, in fact, to clear and
demonstrative knowledge instead of attractive and probable theory; we invite
him as a true son of Science to join our ranks."
BACON (Novum Organum, Prefatio, 1620)
4
ACKNOWLEDGEMENTS
I am very grateful to Dr V.R.S. Hutton for suggesting and supervising the
magnetotelluric project, and for letting me participate in numerous international
field projects: the experience is invaluable. Her great kindness, generosity and
hospitality are thankfully acknowledged. Dr R.A. Scrutton was my second
supervisor and contributed in no small amount to the success of this project:
thanks for everything. Special thanks to Dr Anna Thomas-Betts, my tutor at
Imperial College, for introducing me to the magnetotelluric method and for her
continued support.
I extend my warmest thanks and appreciation to all members of the
Geophysics Department for their help and friendship. I benefitted from
discussions with Drs R. Hipkin and B.A.Hobbs. Special thanks to G.Dawes for
technical support and for developing the improved S.P.A.M. system - It's been
"MT without tears". I thank my colleagues: E.R.G. Hill for helping me
understand inverse theory; Sergio Luiz Fontes for introducing me to digital
filtering and for field help and his friendship; T.Harinarayana, A.Tzanis and R.Parr
for useful discussions; and D.Galanopoulos and Phil Jones for field assistance.
I am grateful to Drs Kathy Whaler and G.iUaiij of Leeds University for
help with my fieldwork. I thank NERC (and especially Mr Valiant of the
Geomagnetic Equipment Pool) for a loan of their AMT system and frequency
analyser.
My thanks also go to the officials of the Locharber Forestry Commision (and
especially Mr Biggin ;, Torlundy office) and owners of various estates (in
particular Loch Eil estates) for their help and cooperation.
I am indebted to the BGS (Edinburgh) and especially Mr D.W.Redmayne for
providing me with historical earthquake records of the Great Glen region; the
illuminating discussions with Redmayne are gratefully acknowledged.
I express my profound gratitude to the Nigerian Federal Government for a
scholarship and the CVCP (London) for an overseas research students award
this study would not have been undertaken without their financial support.
Finally, my deepest thanks go to my family for all their support and
especially to 'Chi' Rosaline for her love, patience and understanding during the
last four years.
CONTENTS
Page
Abstract 3
Acknowledgements 5
Contents 6
List of figures 10
List of tables 12
CHAPTER 1 INTRODUCTION 13
1.1 The quest for deep crustal information 13
1.2 Aspects of geology of northern Scotland 17
1.2.1 Tectonic and geologic setting 17
1.2.2 Geology of the Great Glen study region 19
1.3 History of investigation 21
1.4 Problems for research 23
1.5 Outline of research 25
CHAPTER 2 THE MAGNETOTELLURIC METHOD 26
2.1 Source field 26
2.2 Maxwell's equations 28
2.3 One-dimensional (1-D) MT theory 28
2.3.1 Uniform half-space 28
2.3.2 n-layered half-space 32
2.4 Two- and Three-dimensional MT theory 33
2.4.1 Two-dimensional (2-0) structures 33
2.4.2 Three-dimensional (3-0) structures 36
2.5 Earth response functions 36
2.5.1 Indicators of structure 36
2.5.2 Indicators of structural dimension 39
2.5.2.1 Electrical strike direction and Tipper 39
2.5.2.2 Impedance skew 40
2.5.2.3 IfIlipti6tv andEccntricTy 41
2.6 Estimation of the impedance matrix 41
2.7 Coherence 43
CHAPTER 3 FIELD STUDIES 44
3.1 Field equipment specification 44
3.2 Field instrumentation 45
3.2.1 Magnetic and electric field sensors 45
3.2.2 Data recording equipment 45
3.2.3 Equipment Calib iatior~f 49
3.3 Field procedure and practical considerations 49
3.3.1 Site selection 49
3.3.2 Field set-up 53
3.4 Survey details 53
CHAPTER 4 DATA ANALYSIS AND RESULTS 57
4.1 Data transfer 57
4.2 Data processing 57
4.2.1 Standard analysis 57
4.2.1.1 General window analysis 57
4.2.1.2 Averaging of windows results 58
4.2.2 Bias analysis 61
4.2.3 Cultural noise analysis 61
4.2.3.1 Noise types found in the MT records 63
4.2.3.2 Digital filtering 63
4.2.3.3 Frequency domain window editing 69
4.3 Magnetotelluric field responses 71
4.3.1 Glen Garry profile 71
4.3.2 Loch Arkaig profile 72
4.3.3 Glen Loy profile 0
72
CHAPTER 5 ASPECTS OF DISCRETE INVERSE THEORY 75
5.1 Linear least squares inversion 75
5.2 Constrained linear least squares inversion 77
5.2.1 Inversion with simplicity measures 78
5.2.2 Inversion with prior information 79
7
5.3 Nonlinear least squares problems 80
5.3.1 Iterative least squares inversion 80
5.3.2 Ridge regression 83
5.3.2.1 Properties of ridge solution 84
5.3.2.2 Damping factor and the stability of the regression estimates 85
5.4 Solution appraisal 88
5.4.1 Goodness-of-fit 86
5.4.2 Parameter resolution matrix 86
5.5 Errors and extreme parameter sets 87
5.5.1 Parameter covariance matrix 87
5.5.2 Extreme parameter sets 87
5.6 Geometric interpretation of constrained inversion 89
CHAPTER 6 1-0 MODELLING AND INVERSION OF MT RESPONSES 92
6.1 Forward modelling 92
6.2 Inversion of MT data 92
6.3 Model search 94
6.3.1 Linearising parameterizations 94
6.3.2 The optimization scheme 95
6.3.3 Computational details 98
6.3.3.1 The singular value decomposition (SVD) of a matrix
an overview 98
6.3.3.2 Application of SVD to inversion 99
6.3.3.3 Estimation of damping factors for ridge analysis 99
6.3.4 Convergence characteristics of the optimization algorithm 100
6.4 Model interpretation methods 108
6.4.1 Damped most squares method 108
6.4.2 The a priori information approach 113
6.4.3 Multi-station interpretation 114
6.4.3.1 Joint inversion 114
6.4.3.2 Simultaneous inversion 115
6.4.4 Occam's razor technique 116
6.4.5 Nibett-Bostick (N-B) transformation 117
6.5 One-dimensional models 118
6.5.1 1-D response functions and the existence of solutions 118
6.5.2 Comparison of methods 118
6.5.3 Interpretive geoelectric sections 130
N.
6.5.4 Effective phase sections 134
CHAPTER 7 TWO-DIMENSIONAL NUMERICAL MODELLING STUDIES 136
7.1 Introduction 136
7.2 Model construction 136
7.2.1 Computer modelling technique 137
7.2.1 Programs and execution 137
7.2.1.2 Mesh design considerations 138
7.2.2 Prior information and initial models 139
7.3 2-0 geoelectric models 146
7.4 Inverse calculations 167
7.4.1 Optimal resistivity distribution 167
7.4.2 Errors on the model parameters 167
CHAPTER 8 INTERPRETATIONS AND CONCLUSIONS 170
8.1 Geoelectric structural appraisal fl 171
8.1.1 Relationship with previous geophysical and geological
observations 171
8.1.2 Integrated interpretation of MT and Gravity data 178
8.2 Deep geology of northern Scotland 181
8.2.1 Gross geological overview: problematic observations 182
8.2.2 Evidence of deep structure 184
8.2.3 Regional synthesis: an alternative tectonic view 194
8.2.4 Tectonic summary 199
8.3 Interpretation of high electrical earth conductivity 201
8.4 Discussions 203
8.5 Conclusions 204
8.6 Recommendations and plans for future research 206
REFERENCES 208
Appendix I Glen Garry field results 227
Appendix II Loch Arkaig field results 235
Appendix Ill Glen Loy field results 241
Appendix IV Individual site 1-0 models 262
VC
LIST OF FIGURES
Page
1.1 Seismic and electromagnetic survey map of the British Isles 14
1.2 Map showing the tectonic units of northern Britain and NW Ireland 18
1.3 Simplified geological map of the Great Glen region 20
2.1 Typical amplitude spectrum of magnetic variations in the ELF range 26
2.2 Homogeneous half-space model 29
2.3 Skin depth as a function of period and ground resistivity 31
2.4 n-layer earth 32
2.5 Two-dimensional structure 33
2.6 Rotation of axes by angle e 38
3.1 Block diagram of complete infield MT system 47
3.2 Simplified flowchart of the infield process 48
3.3 Instruments response curves 50
3.4 Map showing the locations of the MT observational sites 54
4.1 Some results of conventional data processing 60
4.2 Example of bias in response estimates assuming noise-free E or H channels62
4.3 Frequency response of a delay line filter 65 :
4.4 Time series for two data windows before and after the application of a delay line filter 66
4.5 Some results of data processing with a simple delay line filter 68
4.6 Results of frequency domain noise analysis for some data windows. 70
5.1 Ridge trace for typical geoelectric model parameters 85
5.2 A two-dimensional simplification of the p-dimensioned interactions in the vector space. 90
6.1 A simplified flowchart of the minimization algorithm 97
6.2 Estimation of the damping factors in ridge analysis 100
6.3 Ridge regression models for synthetic and actual field data 101
6.4 Convergence characteristics of the algorithm 107
10
6.5 A simplified flowchart of the iterative most squares algorithm 112
6.6 COPROD models 120
6.7 Comparison of ridge regression and most squares models for site L18 123
6.8 Some models for site Li 124
6.9 Occam and most squares models for L19 127
6.10 Most squares and N-B results for L8 and L5 128
6.11 Interpretive geoelectric sections for Glen Garry, Loch Arkaig and Glen Loy profiles 131
6.12 Glen Loy invariant phase section 135
7.1 East-west profiles of apparent resistivity across the Great Glen fault for E- and H-polarisations and frequencies 8, 14, 20, 24 and 32 Hz 140
7.2a&b 2-D geoelectric models for the Glen Loy profile 143
7.3a-j Fit of the Glen Loy 2-D response curves to observed data 147
7.4 Fit of the Glen Loy 2-D model to the Loch Arkaig data 158
7.5 A 2-D geoelectric model for the Glen Garry profile 161
7.6 Fit of the Glen Garry 2-D response curves to observed data 162
7.7 Most squares resistivity model for the Glen Loy profile 168
8.1 Some geophysical results for northern Scotland 172
8.2 Location map of the Gravity and MT profiles 179
8.3 Integrated geophysical model for the Great Glen region 180
8.4 Integrated geophysical model for northern Scotland 185
8.5 A comparison of MT/Gravity and seismic structure across northern Scotland 186
8.6 Comparison of the MT/Gravity model and previous models for northern Scotland 188
8.7 The lid of the continental collision zone (orogen) and its frontal wedges model 197
8.8 A model for the deep geology of northern Scotland 200
8.9 Schematic model of fluid circulation systems in regional metamorphic belt 203
11
LIST OF TABLES
Table
Page
3.1 Site specifications 56
6.1 Model resolution matrices for sites Li 1, L16 and Ci 109
6.2 A demonstration of the simultaneous multi-site interpretation technique 129
7.1 Grid size used in 2-D modelling of Glen Loy data 142
12
CHAPTER 1 INTRODUCTION
The resistivity of a material is its ability to impede the flow of current
through it. Rocks, the building materials of the earth, are aggregates of
minerals. The electrical properties of a rock will therefore depend on the
composition of the constituent minerals, the amount and interconnectivity of
pore space between the grains, the presence of volatiles (in connected
channels) and temperature. The resistivity of earth materials varies over a wide
range (see Haak and Hutton 1986, Fig 1) especially when there are temperature
variations or in the presence of specific minerals and volatiles (in connected
pores). This feature enables important and in some respects unique
information to be deduced about the earth's crust and mantle by determining
earth resistivity. The geophysical techniques for earth resistivity determination
(and the depth probed) fall into the following groups:
A.C. Resistivity methods ( 300m)
D.C. Resistivity methods ( 2km)
Magnetotelluric (MT) method ( 600km)
Geomagnetic depth sounding (GDS) and Horizontal gradient method
( 1000km) and
Spherical harmonic analysis of global magnetic time variation data
( 2000km)
They all have wide applications. Of particular importance is the determination
of structural and compositional information about the earth from lateral
resistivity variations.
In this chapter, a brief historical background of the magnetotelluric method
is given and followed by discussions of the known geology and geophysics of
the study area. The project outline and objectives are also given.
1.1 The quest for deep crustal information
Over the past decade, especially in northern Britain, there has been a major
effort to examine the deep structure of the Lithosphere by seismic and
electromagnetic profiling techniques. Some of the various experimental profiles
- completed or proposed - in Britain (including the recent surveys in Ireland)
are shown in fig 1.1. The importance of these investigations cannot be
over-emphasised. These techniques provide good images of structural
patterns in the subsurface. However, despite the vast geophysical observations
in northern Britain the deep geology is still not well understood. The role of
13
0 200
km SHUT
DRUM
MOIST-1
,>16
WINCH I Iq W~)__ GGF Great Glen Fauft
(HOE Highland Boundary fl
'NSUF Southern Uplands Fc
Fig 1.1 Seismic and electromagnetic survey map of the British Isles.
Heavy lines are the British Institution's Reflection Profiling Syndicate's
(BIRPS) seismic traverses (only those relevant to this study are named). MOIST
(Moine and Outer Isles Seismic Traverse), WINCH (Western Isles-North Channel),
DRUM (Deep Reflections from the Upper Mantle) and SHET (Shetland) are deep
seismic reflection lines. The LISPB (Lithospheric Seismic Profiling in Britain)
and the recent seismic refraction experiments in Ireland (Univ. College, Dublin)
are shown by the dashed lines. Beaded lines are the electromagnetic profiles
conducted in Britain by I - Edinburgh University and in Ireland by the Applied
Geophysics Unit, Univ. College, Galway.
14
some of the major crustal dislocations (faults) in the structural evolution of the
regions is at best, poorly understood. One of such faults is the Great Glen
fault in northern Scotland. The present project is designed to probe the deep
structure of this outstanding crustal feature using an electromagnetic profiling
technique. Of the electromagnetic (EM) techniques currently available for
probing the earth, the magnetotelluric (MT) method is the most commonly
used.
Magnetotellurics is a geophysical exploration technique employing
simultaneous measurements of natural transient electric and magnetic fields to
infer the electrical conductivity (the reciprocal of resistivity) distribution within
the earth beneath the site of the surface fields. The history of the MT
prospecting method is comparatively recent, and was first proposed by
Cagniard in 1953, although the Russian scientist Tikhonov (1950) and the
Japanese scientists, Kato and Kikuchi (1950) and Rikitake (1950) had published
on the subject of deep crustal electrical conductivity investigations without
applying the results to practical geophysical exploration.
The method differs from other EM geophysical techniques that require
artificial sources of EM energy to probe the earth. The naturally occuring
interminable variations in the earth's magnetic field induce eddy currents, called
telluric currents, in the conductive crust of the earth that are detectable as
electric field variations at the surface (and within the earth). Cagniard (1953)
showed that if the variation in the magnetic field is assumed to be derived
from a plane EM wave propagating vertically into the earth, the EM impedance
(ie the ratio of the horizontal electrical field in the ground to the orthogonal
horizontal magnetic field), measured at a number of frequencies, gives earth
resistivities as a function of frequency or period resulting in a form of depth
sounding.
This is a wide-band depth sounding technique, the frequency ranging from
<0.001 Hz to >1000 Hz, and is suited for shallow as well as deep structural
investigations. The depth of sounding can be roughly related to frequency by
the use of the skin depth defined as
a z 0.5 (Resistivity/Frequency) 112 Km
where resistivity is in ohm-metres and frequency in Hertz.
The MT method finds its application in the studies of global structures such
as crust, mantle, rift valleys etc. (Madden and Swift 1969; Hutton et al, 1981;
Stanley 1984); in mineral exploration (Strangway et al, 1973; Strangway, 1983);
15
in geothermal resources exploration and evaluation (Hermance and Grillot, 1974;
Hutton et al 1984; Devlin 1984); in basin evaluation for petroleum exploration
(Tikhonov and Berdichevsky 1966; Vozoff, 1972; Alperovitch et al, 1982; Stanley
et al, 1985); and in civil engineering, ground water and archeological problems
(Guineau, 1975). This method has been used in offshore conditions (eg
Cagniard and Morat, 1967; Morat 1974; Hoehn and Warner, 1983) and for
earthquake prediction (Honkura et al, 1977).
Of the conventional deep probing EM techniques, the MT method is the
most advantageous in terms of resolution, sophistication of interpretation, and
ease of logistics. Geomagnetic depth sounding interpretations suffer from a
lack of resolution, and controlled-source studies (an artificial source variant of
MT) are logistically cumbersome and currently simplistic (one-dimensional) in
their interpretation.
The MT method has now developed from its initial reconnaissance
applications into a powerful subsurface mapping tool second only to seismics
in the depth and quality of information it provides. In fact it is now being
argued that a coincident MT study should be made wherever a seismic
reflection survey is undertaken and that the interpretation of reflection images
be constrained by electrical conductivity information (Jones 1987).
Two difficulties beset MT data interpretation : depth resolution of earth
structures and bias effects on the sounding curves. EM sounding experiments
cannot resolve sharp boundaries or thin layers except with ideal observations
(Langer 1933; Jones 1987); the diffusive nature of the energy propagation
"smears out" the real earth structure. Bias effects are a consequence of noise
corruption of the MT signals and produce frequency-independent dc-like
("static") shift or frequency dependent biasing of the apparent resistivity
sounding curves which limits model resolution to a large extent. The phase of
the impedance (not obtained by the D.C. methods) in such instances provides a
useful constraint.
The seismic reflection technique has got its problems too: while it provides
good images of horizontal and sub-horizontal structures, vertical boundaries
are very poorly imaged and may only be inferred from zones in which -
sub-horizontal reflectors are absent, and then not with certainty (Hall, 1986).
Areas of thick sedimentary or volcanic sequences are unfavourable to the
seismic method.
MT is best suited to locating vertical boundaries and for studying deep
sedimentary basins or areas hampered by surface volcanics. Jones (1987) gave
16
an interesting account of cases in which MT results aided the
geological/tectonic interpretations of the seismic sections. Berkman et al
(1984) have successfully used the combined techniques to delineate the
structure of the South Clay basin, Utah. An integrated interpretation of MT,
gravity and aeromagnetic data sets has also been recommended by Prieto et
al, (1985) who showed that in basalt-covered areas, reasonable rock
compositions and regional structural information can be determined from the
combined data sets.
1.2 Aspects of geology of northern Scotland
1.2.1 Tectonic and Geologic setting
The Northern and Grampian Highlands of northern Scotland are separated
by the Great Glen fault zone and constitute the Metamorphic Caledonides of
Britain. The Metamorphic Caledonides are separated from the northwestern
Lewisian Foreland by the Moine Thrust zone and from the Midland Valley to the
south by the Highland Boundary fault (fig 1.2)
The tectonic style of the region is related to Caledonian crustal deformation
of about 500my ago. Several workers have tried to explain the apparent
complex geology in terms of plate tectonics (Dewey, 1971, 1974; Garson and
Plant, 1973; Phillips et al, 1976; Lambert and McKerrow, 1976) but the structural
framework appears to indicate that the Caledonian mountains were not built
according to the simple rules of modern plate tectonics (Brown 1979). It is
believed, however, that the Caledonian orogenic cycle resulted in welding of
previously separated European continents and the North Atlantic continent
(composed of what are now Greenland and North America), together with a
number of trapped smaller crustal units (Dewey, 1969), Scotland being derived
from the marginal portions of the North Atlantic continent (Watson, 1984). The
manner of assembly of the various crustal units (Massifs) and their accretion to
the large continental •margin is largely unknown. The Lewisian foreland is
thought to represent a fragment of the stable North Atlantic shield (Watson
1984).
The geology of northern Scotland is very complex and the distribution of
the major crustal units is shown in fig 1.2. The northwestern part of the
Metamorphic Caledonides is certainly underlain by Lewisian continental crust
(Watson and Dunning 1979). The basement rocks are mainly highly deformed
gneisses and granulites metamorphosed at about 2700 Ma (Scourian) and 1750
Ma (Laxfordian)(see Watson and Dunning, 1979; Johnstone et al, 1979).
17
Lale Caleaon.an th,us;
ate Caieoonran taur
B Barra BS alace Stocearton Moor F.T. Ftannan thrust' G.G.F. Great Glen (aim H.B.F. Htghtano Bounoarn taul I ISLay In tflflt5traflutl 10 lOna L.F. Leannan tautt LM.F. Loch Maree fautt L.S.T; Loch Skerrols thrust M.T. Motne thrust NR North Rona O.H.T. Outer ($ebnaes thrust S.U.F. Southern Uolartos lautt T Ttree II
de Shetlana Is.
...... 1/
•.Orkney is. / • NR . . .
O •
& . . . •....••••
f /
Unst W.B.F. Boundary Q Hebr,des tatia
Fe,,jerg rO
5t*'enttan /
0 50 lOOkn, .1
B • A
' T lo
ve S
jiC1r
I !:
/ /
MIDLAND VALLEY
I !-I j -
TN
POst.CaIeaonn rocks
/ ROCKS OF CALEOONIAN BELT
LAKE NOn. metamorphic \TRICT -
Metamorpruc
ROCKS IN NW FORELAND
fl Lewtsian with cratonic Cove,
RAMP1AN HIGHLANDS
Fig 1.2 Map showing the tectonic units of northern Britain and NW Ireland. The
inset (lower right) shows the distribution of the Moine and Dalradian rocks
(from Watson, 1984).
18
Lewisian type gneisses are tectonically interleaved with metasediments of the
Moine throughout the Northern Highlands and are inferred from similar
basement outcrops in Islay and Colonsay to underlie much of the Grampian
Highlands (Watson 1984). A basement cover of Torridonian sandstone and
younger formations is found in the foreland region. These Foreland rocks are
overidden by Moine rocks at the low-angle Moine Thrust.
The Metamorphic Caledonides are represented mostly by the Moine and
Dairadian rocks, a complex sequence of metamorphosed sediments (psammites
and pelites) which are thought to be of fluviatile or deltaic origin (Johnstone,
1975). Their respective distribution is shown in fig 1.2 (inset). Three
structural/lithostratigraphic units are recognised in the Moines northwest of the
GGF (Johnstone et al 1969) - from west to east, the Morar, Glenfinnan and
Lock Eil divisions with the Sgurr Beag slide (Tanner et al, 1970; Tanner 1971)
and the Loch Quoich line (Roberts and Harris 1983) as successive demarcations.
An initial late Proterozoic (> 750 Ma) deformation and metamorphism affected
these moine rocks which were already crystalline at the onset of the
Caledonian cycle (Watson 1984). Immediately to the east of the GGF is a zone
of mixed rocks consisting of Moinian ('young Moines' or Grampian Division
see Watson 1984) and Dalradian lithofacies. Further east is the Dalradian
assemblage which extends to the Highland Boundary fault. The rocks to the
east of the GGF have not suffered any metamorphism prior to the Grampian
orogeny (Watson, 1984).
Although a great deal of the surface geology of northern Scotland has been
uncovered, the deep structure across the orogen is still poorly understood.
The major Caledonian tectonic boundaries trend NE-SW to ENE-WSW and there
appears to be some geological evidence for appreciable motion associated with
them but no unequivocal model has as yet been developed that is consistent
with the vast geological and geophysical observations in this region. The Great
Glen fault is one of the main tectonic boundaries and its deep structure and
significance form the theme of this study.
1.2.2 Geology of the Great Glen Study region
The area of study lies between Fort William to the south and Fort Augustus
to the north and extends for about 20 Km on either side of the GGF.
The surface geology of the study area is rather complex with a dominant
NE-SW structural trend. Several structural discontinuities occur in the area of
which the GGF is the most outstanding. The GGF is a major crustal feature
19
L
m
I (jnveYes
rt Wil
N
and
1'
' 7 /
m
a,
,' 5 ,ien oy komplex
a jO/
m , /
/ I /
/A vis complex
4 J__ Y -
u dm
---- -- A , tl!JflèI7
'c- '•7i /1,
/ / ds
udrn
ds
/ /
6c leanachan
/ omplex ds
--i Old Red Sandstone
g Igneous rocki
1
a Metamorphic J i' rocks
sdrn 'I
i I Fault
m
Fig 1.3. Simplified geological map of the Great Glen region (after IGS Sheet
62E and Geological map of the U.K. North).
20
which traverses some 160 Km of the Scottish mainland (Harris et al 1978). The
fault zone, Consisting of crushed and sheared rocks as well as indurated rocks,
is about half a kilometre wide and is bordered on either side by extensively
shattered strata. Caught up in the fault zone on the south-east side of Loch
Lochy (fig 1.3), and largely bounded by individual fault planes, are narrow strips
of Old Red Sandstone sediments representing remnants of an originally more
extensive cover to the Caledonian metamorphic rocks. The country rocks are
mostly metasediments but plutonic rocks are common. Moine rocks occur to
the west of the fault and the zone of interfolded Moine and Dalradian rocks
extends eastwards from the fault to the Fort William Slide - a surface along
which at an early stage in the Caledonian orogeny Upper Dalradian units
became detached from and slid bodily over the underlying rocks cutting out
several Dalradian formations (see IGS Sheet 62E). The fault zone is flanked by
outstanding igneous complexes eg Clunes, Glen Loy and Ben Nevis. Outcrops
of granitic gneisses are also common. Dykes and sheets of Ultrabasic to acidic
composition are ubiquitous; a suite of areally restricted (southwest of GGF)
Camptonite dykes have been observed to cut the crush rock of the GGF but do
not penetrate it to the south-east side (IGS Sheet 62E), a feature that perhaps
suggests large-scale horizontal movement along the fault.
As the GGF has been the source of much discussion and speculation, some
geological and geophysical observations on the fault and adjacent areas will be
summarised in the next section.
1.3 History of Investigation
The GGF has been the subject of intensive studies and various workers (eg
Kennedy, 1946; Marston, 1967; Holgate, 1969, Garson and Plant, 1972;
Winchester, 1973; Storetvedt, 1974a,b; Chesher and Bacon, 1977; Harris et al
1978; Van der Voo and Scotese, 1981; Smith and Watson, 1983; Briden et al,
1984; Torsvik, 1984; Hall, 1986) have discussed its long history of movement
and have reached conflicting conclusions as to the nature and extent of
translations along it. The state of stress adjacent to the GGF has been studied
by Parson (1979) and does not support any appreciable lateral motion along the
fault. The course of the fault off the mainland of Scotland has also been
studied. Its precise course to the SW beyond Loch Linhe is almost uncertain,
partly because of probable splays. It has been interpreted as extending
southwestwards beneath the sea for a distance of about 70 miles by Ahmad
(1966). According to McQuillin and Binns (1973) the main fault appears to pass
north of the island of Colonsay whilst a southern branch has been inferred to
21
pass through Islay connecting with the Leannan Fault in Ireland (Pitcher et al,
1964; Dobson and Evans, 1974). From north of Colonsay the main fault was
inferred to continue towards the west or southwest for at least 160 Km by
Bailey et al (1975). Its submarine extension has been traced northeastwards to
the Shetland Islands where it is represented by the Walls Boundary and Nesting
Fault systems (Flinn, 1961,1969). Its total length thus probably exceeds 720 Km
(Harris et al 1978). Correlation of the fault zone with similar faults in
Newfoundland has been discussed by Wilson (1962), Kay (1967), Webb (1969)
and Pitcher (1969) among others. A magnetic high was attributed to the fault
by Avery et al (1968) and Ahmad (1966) which was confirmed by the ridge-like
features observed along the fault on the regional aeromagnetic map of
northern Britain (Hall and Dagley 1970). Hall and Dagley point out that some of
the anomalous magnetic features appear to be offset by the fault and consider
this as evidence of major sinistral strike-slip movement which is in accord with
Kennedy's (1946) interpretation. Recorded seismic activity of the fault has been
reported by Dollar (1949) and Ahmad (1966, 1967).
The Strontian and Foyers granite complexes (at the flanks of the fault) -
interpreted by Kennedy (1946) as probably part of one rock mass based on
outcrop pattern - have been intensively studied by various workers in an
attempt to establish the nature and translation on the fault by geophysical
means. Ahmad (1966, 1967) concludes from seismic, radiometric, magnetic and
gravimetric observations that the Foyers and Strontian granites probably are
not part of one rock mass, that the GGF is an active fault and supports a
dip-slip movement on the fault since all known epicentres lie to the SE of the
fault. The anomaly over the Foyers complex on the smoothed aeromagnetic
map of northern Britain was interpreted (Hall and Dagley, 1970) as due to a
deep-seated (151(m) magnetic body extending to a depth of about 26 Km which
agrees with Ahmad's interpretation, whereas a shallow body extending from the
surface to 0.4 Km was suggested for the Strontian granite.
Sparse heat flow measurements (P.ugh 1977; Oxburgh et al 1980) are
available for the region and point to the deep presence of radiogenic materials
in the Great Glen area and especially near the Foyers Granite and
southwestwards from Loch Tay. Dimitropoulos (1981) constructed a gravity
model for the Grampian region requiring a granitic layer at 7 to 19 Km depths
below the surface between the GGF and Loch Tay fault to the south to fit the
observed gravity anomaly and this adds credence to the previous heat flow
observations.
22
The 1974 LISPB deep seismic refraction investigation was carried out along
a N-S traverse across Britain (fig 1.1) and the results have been analysed by
various workers (Bamford et al, 1977, 1978; Bamford 1979; Faber and Bamford
1979, 1981; Hall 1980). Three crustal layers were identified beneath the
Scottish Caledonides and the lower crustal layer was found to thicken
southwards from the foreland region, but the top of the mid-crustal layer was
not delineated with certainty between the Great Glen and Highland Boundary
faults (Bamford 1979). Bamford and others also provide evidence for a step in
the Moho north of the GGF. These features will be discussed further in chapter
8.
Geoelectromagnetic studies have also been carried out in this region.
Hutton et al (1977, 1980, 1981) studied the crust and upper mantle in Scotland
and show evidence of a deep conductive structure associated with the GGF
and Kirkwood et al (1981) also found a deep-lying conductor at the fault zone.
Mbipom (1980) and Mbipom and Hutton (1983) from a traverse parallel to the
LISPB line showed the presence of a conductor in the GGF area and the lower
crust was found to be distinctly conducting. However, these
geoelectromagnetic studies were based on long period (20 -> 1000 sec)
observations at stations over 15 Km apart and thus suffered from insufficient
shallow depth information and inaccurate spatial resolution of small-scale
features.
High resolution deep seismic reflection profiling in offshore Britain and
especially in northern Scotland (Smythe et al, 1982; Brewer et al, 1983; Brewer
and Smythe 1984; 1 McGeary and Warner, 1985) show spectacular reflections
from the Moho and from thrust zones within the Caledonian fold belt and
foreland. In particular, Brewer et al (1983) and Hall and others (1984) confirmed
the presence of lateral variations in the deep structure in the region of the GGF
and the possible association of zones of high seismic reflectivity in the lower
crust with electrically conducting layers is of considerable interest at the
present time.
1.4 Problems for Research
The exact role of the Great Glen fault (GGF) in the structural evolution of
the region is not well understood. Geological reconstructions suggest lateral
movements along the fault whereas studies of the state of stress adjacent to
the fault (Parson, 1979) in the Fort Augustus area do not indicate lateral
displacements. Earth tremors related to the fault zone are recorded in historic
time and it is still active.
The aeromagnetic data show that there is a marked contrast in crustal
structure across the Great Glen. A NE-SW trend, consistent with the known
geology, is dominant and the magnetic field is predominantly positive and of
deep seated origin. Magnetic profiles across this positive magnetic zone are
interpreted (Hall and Dagley, 1970) to indicate the presence of a thick (-10Km)
normally magnetized (about 3 A/rn) near-horizontal sheet at depths of 8-15 Km
below the surface south of a nearly vertical plane coincident with the GGF; the
apparent absence of any strong regional gravity anomaly associated with the
GGF on land makes it difficult to accept this interpretation as such thick and
magnetic bodies would be associated with density contrasts (Hipkin and
Hussain, 1983). Faruquee (1972) found that the main feature of the gravity field
associated with the possible extension of the GGF south of Mull and southwest
of Colonsay is a broad gravity low. This does not provide a simple solution to
the problem posed by the magnetics. However, one wonders if this magnetic
sheet can be related to Dimitropoulos' (1981) Grampian granite layer.
Previous geoelectromagnetic work (Hutton et al, 1981, Mbipom 1980)
showed the presence of a 10 Km wide crustal zone of low resistivity in the
Great Glen region which connects at depth with a conductive lower crust/upper
Mantle layer. Such a large-scale lateral inhomogeneity would be associated
with gravity contrasts. This apparent dilemma needs to be resolved. Also,
Hutton et al (1977) showed that anomalous conductive zones exist along the
GGF and in the north-west corner of Sutherland which correspond with Garson
and Plant's (1972) ancient plate boundaries in the region. This leads us to ask
if the region is a plate tectonic synthesis.
Bamford and others (1978) show continuity of layered structure across the
fault with a Moho offset to the north of the fault (Faber and Bamford 1981)
while Brewer et al (1983) show the presence of lateral variations in the region
of the fault. The MOIST reflection profile (Smythe et al 1982) which traverses
the offshore extrapolation of the Caledonian Foreland and Orogenic belt (west
of the GGF) -shows lateral variations in the deep structure of the region as
confirmed later by Brewer and others. One wonders if these offshore features
have any land analogues.
Having summarised the various observations on the Great Glen fault and
adjacent regions, we are now faced with the problems of reconcilling
geologically inferred lateral movements on the fault with its present day
seismicity and inferences from known epicentre distribution (Ahmad 1966),
correlation of offshore deep seismic structure with mainland conductivity
24
structures, the possible existence of other features hitherto undiscovered that
may provide the missing link between gravimetric and magnetic data and the
speculative possibility that the region is a plate tectonic synthesis. Most of
these issues will be addressed in this study.
1.5 Outline of Research
The deep structure of the Great Glen fault is still unresolved and
constitutes a major problem of Scottish geology. It was therefore deemed
worthwhile to carry out a high resolution magnetotelluric study of this fault as
it probably holds the key to the deep geology of this region.
The main objectives of the study are: (1) to determine the conductivity
distribution in the neighbourhood of the Great Glen fault and (2) to correlate
any observed conductivity structures with available geophysical and geological
data.
In line with these objectives, a broadband (.016 - 640 Hz) tensorial MT
technique was used to determine the crustal conductivity distribution across
the Great Glen fault. Intensive field experiments were conducted in 1985 and
1986 using state of the art technology. Conversion of observed responses
(experimental data) into earth conductivity structures in line with modern
trends in geophysical data analysis was done by use of efficient computer
programs developed by the author. To prevent accidental discoveries by the
computer and "structural over-interpretation", the problem of resolution and
non-uniqueness was addressed using an iterative most-squares (Jackson, 1976)
technique - never before used in the Magnetotelluric situation.
The two-dimensional results were evaluated in the light of the available
geophysical and geological data and found to be regionally consistent. An
integrated two-dimensional interpretation of MT and gravity data was carried
out and this appeared to resolve the apparent dilemma in the previous Great
Glen gravity interpretations. Extension of this integrated approach to the
adjacent regions indicated the presence of zones of deep crustal/mantle
structures; these findings shed some light on the deep structure of the
Scottish Caledonides. A speculative tectonic model was proposed for the
Scottish highland region.
25
CHAPTER 2 THE MAGNETOTELLURIC METHOD
The general applicability of the magnetotelluric method can be easily
assessed if the differences in spatial behaviour of the electric and magnetic
field components, as well as their different behavioural patterns in two- and
three-dimensional environments are understood. Following a brief description
of the energy source used to probe the earth, the principles of the MT method
are outlined in this chapter.
2.1 Source field
The MT source field is the transient portion of the earth's magnetic field.
The time-varying magnetic field induces current fluctuations in the earth which
have a wide spectruru. A typical average amplitude spectrum of these
magnetic variations shows a minimum at about 1 Hz (fig 2.1) and allows the
source field to be classified into two types of activities, one above and the
other below 1 Hz.
U, 4
x 0.,
U.
Sa 0.01 I—UI- -
"40.001 0.001 0.1 I 10 100 1000
FREQUENCY IN Kz
Fig 2.1 Typical amplitude spectrum of magnetic variations in the Extra Low
Frequency range (after Keller and Frischknecht, 1966).
The main source of MT fields of frequencies above 1 Hz is worldwide
thunderstorm activity, which is extensively concentrated in the tropics and
tends to peak in the early afternoon, local time. There are 3 storm centres
(Brazil, Central Africa and Malaysia) with an average of 100 storm days per
year; their geographic distribution is such that during any hour of the day there
is perhaps a storm in progress in one of the centres.
Some of the thunderstorm energy is converted to EM fields which are
26
propagated with slight attenuation in the earth-inosphere interspace as a
guided wave (Budden, 1961) and at large distances from the source this is a
plane wave of variable frequency. These MT fields penetrate the earth's
surface to produce the telluric currents. Generally, these signals from the
lightning strokes (termed sferics) have higher amplitudes at lower latitudes and
the weak currents induced by these fields in the subsurface have amplitude
peaks at distinct frequencies (the Schumann resonances : 8,14,20,25,32 Hz)
which are used in MT applications. At frequencies of about 2 KHz there is a
strong absorption in the wave guide (Strangway et al, 1973) resulting in
reduction of signal intensity. The signal levels are generally higher in the
summer than in winter (Patra and Mallick, 1980).
Other minor sources of signals above 1 Hz are man-made power
distribution systems which are generally localised and restricted to 50Hz and
harmonics. Spurious signals may be generated in the magnetic field at very
low..frequencies by wind vibration of the coil or ground vibration. Ground
discharge currents from the direct lightning strokes also generate spurious
signals in the electric field. However, some of these spurious signals are
uncorrelated in electric and magnetic field records and are readily detected.
The primary contributors to the source field between 1 and Hz are
pulsations in the earth's magnetic field (Troitskaya, 1967). The pulsations have
some correlation with ionospheric phenomena and are assumed to result
directly from the motion of charged particles above the base of the ionosphere;
current opinion is that they are the magnetic effects of hydrodynamic waves
trapped in the magnetosphere. So far no electrical processes have been
discovered within the earth's crust which would generate activity of this
frequency range and subcrustal processes would have their waves severely
attenuated by the overlying crust. The reader is referred to the detailed
descriptions of pulsations given by Orr (1973) and Rokityansky (1982).
There are latitudinal variations in the amplitude of the signals but this will
be unimportant for small scale MT measurements. The work of Price (1962),
Orange and Bostick (1965) and Berdichevsky et al (1973) seems to suggest a
very widespread source for the low and middle latitudes. According to
Berdichevsky et al (1973), at mid-latitudes the external field changes
insignificantly over distances of 100 - 200 Km, along the meridian and 200 -
500 Km along the parallel. This determines somewhat the size of the region
within which the use of models with a plane wave as the external field is
justified.
27
Thus far MT investigators use only 5 of the quantities available at the
surface for exploration : three perpendicular components of the magnetic field
variations (Hx, Hy, Hz) and two of the electric field variations (Ex, Ey). The
vertical electrical field (Ez) and currents (Jz) are not used.
2.2 Maxwell's equations
The magnetotelluric method of determining the electrical conductivity of the
subsurface strata depends on the fact that natural electromagnetic waves
penetrate into the earth's crust to depths dependent on their frequency and on
rock conductivity. Application of Maxwell's equations enables us to determine
conductivity as a function of depth from surface measurements of electric and
magnetic fields.
Maxwell's field equations in a homogeneous isotropic medium, assuming a
time dependence of the type et and a charge-free space, are
V X E = - 313 = -iwpH
2.1 a at
V X H = J + 3D = ( is + iwc)E 2.1b at
V.B=O 2.lc
V . D = 0 2.1c
Where c (F/rn) is the permitivity, is (mho/m) is the conductivity and p (H/rn) the
magnetic permeability of the medium in which the fields propagate, B is the
magnetic induction, D is the displacement vector, J is the current density
vector, E is the electric field vector, H is the magnetic field vector and w is the
angular frequency of the source field.
2.3 One-dimensional (1-D) MT theory
In a 1-D situation, conductivity is a function of depth only. We shall adopt
the cartesian coordinate represented by the axes x(east), y(north) and
z(vertically downwards), and assume that the conditions of a plane wave
normally incident on the earth are satisfied.
2.3.1 Uniform half-space
Consider a homogeneous half-space model in which the solid earth
conductivity, 6 occupies the half-space z > 0, and z < 0 is free space (fig. 2.2).
The current sheet induced by a time-varying magnetic field flows parallel to
the earth's surface along Ox.
Ef
Fig 2.2 Homogeneous half space model.
Maxwell's equations ,combine to give the wave equation
V 2E = u&aE/at + jica 2E/at 2 22
If we apply the vector Laplacian operator to the three rectangular components
of E and note that Ey = Ez = 0 and a/ax = a/ay = 0, we obtain the expression
3 2EX/aZ 2 = j.tiSaEx/at + j.xca 2Ex/at 2 2.3
The elementary solution of eq. 2.3 is:
Ex = Ae Iz + Be Yz
2.4a
The MT field is known to be quasi-periodic in the frequency range used and for
a harmonically time-varying field we have a general solution:
Ex = Aet + z) + Bet - z) 2.4b
where y = ± (iwj.i - ci.xw2 ) 112 is the wave propagation constant referred to as
the radian wave-length or wave number, and A and B are arbitrary constants
which are evaluated by applying the boundary conditions, i.e., the continuity of
the tangential electric and magnetic components across the interface.
However, at the frequencies of field variations considered in MT work,
resistivity is more dominant than the dielectric constant or the magnetic
permeability and y = (iwi16) 112 .
In the present problem, Ex vanishes as z -> so that
Ex = BeYZ 2.5
From Maxwell's equation (2.1a) we obtain
aEx/az = -iwiHy
Differentiating eq. 2.5 with respect to z and rearranging we obtain
29
Hy = (1/i w1 ) y Be _Yz 2.6
and
Z = Ex/Hy = (iwj.i/6) 1 "2 2.7a
where Z is the wave (or Cagniard) impedance of the homogeneous ground,
defined by the ratio of orthogonal components of E and H.
If however, we consider the magnetic field Hx along OX (fig. 2.2) instead of
the electric field and solve the equation
V 2H = iiSH/t + pc3 2H/3t 2
we obtain
Z = Ey/Hx 2.7b
From eq. 2.7 we can define Cagniard's apparent resistivity
p = i/wiZ 2 2.8
with a constant phase of 45 0 between E and H.
The propagation constant can be separated into real and imaginary parts
= (wj.i/2p)1/2 + i(wp/2p) 112 2.9a
(indicating an exponential decay of amplitude along the travel path and an
oscillatory nature of the wave amplitude), and can be used to define the skin
depth (the depth at which the amplitude of the field has been attenuated by
l/e of its value at the surface) given by
d = 1/real (y) = (2p/w1.1) 1h'2 2.9b
We therefore can calculate the penetration depth of EM waves in a
homogeneous earth for different resistivities using eq. 2.9b as shown in fig 2.3.
all
Nd
z 0 I-
LU z Ui
IL 0
I-.-
Ui a
I tO 100 PERIOD IN SECONDS
Fig. 2.3 Skin depth as a function of period and ground resistivity.
31
2.3.2 n-layered half-space
Let us now consider a model in which the earth is represented by n
horizontal layers where the resistivities of the layers are p 1 ,p 2,..., p,. respectively
and the thicknesses of the top n-i layers are h 1 ,h 2,...h_ 1 as shown in fig 2.4
jh 1 P1
I; 2.
64 P'1
Fig 2.4 n-layer earth
The ratio of orthogonal components of E and H yields the impedance in the
general medium
Z = -(iw.i/y)Coth(yz + In/A/B) 2.10
The constants A and B can be eliminated by considering boundary conditions
or by considering only the ratios of wave impedances at two different points.
If we utilize the ratio of wave impedances and evaluate the impedance Z 2 at
depth z 2 with reference to Z 1 at z 1 in the same medium and solve eq. 2.10 for
In/A/B we find that
Z2 = -( iw3.I/y)[Coth{y(z 2 - z 1 )-Coth 1 (Z 1 y/iwji)}1 2.11
Eq. 2.11 holds for z 1 and z 2 in the same medium. For a homogeneous
semi-infinite medium, with z 2 = 0, and (z 2 - z 1 ) = h 1 , the thickness of the top
layer, then Z2 at the ground surface (z = 0) becomes
Z2 (surface) = (iwi/y)Coth[yh 1 + Coth 1 CyZ 1 (z=h 1 )/iwii}1 2.12
In a layered earth, the continuity conditions which must hold at each boundary
permit us to express the wave impedance observed at the surface in terms of
the wave impedances at each of the lower layers. The general expression for
the impedance at the surface of an n-layered half-space is
Z. = (iw/yi)Coth[y1 h 1 + Coth 1 ((y 1 /y 2)Coth(y 2 h 2+Coth 1 ((y 2/y 3 )...
...+ Coth 1 ((y_2/y_ 1 )Coth(y_ 1 h_ 1 + Coth'(y_ 1 /y))) ... ))}] 2.13
Equation 2.13 is used in the 1-D MT forward problem (discussed in chapter 6)
to calculate the apparent resistivity and phase of a layered earth structure at
32
any given frequency.
2.4 Two- and Three-dimensional MT theory
When the ground conductivity varies in the horizontal direction and with
depth, 6(x,y,z), the relation between horizontal electric and magnetic field
components cannot be expressed by eq. 2.7 but by a tensor impedance to
accommodate lateral inhomogeneities and anisotropic effects (Cantwell, 1960):
that is, the scalar impedance relationship E= ZH becomes
Ex - Zxx Zx Hx
Ey IZVx Zyy IHVI
where Zxy, Zyx are the principal impedances and Zxx, Zyy are the subsidiary (or
additional) impedances due to contributions from parallel components of the
magnetic fields. In this situation, the E fields are dependent on both parallel
and orthogonal H fields and the impedance matrices vary as a function of
measurement position (with respect to geoelectric contrasts) and frequency. If
a structure extends over a great distance only in one direction, this direction is
called the strike. In the MT situation, the criterion for 2-dimensionality is that
extension along strike is great compared to the skin depth of the incident EM
fields.
2.4.1 Two-dimensional (2-D) structures
In a 2-D environment, conductivity is a function of depth and one horizontal
direction, ie. 6 = 6(y,z), as shown in figure 2.5.
2.14
z
Fig 2.5 Two-dimensional structure
Assuming that the source field assumption still holds and an invariance
with respect to the x-axis (fig 2.5), the total EM field splits into two
independent modes, E- and H-polarisations with the E and H fields respectively
polarised in the strike direction. The impedances due to E- and H-polarisations
33
are different (O'Brien and Morrison, 1967) depending on the frequency and the
location of measurements with respect to the resistivity discontinuity.
Starting from Maxwell's equations (2.1 a and b) and neglecting displacement
currents as before, we formulate the appropriate field equations in two
dimensions as follows:
(a) for the H-polarisation case (H=Hx(y,z) and E=(Ey,Ez))
(Ez/ay 3Ey/z) = -iwl.IHx 2.15a
SEz = Hx/3y 2.15b
6Ey = Hx/z 2.15c
and the Hx function satisfies the equation
div (1/6 grad Hx) - y 2/tSHx = 0
(b) for the E-polarisation case (E=Ex(y,z), H=(Hy,Hz)) we have
6Ex = (Hz/ay - aHy/az) 2.16a
aEx/y = iwjiHz 2.16b
3Ex/z = -iWiHy 2.16c
and the Ex function satisfies the equation
div (1/j.t gradEx) - y24iEx = 0
That is, equations 2.15 and 2.16 combine to give the Helmholtz equation of the
form
(3 2 F/3y2 + a 2 F/dz 2) - y 2F = 0 2.17
and satisfy the conditions of the continuity of F and 3F/3n at the relevant
boundaries, where F=Hx or Ex, and y = iwjiS, and n is the normal to the layer
boundary if we consider horizontally inhomogeneous media (Patra and Mallick,
(1980).
It is not always possible to evaluate the response of 2-D structures
analytically. Analytical solutions exist only for simple structures (e.g. Rankin,
1962; d'Erceville and Kunetz, 1962; Weaver, 1963; Hobbs, 1975). In most cases
numerical methods are used to calculate the responses of arbitrary 2-D
resistivity structures. There are four main methods in current use : the
transmission line analogy (Madden and Thompson, 1965; Wright 1969; Swift
1967,1971; Ku et al, 1973), the integral equation (Patra and Mallick, 1980), the
finite element (Coggon 1971; Silvester and Haslam, 1972; Reddy and Rankin,
1973; Wannamaker et al 1985, 1986) and the finite difference (Patrick and
34
Bostick 1969; Jones and Price 1970; Jones and Pascoe, 1971,1972; Williamson
et al 1974; Brewitt-Taylor and Weaver 1976; Weaver and Brewitt-Taylor, 1978)
methods. In all these methods the region to be modelled is divided into a
mesh of elements, with the edges set far enough from any lateral
discontinuities so that the boundary conditions are satisfied.
The finite difference method (which was used-this study) is centred on
solving the Helmholtz EM field equation (2.17) for both E- and H-polarizations.
The main procedure involves setting up a rectangular grid, representing the
conductivity distribution, with variable grid spacings to suit the requirements
near conductivity boundaries and near the surface of the model where the
normal derivative must be estimated for calculating the field components. The
equations for EM fields are represented by a set of finite difference equations
for each point on the grid. Derivatives are replaced with difference quotients
thus reducing the differential equations to a set of linear equations for the field
values at the grid points. Boundary conditions are also reduced to difference
conditions and added to the system of algebraic equations that are solved by
the Gauss-Seidel iterative method.
The finite difference scheme is conceptually the simplest and probably the
most widely used method. From its early days of fixed grid spacings and sharp
transitions in conductivity across boundaries it has developed into a greatly
improved (in terms of accuracy of solutions and ease of operation) modelling
tool. Williamson and others corrected the original formulation of Jones and
Pascoe to allow for variable grid spacings, Brewitt-Taylor and Weaver's smooth
transitions in conductivity helped overcome the early problem of fitting field
values across sharp conductivity boundaries and the work of Weaver and
Brewitt-Taylor led to improved boundary conditions.
It should be borne in mind that the various numerical techniques essentially
approximate otherwise continuous functions by values at discrete points within
a mesh of finite dimension and assume some (commonly linear) functional
relation for the field variations between the grid points. The calculated
responses are therefore prone to errors if the discretization is not good
enough and the solutions become unstable when the overall grid size becomes
too large. It is imperative therefore that attention be paid to the way and
manner in which a problem is discretized.
35
2.4.2 Three-dimensional (3-13) structures
In many MT problems, the conductivity distribution is actually 3-0. In a 3-0
structure, the conductivity is a function of all coordinates, ie. 6 = 6(x,y,z).
Analytical solutions to Maxwell's equations in 3-dimensions are difficult and
intractable and such problems at best are solved by the approximate
techniques. The most successful methods of solution include the differential
equation methods (eg. Jones and Pascoe, 1972; Lines and Jones, 1973 a,b;
Jones and Vozoff, 1978; Pridmore et al, 1981), the integral equation techniques
(Raiche, 1974; Weidelt 1975a; Ting and Hohman, 1981; Das and Verma, 1982;
Wannamaker et al, 1984), thin sheet approximations (Dawson and Weaver 1979;
Ranganayaki and Madden, 1980; Madden, 1980; Madden and Park 1982; Park et
al, 1983; Park 1985) and hybrid techniques (Lee et al, 1981). Analogue scale
modelling experiments have also been used to study 3-0 structures (eg. Rankin
et al, 1965; Dosso, 1966, 1973; Dosso et al, 1980; Nienaber et al, 1981).
2.5 Earth response functions
The MT data recorded on the ground surface provide valuable information
about the earth. Any function that can be derived from such a record is an
earth response function (Rokitvansky 1982). Such a function characterises the
conductivity structure of the earth and could be the impedance tensor , the
apparent resistivity or the phase. The conductivity strike and the
dimensionality characteristics of the earth are auxiliary parameters which
illuminate the nature of the conductivity structures.
2.5.1 Indicators of structure
Eq. 2.14 defines the relationship between the horizontal components of the
electric and magnetic fields in the frequency domain with the time variations of
the E fields given by the convolution of the earth response function Z a, Zb, Z,
Zd say, with the H fields. Put simply,
E = Z.H
2.18a
where the elements of Z are the Fourier transforms of Z a, Zb, Z and Zd. or
Ex = ZxxHx + ZxyHy Ey = ZyxHx + ZyyHy
2.18b
In a 1-D environment, satisfying Cagniard's (1953) tabularity conditions,1
Zxx=Zyy=OandZxy=-Zyx
so that the scalar impedance relationship (2.7) holds good. Cagniard (1953)
KV
defined an apparent resistivity (eq. 2.8) which in practical units is given by
Pa = 0.2TIZxy(yx)1 2 2.19a
and phase
= arg(Zxy(yx)) 2.19b
where T is the period in seconds, and E and H fields are measured in mV/Km
and gammas (or nanoteslas) respectively and Pa is in ohm-metres.
In a 2-D environment, Zxx and Zyy are non-zero and Zxy A -Zyx in general.
The elements of Z vary as the coordinate axes are rotated. When the
measurement axes are aligned with the structural strike Zxx = Zyy = 0, so that
Zxy = Ex/Hy = E 11 2.20a
and
Zyx = Ey/Hx = E 1 2.20b
However, strictly 2-D structures are rare in nature and MT measurements are
made in two orthogonal directions which may not be aligned with the
(unknown) strike.
Suppose our measurement axes (x,y) form an angle 0 (measured clockwise
from the x-axis) with the true strike, we want to determine our field
components in the (preferred) principal anisotropy axes (x,'y'). Let R be the
coordinate rotation matrix for a vector in the (x,y) plane given by
(CosO SinO " R =-Sin0 CosO) 2.21
On rotation of the tensor elements from (x,y) to (x,'y') by an angle 0 about Z
(fig 2.6), the rotated impedance tensor becomes
Z' = RZR 2.22
LI
37
x
z
Fig 2.6 Rotation of axes by angle 0
so that the impedance tensor relationship in the rotated frame is
E' = RZR 1 H 1 = Z'H'
2.23
Expanding eq. 2.22 we obtain (for the general case) the elements of Z' in terms
of the elements of Z, viz:
Z'xx(e) = (Zxx + Zyy)/2 - Zo(0 + 7r) 2.24a 4
Z'yy(0) = (Zyy + Zxx)/2 + Zo(0 + T) 2.24b 4
Z'xy(0) = (Zxy - Zyx)/2 + Zo(e) 2.24c
Z'yx(0) = (Zyx - Zxy)/2 + Zo(0) 2.24d
where
Zo = (Zxx + Zyx) Cos 20 - (Zxx - Zyy) Sin 20 2 2
If the real and in ginary parts of Zo are plotted on an Argand diagram for
varying 0, it is obvious that a 1-D structure will give a single point locus,
whereas a 2-D structure wil generate a straight line and a 3-D structure will
generate an ellipse (Sims, 1969).
In practise, on rotation through 180 degrees, two minima for the diagonal
elements are obtained and the corresponding axes are termed the principal
conductivity axes. The apparent resistivities and phases in these directions are
called major and minor apparent resistivities and phases. These are expressed
as
pmaj = 0.2T I Z'xy 2 4maj = arg(Z'xy) 2.25a
38
pmin = O.2T I Z'yx 2 4min = arg(Z'yx) 2.25b
it is now relatively common to define an effective response function for a
medium which is rotationally invariant (Tikhonov and Berdichevsky, 1966;
Ranganayaki, 1984). The effective impedance, which is related to the
determinant of the system of equations 2.14 is given by
Zeff = (ZxxZyy - ZxyZyx) 112 2.26
It has the physical sense of mean impedance for the medium. The
corresponding apparent resistivity and phase are given by
peff = 1/4wlZxxZyy - ZxyZyxl 2.27a
and
eff = phase of (ZxxZyy - ZxyZyx) 2.27b
In a strictly 2-0 environment, peff = and eff = xv + yx These two
quantities (peff, 'eff) are the geometric mean of parallel and perpendicular
resistivities and phases and have been interpreted by Ranganayaki (1984) as
scalar averages for the medium by analogy with mixtures, where the geometric
mean gives an accurate estimate of the physical properties (Madden, 1976).
2.5.2 Indicators of structural dimension
The earth is complex and in order to obtain a good picture of the
subsurface structural patterns, information from the parameters discussed in
section 2.5.1 are interpreted together with the constraints provided by the
dimensionality indicators. They point to the degree of anisotropy of the earth
and are extracted from the rotated impedance tensor Z'.
2.5.2.1 Electrical strike direction and Tipper
If a well defined conductivity strike can be obtained, then the structure is at
least 2-0. Two principal directions can be found (section 2.5.1) by determining
the angle e 0 and (0 + 90 degrees) at which Z'xy(e) is maximized and Z'yx(e) is
minimized. One of these directions is the strike for a 2-D structure. The
common practice is to maximize some suitable functions of Zxy and Zyx during
coordinate axes rotation. Everett and Hyndman (1967a) maximize IZ'xyl; and
Swift (1967) maximizes I Z'xy 2 + Z'yx 2 analytically using the formula
0 0 = 1/4 arctan (Zxx - Zyy)(Zxy + Zyx)* + (Zxx - Zyy*(Zxy + Zyx) 2.28 Zxx - Zyy I - I Zxy + Zyx I
where * denotes complex conjugate.
39
I Ideally,: Z'xx and Z'yy should be zero at this position but they rarely vanish.
However, Jones and Vozoff (1978) have indicated that even for a 3-D variation,
eq. 2.28 will yield a general strike direction.
Generally, the observed vertical magnetic field H is used to help determine
which of the 2 principal directions is the strike direction. In the 2-D case, the
horizontal direction in which the magnetic field is most highly coherent with H
will be perpendicular to the electrical strike direction (Vozoff, 1972).
The relationship between the measured H and the horizontal components
at a single site can be expressed (Everett and Hyndman 1967b, Madden and
Swift, 1969) in a simple form
H = AHx + BHy 2.29
where A and B are the geomagnetic single station transfer functions calculated
from cross-spectral estimates. These functions can be visualized as operating
on the horizontal magnetic field and tipping part of it into the vertical. Using
the coefficients A and B, Vozoff (1972) defined a quantity, tipper or vertical field
transfer function given by
T = /(A2 B2 ) 2.30
where A and B are given by
A = <HzH><HyHy> - <HzH><HyH> <HxHx><HyHy> - <HxHy><HyH>
B = <HzH><HxH> - <HzH><HxH> <HVHV><HxHx> - <HyH><HxHy>
and * denotes complex conjugate iiiW<>anverage over a ftquencvband.'
In a 1-D structure the vertical transfer function is zero. A is zero for an
x-trending 2-D structure and T can thus be used to assess the 2-D character
of the MT data. The differences in the strike directions determined by
impedance tensor rotation and the vertical-horizontal magnetic field
relationships are thought to be a measure of 3-dimensionality (Vozoff, 1972).
2.5.2.2 Impedance Skew
The impedance skew (Swift, 1967) is the amplitude of the' ratio of
off-diagonal to diagonal elements of the impedance tensor and is expressed as
Skew = Zxx + Zyy 2.31 Zxy - Zyx
It is a measure of the EM coupling between the measured E and H fields along
coincident directions (Vozoff, 1972). There will be no coupling in a 1-D
all
situation or if our measurement axes coincide with the principal axes of a 2-D
structure or if our measurement site is located over a point of radial symmetry
in a 3-0 structure (Word et al, 1970; Kaufman and Keller 1981). It is
rotationally invariant; the parameter combinations (Zxx + Zyy) and (Zxy - Zyx)
are independent of e and can be used to define two invariant quantities
Ii = Zxx + Zyy 2.32a
12 = (Zxy - Zyx)/2 2.32b
(see Berdichevsky and Dmitriev, 1976) which have found use in 1-D MT
modelling studies. Skew is zero for ideal 1-D and 2-D cases and is non-zero
for general 3-D structures. However, in practical field situations, skew is
non-zero and there is no concensus as to what value it should take in a 2- or
3-D structure (see Kao and Orr, 1982). However, from 3-D modelling studies
Reddy et al (1977a), Ting and Hohman (1981), and Park et al (1983) suggested
upper limits of 0.4, 0.12 and 0.5 respectively for the onset of 3-D behaviour. A
value of < 0.4 is generally used in routine studies as indicative of
2-dimensionality.
2.5.2.3 Ellipticity and eccentricity
Ellipticity (Word et al 1970) is a 3-0 parameter related to the principal radii
of the Z rotational ellipse, defined as
= IZxx(0 0) - Zyy(00)l/lZxy(e0) + Zyx(e 0)I 2.33a
B is zero for 1-D and 2-D cases. It is used as a semi-quantitative measure of
3-dimensionality of the structure and the degree of coupling between individual
3-0 features (Word et al, 1970).
Eccentricity is an indicator of a 3-0 structure. Word et al, (1970) defined
eccentricity of the rotation ellipse as
8(0) = (Zxx - Zyy)/(Zxy + Zyx) 2.33b
It is dependent on the rotation angle and is zero for a 2-D structure when
evaluated at the strike direction. Thus 8(e 0) = 8.
2.6 Estimation of the impedance matrix
The estimation of the impedance tensor elements is usually done in the
frequency domain. Sims and Bostick (1969) and Sims et al (1971) have
discussed a classical least squares spectral analysis procedure for optimizing
the estimates of the tensor elements from a large number of independent
record sets.
41
The natural MT signals are generally contaminated with noise which add
onto the spectral estimates of the complex amplitudes of the E and H field
variations at a given frequency degrading the relationships
Ex = ZxxHx + ZxyHy
Ey = ZyxHx + ZyyHy
The method of Sims and Bostick (1969) and Sims et al (1971) uses the
complex conjugate of each component and minimizes the noise power on the
magnetic or electric channels to yield four cross-spectral equations (for each
of the two matrix equations). Any two of the four equations (6 possible
combinations) may be solved simultaneously for the relevant tensor element so
that six distinct equations emerge for each of the tensor elements. For
example, six different estimates of the element Zxy may be computed using the
equations
<Zxy> = <HxE><ExE> - <HxE><ExE> 2.34a <HxEx><HyEy> - <HxE><HyEx>
<Zxy> = <HxE><ExH> - <HxH><ExE> 2.34b <HxEx><HyHx> - <HxH><HyE>
<Zxy> = ExH i. - <HxH><ExE> 2.34c <HxEx><HyHy> - <HxH><HyEx>
<Zxy> = <HxE><ExH> - <HxH><ExE> 2.34d <HxEy><HyHx> - <HxH><HyE'>
<Zxy> = <HxE*y><ExH> - <HxH><ExEç,> 2.34e <HxEy><HyH> - <HxH><HyEr>
<Zxy> = <HxH><ExH> - <HxH'><ExH> 2.34f <HxH><HyH> - <HxH><HyH>
where * denotes complex conjugate and <Zxy> is one measured estimate of
Zxy.
The Zij are usually assumed to be slowly varying functions of frequency
(although this may not always be true as in the case near vertical conductivity
boundaries) so that <AB> may be regarded as averages over some finite
bandwidth or the cross-power spectrum between A i and B at some centre
frequency. Two of the equations (2.34c and d) are relatively unstable in the
1-D case where the fields are unpolarised since <ExEi>, <ExH>, <EyHç'>
and <HxH> tend to zero. The other four equations are stable in the absence
of highly polarised incident fields (Sims et al, 1971). Eqs. 2.34e and f are
42
biased downward by random noise on H(and not by noise on E) and eqs. 2.34a
and b give estimates that are biased up by random noise on E (and not by / '. C.., noise ,On H). Eq. 2.34f is most commonly used in MT data analysis where it is
assumed that the H field is less contaminated with noise than the E field. The
remote reference technique (Gamble et al., 1979) uses eq. 2.34f but with H' and
Hx obtained from a remote station with (assumed) independent local noise
contributions.
2.7 Coherence
Let S(t) and X(t) be two time series and S(f) and X(f) be their respective
Fourier transforms. The coherence between the two signals is given by
C = <S(f) X(f)><S(f) X*(f)>* 2.35 <S(f) S(f)><X(f) X'(f)>
where 0 K C1 for all frequencies (Bendat and Piersol, 1971). C will beSX
unity if the two signals are perfectly correlated and will be zero if they are
uncorrelated.
Real signals contain additive noise. Now, let us consider S(t) as a linear
combination of two signals U(t) and V(t) so that
S(f) = Z(f)U(f) + Z(f)V(f)
2.36
with an expected value (f) given by
(f) = <Z(f)> U(f) + <Z(f)> V(f) 2.37
where <Z n > and <Z> are averaged values of the transfer functions in
equation 2.36.
The coherence between a signal S(f) and its predicted value (f) is known
as the Predicted Coherence defined as
c= I <z>P + < ZV>PSVI2/PSSI <Z> I 2 P UU + I <Z,> f 2P + 2Re(<Z><Z>P)}
where PAA = <A(f) A*(f)> and PAB = < A*(f) B(f)>.
Equation 2.36 is very similar to the MT matrix equations 2.18b. If we
substitute the values of the impedence estimates (section 2.6) into eq. 2.18b
we obtain the predicted values of the two electric fields Ex and Ey. The
coherences between the pairs (Ex, x) and (Ey, Ey) are known as their
predictabilities (Swift, 1967) and for Ex, say is expressed as
Coh(Ex,Ex) = I <ExE> I/[<ExE><xE>] 112 2.39
In MT work, the predicted coherence is preferred to the the ordinary coherence.
43
CHAPTER 3 FIELD STUDIES
A description of the audiofrequency magnetotelluric (AMT) data acquisition
system, its and field installation are now given, followed by
information about the survey sites.
3.1 Field equipment specification
The most remarkable practical interest of the MT method comes from the
fact that the depth of penetration is linked to the measurement frequencies,
and if use is made of a sufficiently large range of frequencies one can explore
electrical crystalline basement or sedimentary subsoil of great thickness. The
aim of this project is to obtain depth and resistivity distributions across the
Great Glen fault. The depth of interest is from about lOOm to 30-60Km.
Previous studies in the region have shown that upper crustal resistivities are in
the range 1000 -> 10000 Ohm-m and that the lower crust/upper mantle is
characterised by resistivities in the range 50-500 Ohm-m.
Using the above information, the potential maximum and minimum
measurement frequencies may be estimated through the skin depth equation
(2.9b). Selecting conservative resistivity values of 100-3000 Ohm-rn as
possible crustal averages for the region and a frequency range 600-0.01 Hertz,
skin depths are approximately 204m-1.1Km at 600Hz and 50Km-274Km at
0.01Hz. However, since the skin depth is practically several times greater than
the actual maximum depth from which information might be derived and
considering an overestimation by a factor of about 5 (Vozoff et al, 1982), we
expect information from about 40-220m to 10-55Km. It was thus considered
satisfactory to employ the range 0.01-600Hz in this study.
The nature of the MT signals was also considered when specifying the type
of equipment to be used. The main characteristics of the signa's are:
a practically continuous spectrum of frequency; only signals with
frequencies of interest need be retained, and
an amplitude that is frequency dependent and also varies as a
function of time (or season). Values less than 10 mV/Km (electric
signal) and a few milligamma (magnetic signal) are not uncommon
and we therefore need to be able to detect (with precision) and
amplify the signals.
Moreover, the high frequencies of interest dictate that the signals be
44
received and stored at a suitable rate. For all these reasons, the desired MT
measurement system must possess an excellent signal-to-noise ratio, strong
amplification, ability to process digitally and record data for a broad range of
frequencies.
3.2 Field instrumentation
The basic MT, instrumentation consists of the signal detecting system
(magnetic and electric sensors) and the data recording equipment.
3.2.1 Magnetic and electric field sensors
The magnetic sensors used in this study consisted of two sets (CM 16 and
CM11E) of three induction coils (sensitivity 50 mV/y) supplied by Societie ECA
- France (the sensors were developed by Dr G.Clerc, Garchy, France). There is
a flat response between 3 and 800 Hz for the CM16 coils and 0.012 and 100 Hz
for the CM11E coils. Each coil consists of turns of fine copper wire wound on
a core of high magnetic permeability, and a preamplifier with low-noise
connections and components, all enclosed in a waterproof casing. The three
orthogonal magnetic fields were measured with separate coils. For the electric
field measurements, polarizable electrodes (copper rods) were mostly used.
The copper electrodes were about 30cm long and 2cm in diameter; they were
tapered at one end for easy insertion into the ground and each had a (clip)
cable-connection at the other end. Petiau and Dupis (1980) have shown that
non-polarizable electrodes have the lowest noise characteristics and are
superior to the polarizable types at frequencies below 10Hz but that both types
produce similar results at frequencies above 10Hz. However, in this study
"Dupis" dyke electrodes were used only at a few sites for measurement below
10Hz. Each of the "Dupis" electrodes consisted of a lead tube in lead chloride
solution housed in a plastic tube with a porous base. The dimensions are
about the same as for the copper rods. The electric field in the earth was
measured with 2 pairs of electrodes placed about 50-100m apart.
3.2.2 Data recording equipment
The Short Period Automatic Magnetotelluric (S.P.A.M.) MKll real-time data
acquisition and analysis system developed at Edinburgh University by Dawes
(1984) was used to record data digitally on cartridge tapes over the frequency
range 0.016-640Hz. The S.P.A.M. MKll is a portable battery operated, digital
7-channel system and incorporates full tensorial analyses, a remote reference
facility and one-dimensional inversion to resistivity as a function of depth. A
detailed description of the S.P.A.M. MKII system can be found in Hutton et al,
45
(1984) and only the essential features shall be recounted here. A block
diagram of the complete in-field system is given in fig 3.1.
Two versions of S.P.A.M. were used in this study. In one version (belonging
to Edinburgh University) the frequency sounding bandwidth was from 640 to
0.09Hz, while the other version (on loan from N.E.R.C.) the frequency bandwidth
was 128-0.016Hz. The total S.P.A.M. frequency range is divided into 4
overlapping bands to prevent the risk of large amplitude signals saturating any
of the channels or of signals of weak amplitude being obliterated by the
background noise. The gain can therefore be optimized. The main units of the
system are (i) a microcomputer (LSI 11 with a 64 Kilobyte RAM), a
programmable amplifier/filter bank and a power regulator; (ii) twin cartridge
(programs and data) storage deck; (iii) alphanumeric pocket VDU and keyboard;
40 column miniature electrosensitive printer for listing and plotting results;
sensor distribution box and (vi) a power distribution box.
The field programs are read off the program tape by the computer and any
useful results written onto the other tape. These 4-track TU58 tapes can each
hold sufficient data for all the 4 bands recorded at each field location. The
main program facilities are : software band selection; instrument response
corrections; automatic gain adjustment and signal selection, with manual
override; alteration of signal selection criteria, and continous or switchable
display of results (which can be listed on the printer or stored on tape).
The sensor distribution box contains the telluric preamplifiers and
connections to the induction coils. The signal from each electrode pair is
connected to a preamplifier with a gain of 40. The signals from the induction
coils are brought to the sensor distribution box by 3m long connecting cables.
A 50m individually shielded multiconductor cable powers the box (the telluric
preamplifiers and the coil amplifiers in the coil housings) and takes the electric
and magnetic signals back to the amplifier/filter bank. The maximum allowable
(but changeable) input range of the analog to digital converter (ADC) is ±5V
and all incoming signals are tested to see that they did not exceed this range.
All the 7 channels have switchable 50 and 150Hz notch filters to eliminate
noise from power transmission lines. Each amplifier/filter unit has automatic
(or manual) gain adjustment with hardware band selection. From the
amplifier/filter bank the signals are sent to the multiplexer - ADC unit where
the channels are sequentially sampled and digitized. The sampling frequency is
different for each frequency band. Each data set is called an 'event' or
'window' and consists of 256 samples for each of the 7 channels.
EN
tell I in€
corn.
/-1
0
U
-C ..
Power
N Signal
14
Battery
1charger Test
Scope
A
Programmable
Anipif lent liter
Bank
7 channels 4 bands
L
Test Signal
Generator
Fig 3.1 Block diagram of , complete in-field MT System (after Dawes, 1984)
START
Initialize
Digitize I
Tim.
ly
Domain
AnasIi
Nt Fr,qu.ny Do.nii Anolysit.
Lwrit.ta to Tap.
J<;it)
Includ. Data
In R.sultt
stop
r ist. Plot 4 FII.]
R.guIt
(_S TOP '\
Fig 3.2 Simplified flowchart of the in-field system (after Dawes, 1984)
48
The data window is transferred in real-time into computer memory for infield
analysis and is stored if it satisfies certain preset conditions. The computer
then initiates the acquisition of the next window. Depending on the quality of
the data, about 24-30 data windows are considered adequate for a particular
frequency band. A simplified infield flowchart is shown in fig 3.2
Infield analysis enables a very quick appraisal of the data being collected
and basically involves converting the data into apparent resistivities, phases
and other vital information as functions of frequency. Essentially, by the Fast
Fourier transformation of the data, the intensity of each signal is obtained over
a range of frequencies and then corrected for the instrument response to yield
the earth response functions desired. Data analysis will be discussed in detail
in chapter 4.
3.2.3 Equipment Calibration
The MT instruments described above required calibration before and after
each field campaign to obtain their frequency response characteristics for
accurate determination of the earth response functions. Facilities for precise
calibration of the induction coils are lacking at Edinburgh University but this
task was undertaken at Centre de Recherches Geophysiques (C.R.G.) in Garchy,
France. The calibration data from Garchy were used in the present study and
are plotted in fig 3.3a.
The telluric pre-amplifiers and the S.P.A.M. filter/amplifier units were
calibrated by the author in Edinburgh using a frequency analyser. This analyser
has an internal signal generator and outputs the instrument response functions
(amplitude and phase) for a range of frequencies. The telluric pre-amplifiers
frequency response curves are shown in fig. 3.3b. The S.P.A.M. amplifier/filter
response curves for one of the 7 identical channels are shown in fig 3.3c for
the 4 overlapping frequency bands.
3.3 Field procedure and practical considerations
3.3.1 Site Selection
Ideally MT soundings are conducted in open flat-lying areas. Since the
region of study is mountainous with very difficult terrain conditions,
measurements were restricted to the more accessible wooded forests and
farmlands inbetween impressive topographic features. The open ground except
on farmlands is mostly boggy (or water logged). However, care was taken to
locate the sites as far away as possible from topographic features. A chosen
site would preferably be an open or clear area for 60-100 metres square,
49
Fig 3.3. Instrument response curves.
a. Calihiation'curves for the induction coils
b Ca1ibratiôn curves for the Edinburgh University and NERC telluric
preamplifiers.
c. The S.P.A.M. amplifier/filter response curves of one of the seven identical
channels for four overlapping frequency bands. The frequency bands for the
Edinburgh University and NERC S.P.A.M. systems are respectively 640-66 Hz
and 128-16Hz for band 0; 66-8 Hz and 16-2 Hz for band 1; 8-1 Hz and
2-0.25Hz for band 2; and 1-0.09 Hz and 0.25-0.016Hz for band 3.
50
1.00
0.75
I
t rmw
CHIS •0 _____ _____ _____ _____
25
as
0.00
1
-J a a
5 ,
0.
1009.000 100.000
90
95
0
-95
99
us
o a
-'15
-90 10.556 1.060 0.106 0.510 0.099
FRLOUUCV IN HZ
hi 0
a a iI]:
0005.568 160.005 19.506 1.560
0.106 0.015 0.051 FIILOUCIICY IN HZ
Cl
51
NERC—TELL 90- ---
( LI-i CD
7-1 0
LI-i (1) cr 0
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FREQUENCy IN HZ
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cr- to
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a
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51
ts
-3
-5
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- -2
Q. €
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-35
BRND 3 BRNO 2 BRND 1
Fr'oqugnoy in Oot•ave8
away from trees, steep ground, fences and telephone lines, and with the
nearest power transmission line at about 400m to 1Km distance. At some
locations however, measurements were made in forest clearings of about 50
square metres.
3.3.2 Field set-up
The sequence of operations in setting up an MT site is described below.
(1) The centre for the sensor systems is located. The sensor box is left at this
position. A prismatic compass is used to lay out 25-50m electric cables in
four orthogonal directions (preferably magnetic north, south, west and east).
The electrodes are planted in the ground at the end of these lines and the
cables connected with the clips at the top. The contact resistance between an
electrode pair (N-S, or E-W) is measured. A typical resistance is about 2-5kft
Values greater than 100 do not give good results and if necessary the
electrodes are checked to see that they are firmly planted in the ground and
that the cables are properly connected. (2) Shallow trenches are dug for the
two horizontal induction coils (since they are very sensitive to wind vibrations)
preferably in the E-W and N-S directions. A deep hole is dug with the aid of
an auger for the vertical coil. The horizontal coils are aligned and all are
levelled with a spirit level after ensuring that they are adequately supported.
As a precaution, the same coil is used for the same magnetic element
throughout the field campaign. The coils are connected to the sensor box with
3m long cables. Care must be taken that the cables do not vibrate in the wind
and so clods of dirt or pieces of rock are placed along their lengths. (3) the
- multiconductor cable is connected to the sensor box at one end and to the
S.P.A.M. system (placed in the back of a Landrover) at the other end about
50-100m away. (4) The whole system is earthed and then tested (initial system
CalibrTion) before the measurements commence.
Only a crew of 2 is needed to install the various systems and typical set-up
time is 30-40 minutes. Measurements can be made at one or two sites in a
day depending on the signal activity and required data quality. In this study
however, each site was occupied for at least one day.
3.4 Survey details
In two fieldwork campaigns in 1985, MT measurements in the frequency
range 0.09-640Hz were undertaken at 22 sites by the author. In 1986, together
with D.Galanopoulos and Drs K. Whaler and C. Stuart I of Leeds University,
additional soundings were performed at 12 sites in the frequency range
53
N
MT profile
± MT site 1986
. MT site 1985
;
Fig 3.4. Map showing the locations of the MT observational sites.
54
0.016-128Hz.
The MT surveys consisted of 7 tensor soundings along the Glen Garry (GG')
profile, 6 along the Loch Arkaig (AA') profile and 19 soundings along the Glen
Loy (LL') traverse. The observational sites are shown in fig 3.4. The average
station spacing was about 2Km (1Km along LL' in 1986) which was considered
adequate for accurate spatial resolution of possible short wavelength
anomalies. The S.P.A.M. MKII 7-channel system was used with ECA induction
coils for all the soundings. Remote reference techniques (Gamble et al, 1979)
were not used in this study. Only data which satisfy certain preset conditions
(predicted coherencies, signal-to-noise ratio and frequency content) were
accepted. The minimum coherency criterion used was 0.8. The accepted data
sets were initially analysed in real-time using standard impedance tensor
techniques and the raw data and computed result output to a cartridge
recorder and to a small Brother HP printer enabling immediate scrutiny of the
results. Thus the data quality was controlled, and the success of the sounding
assessed before leaving any field location; it was also possible to ascertain if
the station spacing was satisfactory or whether a greater station density was
required.
The Glen Carry traverse runs approximately •east-west with the extreme
sites at about 20Km and 1Km to the west and east of the GGF respectively.
The Loch Arkaig profile extends westwards from the GGF for about 20Km and
all but one of the sites lie within 10Km distance from the fault and on the
north-side of Loch Arkaig. The Glen Loy traverse runs across the fault in an
approximately northwest-southeast direction. It presented the only easy access
to the region east of the fault and together with the westernmost AA' site 607
(projected) forms the most extensive (38Km long) and necessarily
representative traverse across the Great Glen study region. It has the closest
network of stations (see fig 3.4) around the fault. The British national grid
references of the sites are given in table 3.1 (sites with less than two band
recordings neglected).
55
Site Number Profile Code Grid Reference
553 Gi NN 117 998 574 G2 NN 173 986 550 G3 NH 203 006 551 G4 NH 231 002 552 G5 NH 281 009 555 G6 NH 337 051 556 G7 NN 318 976 573 Al NN 122 913 572 A2 NN 150 896 571 A3 NN 174 887 570 A4 NN 194 887 557 A5 NN 182 901 607 Li NM 988 915 560 L2 NN 089 849 606 L3 NN 097 847 605 L4 NN 107 845 603 L5 NN 111 840 601 L6 NN 122 828 559 L7 NN 133 833 602 L8 NN 136 823 558A L9A NN 146 827 558B 1-913 NN 146 825 561 L10 NN 160 831 609 Lii NN 141 797 562 L12 NN 165 821 611 L13 NN 155 789 563 L14 NN 173 804 565 L15 NN 222 823 608 L16 NN 233 798 604 L17 NN 253 795 566 L18 NN 285 821 610 L19 NN 350 797
Table 3.1. Site specifications
iM
CHAPTER 4 DATA ANALYSIS AND RESULTS
This chapter describes the procedure adopted in processing the MT field
records to yield the response functions needed for qualitative and quantitative
(computer modelling and inversion) studies. For many purposes the infield
standard automatic data processing is sufficient. However, at the base
laboratory in Edinburgh the various "raw" data sets were reprocessed on the
mainframe computer (EMAS) using additional or different data selection criteria
so as to obtain more reliable estimates. Inspection of these results showed
that certain sections of the data were contaminated with noise. The
impedance tensor elements were then averaged (Sims et al, 1971) to minimize
bias effects and digital filtering techniques were applied to enhance further the
quality of the data.
4.1 Data transfer
On return to Edinburgh, the "raw" field data that were initially stored on
cartridge tapes (TU58) were transferred onto floppy discs and then to a 1/2"
magnetic tape. The data were finally transferred from the magnetic tape onto
the local network for a very rigorous analysis.
4.2 Data Processing
Basically, MT data processing is the conversion of the recorded time series
for the 5 field components to earth response functions. Hermance (1973) has
reviewed the different methods for MT data analysis and details of the
techniques are described in Vozoff (1972), Sims and Bostick (1969) and Sims et
al (1971). The various formulations will not be reproduced here. The classical,
and by now standard, tensor impedance analysis techniques (see section 2.6)
were used in this study.
4.2.1 Standard analysis
4.2.1.1 General window analysis
Time series analysis was done using the data analysis program originally
written and described by Rooney (1976) and re-written for computational
efficiency by G.Dawes. Analysis was done in the frequency domain using
equation 2.34f. The following operations are executed for each of the windows
of the 4 frequency bands by the program.
(i) The time series are first scaled using the recording gains. The field
57
components are optionally rotated into desired coordinates. Instrument
correction tables are then set up.
The data are reduced to zero trend and normalized to unit standard
deviation (Bendat and Piersol, 1971) using the least squares method.
A cosine bell window (Harris, 1978) taper is applied to the first and last
10% of the data set.
The data set is augmented with zeros to a power of 2 to satisfy the
requirement of the Fast Fourier transform (FFT) algorithm (Cooley and Tukey,
1965). The complex Fourier coefficients of each component are then calculated
using a general FFT routine.
The Fourier transform coefficients are corrected for the instruments
(telluric preamplifier and CM16 & CM11E coils) responses.
Auto- and cross-spectral estimates are then calculated and
band-averaged over 8 neighbouring frequencies so as to reduce the variance of
the estimates.
The initial results of the analysis are then calculated using the
smoothed spectral estimates. These are the impedance tensor elements using
eq. 2-34f, the Cagniard resistivities and phases (i.e. in the measurement
directions), polarisation, predicted Hz, Ex and Ey coherencies, Hz transfer
functions (tipper coefficients); these are saved in a file.
These procedures are executed for each recorded window in turn.
4.2.1.2 Averaging of window results
A comparison was made (Sule, 1985) of the three conventional methods of
averaging the individual data windows (auto- and cross-spectral averaging
assuming a normal distribution, impedance tensor averaging assuming a
lognormal distribution and apparent resistivity averaging assuming a lognormal
distribution) and it was found that impedance tensor averaging is practically
the most convenient. The impedance tensor averaging technique was thus
adopted in this study.
The unrotated tensor elements obtained from general window analysis are
subjected to further statistical analysis and winnowing leaving only reliable
estimates for averaging - this minimizes the variance of the resultant earth
response functions.
The averaging program of Rooney (1976) was adapted for short period MT
data by G. Dawes. Only estimates with minimum predicted coherencies of 0.8
are accepted subject to other conditions. For the pairs Zxx, Zxy and Zyx, Zyy
and the tipper coefficients, the predicted Ex, Ey and Hz coherencies are used
58
respectively. Those with small signal to noise ratio are rejected and it is also
required that each frequency set possesses a preferred sign of the off-diagonal
elements of the impedance tensor. Outliers (all estimates lying outside ±2.2
standard deviations from the mean value) are rejected. A new mean and
standard deviation are then calculated and the procedure repeated. The
impedance tensor elements derived from windows which satisfy the coherency,
power, sign and outlier rejection criteria are then averaged over all the 4
frequency bands assuming a lognormal distribution (Bentley, 1973) and the
tipper coefficients averaged assuming a norma! distribution. These are then
subdivided to give equispaced estimates on a logarithmic scale (usually 8-10
per frequency decade/band).
Several parameters are then computed (as functions of frequency) from
these mean estimates. They include: the apparent resistivities and phases
(unrotated and rotated), the azimuth of the principal impedance axis, and the
impedance skew values. In this study, apparent resistivities and phases were
also obtained with axes parallel and perpendicular to N49E (magnetic), the
direction of the GGF, thus providing the E- and H-polarization field response
curves required for 2-0 modelling. The effective MT responses for 1-D
modelling were also obtained. Vertical magnetic field information is both
scanty and noisy and was deleted from the interpretation process.
The data from 32 stations were analysed using the methods discussed
above. Some examples of the apparent resistivity and phase results from
standard data analysis are shown in figure 4.1. The error bars assigned to the
parameter estimates are ±1 standard deviation about the mean assuming a
normal distribution (lognormal for those on log scale) (Bentley, 1973). In this
figure are shown the apparent resistivity and phase information obtained at
various frequencies. Note that there is no overlap between the first segment
(band 0) and the rest of the curve. This discrepancy can only be explained in
terms of noise corruption of the data. This feature was found to be common
to the high frequency (band 0) data at some sites and at others there is a large
scatter in the values of the responses at high frequencies. The lower
frequency data at some sites were also degraded in quality by noise
contamination (see fig 4.1). This seems to suggest that the data window
acceptance criteria used in routine analVsis are not effective in eliminating
noisy records since any coherent noise with good power characteristics would
be accepted as a good signal by the program. A different way of noise
detection and elimination seems imperative.
59
a a a 8
z a a
3-
a I- 0
hi
I- Z a (U a
cr CL a-
0
0 a
a I.
a, (Ii hi
3 0 Iii (3
w C) Cr = a a. -
a
21 I I 1 III I I
r
11
a
.1666.66 160.56 15.66 1.66 0.10 0.61
- FRFQIIFNCY TN H?
V
0 0 U
us
r
a, hi
(U a C) cc = I a. ,
C P01.
EL I ti ._.•_•
cr01.
pow
II
1550.60 160.06 10.50 1.06 6.16 6.61 FR0IIFwr.y TN H
C P01.
F I I =8
3- I-
I .0 I 1• U
a, (U '- z 0 -
a-
U a a
iu a an (U 0 .1 I tii a Ij I
'EjJflJJflhI'1 t 8 I
I a
ft
I
MTTT
B
Till
a I-
Fig 4.1 Some results of conventional data processing
4.2.2 Bias analysis
To assess the bias effects on the response curves it was found necessary
to reprocess all the data sets using the 6 different methods of estimating the
impedance tensor elements (i.e. eqs. 2.34 a-f). The effect of noise on the data
will be to elevate or decrease the apparent resistivity estimates. If the noise is
mainly associated with the telluric measurements the impedance estimator
2.34a would produce the maximum values for the apparent rèiiiVii!On the
other hand if the noise is in the magnetic channels then eq. 2.34f would
produce curves that are biased downwards. The range between these
extremes is taken to characterise the bias in the data for each site. For each
site the 6 different response curves were inspected for bias effects. While the
curves derived with eqs. 2.34a,b were generally shifted upward and those from
eqs. 2.34d, f, shifted downward with respect to the others, the bias range was
small (less than half a decade) especially at high frequencies as shown by the
example from one site in fig 4.2.
It was envisaged that an approach to the noise problem would be to
process the "raw" data using an acceptance criterion that is based on the
scatter between the 4 different stable equations (2.34 a,b,e and f), the stability
criterion of Sims et al (1971) defined as
[lZ(a)l . lZ(b)l] 4.1 S = ------------------
[IZ(e)l . IZ(f)l]
where Z(a), Z(b) etc refer to the estimates from eqs. 2.34a,b etc. S will be unity
if the 4 estimates are identical and will increase as the estimates diverge. A
value of O?< <1.3 was used but it did not improve the mismatch between
the high and.moderate frequency segments of the response curves. This time
consuming approach was therefore abandoned. However, to provide unbiased
response curves (i.e. with minimized frequency independent shifts), the 4 stable
solutions were averaged (Tikhonov and Berdichevsky, 1966; Sims et al, 1971)
and only these mean response functions have been used in subsequent studies.
The frequency-dependent noise problem still remains.
4.2.3 Cultural noise analysis
It is by now obvious that the automatic standard or bias analysis
procedures are not of much help in the presence of certain types of noise. It
was therefore decided that each data window of a given set be investigated
and analysed as appropriate.
61
C C C C C
C a C
3.- I-
3. ..4 C I- C U)
a, ma dc
I- C
Uj
cc a. a-
C
N,
Ui C Sd
(3 N, Ui C a Ui
N, a.-.
C
C C C
x a 3.- 5- I- 3.
I i- Ui Ca Ui 07 cc
N, 0_S.
C C C C C
— C I— C
C
a, Ui
La
cc cL
I-.
a-
C
N, I-
a) Ui C Ui cc (3N, Ui (3
cr N,
0.
U
C C C C U
gg
E NOISE - FREE Ia) mR
Sen
Ir
I • .-.•-••
______________ •: : _____________
[1
!
MINOR
3.
A 0) Ui
I
I- C
Sd C
cc a-a. cc
C
Sd Ia (3 Ia a
Cr
CLN, -
C 1655.65 165.55 16.66 1.56 5.16 5.01
FRFOISFIIf.Y TN H
H NOISE-FREE WIflR
S.
I I •
P
I
1 [1 •' -
C i.ce 100.00 1I.e6 1.15 5.15
FRFQPIFP1IY IN H
Fig 4.2 Example of bias in response function estimates assuming noise-free
E or H channels.
-
62
4.2.3.1 Noise types found in the MT records
Examination of some data windows revealed the presence of two main
types of noise:
(I) impulsive signals which may be correlated or uncorrelated in the
corresponding components. Such noise affecting the telluric components may
be caused by electric sources (leakage currents, electric fences, etc) and in the
magnetic channels such noise may be due to movement of vehicles or heavy
machines (e.g. agricultural and mining). It should be remarked that extensive
quarrying activities and road construction work were going on at the Glen
Garry Forest at the time of the 1985 investigations and that the worst noise
was observed in the 1985 data sets.
(ii) Strong coherent near-stationary perturbations in all components. The
commonest of these signals are 50Hz and harmonics affecting the bands 0 and
1 data. The source of these is the electric mains. Other less stable
perturbations (5-6Hz, 35-36Hz and 106-110Hz) are also present in some of the
1985 data and may have been caused by motors working at or close to 300,
2100 or 6480 rpm respectively. It was thus obvious that the infield (S.P.A.M.'s)
50Hz and 150Hz analog rejection (notch) filters did not eliminate completely the
electric mains noise at some locations.
4.2.3.2 Digital filtering
The remarkable time-invariance of some of the high frequency noise
suggested that a simple comb (delay-line) or notch filter could be used to
remove them. The impulsive noise (spikes) could also be eliminated using a
suitable de-spiking filter. These filters are routinely used in geophysical data
analysis and the reader is referred to the excellent text on this matter by
Kanasewich (1981).
Using the Z transform method (i.e. letting eW=Z) we can design a delay line
filter to remove unwanted near-stationary signals. The Z transform of the
impulse response of a simple subtractive delay-line filter is given by
W(z) = 1 - Zn
4.2
where n is the number of digital time units by which an input signal is delayed.
If in the z plane the input data X(z) is convolved with the impulse response
of the filter W(z) we obtain an ouptut whose z transform is given by
Y(z) = W(z) . X(z)
63
or
Y(z) = (1 - Z) . X(z)
4.3
The equation for use in digital filtering on a computer is
v i = x - x_
4.4
where X i is the ith sample of the 5 time series of a data window, Y i is the
recovered signal sample for the 5 components and n is the delay unit given by
n = [Sampling frequency/perturbation frequency]
4.5
The operation represented by equation 4.4 effectively cancels 50Hz (and other
perturbations) and harmonics in synthetic data, but the simple comb filter (eq.
4.2) is not a very good filter in that it distorts the spectrum at all other
frequencies between zero and F nq where there is a gain of two. F q is given
by
F nq = RF (2K + 1)/2, K = 0,1,2...
4.6
where RF is the rejection frequency. This behaviour is illustrated in fig 4.3.
However, we are only interested in the ratios between corresponding field
components and this problem is therefore trivial.
64
Fig 4.3 Frequency response of a delay line filter
65
rH
55808 3
FILE r AH55800 STATIONs 558 BAND , 0 NCHAN • q NSR?IP t 256 SARTE , 20q8 HZ
CL XTJCK 0.02 SEC
U U U
0 0
0
U U I U U U
19 I-
44WW4 55800 2 55808 3
O2 sec
FILE i flH55808 STATIONs 558 BAND I 0 NCHAN NSAMP s 256
b SARTE 2Dl8 HZ I XTICK v 0.02 SECI
MJ —
IiJ U
A .A C
U.
0 AA i'\.. ftvJII UI
0I vw.— . I III 10 55808 2
.O2sec,
Fig 4.4 Time series for two data windows (a) before and (b) after the
application of a delay line filter.
Digital notch filters (see Kanasewich (1981) for a treatment of design
considerations) were also tested on synthetic data and found to be satisfactory.
Digital filters were then built into the existing MT data analysis program and
the various data sets reprocessed. A remarkable improvement in the quality
and degree of overlap between adjacent data bands was achieved by
application of the delay-line filter. In figs 4.4a & b are shown respectively the
time series for 2 windows of a high frequency band before and after the
application of the delay-line filter. Unwanted single frequencies were removed
by notch filtering. In figure 4.5 are shown the results of analysis for some
sites after the application of a simple subtractive delay-line filter (compare with
the results from conventional analysis shown for the same sites in fig 4.1).
Note the considerable improvement achieved for the band 0 data using the
delay-line filter.
The low frequency data are generally noisy (a consequence of the
polarizable electrodes used for the field recordings). However, the subtractive
delay-line filter appears to be ineffective at low frequencies and Fischer (1982)
suggested an additive delay-line filter (not used in this study) given by
V I = X + X +
4.7
It was observed that no amount of digital filtering was helpful if the data were
contaminated' with irregular noise. A different strategy had to be adopted for
such cases.
67
a C C U C
x a D C
I-
C I— C U) C
U) Ui
z a Ui a
Cr
Cc
a- a-
C
a a
a
U) Ui LLI
CJ
a
a Ui a a
C C
a
U) Ui
Ui cc cc
I-
0 a-
a
a
0)
Ui cc to a Ui a a
bi
a —
C
EPOL -
J 1 !111 I n3E tI
II!T;IJII1hI I
£ POL
I I I
a C
C
Cr
a Ui a
Cr
ILa -.
a
a
0
a) Ui ir
I—
a- - a-
a
r a,
a 'ii a
Ui Co
a a- —
a
boo too to
çP.EQqENC IN 14z
E Pal
I 11 I i
--
11 11 II I} •
F Pft1
MENEM
NONE 111
IIh IINIIU
MINIM 1000 ID 10 1 o1 001
FR.EaFsc1 iN Hz
Fig 4.5 Some results of data processing with a simple delay line filter.
API
4.2.3.3 Frequency domain window editing
An additional program (written by G.Dawes) was used to inspect the
different stages of analysis, window by window, in order to separate the
distinctly corrupted and the uncorrupted data. This manual window editing was
begun by inspecting each window in the frequency domain for a given
frequency band. From the inspection of the cross-spectral amplitudes for the
electric and magnetic components and the Cagniard resistivities along
measurement axes one can establish which of the signals is significantly
contaminated with noise and may either delete the affected windows from the
analysis process or employ the appropriate impedance estimator. In fig 4.6 are
shown the results of analysis of nine windows of a noisy frequency band. Each
window in fig 4.6 has four sets of analysis against frequency: the Cagniard
resistivity from Ex and Hy (scaled to .1 to 1Oc2m); the averaged cross-spectral
amplitudes for Hy and Ex, with E normalised to H; the cross-spectral
amplitudes for Hx and Ey and the Cagniard resistivity from Ey and Hx.
Windows which have much greater power than normal in some or all of the 4
field components and with steeply rising (or decreasing) resistivity curves are
rejected. In this case it can be seen that the noise is strong in Ey and Hx and
that the associated curves rise too steeply; the whole frequency band was
rejected. For other sites, it was possible to distinguish between the heavily
contaminated and the less disturbed windows.
The existing data analysis program was therefore modified by the author to
allow for data-dependent processing, viz: (1) manual intervention so that only
selected windows (after prior visual inspection) can be analysed,(2) any of the
6 (or an average of 4) methods of impedance tensor estimation can be used to
determine the response functions and (3) digital filtering techniques can be
applied repeatedly to noise-degraded data.
The mean results obtained using this modified analysis procedures will be
discussed in the next section.
NMI
10.0 1.0 a. I
to .1
E>.c 10
10 AT
10 .....
*0.0 1.0 0. 1
- -.-------.-- -
10.0 1 1.0
HV •iHIllIIIIIl 1 hll
nT H.
10
1 . 1 1 1 •1 i
15
10 • 0 1.0 0. 1
H IM III III
10.0 1.0 0.1
HV
10
10 Al HW
10. 0 1.0 0. 1
I-
C 19 0 S S
rvi I i frLJcJ 5533
Fig 4.6 Results of frequency domain noise analysis for some data windows.'
For each of the windows (labelled 19 - 25) are shown the Cagniard apparent
resistivity (scaled to .1 to lOQm) for the N-S and E-W directions, and the cross
spectral amplitudes of orthogonal E and H fields as functions of frequency.
70
4.3 Magnetotelluric field responses
Data from 32 sites have been processed with attention being paid to the
biasing of the impedance tensor. The mean tensor response functions
(presented in appendices I, II and Ill) are qualitatively assessed in this section.
For each site, plots are given of the frequency dependence of the major and
minor apparent resistivity and phase, azimuth of major impedance and skew
factor with an indication of coherence and the number of averaged (mean)
estimates. The azimuths refer to magnetic coordinates and 0, 90, -90 degrees
denote respectively magnetic north (N), east (E), west (W). The azimuths
indicate the preferred direction of the electric field and have a 90 degrees
ambiguity. For a simple 2-D structure, the major principal direction will be
parallel to strike if on the conductive side of the lateral inhomogeneity and
perpendicular to strike on the resistive side of the contrast (Reddy and Rankin,
1972; Reddy et al, 1977b). Site names and profile code are as given in table
3.1.
In general, the regional geoetectric profile is characterised by at least 3
geoelectric units:- an upper unit of moderate resistivity, a highly resistive
intermediate unit (resistor) and a lower unit of low resistivity (conductor). The
results for each survey profile are presented in an appendix. For each profile,
the individual site results are presented starting from the west through to the
east. The low frequency (<1Hz) data are generally noisy because the signal
strength is notoriously low in the range .1 to 1Hz (the so called "dead band")
and also due to additive noise from the polarizable electrodes used in this
study. The results for the various profiles are now discussed.
4.3.1 Glen Garry profile
In appendix I are shown the results for sites 553 (Gi), 574 (G2), 550 (G3),
551 (G4), 552 (G5), 555 (G6) and 556 (G7). Site 555 has been projected onto
this traverse from about 51(m to the north (see fig 3.4). The quality of the data
based upon scatter, error bars, and data point continuity is generally good at
most sites in the frequency range 612 to 0.5Hz. Lower frequency data are of
good quality only at sites 551 and 574. Apparent resistivities at most sites are
only moderately anisotropic i.e. the values in the principal directions differed by
less than half a decade but the phase information seems to suggest a 2- or
3-0 environment. Lateral variations occur between G2 and G3 and to the east
of G5 and there appears to be an indication of a decrease in depth to the basal
conductor at G3. The skew factors are generally low in value and appear to
increase from 0.1-.2 at those sites far away from the GGF (and at G7) to about
71
0.24-0.4 near the fault.
The azimuth is about 45 degrees E of N (magnetic) at the western margin
of the profile and changes to about 45 degrees W of N near the fault. This is
the expected behaviour from MT theory assuming that the Great Glen fault
zone is of lower resistivity than the surrounding crust. This is thus the first
signature of the fault on the MT records. Note the rotation of azimuth at site
556 from about 40 degrees W of N between 612 and 13Hz to about 10-20
degrees E of N at about 1Hz.
4.3.2 Loch Arkaig profile
The results for the Loch Arkaig stations 573 (Al), 572 (A2), 557 (AS), 571
(A3) and 570 (A4) are shown in that order in appendix II. The results are
generally of poor (although usable) quality. The high frequency data were
corrupted by 5-6Hz, 32-36Hz and about 100Hz noise. In addition, most of the
apparent resistivity curves appear to have been shifted along the vertical axis
between 100Hz and 10Hz. This might be due to near-surface inhomogeneities
- the so called static distortion (Berdichevsky and Dmitriev 1976; Hermance,
1982; Park • et al, 1983). Near-surface resistivity variations of limited extent
typically cause vertical displacement of both the major and minor resistivity
curves without changing the phase information. In this situation, the major
resistivity or the effective resistivity curve provides some useful information
(Berdichevsky and Dmitriev, 1976). It is obvious from these plots that the
apparent resistivities are generally anisotropic.
At site 573 (located in an area of E-W camptonite dykes) the azimuth is 0
or 90 degrees between 612Hz and 1Hz and rotates to the regionally consistent
strike of 45 degrees E of N at lower frequencies. The telluric field appears
polarised in the NE-SW direction (45 degrees E of N) at site 572 and may
suggest the presence at depth of a low resistivity linear structure such as a
fault. East of this station, the azimuth is 45 degrees W of N, i.e. perpendicular
to the Great Glen fault axis. This is as expected from 2-D MT theory. These
sites therefore require at least 2-D interpretation.
4.3.3 Glen Loy profile
The Glen Loy traverse is the most interesting in that it transects different
structural units and with sites distributed in a manner that permits accurate
spatial resolution of short wavelength features in the vicinity of the GGF. In
appendix III are shown the results for sites 607 (Li), 560 (L2), 606 (0), 605 (L4),
603 (L5), 601 (L6), 559 (L7), 602 (L8), 558A, 558B (0), 561 (1_10), 609 (Lii), 562
72
(112), 611 (03), 563 (L14), 565 (L15), 608 (L16), 604 (L17), 566 (L18) and 610
(L19) in that order. In general the data imply a strong 2-0, anisotropic,
electrical earth with 3 to 4 distinct layers.
The quality of the data is good at some sites but noise-degraded sections
of the sounding curves are a common feature (the data acquired in the 1985
fieldwork using the Edinburgh University S.P.A.M. MK II system appear
contaminated by 5-6Hz and 32-36Hz noise). The difficult terrain conditions
limited the experiments to accessible areas which are mostly inhabited or
farmed.
Site 607 is located within the Glenfinnan Division (Johnstone et al, 1969) of
the Moines and is the least anisotropic of all the observations with skew values
which decrease from 0.4 at around 100Hz to about 0.2 at frequencies below
5Hz. The azimuthal values are scattered and this feature together with the
somewhat isotropic resistivities suggest that 1-D interpretation of the results
could provide a reasonable indication of the actual resistivities at this site.
Sites 560 and 606 are respectively located to the west of and at the
periphery of the Glen Loy Gabbro Complex. Their azimuthal values vary from
30 degrees E of N at high frequencies to a northerly (or westerly) trend at
mid-frequencies rotating back to 30-40 degrees E of N at frequencies below
1Hz. At 560 the skew values are generally low (about 0.2) at frequencies below
100Hz and the resistivity anisotropy is less than half a decade at high
frequencies. The L3 (606) curves suggest the presence of near-surface
inhomogeneities. The skew values are about 0.1 and 0.3 above and below 1Hz
respectively. The anomaly is suspected to be 3-D here.
The sites to the east of L3 and west of L12 (562) are situated within the
Glen Loy intrusive complex and exhibit anomalous features. The major
apparent resistivity at site 605 shows only little frequency dependence (a
feature seen at 560 and 606), fluctuating around 10000 2m above 0.1Hz. The
azimuthal direction is persistently N60 ° E. A geoelectric contrast is inferred in
the neighbourhood of this site. Sites 603 and 601 are slightly anisotropic with
low skew values (0.1 - 0.2) and with the same azimuthal behaviour as at 605.
The resistivity variations at the other sites to the west of L12 are similar and
the azimuth varies from N30 ° W to almost E-W. Sites 558A and 558B are only
tens of metres apart and were occupied at different field seasons. Only site
55813, the repeat sounding, will be interpreted in this study. The nature of the
anomaly in the Glen Loy complex appears to be 3-D.
Sites 609 and 562 are situated in the Great Glen fault zone. The apparent
73
resistivities are strongly anisotropic and the telluric field is strongly polarised in
the direction of the fault axis, N45 0 E as expected from MT theory assuming
that this zone of thoroughly comminuted and indurated rock is conductive.
Skew factors are about 0.2. The "dead band" data are noisy. The structure is
strongly 2-D here.
The dead band is noisy at all sites east of the fault zone except at 566
which was measured in the early hours of the morning (about 0200 hrs) but the
high frequency data are of good quality. Site 611 is located close to a zone of
shattered rocks bordering the fault zone and the apparent resistivity is very
anisotropic with the electric field polarised in the NE/SW direction. Site 563
shows anisotropic character and the azimuthal directions vary from about
N55 0 W above 5Hz with skew values of about 0.2 to E-W below 5Hz with skew
values greater than 0.4. The azimuthal direction shows a rotation from N40 ° W
above 1Hz to about 10 0 -30 0 E of N below 1Hz at site 565 with a comparable
frequency dependance of the skew factors as at 563. The level of cultural
noise at site 565 precluded bands 0 and 3 measurements.
Site 608 is located within the Fort William Slide zone and site 604 just
outside it. The data are at least 2-D in character (large skew values, strong
anisotropy, very different phase data for the two principal directions). Sites 566
and 610 are situated to the east of the surface position of the Fort William
Slide. The apparent resistivites are anisotropic but the skew values are
generally only about 0.2. The azimuth shows abrupt 90 degrees rotation from
N40 ° W to about N45 0 E at 1Hz suggesting a possible different trend in the
basement rocks or the presence of a complex structure at depth. Lateral
variations in apparent resistivities occur in the vicinity of the Fort William Slide
and there appears to be an indication of a decrease in depth to the lower
geoelectric conductor in the region of the slide. Two-dimensional
interpretation (at least) is requried for these sites.
74
CHAPTER 5 ASPECTS OF DISCRETE INVERSE THEORY
Inverse theory is an organised set of mathematical techniques for reducing
experimental data to useful information about the physical world. It provides a
formalism in which many questions fundamental to geophysical data analysis
may be addressed. For instance, consider the problem of determining the
resistivity distribution of the subsurface from MT data. The resistivity
distribution within the earth is a continuous function of depth which can only
be uniquely determined if the MT response functions are known for all
frequencies in the range [0,a'], a problem in continuous inverse theory.
However, geophysical data with associated uncertainties, are acquired over a
limited bandwidth, and we seek the minimum set of parameters that describe
the earth. The problem is therefore discretized. Discrete inverse theory is
concerned with the approximations of otherwise continuous functions with a
finite number of discrete parameters.
The underlying concepts of some of the procedures used for deriving useful
information from a finite set of observations are outlined in this chapter. In
this short account, the least squares formalism is adopted because of its
mathematical robustness when the experimental data are, in the words of
Jackson (1972), "inaccurate, insufficient, and inconsistent". A complete survey
of the art of geophysical inversion can be found in the book by Menke (1984);
and Jackson (1972, 1976, 1979), Aki and Richards (1980) among others have
published excellent treatments of the underlying mathematical concepts while
Lines and Treitel (1984) give an easily digestible review of least squares
inversion in geophysics. A comprehensive treatment of the inverse methods in
remote sensing is given in Twomey (1977) on which some of the material in
this chapter draws heavily.
Many physical inverse problems are nonlinear. To explain how to tackle
nonlinear problems, it is instructive to consider first the simpler and
better-understood linear inverse problem. The same ideas are then extended
to the nonlinear case.
5.1 Linear least squares inversion
Consider a physical system (such as the earth) that can be characterised by
P unknown model parameters which represent members of a vector
m = col (m 1 , m 2.....m)
5.1
75
•Suppose that N measurements are performed on this system in a particular
experiment, we can represent the observed data by the vector of random
variables
d = col (d 1 , d 2.....d 11 ) 5.2
Assuming that a theory exists for predicting the outcome of the experiment, let
F be the set of n (usually nonlinear) functions of the model parameters that
describes the theoretical predictions. We also assume that the errors in the
observations are additive, Gaussian, statistically independent, with zero mean
and of unit variance. In general, the observations are related to the physical
model by
d= F [m] + e
5.3
where e is the vector of additive noise (errors) expressed as
e = col (e 1 , e2..... e)
5.4
Our goal is to discover the p variates, m that describe our physical system (i.e.
the solution of equation 5.3). If the functions F (m) are linear in m, we may
write the problem in matrix form
d 1 = Gm + e, =1,2 ..... n, j=1,2 .... p
or
d = Gm + e
5.5
where the n x p matrix G is a linear functional of the parameters and is
sometimes called the design matrix.
As usual in physical sciences, the number of data points exceeds that of
the model parameters (an overdetermined system) and the resulting system of
equations is solved in the least squares sense.
We may rewrite eq. 5.5 as
e=d - Gm 5.6
The method of least squares estimates the solution of an inverse problem of
this kind (5.6) by finding the model parameters that minimizes a particular
measure of the length of the estimated data, namely, its euclidean distance
from the observations.
We define a quadratic function Q by the formula
Q = e T e = (d - Gm)T(d - Gm) 5.7
76
(here and in future formulations aT signifies the transpose of a) and further
define the least squares estimator, fi, to be the estimator that minimizes Q, the
sum of squares of the errors. Minimization is effected by setting to zero the
derivatives of Q with respect to each of the p parameters m j and solving the
resulting equations, that is,
I = (dd - 2mTGTd + mTGTGm )
3m 1 3m =0
iI = - 2GTd + 2GTG = 0 5.8 3m mm
This yields the matrix equation
G T G fii = G T d 5.9
the so-called "normal equations" for the parameters m. This may be solved
for ft providing that (GTG) exists:
fri = (GTG)GTd 5.10
Eq. 5.10 is known as the unconstrained least squares (or Gauss-Newton)
solution of eq. 5.6. A solution of eq. 5.5 in the above absence of errors and
when n = p (i.e. G is a square matrix) is
m = G 1 d '. 5.11
The least squares method uses the Lanczos inverse (Lanczos, 1961) instead of
the direct inverse of 0 and is given by
G 1 = (GTG)_ 1 GT
5.12
G 1 is also known as the natural generalized inverse of G.
5.2 Constrained linear least squares inversion
In many physical problems it is possible to generate a set of different
solutions that adequately explain some experimental data especially in the
presence of measurement errors. Ultimately one solution has to be selected
and interpreted. To do this we often have to add to the problem some
information not contained in the equation d = Gm+e. This extra information is
referred to as a priori information (Jackson, 1979) and can take many forms. It
is noted here that the matrix G T G may be singular or near-singular. This
situation produces some undesirable effects but the difficulty can be obviated
by using a priori data. .
77
5.2.1 Inversion with simplicity measures
A very effective way of inverting a finite collection of inexact data is to
impose the constraint that the desired solution be smooth. If it is desired that
the model parameters vary slowly with position, say, then we may choose to
minimize the difference between physically adjacent parameters (m 1 - m2), (m 2
- m 3 ) .... (m_ 1 - m r). These first differences can be written in the form
ft—i 7 mi
I .1 m2 Dm=
L •i :j 0 m 513
where 0 is the difference operator known here as the smoothness matrix and
Dm is the smoothness of the vector m.
We may quantify the roughness of the solution by considering instead the
second differences so that D is of the form
ro D2 = 1121
L -i 2 -1
5.14
To gauge the smoothness of our least squares solution we use a quadratic
measure q(m) given by
q(m) = mTDTDm 5.15
Let us define a symmetric matrix, H, given by
H = D T D 5.16
We may state the constrained problem as : minimize q = mTHm under the
condition I d - Gm 12 < Q1 (or specifically I d - Gm 12 = Q1) where 0 1 is the
maximum tolerable sum of squared residuals.
Using the method of Lagrange multipliers we find an unconstrained
extremum of the function
= (d - Gm)T(d - Gm) + m TH m 5.17
where 0 < R K is an undetermined multiplier.
This requires that for all
3(dTd - m T G T d - dTG m + mTGTGm - Q1 + mTHm) = 0 5.18 a m
78
so that
(GTG + BH)m = G T d
from which we obtain the solution
m = (GTG + BH)_1GTd 5.19
Equivalently we can minimize I d - Gm 12 holding q constant. Proceeding as
before, we find an absolute extremum of the Lagrange function
= mHm + 1(d - Gm)T(d - Gm) 5.20
If we assign a value Q2 to q it follows that
[mHm - + 1/B(dTcJ_2mTGT cJ + mTGTGm)] = 0 5.21 am
so that
(GTG + H)m = GTd as before, B being undetermined.
When H is an identity matrix we obtain a special constrained inversion formula.
If in this case the constraint equations are arranged to form an expression
equal to zero (Marquardt, 1970; Twomey, 1977, pp.134-135) such as
1 m 1 0 1 m 2 0
Dm=
1 rn 0 5.22
and a weight B is assigned to each of the constraint equations and a weight of
unity to the actual n data equations, we can solve the augmented problem in a
least squares sense. Since the right-hand side of the constraint equations is
zero, they do not contribute to the right-hand side of the final equations. The
special least squares solution in this circumstance is
m = (GTG + Bl)_lGTd 5.23
This important formula will be used later when dealing with nonlinear problems.
5.2.2 Inversion with prior information
We can incorporate previously obtained information about the model
parameters in our inversion process. This could be in the form of results from
previous experiments or quantified expectations dictated by the physics of the
problem. Generally, these external data help single out a unique solution
among all the equivalent ones. The procedure is an extension of the solution
r1]
simplicity technique. We employ linear equality constraints that are to be
satisfied exactly and of the form
Dm = h 5.24
where D is an identity matrix as in eq. 5.22 and h is a (p x 1) vector of a priori
values of m. We wish to bias m towards h 1 . We simply minimize
S = (d - Gm)T(d - Cm) + B (m - h)T(m - h) 5.25
Setting the derivatives of S with respect to the model parameters to zero we
find that
GTG m - G T d + Bm - Bh = 0
or
(GTG Bl)m = (G Td + Bh) 5.26
from which we obtain the biased solution
m, = (CTG + BI)_l(GTd + Bh) 5.27
However this procedure should only be used when it is reasonably justified for
it produces undesirable effects when h is unrealistic (see Twomey, 1977, p.138).
In the cases discussed so far, the constraints are implemented by arranging
the constraint equations as rows in Cm = d. The parameter B is chosen by
trial and error, and we seek a B value that produces a predicted residual of the
same order of magnitude as the actual misfit of observed and calculated data.
It is also possible to solve the constrained equation directly for B and the
constrained solution m̂ c (see Meyer, 1975, p.398).
5.3 Nonlinear least squares problems
In nonlinear problems the experimental data are related to the model
parameters via a nonlinear forward functional F and the general problem is
expressed as
d = F (m) + e
where the various quantities have been previously defined.
5.3.1 Iterative least squares inversion
In solving nonlinear least squares problems, a Taylor series approximation is
often used to linearize the problem so that the simple least squares method
can be used to solve the linearized set of equations. We wish to minimize
a0 = (d - F(m))T(d - F(m)) 5.28
Differentiation as before yields a system of nonlinear equations that are difficult
to solve. The general procedure is to start with a first approximation to the
actual model parameters and then successively improve the values of the
estimates until Q 0 is minimized.
We may assume that the model response function F(x) is approximately
linear around some starting model m (j=1,2 ..... p) such that perturbation of the
model response about m ° can be represented by the first-order Taylor series
expansion:
F(m) = F(m °) + aFf (m - m3 5.29 j=1 am
Define an n x p matrix of partial derivatives or Jacobian, A, computed by the
forward theory and with elements
A1 = 5.30a a m
and a (p x 1) parameter change vector x with elements
x = m - m9 5.30b
Eq. 5.29 can now be re-expressed as
F(m) = F(m °) + Ax 5.31a
Let us define further an (n x 1) discrepancy vector, y, containing the differences
between the initial model response F(m °) and the observed data
y = d - F(m °) 5.31b
Substituting eqs. 531a,b in eq. 5.28 we find that
= ee = (y - Ax)T(y - Ax) 5.32
We need to solve for the x that satisfies
y=Ax 5.33
Using the least squares method it follows that
DQ= 3(T - xTATy - YTAX + xTATAx) = 0
ax 3x 0 = ATAx + xTATA_yTA_ATy 5.34
which is equivalent to the vector equation
ATAX - ATV = 0
or
[:11
ATAx = A T V 5.35
Assuming that EATAF exists, we have from eq. 5.35
x = [ATA]_lATy 5.36
which is the parameter correction to be applied to our initial model, m °. The
successive application of this procedure minimizes eqs. 5.28 or 5.32. Let x °=m °
and x I=x0+(ATA)_IATy. The iterative formula can be written as
xK4 i=x 1 +[ATA]_ 1 ATy 5.37
where the Jacobian A is evaluated at xk.
The main drawbacks of this technique are that a good approximation to the
best estimate is required for the procedure to converge and that the matrix
A T A may be singular or near-singular producing some undesirable effects.
Even if A T A is non-singular, the solution may diverge or converge very slowly.
Fortunately, techniques have been developed to overcome some of these
problems. It is remarked here that the (n x p) matrix, A, can be factored
(Lanczos, 1961) into a product of three other matrices
A = UAVT
where U(n x p) and V(p x p) are the data space and parameter space
eigenvectors respectively and A is a (p x p) diagonal matrix containing at most
r non-zero eigenvalues of A, with r < p. The eigenvalues of A T A are explicitly
related to those of A and A T A is said to be ill-conditioned if the eigenvalues
are small.
To prevent unbounded solution growth when A T A is ill-conditioned,
Levenberg (1944) suggested a method of "damped least squares" to damp the
absolute values of the increments during the successive applications of Taylor
approximations; arbitrary positive weights are added to the main diagonal of
ATA . Levenberg also showed that the directional derivative of the residual sum
of squares has a minimum when the weights are equal. The idea of adding a
biasing constant to the diagonal of A T A was later used by Marquardt (1963,
1970) and Hoerl and Kennard (1970a) to propose very useful nonlinear least
squares algorithms. This technique is commonly known as the
Marquardt-Levenberg or ridge regression method. This approach is just one of
a class of damped approximate inverses (Jupp and Vozoff, 1975).
82
.5.3.2 Ridge regression
In this method, we minimize some combination 4) of the prediction error and
the solution length. We form the function
4) =Q1 + BQ2 = e T e + B(xTx - L) 5.38
where we have placed a bound, L, on the energy of the parameter increments
and B is an undetermined Lagrange multiplier which determines the relative
importance that is given to Q1 and Q2. Here - is referred to as the damping
factor. Minimization of 4) in a manner exactly analogous to the least squares
derivation (section 5.3.1) yields the normal equations
(ATA + l)x = ATy 5.39
from which we obtain the ridge estimator of the increments,
XR = (ATA + l)_lATy 5.40a
This is similar to the noniterative constrained linear inversion formula for
the case H=l (eq. 5.23) and may be regarded as its nonlinear, iterative analogue.
The process can be iterated until a solution to the general inverse problem is
obtained. If our starting model is x ° such that the first linear approximate
estimate of the parameters is x=xO+x R, nonlinearity is dealt with using the
iterative formula
x =x (+(ATA + Bl)ATy 5.40b
Note however that we can base our, measures on solution length on the
estimate x 1 instead of simply limiting the perturbation length XR. To do this,
we treat the estimates xk as the a priori data in the nonlinear equivalent of the
formulations described in sub-section 5.2.2.
Examination of eqs. 5.36 and 5:40a shows that the latter is an effective way
of dealing with singularities or near-singularities in A TA. Ridge regression is in
effect a hybrid technique in the sense that it combines the steepest descent
and the least squares methods. The steepest descent method dominates when
the starting model is far from the solution while the least squares method
becomes effective as the solution is approached. Marquardt (1970) showed
that ridge regression is similar in character to generalized inversion, and that
(ATA + BI)AT approaches the generalized inverse of Penrose (1955) as
B -> . Unlike Hoerl and Kennard (1970a), Marquardt also interpreted the
matrix BI as the covariance matrix of a set of a priori data. The properties of
the ridge solution have been discussed in great detail by Marquardt (1970a) and
83
Hoerl and Kennard (1970a) and will be discussed only briefly here.
5.3.2.1 Properties of ridge solution
Since the ridge solution R is a biased estimate of the least squares
solution its mean squared error is defined by application of the expectation
operator as
MS E(xR) = E[(x_,l) T(x R_,)]
= trace{Var(x R )+(E(x R)-) T (E(x R)-)
= variance + (bias) 2 5.41a
Let A 1 , A2 ..... A r .... XP be the eigenvalues of the matrix ATA. We can express
eq.5.41a in terms of the eigenvalues of A T A and B. The variance of XR is
Var(xR)= 5 2(ATA+ Bl)ATA(ATA+ Bl) 1 = S 2VA +VT 5.41 b
where (A +),=X/(X+8) 2 and the data are assumed to be Uncorrelated and oft'
equal variance 62 -
We have that trVar(xR)=6 2 X/(X+B) 2 j= 1
Where tr denotes the trace and there are r non-zero eigenvalues of ATA.
Since E (XR) = (ATA + al)-'A T AZ = z.Z we obtain from eq 5.41a
(Bias (XR))2 = T(Z - l)T(Z - l)
= B2T(ATA + Bl)_ 2 = B 2xTVA 2VT 5.41c
where (A) = (X + B). Substituting y T = (Y1'Y2'...Yr) = TV we find that
MSE (XR) = 2 A( + BY2 + y 2 B 2 () + BY2 5.41d j=1 j=1
It can therefore be demonstrated that the variance of XR decreases as B
increases while the squared bias increases with B. However, Hoerl and
Kennard (1970a), Marquardt (1970) and Marquardt and Snee (1975) showed that
there exists a value of B > 0 such that the mean square error (misfit) of the
ridge estimator is tess than for the least squares estimator. While this
provides a justification for using the ridge regression method in this study, it is
widely known in geophysical modelling that the best fitting solution is not
necessarily unique.
84
0
a E
0.00 I,
0
-2.00 0 16 a, 0
C 0
-4.00
-6.00
8.00
4.00
2.00
,00
Damping factor P .
.5.3.2.2 Damping factor and the stability of the regression estimates
Hoerl and Kennard (1970a) suggested a subjective graphical technique for
selecting the damping factor, B. This involves the use of a ridge trace (plots of
the components of XR against B. The ridge trace is examined and a value of B chosen in the region where the estimate have stabilised. An example of a
ridge plot is shown in fig 5.1.
Fig 5.1 Ridge trace for typical geoelectric model parameters
The ridge trace can also be used in the alternative solution stabilization
technique. This involves deleting those parameters with small eigenvalues (see
Wiggins, 1972) - the singular value truncation method. From the ridge plots,
those parameters whose estimates XRJ are unstable with either sign changes or
that rapidly decrease to zero are deleted from the inverse problem. We retain
only those XRj that hold their predicting power, or become important as a small
bias is added to the problem.
However, for automatic inversion the common practise is to set B first to a
large positive value thus taking advantage of the good initial convergence
properties of the steepest descent method and thereafter B is multiplied by a
85
factor less than unity after each iteration so that the linear least squares
method predominates near the solution. A variant of this procedure (Johansen,
1977) assumes as B the smallest eigenvalue of A T A matrix; if divergence occurs
it is substituted by the next largest eigenvalue until the solution is obtained.
5.4 Solution Appraisal
One question that is fundamental to physical data analysis is, how
representative of the real physical system is our reconstructed least squares
model? Inverse theory, in addition to the model parameters, provides us with a
plethora of related information which can be used to gauge the "goodness" of
the least squares solution to the inverse problem. Two of such "auxiliary
parameters" are discussed below.
5.4.1 Goodness-of-fit
Assuming that our data d 1 (or y 1 ) are normally distributed about their
expected values and with known uncertainties S, we can assess the fit
between the observed and calculated data by calculating the statistical
parameter, Q, defined as
N
= (d 1 - F(m)) 2/6 2 5.42 1=1
For N independent observations and p independent parameters, Q is distributed
as x2 with (n - p) degrees of freedom. For an unconstrained solution we reject
the fit at the QT level of significance (see Meyer, 1975 p.397) if
Pr{X2 (n - p) ~ Q} < QT 5.43
For a constrained solution with L independent constraint equations the number
of degrees of freedom is n - p + L, and we reject or accept the fit based on Q T
as given by (Meyer 1975, p.397)
QT = r Cx2(n - p + L)>Q} 5.44
by comparing the calculated value with Q(X2;n - p+L) from x2 tables.
5.4.2 Parameter resolution matrix
For a linear system, we can assess the quality of the model derived from a
given data set by examining the parameter resolution matrix (Jackson, 1972)
given by
R = HA = (VA1UT}(UAVT} = VVT 545
where U and V have been previously defined and the matrix H is the inverse
M.
used. R is of dimension p x r where r, the number of non-zero eigenvalues, is
the degree of freedom of the problem. R is evaluated at an acceptable model.
If R = I, ie p = r, then each model parameter is uniquely determined. The
deviation of the rows of R from those of the identity matrix, I, measures the
lack of resolution for the corresponding parameters.
For a nonlinear system we can obtain the resolution of a linear problem
that is in some sense close to the nonlinear one. For the constrained inversion
process, the parameter resolution matrix is given by
R = VVT+ VA 2VT = VA 2VT 5.46 81 A+8l
where B is as previously defined.
As usual in the physical sciences, we may want to estimate the range of
values which the model parameters might have as a result of the uncertainties
associated with the experimental observations.
5.5 Errors and extreme parameter sets
An important aspect of data analysis is the determination of bounds on the
various model parameters that are consistent with the data and the associated
errors. For the sake of clarity we shall adopt the notations of section 5.3.
5.5.1 Parameter covariance matrix
The simplest form of error estimation is the determination of the limits of
the parameters from the covariance matrix, C. The parameter covariance matrix
depends on the covariance of the data and the way in which we map the data
errors into parameter errors (Menke, 1984 p.65). If the data are uncorrelated
and of equal variance 62, then
C = [(AA)_'A] 6 21 [(ATA)AT]T = 6 2 [ATAF 1 5.47
The square roots of the diagonal components of this matrix are generally
referred to as standard deviations of the least squares estimates and may be
used to estimate the bounds of the model parameters. The off-diagonal
elements, the covariances, of C indicate the correlations between the
parameters.
5.5.2 Extreme parameter sets
We may elect to determine a solution with the maximum tolerable sum of
square residuals. One method of extremal inversion is the Most Squares
technique of Jackson (1976) in which a value is determined for each parameter
which is maximum (or minimum) under the constraint that the misfit of the
[iP4
observed and calculated data is equal to some desired value. We extremize the
linear objective function x T b under the constraints
y - Axi 2 = QT 5.48
where the functional b is a vector of zeros with the kth element (to be
maximized) equal to 1, i.e. bT=(0,...0,1K,0,0) and x is the vector of parameters.
Introducing the Lagrange multiplier j.i, we have that
[x b + 1/2p (xTATAx - 2xTATy + QT)] = 0 5.49
3x
or
ATAx = ATV - i.ib
from which we obtain the Most suares solution
xm = [ATA]_l [ATV - pb]
5.50
Thus the value of 1.1=0 corresponds to the least squares solution. However, eq.
5.50 must satisfy eq. 5.48, therefore
= jj 2 bT[ATA]_Ib + pbT[ATA]_1ATy + VTV
- VTAEATA] -lb - VTA[ATA]_IATy 5.51
so that
1.1 =(ci- - yTy + yA [ATA1_1ATy'1'2 Qi s"1/2
\ b[AA]_1b (bTEATA]_ 1 b) 5.52
where QLS is the sum of the square residuals of the optimal least squares
solution.
Whenever QT > QLS there will be 2P solutions (for P parameters) since
there are two solutions for li for each parameter. If the errors on the data are
assumed univariant and uncorrelated, it is expected that QT will have a value
equal or close to the number of the data, ie. 0T n.
For cases in which the matrix ATA is ill-conditioned Jackson (1976)
suggested a modification of (5.50) using instead H, the generalized inverse of
A. H and A satisfy the Penrose (1955) conditions. Thus from
AHA = A = UAVT(VA 1 UT)UAVT
we find that
H = VA 1 U T
so that
88
Xms = H (y - i HTb)
5.53
and
Ii = ± [(Q - QLs)/b THH Tb] 112 5.54
Thus the Most squares method, in effect, provides us with an envelope of
solutions that is maximally consistent with the observed data where the ±
refers to the upper and lower envelopes.
We can compare directly the most squares and the least squares solutions.
Eq. 5.50 can be expressed as
Xms = (ATA)ATy - 1J (AA)_ 1 b
= XLS ± ((OT QLs)/bT(ATA)-ib)i'2(ATA)_lb
where XLS is the least squares solution. The most squares solution envelope
may thus be interpreted as the confidence limits of the least squares solution.
5.6 Geometric interpretation of constrained inversion
Recall that constrained inversion requires minimization of some combined
function
= I y - Ax 12 + BXTHX = 1 + aQ2 5.55a
or
= B I y - Ax 12 + xTHx = BQ1 + 2 5.55b
Let H = I so that Q2 = Lx with an absolute minimum only when
= x2 = ... = x = 0. In the constrained solution process we first determine a
solution for a particular value of the Lagrange multiplier B. Q, and Q2 are then
calculated. The process is repeated for several values of B. From these
calculations we can examine the path of the solution in function space. Such a
solution trajectory is illustrated in fig 5.2. In this figure are shown the Q,
contours (solid lines) and the Q2 contours (broken lines) which in the general
situation would represent the (p-1)-dimensioned hypersurfaces of constant Q,
or 2• 1 attains a minimum at A and Q2 at B.
Fig 5.2 A two-dimensional simplification of the p-dimensioned interactions in
the vector space (after Twomey, 1977).
If eq. (5.55a) is minimized, the solution travels from A through C to B as
is varied from zero to an infinitely large value and the direction of travel is
reversed when eq. (5.55b) is minimized. If, however, we assign a fixed value a
to Q1 and minimize (5.55a) or equivalently minimize (5.55b) holding Q2 at a
constant value of y, a solution C is obtained (the point at which the Q1 = a
and Q2 =y contours touch in fig 5.2) and it solves both problems. Note that
the solution C 1 also satisfies the condition Q2 =y eventhough it may be a false
or undesired solution, a situation that may arise if our initial guess model is
very far from the true solution.
However, most practical problems are formulated as the search for the
smoothest solution among all possible solutions with Q1 ~ QT, where QT is the
maximum permissible error. Here again cIT may be such that an undesirable
solution may result. To illustrate this point let us minimize Q2 under the
condition Q 1 (Q-- = a") and with reference to fig 5.2. The surface a" encloses
the point B where Q 2 is smoothest and therefore solves the problem; C' is also
a possible solution for the problem. These may not be the desired solutions to
the above problem.
It cannot be over-emphasised that a suitable choice of Q 1 , not so large as
to enclose the point B, and a realistic step length for the undetermined
multiplier B during successive (le. iterative) application are essential ingredients
in the constrained interpretation process. If we use very coarse steps in the
90
factors we may overshoot the true solution C say, and obtain C 1 as the best
solution to our problem (see also Twomey 1977, p.148).
In geophysical data modelling, our search is often confined to the
neighbourhood of Q1 = A especially when we begin our iterative search with a
good approximation to the true solution and we seek the solution with the
smallest misfit between actual and calculated data. The applications (with
necessary modifications) of the various concepts outlined in this chapter to
practical magnetotelluric problems will be discussed in chapter 6. It is
remarked here that most of the current research in nonlinear inversion falls
into three main classes: (i) approximate methods of solution, (ii) simpler and
more effective numerical procedures for solving the normal equations and (iii)
methods for estimating errors or bounds on the parameters of the nonlinear
problem. These trends in conventional data analysis will be reflected in the
work presented in chapters 6 and 7.
91
CHAPTER 6 1-D MODELLING AND INVERSION OF MT RESPONSES
Theoretical modelling techniques are used as tools to improve the
understanding of the relationship between the MT response functions and the
various subsurface resistivity discontinuities that have generated them. The
basic theoretical background is contained in chapter 5, and the present chapter
elaborates both those aspects of it which were utilised in the present study
and those which have been modified and/or extended by the author. The
eigenstate analysis of Lanczos (1961), also known as singular value
decomposition (SVD), was used extensively in this study.
6.1 Forward modelling.
Traditionally, 1-0 interpretation of MT field data involves comparison with
theoretical master curves (Cagniard, 1953; Yungul, 1961; Wait 1962b; Srivastava,
1967). These curves are generated by application of equation 2.13 assuming a
horizontally homogeneous model with 1-D resistivity distribution. This
constitutes the forward approach : given some information on the values of the
set of parameters (here, the number of layers and their resistivities and
thicknesses) a theoretical relationship is used to derive the values of some
measurable quantities (here, apparent resistivities and phases).
Forward modelling by interactive computing is a more versatile extension of
the original curve-matching technique. The theoretical curves generated for an
initial model are displayed together with the field curves on an interactive
terminal. The model parameters are adjusted and the operations repeated until
an acceptable visual fit is obtained between the field and theoretical curves.
6.2 Inversion of MT data
An important procedure which has been commonly adopted in recent years
for MT data interpretation is the inverse problem in which a conductivity
structure is directly retrieved from the data given some information on the
values of some measured quantities (field data) we use a theoretical
relationship to derive the values of the set of parameters that explains our field
observations as discussed in chapter 5.
The inversion of MT data for subsurface resistivity distribution is a
non-unique and nonlinear process, and this poses an interpretation problem.
The non-uniqueness stems from the fact that an infinite number of completely
different resistivity structures exist that satisfy a finite collection of MT data.
92
The inversion of actual field data is complicated by additional sources of
non-uniqueness errors on the data, the limited bandwith of observations, bias
eftects due to lateral variations in the earth's resistivity structure and cultural
noise, and the fact that electromagnetic (EM) sounding data cannot resolve
sharp boundaries or thin layers (since the real earth structure is "smeared out"
as the propagating energy becomes diffused).
A variety of methods exist for inverting MT data. The commonest
technique is the layered earth (parametric) inversion in which a succession of
layer resistivities and thicknesses which reproduce the observations is sought
(eg. Wu, 1968; Nabetani and Rankin, 1969; Laird and Bostick 1970; Jupp and
Vozoff, 1975; Jones and Hutton, 1979b; Larsen 1981). This method requries a
reasonable initial guess to the resistivity structure to enable the algorithm to
converge rapidly. Direct or exact schemes that do not require a first guess
model have been developed by Parker (1980), Parker and Whaler (1981) and
Fischer et al (1981). Other workers (eg Parker. 1970, 1977; Oldenburg, 1979;
Hobbs, 1982) have developed inversion schemes that produce a resistivity
structure that is isotropic and a continous function of depth. Along similar
lines, Niblett and Sayn-Wittgenstein (1960), Becher and Sharpe (1969) and
Bostick (1977) have developed techniques to obtain a continuous resistivity
function which approximately reproduces the data. Most of the methods
referred to above seek a model that either fits the data best or to within an
expected tolerance.
Considering the nature of the MT data, it is not expected that any single
model will satisfy the recorded data uniquely in the absence of a priori
information. Partial remedies to this problem are provided by the random
search (Monte Carlo) and linear (Backus-Gilbert) methods. However, the Monte
Carlo method (eg Jones and Hutton 1979b), in which a large number of
randomly generated models are tested against the data, is too consumptive of
computing time. Moreover, such a random search can never be exhaustive. A
deficiency of the Backus-Gilbert (Backus and Gilbert 1968) linearized appraisal
is that the unique averages refer only to those models that are linearly close to
the constructed model and thus it cannot illuminate the true variability of
model space (Oldenburg, 1983) and its 'averaged parameters' may not have
physical significance (eg. see Jackson 1979, p.144). Oldenburg (1983) has
developed techniques for appraising the non-uniqueness inherent in the
inversion of MT data for a subsurface resistivity which is a continuous function
of depth, and Oldenburg et al (1984) favour the use of a variety of
93
interpretation algorithms to explore the range of acceptable models as a means
of appraising the non-uniqueness. As a way of accounting for non-uniqueness,
Constable et al (1987) have developed an algorithm for generating smooth
models which are devoid of the sharp discontinuities that typify conventional
least squares inversion. It is clear from the above discussion that there is no
unequivocal method as yet for resolving non-uniqueness in MT inversion.
In the subsequent sections of this chapter, the well developed tools of
generalized matrix inversion (Lanczos, 1961; Jackson, 1972; Wiggins, 1972) are
used to invert MT data from the Great Glen region. A technique, similar to
Jackson's (1976), is developed to quantify the non-uniqueness inherent in the
inversion process. In the use of alternative approaches we incorporate a priori
information (Jackson, 1979) to produce acceptable models. The various
algorithms are characterised by numerical stability and rapid convergence. The
full inversion process is discussed under the two headings :- Model Search and
Model Interpretation methods.
6.3 Model Search
The solution to a linear inverse problem can be obtained in one step. For
the non-linear MT problem, we need to linearize and iterate towards a solution.
Since the inherent nonlinearity might be too strong for the linearized technique
to be applicable, a variety of stabilization techniques are employed in the
solution process.
6.3.1 Linearising parameterizations
Parameterization is the process of selecting variables to represent data and
model parameters. In determining subsurface resistivity distributions the model
parameters are the layer resistivities and thicknesses or alternatively
resistivity-thickness products and the data are the field observations and the
model responses calculated from the forward theory. The apparent resistivity
is well known to display a lognormal distribution rather than a normal one. It
can be demonstrated that the nonlinearity is reduced when the logarithms of
the apparent resistivity data are considered instead of the apparent resistivity
data themselves. In fact such transformed variables are conventionally used in
MT work; the logarithm of model parameters are used in the present inversion
process, ie. m=x=(logp1,Iogp2 ....
logp,logT 1 ,logT2,...IogT_ 1 } where 2C-1=p, the
number of model parameters and p; and T 1 are the ith layer's resistivity and
thickness respectively.
.6.3.2 The optimization scheme
Model search involves finding a solution that minimizes a suitable cost
function. It is desired that this solution be both numerically and statistically
stable. Statistical stability is important here because of the differing standard
errors associated with each observation.
Assuming the data errors 6 i to be statistically independent, let us define a
(n x n) diagonal matrix, W
W = (1/61, 1/62, 1/6 3 .....
For the sake of clarity we shall keep the same notations as in section 5.3. Let
us define further the quantity Q, that measures the goodness of our estimated
solution as a possible subsurface resistivity distribution
a = e T e = [Wy 1 - WA 1 x]2 j=1.....p. 6.1
Q is known to be distributed as x2•
We state the constrained optimization problems as : Given a maximum
admissible misfit Q1, find in the set of possible solutions (on account of
observation errors and model errors) with Q < QT, an acceptable solution which
is the smoothest as judged by the measure xTHx.
The optimization scheme is simple. We minimize
0 1 = (Wy - WAx)T(Wy - WAx)
Subject to the constraint
= XTHX
where the smoothness measure Q2 represents the squared solution step-length
when H = I. Note that each datum is scaled by its associated error to prevent
undue weight being given to poorly estimated data points. The cost function
to be minimized is
4' = C(Wv - WAx)T(Wy - Ax) + B(xTHx - a2)) 6.2
Recall that H = DTD. Rearranging the constraint equations to form an
expression equal to zero and proceeding in the manner described in section
5.3. We obtain the scaled least squares normal equations
[(WA)TBl 2DT]F WA lxi = [ (WA)T2DT1 1 01wy 6.3 oJ j
which may be solved for the constrained solution
95
x = [(WA)T(WA) + BHF 1 (WA) T(Wy) 6.4
When H = I, we have a member of the family of ridge estimators (Hoerl and
Kennard, 1970a; Marquardt, 1970). As in section 5.3 nonlinearity is dealt with
using an iterative procedure of the form
= xk + [(WA)T(WA) + 131-11 -1 (WA)T(Wy) 6.5
If the matrix D is a first difference operator (eq. 5.13) we in effect obtain a
smooth model (Constable et al, 1987). There are two ways of implementing
this algorithm : we either augment the actual data equations with extraneous
data (the constraint equations) as shown in equation 6.3 and with B chosen by
trial and error or add a constant value of to the main diagonal of A T A
(directly augmented singular values) with automatic selection of B; these two
procedures are used respectively by the optimization routines SMTHF and
RIDGE in the inversion program MTINV which has been written by the author.
A flowchart of this optimization scheme is shown in figure 6.1. To speed
further the convergence of (6.5) we employ a technique for scaling the
Jacobian matrix. Each column of the matrix A is scaled by the root mean sum
of squares value of the coefficients (Marquardt, 1963). The scaled matrix As is
= A 1/S, = D -1 A jj
N where S= 1/N I (A 1 ) 2
i= 1
and the diagonal matrix 0 is of the form
Si .0 0 =
•O 0 S
Let Z = WA, where W is the matrix of observation errors. The least squares
estimate of the scaled parameter change vector x is
x = [(DZ)T(DZ) + BHI1 (DZ)T (Wy)
= DEZTZ + BHF 1 ZT(Wy)
= D[(WA)T(WA) + BHF1 (WA)T(Wy) = DXR
To obtain the original parameter perturbation we then rescale the resultant
solution in the form
xR = D1x
START ;
MARQUARDT-
LE V EN BE RG
MINIMIZATION
INITIAL VALUES X°
-F-- CALCULATE
JACOBIAN :
CALCULATE t —
SVD OF
12 MINIMIZATION
WITH SMOOTHNE
MEASURES
CONVERGE
ES
T ESTIMATED VALUES Xr
STOP
AUGMENT JACOBIAN
WITH CONSTRAINT EQUATION S
Fig 6.1 A simplified flowchart of the minimization algorithm. 1 and 2 refer to
optional minimization techniques.
97
However, scaling is done only when the coefficients in one row are widely
different from those in another row. It is avoided when all values are about
the same order of magnitude since the additional round-off error incurred
during the scaling operation itself may adversely affect the accuracy. As a
pre-scaling check, the largest and smallest elements of each column are
initially ouptut by the scaling routine SCALEJ in the program MTINV thus
permitting user intervention.
Additional stabilizing side constraints come into play during the solution
process. For physical reasons bounds are placed on the size of the
perturbations using the "smoothness criterion" of Jackson (1973) defined as
p S=[1/p I (x) 2 ] 1"2 1
j= 1
At each iteration any perturbation x greater than 1 is regarded as unsuccessful
or multiplied by a factor (<1) decreasing the length without changing the
direction. While this operation prevents the solution from wildly
"over-shooting" the linear range, it does slow the convergence. Another side
constraint is that the model parameters are non-negative scalar functions.
6.3.3 Computational details
6.3.3.1 The Singular value decomposition (SVD) of a matrix : an overview
Lanczos (1961) discussed the extension of eigenvector analysis to a real n x
p matrix. For such a matrix, A, there is an n x n orthogonal matrix U and a p x
p orthogonal matrix V such that UTAV assumes one of the two following forms:
UTAV ={A if n > p 10 J
UTAV = [A 01 if n < p
This decomposition is known as the Singular value (or spectral) decomposition
(Lanczos, 1961) of A. The columns of U and V are the left and right singular
vectors of A respectively (ie. pairwise orthogonal unit vectors). A is an r x r
diagonal matrix, its diagonal entries X 1 .....X,. are the positive singular values of
A. The number r is the rank of A (or the potential number of degrees of
freedom in our data).
r < mm (p,n) and X1 ~ X2 ~ .. . ~ X r > X r+1...Xp 0
However, when n >> p, as in most geophysical problems, it is impractical to
store all the matrix U. U is often partitioned in the form
u =[u1 u 0]
where U 1 is n x p so that we have the factorisation
A = U 1 AVT
For simplicity we shall refer to U 1 as U.
The SVD method has been shown to be more robust numerically than most
other methods of solving the normal equations for x (see Stewart, 1973;
Lawson and Hanson, 1974; Lines and Treitel, 1984). The SVD algorithm used in
this study is a variant of the one given by Golub and Reinsch (1970) and was
translated from ALGOL by RLParker.
6.3.3.2 Application of SVD to inversion
The least squares solution given by eq 5.36 can be written in terms of the
SVD of A as
= [ATA]_ 1 ATV = [VA_2VT]VAUTV = VAUTy 6.6
where we have assumed that r = p and the inverse of A T A (i.e. VA_2VT) exists.
In ridge regression (the present application), we replace the l/X in the A 1
matrix by the element
=x/(x+ ) 6.7
If we drop the weighting term W in equation 6.4 and consider the case where
H = I, the ridge estimator in terms of SVD is
XR = (ATA + I)_lATy = VAUTy 6.8
where Ais related to A 1 by the transform (6.7).
The advantages of using these transformed variables are obvious when X i
is much larger than B, X + B differs very little from X i and when X i is close to
zero, A T A will not be so close to singularity.
6.3.3.3 Estimation of damping factors for ridge analysis
The optimum damping factors are determined automatically in ridge
regression. The largest and smallest singular values of A T A are respectively
multiplied by 10 and 0.1, and the range divided into 10 values, Q k. This
procedure is illustrated graphically in fig 6.2.
640.00
480.00
CM
320.00
a
J 160.00
0.00
Index numbeç ic
Fig 6.2. Estimation of the damping factors in ridge regression. QS = X/10 and
QL = 1 0XL where X. and XL are respectively the smallest and largest singular
values of the problem.
The formula for the Kth Q is
QK = (100 Qs - L + (QL - Q)K2)/99 K = 1.....10 6.9
The values of Q are squared to give the damping factors. It is clear from fig
6.2 that this technique involves coarse steps in 8 in the region of BL. These
become finer towards B B is initially set to the largest value and then the
successively smaller values are used until divergence occurs (ie. solution
becomes unstable). If no divergence occurs a value of zero is assigned to B
(yielding the unbiased estimate). This "down range" search for solutions with
minimum residuals is performed by the routine RIDGE and the solution before
divergence is used as the next iterate in the main (calling) routine MTINV. The
extra computation time is minimal ; once the elgenvectors and singular values
of the A T A matrix have been obtained by the main program, ridge regression
estimates can easily be calculated for any value of B.
As noted in section 5.6 coarse steps in B may lead us past the true
solution. On the other hand, very fine steps in B involving multiplication by a
constant value <1 will slow down the convergence. The approach adopted
here can be interpreted as intermediate between these two extremes and this
feature together with the side stabilizing constraints ensures that the algorithm
is characterised by rapid convergence and numerical stability.
6.3.4 Convergence characteristics of the optimization algorithm
To assess the applicability of MTINV in geoelectric interpretation it is
instructive to apply the inversion methods first to synthetic data generated
using equation 2.13. We generate the responses, at discrete frequencies, of
100
Fig 6.3. Ridge regression models for synthetic and actual field data.
(a)&(b) 3-layer models for synthetic data
4-layer models for synthetic data
3-layer model for actual field data
4-layer model for actual field data
S = starting model, F = final model. The parameter CHISQ represents the
minimum achievable misfit in models (a) & (b) and the expected tolerance in
the other models. DF is the damping factor used in (c). Ri and Di are
respectively the bounds on the model resistivities (in c2m) and depths(m) to
interfaces. The Niblett-Bostick transformation are also shown with error
crosses. Details of the synthetic models are given on page 107.
101
U
U
a!
I-
U I-. 0
0 I-.
a, Ui
Z U U
CC
a- a-
a
0
I-
a,. Ui 0 Iii •
(3 . lii a
Ui a, cc
a--.
a
C
P -4-
01 1001 I
02g1,S1
0 0 a a U
10 RIDGE REGRESSION MODEL FOR. SITE RAiD TEST CHISQ.1 NFREQ.38 3 LAYERS AMPPttRSE FIT
RESISTIVITY OHM.N
S
I- a- Lii C
iaoo.ee IOLIO ta-oe 1.00 0.16 0.01 FREQUENCY IN HZ
1D I9IDGE J9EGI9ESSION MODEL FOR SITE RR1O TEST
D
0 0
CHISQ=O. 1 NFREQ=38 3 LAYERS RMF+PHRSE FIT
RESISTIVITY OHM.M
cocoa
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0 0
I- I-
0 I— 0 U) -I Ci) Lii cc
I- 0
'C 'C
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0
0 a,
U, I...
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Lii 'C LO Lii C
Lii
Cr = U,
-
0
1.00
FREQUENCY IN HZ
I- a- Lii C
I .
-I-
1D RIDGE REGRESSION MODEL FOR SITE RR1D TEST CHISQ=7.24 DF=0.3815 NFREQ=38 '1 LRYERS
0 0 0 -.4
0
0 -0
RESISTIVIT'( OHM.M
S
1889
01 934 107
i197
02 123 988:
25
03 28280 341
217
I6.8 10.0 1.00 PREQUNCY IN HZ
S
a- U.' C
10 RIDGE REGRESSION MODEL FOR SITE L10 INVARIANT
WI WI WI WI C
a —
I- I— I-.
> WI
I— WI
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CIIISQ-15. 83 NFREQ.'29 3 LPYERS RNP+PHRSE FIT
RESISTIVITY OHM.M
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02 17362 171
51
0 0 a 0 0
x
a —
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0 I— 0
0
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I-
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0
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01 913 913
* %S26 qn20 39 5%92$ 5U29
10 RIDGE REGRESSION MODEL FOR SITE L5 CHISQ- 143,99 NFHEQ-34 L LRYEBS RNPPHRSE FIT
RESISTIVITY 0HN.l1
S
a. 'U a
C C
S
19.00 1.co 8.10 e at- FREQUENCY IN HZ
EFFECTIVE
Li -
t.fl -J
240O If., uJ Cr
cc looc
U.,
° 80c
LI)
barely perceptible.
4800
4000
3200
ITERATION NUMBER
two models that typify the situation in northern Scotland (as determined by
previous studies), A and B shown below
Model A (3 layers)
1000 Ohm-rn (1 Km) 10000 Ohm-rn (10 Krn) 1000 Ohm-rn (a')
Model B (4 layers)
1000 Ohm-rn (1 Km) 10000 Ohm-rn (10 Km) 200 Ohm-rn (30 Km) 2000 Ohm-rn (a')
We add 10% noise to the calculated responses to reflect possible field
conditions. The minimization routine is applied to these data sets using widely
different starting models. Two 3-layer models (derived from Model A data)
with the minimum achievable error are shown in figs 6.3 a&b. In fig 6.3c are
shown two recovered 4-layer models (from Model B data) with a prescribed
error: in this case an optimum model is first sought and then the model
parameters are changed in opposite direction until the error condition is
satisfied. These results are satisfactory. The Bostick (1977) transformation of
the data are also shown for comparison. The results are compatible. Two
applications to actual field data are shown in figs 6.3 d&e.
The convergence information is represented in fig 6.4. The error for 10
iterations are shown in order that we can appreciate the convergence rate; the
sum of squared residuals decreased by a factor of about 13 in only two
iterations. Notice that the decrease in error between iterations 4 and 10 is
Fig 6.4 Convergence characteristics of the algorithm
107
Comparison of the ridge regression method as proposed by Hoerl and
Kennard (1970a) and the generically related procedure of augmenting the actual
data with extraneous data showed that the latter on account of computational
cutbacks (damping factor determined post facto) is about 40% less
consumptive of computing time but the former is preferred when the starting
model is very far from the true solution.
6.4 Model interpretation methods
It is obvious from the previous section that we have at our disposal
efficient routines for minimizing a functional subject to some constraints.
However, the optimal solutions may not be mathematically unique. We may
examine the parameter resolution matrix of any acceptable model to see what
features of the model are important. The exact damped resolution matrix is
given by equation 5.46.
The model resolution matrices for optimal models derived from actual field
data are shown in tables 6.1a-c. The rows of the matrices show the extent to
which each model parameter can be resolved. It is obvious that the main
diagonals of the matrices are delta-like le. R = I. However, the MT problem
typically admits an infinite number of solutions. In this situation, the favoured
approach is to incorporate a priori information in the interpretation scheme to
yield solutions that are coherent with the geological background and other
geophysical data. In this study the matrix A T A is always of full rank and R = 1.
6.4.1 Damped most squares method
This is an iterative extremal inversion technique never before applied in the
MT situation and is used here to quantify the non-uniqueness inherent in the
inversion problem. We formulate the constrained extremal inversion problem
as follows: given an optimal solution to an inverse problem, x 0 with error Q0 ,
and a specified maximum tolerable residual error 0T' search around this
solution for other solutions which fit the data to the expected tolerance; or
equivalently, extremize the linear parametric function xTb under the constraint
(Wy - WAx)T(WV - WAx) + KxTHx = QT 6.10
where the quantity KxTHx is a penalization criterion that measures the
solution's departure from smoàthness and helps stabilize the iterative solution
process.
108
Q
PAHAIIETER SENSITIVITY ANALYSIS FOR SITE 'Lii EFFECTIVE
RESISTIVITY DEPTH (BEST FIT tIODEL)
369.41 1136.29
1139.16 20532.06 298.25
flO DEL RE 1.000004
0. 4940957E-06 0.22351 74E-07 0.96335 13E-06 0. 0000000E+00
5OLUTION HATRIX 0. 4940957E-06
1.000003 0. 2123415E-06
-0. 2476736E-07 0. 1266599E-06
0.2235 1745-07 0. 21234 15E-0S
1.000000 0. 1008 157E-06
-0. 125 1698E-05
0.96335 13E-05 0. 0000000E+O0 -0.2476736E-07 0. 1266599E-06
0. 1008 157E-06 -0. 125 15985-05 1.000005 0.63097105-07
0.63097105-07 1.000901
b
PARAIIETER SENSITIVITY ANALYSIS FOR SITE 9 L16 EFFECTIVE
RESISTIVITY DEPTH (BEST FIT 1IODEL)
843.80 1946.86
7565.90 5181.04 62.34
tIODEL RESOLUTION HATRIX
1.000007 0. 1352059E-05 0. 7312847E-06 -0. 7859439E-05 0. 4786880E-06 0. 1362059E-06 0.9999988 -0. 6312054E-07 -0. 2980232E-06 0. 9685755E-07 0. 7312847E-06 -0. 8312054E-07 1.000006 0. 4463946E-Ô6 0.29281235-05
-0. 78594895-05 -0. 2980232E-05 0. 4463946E-06 1.000005 0. 32619575-05 0. 4785880E-05 0. 96857555-07 0. 2928123E-05 0. 3261957E-06 1.000006
'C
PARAIIETER SENSITIVITY ANALYSIS FOR SITE I Gi EFFECTIVE
RESISTIVITY DEPTH (BEST FIT tIODEL)
7542.53 2259.46
11877.24 16684.60 3871.74
1IODEL RE 1.000003
-0. 125 1698E-05 -0. 1788139E-05 0. 9536743E-06 0. 5960464E-07
3OLUTION HATRIX -0. 12516985-05
1.000002 0.2384 186E-06 0. 1110137E-05
-0. 4768372E-06
-0. 1788139E-05 0.9536743E-06 0.5950464E-07 0.23841865-05 0. i110137E-05 -0.47653725-05
1.000004 -0. 83446505-05 0. 3576279E-05 -0.83446505-06 1.000004 -0. 7040799E-06 0. 3575279E-05 -0. 7040799E-06 1.000003
ble,6:i Mode!resohJtionma.t.i;.e 6.ani.t (c)P 1.TIie - - )
First 3 nienbers of the rows and oliimnc of tile matrices correspond to the
revities s,st u and the otherstotJie thicknesses 6Lthe model layers
109
The functional b consists of zeros except at the ith position since we wish to
extremize the jth component. The inner product x T b thus represents a
projection of the unknown vector x along a known vector b. For the sake of
clarity, we shall drop the error term W in equation 6.10 but keep in mind that
the equations have been scaled by their associated standard errors.
We proceed in the same manner as in subsection 5.5.2 and find the
constrained solution at the extremum. The minimizaiton is straightforward.
Introduclfl9 the Lagrange multiplier 1/2j.i, we have that
a [1/2p (xTATAx - x T A T y - YTAX + y T y + Kx THX - QT) + xTb] = 0 6.11
ax
0 = ATAx + x T A T A + KHx + KxTH - 2ATy + b
or
[ATA + KHI x = [ATY + pb]
from which we obtain the damped most squares solution
XD = [ATA + KHI1 (ATY - j.ib) 6.12
Here we will consider the case where H = I so that
XD = EATA + Ku1 (ATV - pb) 6.13
where K is the biasing constant added to the main diagonal of A T A during the
search for the optimal solution.
When the quadratic constraint (eq. 6.10) is satisfied, we have that
QT= xT [ATA + Kl]x - 2xTATy + yT
= (vTA - pbT) (ATA + KI) -1 (ATV - ib) + T_2 [(yTA - jJ bT) (ATA + Kl)1 ATV]
After some algebra we find that
0T = V T V + 112bT (ATA + Kl)_lb yTA(ATA + Kl)_ 1 ATY
from which we obtain
11 =((QT_vTv + YTA(ATA + kl)_1ATy)/bT(ATA + kl)_1b) tA
= ± ((QT - Q0)/bT(ATA + kl)b) 1/2 6.14
Provided that QT > Q0, there exist. two solutions for p yielding 2p solutions for
the p-parameters which are maximally consistent with the data. The expected
value of QT is close to N, the number of observations.
Using the singular value decomposition of A, we re-write eq. 6.13 as
110
XD = VAUTy - pVA 2 VTb 6.15
where
AD = diag ((+ K)/X, j=1...,p)
The initial iteration of equation 6.13 gives the first linear approximation to the
extreme model Xm
x!,, = x0 + x0
The minimization at iteration K+1 is
K+1 - K + (ATA + Kl)1 (ATy - iib) 6.16 X m X m
where the Jacobian, A and p are evaluated at x. We may re-express eq. 6.15
as
x0 = VAUTy ± I p I VA 2 VTb 6.17
where the first term on the right hand side is the 'ridge' solution and I P I denotes the absolute magnitude of p So that x 0 may be interpreted as
providing the confidence limits of our optimal solution.
The damping of the singular values corresponds to the a priori assumption
that the parameter change vectors are small. The damping ('filtering')
procedure therefore confines the parameter change vector to a region where
the linearization yields meaningful approximations to the nonlinear function
(Marquardt, 1963). This approach was adopted since Jackson's (1976)
prescription for dealing with singularities in A T A was found to be inadequate
for practical MT data. The use of the fomuIa. (eq. 6.16) ensures that the
solutions do not oscillate wildly and that the condition given by eq. 6.10 is
satisfied. A flowchart of this interpretation process is shown in fig 6.5. One
pitfall in most squares interpretation is that a negative model parameter may
be reached before the stipulated condition is satisfied. This situation arises
when the error parametric surface is open in certain directions so that the
parameter can be changed without any perceptible change in the sum of the
squared residuals. To arrest this likely situation, an upper bound is fixed for
the parameter changes (33% of the actual value of the parameter) in the
algorithm, leading to a trade-off between convergence rate and stability. This
side constraint also ensures that the sought quantities do not change their sign
too soon in pathological situations; the search is terminated in any one
direction if the parameter assumes a negative value. This scheme is executed
by the routine MOSTSQ in the interpretation program MTINV.
111
START OPTIMAL MODE
INITIALIZE VARIABLES
Q T =QO XFACTOR. JO
=J.I, m=X0 0r=°o
FORM PROJECTION
VECTOR b;
CALCULATE I 2 I AUGMENT
I JACOBIAN, A A WITH H
CALCULATE SVD jc OF A
OBTAIN J, Xd -J
0
ROLTEOF 0
ICHANGE OF
NEW ESTIMATE 0 r9=rrj+Xd
NO LCULATE NEW ERROR . .. GE.QT
r
OR . ± CHANGE SIGN. LLPARAI€rER SOLUTIONS OF
OBTAINED 7
YES
STOP
Fig 6.5 A simplified flowchart of the iterative most squares algorithm. 1 and
2 refer to optional optimization procedures.
112
The main advantage of the damped most squares interpretation method is that
the errors on the data and the non-unique nature of the solution are accounted
for using the class of solutions which are maximally consistent with the
observations. In this way the interpreter can decide, at a glance, what features
of the model are important. It is hoped that this procedure will prevent the
over-interpretation of sparse data. The method also forms a compromise
between the time-consumptive Monte-Carlo and the faster single-model
interpretation methods.
6.4.2 The a priori information approach
A priori knowledge about the values of some parameters can be
incorporated right from the start in the optimization scheme to yield a unique
solution (Jackson, 1979). For instance, we may obtain values for certain layer
resistivities or thicknesses from borehole logs or other geophysical or
geological observations and wish to retain these values in our final result. In
the same vein, for an MT site A with a sparse or biased data set we may
presume that the resistivity distribution at a neighbouring site B (with good
quality data) is similar and incorporate its resistivity-depth information in the
interpretation of site A responses.
Let the parameter vector m = (m 1 ,m 2 .....m) characterise the resistivity-depth
distribution under the site whose data we wish to invert. Let the vector c =
(c 1 ,c2 .....c 1,) be the known results from prior geophysical observations etc. This
interpretation problem is the same as posed by equation 5.25. The differences
between the vectors m and c in the present problem are represented by the
vector h in equation 5.25. Here we minimize
(Wy - WAx)T(Wy - WAx) + B(x - (m - c)) T(x - (m - c)) 6.18
The solution for the parameter increment is
Xa = [(WA)T(WA) + W]_1 {(WA)T(Wy) + Bh} 6.19
where h• = m - c.
Equation 6.19 is applied iteratively to minimize eq. 6.18. In the first
iteration, however, h is a null vector since our initial best guess model m 0 is
the a priori model c. The minimization at iteration k+1 is
mk = m k + [(WA)T(WA) + aI]_(WA)T(W y) + B(mk - c)} 6.20
In the implementation of this algorithm we write the a priori data in the form
given by equation 5.24. The augmented data equations lead to the alternative
113
leastsquares normal equations of the form given by eq. 6.3 (where h is a null
vector). It follows from this that the ridge regression method uses a different
a priori model at each iteration. This procedure is optionally executed by the
routine SMTHF in the interpretation program MTINV. A value of unity is
assigned to Bin practice.
It is normal practice to invert the amplitude and phase data simultaneously.
Often we find that apparent resistivity curves have been shifted staticly. The
inversion of the phase data only with a priori parameters can therefore yield a
much more realistic geoelectric structure than would be obtained from
incorporation of the apparent resistivity data.
6.4.3 Multi-station interpretation
The interpretation ofdata sets from sites along a traverse poses a further
problem since we often find zones in which the electrical properties are in total
contrast to those of the adjacent regions (ie. anomalous). Such anomalies may
be artifacts of the inversion algorithm used or may have resulted from
over-interpretation on the part of the interpreter when faced with a catalogue
of equivalent models. The non-subjective interpretation method adopted in
this study involves inversion of the various data sets either jointly or
simultaneously.
6.4.3.1 Joint inversion
The independent data sets are combined and inverted to yield a
representative structure. If this structure appears unacceptable then the data
sets do not characterise the same geoelectric structure. For the acceptance
criterion we use the joint misfit defined as
= I lWv - WA*xl 1 2
where
Vi Al
y * = . A =
Ak
K is the total number of sites and I I . I I denotes the usual Euclidean norm.
This procedure is an optional facility in the interpretation program MTINV.
114
6.4.3.2 Simultaneous inversion -
We are interested in obtaining a laterally smooth profile. We start off with
a smooth profile (ie. the same model for all sites) and invert all the various
data sets simultaneously with the constraint that the differences in layer
parameters beiween physically adjacent sites be minimal and that the solutions
be statistically stable.
This technique is now demonstrated using a two-site example. Extension
to higher dimensions is straightforward. The data and constraint equations are
partitioned in the form
A x
A1 0 x 11 Vii
I X 21 V21
I pi V ! 1 x 12 V12
0 A2 X22 V22
1 -1 1 I -. •-
xp2
ii Vn'2
-1 h l h2
h
where A is an (n+n'+p)x(p+p) matrix containing the partial derivatives A 1 ,A2
plus the px2p smoothness matrix for the two sites, x contains the solution
vectors x 1 and x2 each of dimension pxl and the vector y contains the
discrepancy vectors y 1 (nxl) and y 2(n'xl) plus the augmenting data h 1 .....h which
are the desired differences in the values of the layer parameters. In this
example h is a null vector.
We state the optimization problem as the search among all possible
solutions with I y-Ax 12 <CIT (the maximum tolerable error), for the
smoothest solution as judged by 2
We form the Lagrange function
(x 1 ,x2) = (x1 - x2)T(xi - x 2) + 1(Eyi - A1x11]2 + Yk2 - AkI2x2]2 - T) 6.21 j=1 i=1 j=1 k=1 j=1
115
where p is an undetermined multiplier and the numbers 1 and 2 indicate the
site contributing the data; these data have been standardized but the W term is
neglected for clarity. Setting its derivatives with respect to the model
parameters to zero we obtain for the first linear approximation
aJx 1 ,x2) = x 1 - x2 + .i1(ATAixi - AJy 1 ) = 0 axjl
giving (AJA 1 + pl)x 1 = A 1Ty 1 + px2
or
34 (x 1 ,x7) = x2 - x 1 + jf1(AA2x2 - Ay2) = 0
3 xJ 2
giving (AA2 + pl)x2 = Ajy 1 + lixi
so that
x i = (A[A1 + 111)_ 1 (AJy 1 + px2 ) 6.22a
and
x2 = (AA2 + iilY 1 (Ayz + px 1 ) 6.22b
The above equations (6.22a&b) must be solved simultaneously for the solutions
and may be applied successively to recover the desired profiles. It is clear
from equations 6.22a&b that the solutions are coupled via j.i. If the constraint
equations are weighted more heavily than the data equations then the
differences between the values of the parameters are minimized at the expense
of increasing the prediction error of the other equations. A very small value of
p would produce the usual rough models. p is determined by trial and error
until an acceptable total sum of squared error is obtained. This technique may
be interpreted in simple terms as moving a resolving kernel across the surface
instead of down the geoelectric section as is customary in geophysical
modelling. With appropriate constraints one might deduce a less complicated
resistivity structure for the traverse than would be obtained using conventional
interpretation methods. The interpretation program MT4INV handles a
maximum of 4 sites; modification to interpret > 4 sites is trivial.
6.4.4 Occam's razor technique
This is an interpretation technique that produces a very smooth layered
structure. The mathematical formulation is as given in section 5.2 and applied
in subsection 6.3.2. The constraint equations are of the form given by eq.5.13.
116
The method is conceptually similar to the multi-site lateral smoothing
technique the difterence being that the smoothness measures are applied down
a geoelectric section here. Unlike conventional simple layered inversion we
allow for a large number of layers (dependent on the number of field
responses) with the constraint that the differences between the values of
physically adjacent parameters be minimal (ie. "shaved off by Occam's razor").
The 1 desired misfitj is fixed at or near the number of field data Operationally,
a starting model is constructed by the routine PREP (called by MTINV) using a
simple data transformation and the step at each iteration is optimized by the
routine SMTHF until a model that satisfies the prescribed misfit condition is
found. Only a few iterations (<5) are requried in general and computational
time is minimal (<6 sec. CPU).
The advantage of this technique is that the solution does not require or
depend on a user-supplied preferred (or guess) model but on the data.
Constable et al, (1987) applied a similar technique to practical EM sounding
data and termed it Occam's inversion.
6.4.5 Niblett-Bostick (N-B) transformation
The N-B transformation (Niblett and Sayn-Wittgenstein, 1960; Bostick 1977;
Jones 1983) is a simple semi-continuous mapping of MT data into
resistivity-depth information using an algorithm based upon the asymptotic
response of the impedance data in a horizontally layered model with infinitely
conducting or infinitely resistive substrata. Parameters for a single layer are
provided, by this technique, for each resistivity and phase point, so that a
continuous resistivity-depth curve may be obtained for the resistivity and
phase independently. The relevant expressions are
h = [Pa / ( u0w)] 1 "2
and
p (h) = Pa U7T1"24)) - 11
where w is the angular frequency, h is the penetration depth in a half space
medium of resistivity equal to the observed value Pa at the particular frequency
and p is an alternative expression that incorporates phase (4, in radians)
information for the N-B resistivity at depth h (Weidelt et al, 1980).
117
6.5 One-dimensional models
6.5.1 1-D response functions and the existence of solutions
Weidelt (1972, 1986), Parker (1977, 1980, 1983) and Parker and Whaler (1981)
provide us with conditions that a given MT data set must satisfy for a 1-D
solution to exist. While the Great Glen region is at least 2-D in character, it is
now well-known from 2-D and 3-0 modelling studies that we can recover
representative 1-D profiles in regions of complex structures (Tikhonov and
Berdichevsky, 1966; Berdichevsky and Dmitriev, 1976; Ranganayaki 1984). The
effective response functions given by equations 2.27a&b have been shown by
Ranganayaki (1984) to be very useful in 1-D modelling studies and were
considered invertible in this study.
6.5.2 Comparison of methods
It is almost customary to interprete the 1979 COPROD data (Jones, 1979) as
a check on any 1-0 inversion scheme. Parker (1982) shows that any model for
the COPROD data can only be constrained to a maximum depth of about
3001(m. The model obtained using the Occam's razor technique is shown in fig
6.6a. It is noteworthy that the Occam model shows no structure above 5Km
and below 3001(m where there are apparently no constraining data. The Occam
model is superposed on the 'smoothest' model derived by Constable et al
(1987) for the COPROD data in fig 6.6b. The Monte Carlo model of Jones and
Hutton (1979b) for the same data is also shown in this figure for comparison.
In fig 6.6c the 'smoothest' and Monte Carlo models are superposed on the
most squares models for the COPROD data for comparison. Notice the high
degree of concordance between the various models. Also, it can be noted that
as might be expected the most squares models cover a zone within which all
the other structures lie.
For most of the Great Glen observation sites the effective data sets were
inverted using the multi-purpose user-friendly interpretation program MTINV.
A typical value for the time for a detailed interpretation (ie. model search and
most squares analysis) for a 4-layer problem is 45 seconds (CPU). For a
smooth (Occam) model with <20 layers the run time is <10 seconds (CPU).
The resultant models for five selected sites are presented in figs 6.7-6.10 and
table 6.2 to illustrate various aspects of the different procedures.
A comparison of the results obtained using the ridge regression method,
the most squares technique and the N-B transformation of the data is shown
in figs 6.7a&b. The same error condition has been applied in the two former
118
methods. The most squares models clearly cover a range of possible models
and thus provide a measure of the non-uniqueness. The result of the joint
interpretation of E- and H-polarisation data for site Li is shown in fig 6.8a.
The low frequency data (< 1Hz) are typically noisy but 1-0 interpretation could
provide a realistic estimate of the resistivity distribution under this location as
also suggested by the most squares models for the joint data set (fig 6.8b).
The Occam model (amplitude fit only) with superposed N-B results for Li is
shown in fig 6.8c. Note the general agreement between the Occam and N-B
results. However, the Occam model does not fit the phase data satisfactorily
at the low and very high frequencies. An independent interpretation of the
phase data alone produced an unacceptable model as shown in figure 6.8d.
This suggests that simultaneous inversion of phase and amplitude data is
required for site Li.
The adequacy of interpreting the amplitude data only for site L19 is
suggested by the Occam and most squares models shown in figs 6.9 a&b.
Notice the excellent fit of both the amplitude and phase data for this site. The
simultaneous multi-site interpretation technique is demonstrated for two sites
L8 (Si, 33 frequencies) and L5 (S2, 34 frequencies) in the computer printout
from MT4INV (table 6.2). The individual station (L8 and L5) interpretations
(using MTINV) are presented in figs 6.10 a&b for comparison.
119
fig 6.6. COPROD models
Occam model.
Comparison of Occam, Monte Carlo (Jones and Hutton 1979b) and
!smoothest' (Constable, Parker and Constable 1987) models.
(rl Comoarison of most sauares, Monte Carlo and 'Smoothest' models.
in
a
120
COPROD Model, rms 1.0
2,5
2.4
- 2.3 C
2.2
2.1
0
102 iO 3
0 C.,
80
-'
60
40
20
0
0 Ii.
0 01
102 iO
Period (s)
121
OCCflM MODEL FOR SITE 1979 COFFiOD
a CH1SQ.14.75 HFREQ15 15 LAYERS AMP FIT
V
z zCp
30- S.- b-I
I-I
fc cc
fas
uj
ui
0
e.te 6.e1 e.ee FREQUENCY IN H
RESISTIVITY OIIK.H
z
I-0 tlJ a
LOG10 resistivity (t2.m)
N N N N N C) C) 0 F',) 0) 0)
I I I
1iii -
im
CA
LI:c:::L. .-,
a
I-.
NJ i- NJ
'U
I-
I! le
i0
V
I-. IL 'U
i.e. FREQUENCY IN HZ I.e.
RESISTIVITY OIfM.M
MOST—SQUARES SM MODELS FOR SITE 1979 COPROD CHISGuS0.6a NFREQ.15 ' LAYERS RNPPHA5E FIT
C
C C
U C
C
I-
U, hi cc
z 0 hi! cc
a: a-a-a:
C
C a,
U,
U, hi hi cc
D
hi U, a:
U, -.
0
FREQUENCY IN HZ
LU
r - __
(1, hi
Uj
a: a- a- a:
C
U, hi C Ui
LO U' hi D
Ui (fl U a: r
-
CHISQ=28. 27 NFREQ=31 4 LAYERS AHP+PHRSE FIT
RESISTIVITY OI*LM
10 MOST-SQUARES RR MODELS FOR SITE L18 INVARIANT CHISQ28.27 NFREQ=31 t LAYERS AHP+PHRSE FIT
C RESISTIVITY OHH.M
C C C C
I- a- hi
1600.06
I6.5 I.60. 1.00 0.10 6.01
FREQUENCY IN HZ
Fig 6.7. Comparison of (a) Ridge regression and (b) Most Squares models for
site 118.
123
Fig 6.8. Some models for site Li.
Joint interpretive regression model
Joint most squares models
Occam and N-B results
Independent interpretation of phase data
L
124
x
I- a. lU 0
a
'a-
1000.00 i.00 IG.w 1.08 - 0.10 0.01 FREQUENCY IN HZ
z 0
r Lu cc cr 0 a-
a, LLS
cc 0 Ui
Lu
cr
jL
0
a
JOINT INTERPRETIVE MODEL FOR SITE Li
a CHISQ-84.20 NFREQ-84 3 LAYERS AMP FIT
RESISTIVITY OHM.M
MOST-SQUARES SM MODELS FOR SITE LI JOINT POL CHISQ7I8.26 NFREQ64
b 3 LAYERS AMP FIT RESISTIVITY OHM.M
0
II
- V
(n
a' Ui
V
Ui CC
U- U-
a,
cr
Ui V
Ui 0
Ui 0
a, V
V —
0 Ui 0
1000.00 100.00 10.00 1.00 0.10 0.01 FREQUENCY IN HZ
125
ir
I— a-
______ Cli
FREQUENCY IN HZ
= 0 0
Cl-
>.
- 0 Cl, 0
Cl, w
cc cr
I- 0
a- a-
0
ui cc
N
Cl,
Cli
Cli Cl) cc x a.. -.
C
OCCAM MODEL FOR SITE Li EFFECTIVE Es CHISQ=31 .89 NFREQ=32
16 LAYERS AMP FIT
9ESISTIVIT'r OHM.tl
OCCAM MODEL FOR SITE Li EFFECTIVE
d CHISQ22.74 NFREQ=32 16 LAYERS PHASE FIT
C
RESISTIVITY OHM.M
I ••
cr
CL CE
0
ui uj
fr &1J Cc
cc
E
= I-a- Cli
1000.60 100.00 10.00 1.00 6.10 0.01 FREQUENCY IN HZ
-
126
2
z - a
I-
- a I- a
Cl, 'U
I- z a
Cr Cl- 0 cc =
I-
2 'U a
• 0
N
Cl, Lu a L.J
Lli a Lu a 'I, Cr = a -
a 1006.00 100.06 10.00 1.60 6.10 0.01
FREQUENCY IN HZ
CHISQ28.24 NFREQ=30 15 LAYERS AMP FIT
a
RESISTIVITY OHII.H
Uj Cr
Cr
Cr
Uj
CM Uj Uj
CD LU -f __
io.ac io.a i.ø a.i 0.01 FREQUENCY IN HZ
10 MOST-SQUARES SM MODELS FDA SITE L19 b
CHISQ=27.36 NFREQ=30 LI LAYERS AMP FIT
RESISTIVITY OHM.M
EFFECTIVE
Fig 6.9. (a) Occam and (b) Most squares models for 1-19. The N—B results are
also shown for comparison.
127
cn
Cc 0.
z Dl 12
I-
w
02 21
1005.00 100.05 10.00 1.50 5.15 0.01 FREQUENCY IN HZ
,
cc
cc
Uj cc to ca
cn CE
Dl 570
uJ
02 376
03 7182 155040 100.00 10.00 2.06 0.10 0.61 FREQUHCT IN HZ
ii rIuuL rUM biIL Ui U-F
CHI5Q.77.1I3 NFEQ.33 3 LATEJiS AMFPHR5E FIT
U
ESI5TIVITT OHII.M
10 MOST-SQUPRES SM MODEL FOR SITE LS EFFECTIVE CHISO=36. 61 NFREQ=34 U LAYERS AMP+FHRSE FIT
RESISTIVJTT OIIM.M
Fig 6.10. Most squares and N-B results for (a) L8 and (b) L5.
128
lu riui — uMfl.3 I1 IIUULL r um b 1I t Ui U- I- LCTIVE
0. CHISQ.77.43 NFFiEQ33 3 LAYERS AMPPHR5E FIT
RESISTIVITY OHM.M to teo 000
E
cn LU
CC Cc
cc Uj
Uj
)
Uj cn cx
1000.00 100.00 10.00 1.00 0.10 0.01 FREQUENCY IN HZ
10 MOST-SQURI9ES SM MODEL FOF SITE L5 EFFECTIVE b CHISQ38.6I NFREQ34
4 LAYERS RMP+FHASE FIT RESISTIVITY OHM.M
(0 -
200 (000 20000 I I
Rl
cn q6!
I:
1000.00 100.00 1.00 1.00 0.10 0.01 fill
FREQUENCY IN lIZ
00000
F•1t•
7200
02 20
2590
01 570
02 370
03 7101
Fig 6.10. Most squares and N-B results for (a) L8 and (b) L5.
128
START MODELS
RESISTIVITY DEPTH(S1) RESISTIV DEPTH(S2)
2000.00 1500. 00 2000.00 1500. 00
8000.00 30000.00 8000.00 30000.00
300.00 300. 00
EXACT SUM OF SQUARED MISFIT(SSQ), FIELD/MODEL DATA 1497.216
ITERATION 1 RESISTIVITY EEPTh'(Sl) RESISTIV. DEPTH(S2)
1975.35 1457.78 2628.73 987.47 5072.25 27922.00 6860.70 36013.31 1098.23 696.14
PREDICTED SUM SQUARED DEVIATIONS 75.67929
EXACT SUM OF SQUARED MISFIT(SSQ), FIELD/MODEL DATA • 151.8023
FINAL MODELS AFTER 1 ITERATION
RESISTIVITY DEPTH(81) RESISTIV. DEPTH(S2)
.1872.60 .1299.61 2178.69 599.54 • •.5146.33 29633.41 7499.34 39653.50
906.58 565. 53
PREDICTED SUM SQUARED DEVIATIONS 75.02228
EXACT SUM OF SQUARED MISFIT(SSQ) FIELD/MODEL DATA
76. 09239 C
PREDICTED SUM SQUARED DEVIATIONS 75.31596
Table 62 A he dmonstration Of t simultaneous multi-site interpretation
chnique Model resistivities are in c2m arid depths in metres SSQ is the - ..,•.
•-•. ..- . .. :., . • . . . •r •• .
ctual suared misfit between ? field and calculated model data and the 1
predicted value is the squared residuals Iv - AxJ
129
The individual station results for the three traverses are presented in appendix
W. The interpretive composite sections derived from the 1-0 results are now
presented and discussed.
6.5.3 Interpretive geoelectric sections
For each traverse the 1-D most squares results are 'pieced together' into a
coherent structure that can be interpreted as characterising the resistivity
distribution of the region and by implication its geotectonic structure. These
are called the geoelectric sections. Only the top 40Km is shown except for the
Glen Loy traverse.
The interpretive section for the Glen Garry traverse is shown in fig 6.1 la. A
1-2Km thick superficial layer (probably consisting of weathered igneous and
metamorphic rocks) with resistivity in the range 1500-30002m is underlain by a
highly resistive (4000-110002m) ?crystalline upper crustal unit that is thickest
near the Great Glen fault (20-30Km) and under Gi (interpreted here as due to a
plutonic body which outcrops in the area). Between G2 and G4 this unit is
only 6-9 Km thick. A lower resistivity (300-15000m) unit underlies the resistive
upper crust. The depth to this low resistivity unit appears to be shallow (-6
Km) in the neighbourhood of G3. However, 2-D modelling is required to
ascertain the presence of any lateral inhomogeneity. Recall that this feature
was also inferred from qualitative studies of the data (chapter 4).
The Loch Arkaig geoelectric section (fig 6.11b) shows a less complicated
structure than the Garry section. Three crustal layers are distinguished. A
supercrustal layer, 1-2 Km thick and of 1000-32000m resistivity can be traced
continuously from Li to A4 west of the GGF. This layer probably consists of
weathered Caledonian metamorphics and is underlain by a highly resistive
(2500-70002m) layer, about 13-16 Km thick. A lower crust of 200-5002rn
resistivity forms the basal unit of the structure in this area.
The much longer Glen Loy geoelectric section is shown in fig 6.11c. Each
geoelectric unit exhibits a marked horizontal inhomogeneity with irregular
bounding surfaces that possibly reflect the complex tectonics of the region.
The upper crust is in general very resistive (2000-2200092m) and appears to
thicken from 10-15 Km at the traverse extremes to about 35 Km over a 12 Km
wide zone that is asymmetric about the GGF. A deep trough-like structure of
low resistivity (100-10000m) occurs at the position of GGF. The lower
crust/upper mantle layer is of low resistivity (50-4002m) but its thickness is
not well-determined.
130
Fig 6.11. Interpretive geoelectric sections for (a) Glen Garry (b) Loch Arkaig
and (c) Glen LOV profiles. The Vertical bars in fig 6.1 ic represent the most
squares limits of the 1-D models.
131
m
m
m
0km
18km
20km
38km
IiOkm
INVARIANT INTEI9PI9ETIVE SECTION : GLEN GARHI TRAVERSE a CGF
0km -
CJ (fl C ( (
J (I) D N
4uuu.lb Io ,'2l(m .0km
6000.0 5500..Q.. 5000.0
10km - 11800.0 I ..
._J - 8000.0
700.0 10km .
1500.0 6000.0
- ?' 750.0 20000. -
/
20km ' 3000.0 2000.
' -
.20km
5000-a 38km . 350. '? .30km
110km qokm
INVARIANT INTERPRETIVE SECTION : LOCH ARKAIG PROFILE b
GGF
(A)
Ic
10
20
30
E
40
a. LU 0
50
Ell
70
(A)
(Al
INVARIANT GEOELECTRIC SECTION : GLEN LOY TRAVERSE C
GGF t -
I °°° I I I I III!! H 8003000 ___40008000 ___8000+1)
An intermediate layer, about 10-15 Km thick, with resistivity in the range
900-15002m appears to be present from about 8 Km east of the GGF. This
layer is separated from the zone of thickened crust by a low resistivity
dyke-like body. The surface layer is thickest in the fault zone and its
immediate surroundings and is shallower than 2.5 Km elsewhere. This
structure will be compared with the regional 2-0 model and interpreted
tectonically in chapter 8.
6.5.4 Effective phase sections
Ranganayaki (1984) demonstrated the usefulness of pseudosections of the
phase of the eftective impedance for structural interpretation in 2-D
environments. Traditionally, pseudosections provide a general picture of
resistivity or phase variations with frequency; only qualitative inferences can be
made from such sections. In this study however, the phase data are mapped
onto their appropriate depths via the Niblett-Bostick transformation to give a
more realistic structural pattern. Only the Glen Loy section (shown in fig 6.12)
will be considered here. In a homogeneous medium the effective phase is 90
degrees, and it is higher or lower than this value in conductive or resistive
strata respectively. It is obvious in fig 6.12 that the top 10 Km (about 30-40
Km near the GGF) of the crust is resistive and is underlain by a conductive
layer. The phase contours show lateral variations in structure especially to the
east of the GGF. Note the indication of a buried vertical contact in the
neighbourhood of sites L16 and L17. Thus this new method of presentation of
the phase data provides an equivalent interpretive cross-section to that
obtained from the more rigorous data inversion (fig 6.1 lc). Additional
structural information are also provided by the phase section. Notice the
indication of structural discontinuity at the near-surface in the vicinity of L17.
This probably corresponds to the Fort William Slide. The underlying structures
appear severely contorted. The western part of the region appears to be
relatively less deformed. The Glen Loy Gabbro may be funnel-shaped with a
bottom depth of about 25 Km; and appears to be fault-bounded at the
near-surface in the vicinity of L4 and L5. The lack of observations between L2
and Li militates against any detailed interpretation west of L2. A general
eastward dip of the deep structures in the vicinity of the GGF is apparent.
134
I40km
50km
I t o
t0km
Ito
50km
: GLEN LOT THRVHSE
-4
-j
GGF C\i C!) ' in (ON000)
in co r- co — -4 —
0) '-4
-J
0km - 111111 I,111,1.ji <2Km>4 0km
70
nr 10km - 10km
20km
20km C) C,,
130 \
75.
30km
30km
60km Fig 6:12 Glen y. e:..o intervat is 5 degrees. The 60km
pecked line's correspond toapproximate bondaries between resistive <
nn°i ...,,i nn°i
CHAPTER 7 TWO-DIMENSIONAL NUMERICAL MODELLING STUDIES
The structural complexity of the Great Glen region requires the use of 2- or
3-dimensional modelling procedures for realistic quantitative interpretations.
Fortunately the main structural elements are, to a fair approximation,
two-dimensional with a dominant N49 ° E (magnetic) trend. The relevant MT
responses have been obtained with axes parallel and perpendicular to this
direction (see Chapter 4). The determination of 2-D models along with a
limited search for the range of possible models that fit the measured
responses for the various traverses will be discussed and the models presented
following an overview of the conventional techniques for deriving 2-D
geoelectric information.
7.1 Introduction
The search for a 2-D model proceeds either through forward calculations in
which the MT response of a prior resistivity model is obtained and compared
with measured data or by inverse calculations in which a model is discovered
from the field data. These techniques are plagued by the non-uniqueness
problem.
Direct inversion of 2-D MT data is difficult and highly non-linear. The
commonest way to solve the inverse problems at the present time is to
linearize them and apply the tools of generalized linear inversion (e.g. Weidelt,
1975; Jupp and Vozoff, 1977; Oristaglio and Worthington, 1980; Hill, 1987). The
greatest disadvantage with the linear inverse methods is that a realistic first
guess is required and even with a good initial model the iterative computations
do not always converge.
The forward solution process is cheaper and most widely used in MT work.
This process allows the modeller to have a feel for the structure and to
incorporate a priori information so that the ultimate model will be (at least to
some extent) consistent with the known geology and any available geophysical
data.
7.2 Model Construction
• Two-dimensional model construction is a difficult task and the search for
acceptable models can never be exhaustive for the same reasons as
enumerated in the previous chapter. The strategy adopted in this study was to
explore a limited range of models that are consistent with the field
136
observations. For this purpose, a finite difference scheme was adopted with
the Glen Loy geoelectric section as the starting model. The resultant models
were analysed for structural similarities with the available geophysical models
for the region and the known geology. Inverse calculations were then
performed on the most appealing models to determine the bounds on the
parameters.
7.2.1 Computer modelling technique
7.2.1.1 Programs and execution
The finite difference computer modelling program based on the method of
Brewitt-Taylor and Weaver (1976) and written by Brewitt-Taylor was used in
this study. A good account of the finite difference method and its application
to the 2-0 EM induction problems has been given by Brewitt-Taylor and
Weaver (1976) and the underlying theory will not be reproduced here. Rather,
the essential features of the numerical modelling technique will be recounted.
In the main, the region to be modelled is divided into a mesh of rectangular
elements (or cells) of variable sizes (the intersections of vertical and horizontal
grid lines form the grid points or nodes). Conductivity values are allocated to
the centres of these rectangular cells allowing for a smooth conductivity
variation between neighbouring cell centres. Using central difference formulae,
the finite difference representation of the Helmholtz equation (2.17) is obtained
for smoothly varying conductivity functions in which the value at each grid
point is given by a weighted mean of the specified values of the four ambient
cells. The applicable boundary conditions are as follows:
The conductivity is only a function of depth at infinitely large horizontal
distances (i.e. tabularity conditions prevail far away from the region of interest).
A constant magnetic field is assumed at the air-earth interface in the
H-polarization case and for the E-polarization case it is assumed that there is
an intervening thick air layer at the top of which the magnetic field is constant.
The fields vanish at great depths. A zero field value is assigned to each
of the nodes at the lower boundary and the 1-D finite difference solution
obtained; the field values so derived are assigned to the side boundaries.
The tangential components of the electric and magnetic fields as well as
the normal components of H are continous across internal conductivity
boundaries.
The reader is referred to the excellent paper on the improved boundary
conditions for EM induction problems by Weaver and Brewitt-Taylor (1978). For
137
each polarization and with the relevant boundary conditions, the finite
difference formulations are derived for the interior points of the grid and the
field solutions are obtained. Once a solution (E or Hx for the E- or
H-polarization respectively) has been obtained, the other necessary field
solutions can be computed via eqs. 2.16 b & c or 2.15 b & c as appropriate.
From the field values the surface responses (apparent resistivity and phase)
may be obtained.
The 2-D modelling process consists of two main stages:
The data preparation stage. The problem is defined by the grid
dimension, geometry and properties (relative permeability and permitivity, and
the conductivity of the subregions), the unit of distance, the frequency (or
period), the field components for evaluation, and the boundary conditions. The
information is read by the 2-D EM data preparation program PREP and
assembled in a form suitable for the next stage.
The problem solution stage. The main arithmetic program (ARITH)
utilizes the information provided by PREP to solve the 2-D induction problem.
The field values at the grid points are written onto files.
The form of presentation of the results depends on the user. The method
adopted in this study was to display the surface responses together with the
actual field data on the same plot for visual comparison using a computer
program written by the author. The above procedures are repeated for each
desired frequency and for each different model. In this study 9-11 equispaced
(on a logarithmic scale) frequencies were used. ARITH was modified by the
author to calculate the solutions at more than one frequency going through the
first stage only once. A reworked (more accurate and computationally efficient)
version of ARITH (Hill, 1987) was used in the final stages of the modelling
studies.
7.2.1.2 Mesh design considerations
The accuracy of the numerical solutions to the 2-D problem depends
strongly on the level of discretization used as shown by Muller and Losecke
(1975). However, not much has been published in the literature on grid design
but the following general rules (see Wannamaker et al, 1985) were applied in
the grid design.
The ratio of the size of adjacent mesh elements should not exceed 2 to
4.
Near a boundary the element dimensions should be less or equal to a
quarter skin depth in the medium where the element resides; at least 4 such
138
elements are necessary on either side of the contrast.
Far away from vertical contrasts (2 or 3 skin depths) the size of the
elements may be increased to about one skin depth of the medium of
residence.
At least 3 elements in the horizontal direction and 2 elements in the
vertical direction are needed to define a resistivity subregion.
Vertical grid spacings may be increased logarithmically with depth since
the fields decay exponentially. The same applies above the ground surface but
only a few elements (8-9) are needed because of the long wavelength of the
field in the air. Within the earth however, the maximum vertical spacings are
kept to 1 or 2 skin depths in the appropriate medium. Near the bottom, the
dimension of the elements is kept uniform.
7.2.2 Prior information and initial models
From the surface geology it is known that the GGF is a major crustal
dislocation and that the zone of crushed rocks is about half a kilometre wide
(see fig 1.3 and IGS Sheet 62E). The dominant structural trend as defined by
geology and aeromagnetics (Hall and Dagley, 1970) is NE-SW. The GGF is
probably a deep feature as suggested by past geoelectromagnetic work
(Mbipom 1980; Mbipom and Hutton 1983) and deep seismic reflection profiling
(Brewer et al, 1983) and its total length possibly exceeds 720Km (Harris et al,
1978). A structure with such characteristics may be assumed to be
2-dimensional.
MT studies of dyke and fault structures have been carried out by d'Erceville
and Kunetz (1962) and Rankin (1962) among others and their theoretical
responses are known. If the Great Glen fault zone is likened to a dyke and
assuming that a resistivity contrast exists across its borders, then the recorded
MT data along a profile across it would exhibit the known MT behaviour across
a dyke.
The actual apparent resistivities obtained at frequencies of 8,14,20,24 and 32
Hz for the E- and H-polarization cases are plotted in fig 7.1 against the station
distance from the fault axis for the Glen Garry and Glen Loy profiles. As
expected from MT theory, there is greater discontinuity in the apparent
resistivity values for the H-polarization than for the E-polarization data at the
fault zone. This provides a justification for the 2-0 interpretation of the field
data and a first quantitative indication of the basic structure of the fault - an
outcropping "dyke-like" body. The spatial resistivity plots for the Glen Loy
traverse also show that the resistivity (200-400 Ohm-rn) of the fault zone is an
139
GLEN LOT PROFILE H-POL (NL1W-4IRGNETIc)
I ------
K
/
V
0 0
GLEN CARRY PROFILE H-POL (N41w-MRGNETIC
•0
0
a
I: -'---- 0
0 0
GLEN LOT PROFILE E-POL (N'LSE-NRGNETIC)
• * /2 £
*
0 e Hz A III Hz + 26 Hz X 2,1 Hz
16 8 8 IL 2 0 2 IL 6 8 10 12 IlL NW SE rnun UflLIII ULr1 rHULI IIPIJ
0
GLEN GARRY PROFILE E-POL (N9E-HRCNETIC)
Mc
0
a- I-a-.
a,0 -l u, w
I- z ILl
cm 0-.
Cr0.
0
- DISTANCE FROM GREAT GLEN FAULT (KH)
Fig. 7.1 East-west profiles of apparent resistivity across the Great Glen fault for
E- and H-polarisations and frequencies 8,14,20,24 and 32 Hz. The direction of
the electric field in each case is indicated on the top right-hand corner of the
plot.
140
order of magnitude less than that of the surrounding crust (2000 -> 10000
Ohm-rn) in the E-polarization case (Meju and Hutton, 1987). Also, the low
resistivity zone appears to be about 2Km wide instead of about 0.5Km as
defined by surface geology. Notice that somewhat similar structures are
apparent at about 12Km west of the GGF on the Glen Gary profile and at about
6-7Km east of the GGF on the Glen Loy traverse.
The geoelectric section for the Glen Loy traverse (fig 6.11c, chp. 6) also
showed features that are common with the above observations and was
considered suitable as a 2-D starting model. The structure was extended over
great distances to the left and right of the traverse extremes. The problem
was well discretized with a grid size of 60 x 62 elements (see table 7.1). The
Glen Loy data sets were considered first and the fault zone was modelled as a
4Km deep trough-like conductive structure. The fit of the calculated response
to the observed data was only reasonable for the H-polarization (H-pol) data.
The conductive structure was made to extend to 10Km depth and about 21(m
wide. This provided a better fit to the H-pol data but the E-polarization (E-pol)
data were still poorly fitted especially at frequencies below 10Hz. The 10Km
thick resistive upper crust was made thicker (20Km) and this proved a better
model than previous ones (as judged by the E-pol fit). The conductive dyke
representing the GGF was later found to connect with the lower crust and this
basic structure was retained in all the subsequent models. The dyke was
modelled as an inclined or a vertical structure and as splayed structures, and a
single vertical structure extending from the surface to at least 20Km depth was
favoured. Other features were added onto this basic framework with necessary
adjustments until a reasonable fit was obtained for both polarizations. Another
conductive dyke was found to exist at about 6-7Km east of the GGF and buried
under a 4-7Km thick resistive cover. It should be mentioned here that all these
modifications to the preliminary models were done keeping in mind the
trade-off between thicknesses and resistivities.
Having obtained a satisfactory model for Glen Loy, the Glen Garry profile
was modelled next. A satisfactory model here means that the model response
curves lie within the zone defined by the error bars of the E-pol and H-pol
data for most of the sites. A comforting feature was that the same general
141
Table 7.1
Grid size used in the 2-D modelling: Glen Loy profile.
Number of Y-grids=60 and Z-grids=62
V-Grid positions(km)
1 -500 13 -10.1 25 -0.30 37 1.45 49 10.90 2 -350 14 -9.60 26 -0.20 38 1.55 50 13.60 3 -100 15 -8.40 27 -0.10 39 1.90 51 14.50 4 -44.0 16 -7.20 28 0.00 40 3.10 52 14.90 5 -20.0 17 -6.80 29 0.20 41 5.50 53 15.10 6 -14.4 18 -6.40 30 0.40 42 6.30 54 15.40 7 -12.6 19 -6.00 31 0.60 43 6.70 55 16.30 8 -12.0 20 -5.40 32 0.80 64 6.80 56 20.00 9 -11.4 21 -3.60 33 0.95 45 6.90 57 50.00 10 -11.0 22 -1.40 34 1.10 46 7.00 58 120.00 11 -10.7 23 -0.90 35 1.25 47 7.40 59 350.00 12 -10.4 24 -0.40 36 1.35 48 8.60 60 500.00
Z-Grid positions (km)
1 -145.0 14 0.70 27 6.31 40 33.90 53 62.40 2 - 46.0 15 1.60 28 10.90 41 38.60 54 62.50 3 - 16.0 16 1.90 29 20.00 42 40.16 55 62.60 4 - 6.50 17 2.00 30 23.00 43 40.66 56 62.80 5 - 3.20 18 2.10 31 24.00 44 40.83 57 63.40 6 - 1.40 19 2.22 32 24.30 45 41.00 58 65.20 7 - 0.44 20 2.50 33 24.40 46 41.60 59 67.40 8 - 0.22 21 3.40 34 24.50 47 44.00 60 85.00 9 0.00 22 3.70 35 24.60 48 51.00 61 250.0 10 0.10 23 4.00 36 24.9 49 58.00 62 500.0 11 0.20 24 4.30 37 25.8 50 61.00 12 0.30 25 4.60 38 28.5 51 62.00 13 0.40 26 5.00 39 31.2 52 62.30
model (apart from minor supracrustal differences and the presence of a
concealed dyke at about 10-11Km to the west of the GGF) produced a
satisfactory fit to the various data sets and this attests further to the
2-dimensionality of the region. On the basis of structural similarities between
the two profiles, a concealed (4-7Km depth) conductive dyke was introduced in
the Glen Loy model at about 10-11 Km west of the GGF and this improved the
fit of the observed and calculated responses.
The noise-degraded Loch Arkaig data were plotted against the Glen Loy
model and a satisfactory fit was observed. No separate computations were
done for this profile. The Glen Loy model was then accepted as typical of the
region and other equally acceptable models were calculated for the region.
142
Fig 7.2 a & b. 2-0 geoelectric models for the Glen Loy profile.
143
2-D GEOELECTRJC MODEL :. GLEN LOT TRAVERSE GGF
CD C\L C!) U) (O N CO
- CJ (V) V U) CON CO C)'-.
I 1 I LI Id 1.11111 Li I 1. 1.1 JôriJ lOno-2crin
a, -j
80000
<2Km>1 0km
10km
-I
C..
0 1. .C)
50 30km
200
0km
300 50km
60km
i 2000
mnrfrI re+.:e;..-
2-0 GEOELECTRIC MODEL : GLEN LOT TRAVERSE GGF
D C4 cn =r U) CO N CO
— C\J C) v U) CON CO 0)---•— —. — — — _J __I _I - - _J - - _j - _I J
1 I II 1.1 111111! lii 1 LI 2000 4Wf1115002500 —
C) —j
<2Km> Ok
Li 6
Ui
0 I C' 'cv)' L 201
50-80 L - - - -- 301
200
(6I
300
L501
L601
I 200O-•.
model resjstjvjtjes in NL
7.3 2-0 geoelectric models
Two acceptable and equivalent 2-D models for the Glen Loy traverse are
shown in figs 7.2 a & b. Only the relevant portion of each model is presented
even though the actual dimension used in the 2-D calculations was much
larger. An outcropping low resistivity (150-200 Ohm-rn) "dyke-like" body,
about 1Km wide, occurs at the position of the GGF and is bordered on either
side by thickened (30-451(m) resistive crust. This low resistivity zone connects
with the lower crust at depth. Two other similar dyke structures are concealed
(about 4-71(m depth) at 6-71(m east and 10-12Km west of the GGF. A
superficial layer of variable thickness (1-21(m) and resistivity (1000-3000
Ohm-rn) overlies a very resistive upper crust. The upper crust has a resistivity
of 8000-80,000 Ohm-rn and thickness which varies laterally from about 20Km
at the traverse extremes to about 40Km near the fault zone. The lower crust is
characterised by low resistivities (50-300 Ohm-rn) and the whole structure is
underlain by a semi-infinite layer of 2000 Ohm-rn at about 63Km depth which
is in accord with the depth derived from previous EM induction studies in the
region (Mbipom, 1980). The fit of the model curves to the observed E- and
H-polarization data for all the sites was good and is shown in figs 7.3 a-j. Note
the good fit in both amplitude and phase for the E- and H-polarizations at
most of the sites. However, the E-polarization data for those sites situated
over the Glen Loy Gabbro, an obviously 3-D mass, appear to have been
elevated staticly as suggested by the near parallelism of the computed and
observed response curves at all frequencies. Three-dimensional modelling
(beyond the scope of the present study) would be appropriate in studying such
anomalies. Sites L16, L17 and L19 also appear to show 3-D features affecting
mostly the E-polarization data.
In figs 7.4a & b are shown the fit of the Glen Loy model responses and the
Loch Arkaig data. The degree of fit is satisfactory.
A model for the Glen Garry traverse is shown in fig 7.5 to illustrate the
differences in superficial crustal structures between this area and the Glen Loy
region. The superficial layer exhibits a marked horizontal inhomogeneity. It is
of variable thickness (400m-2Km) and has resistivities in the range 1000-4000
Ohm-rn. It appears to deepen towards the fault from the traverse extremes.
The fit of the model responses to observational data was good and is shown in
figs 7.6 a - d.
146
Fig 7.3 a-i. Fit of the Glen Loy 2-0 response curves to observed data. The
apparent resistivity and phase information are presented for different
polarisations (E & H) at each station. The station position with respect to the
fault axis is indicated at the top of each set of plots.
147
Ci) tLi a:CY)
0 a-cr:C'j
D -J
-4
LIE .11)
E = D .
U) Lii a:CT)
a- a- cr c.J
0 -J
STATION 01ST. = —19.2KM
0
_ro) 0)
co Lii
=(T) = C) a- -
1000 lao rio I b.i b.o 1000 lao rio I b.i b.o FREQUENCY IN HZ FREQUENCY IN HZ
STATION 01ST. = —7. 14KM
L1) LI,
L2E L2H c,zr
U, U,
a: c) Lii Lii • a i
a: ,
a- - a- - 0 0 _J ._.
to D 0)
coCD uj M M
CD
Jl19kNNs Lii Lii cr D Ji cn = (Y) cr
=cY). a- - CL
tD iooa lao rio I b.i b.o boo lao rio I b.i b FREQUENCY IN HZ FREQUENCY IN HZ
L3H U)
=
U, LIJ cCCY)
ctC'J
(3 CD —J
tD 0)
(3 Ui SID to Iii in ct 10 =CT,
.0
L3E
an _1iIB.1.II
Cl, Lu
0 czCSJ
(3 CD —J
STATION 01ST. = -6.6KM
—' U,
—co
(3
90 co lU
cro =(f)
1666 160 10 1 b.i b.0 FFIEQUENCY IN HZ
STATION DIST. = -5.6KM
_Lfl Ll)
L4E L LIM
___• U.-,...
cn uj
ID
O)
(3 (.3 uj
U I, ecD
—Co no
Ui LU
=CY, CL
1006 100 16 1 0.1 b.0 FREQUENCI IN HZ
MR
4 1 f%
STATION 01ST. = -5.0KM
' -' U,
L5E
LLJ
=
CL a- a:C'i
D -J
—U,
= D .
U) u-I
a-a-crcJ CD D -J
, • 0)
u-i
CT
a-
1000 100 10 1 b.i b.a FREQUENCY IN HZ
ID —C)
CD ui
Co
Lu
=CT) (-
1060 160 16 1 b.i b.a FREQUENCY IN HZ
STATION 01ST. = -3.5KM
-' U,
E = D .
U, Lu a:CY)
a- a- c C'J
(.3 D -J
E = 0
U) Lu
CT)
a- c'J
CD 0 -J
W. (D I LW
LLI
a-I
1006 106 10 1 0.1 0.0 FREQUENCY IN HZ
.0)
(3 Lu
— Co
Lu (1) Cr D
Cl, a-
1600 160 io i b.i o.o FREQUENCY IN HZ
150
STATION 01ST. = -2.9KM
L7E ,..Lt)
1 ' -I
I L7H
I l
1 I W Ui I
cc Cr) 4 .1 1 • I I aI a. CL a- czC\J.l ctc'J CD CD 0 0 _J I —i _•
• 0)
ui
MO
WiT
ED I I* a- i a- I
o I
1000 100 10 1 071 b.a D
1000 100 10 1 0.1 b.o FREQUENCY IN HZ FREQUENCY IN HZ
STATION 01ST. = -2.2KM
LI)
L8E
c J
cc Cr)]
a. a- C'J Ci 0 —J
- I 0 0)
,14)
L8H
D .
Lu
CD 0 —J
0 0)
LD LLJ
LAJ •I
Ld
cr 0
DI DI 1000 100 10 1 0.1 0.0 1000 100 10 1 0.1
FREQUENCY IN HZ FREQUENCY IN HZ Ej
151
Li OH = D .
U, Ui
a- - a-crC'd CD 0 -J
0 a,
STATION 01ST. = —1.5MM
—U, 1
p I L9E I I e . 1 a
ujil
0 a-
CJ
CD D -J t-- —I __
-' U)
co IjJ
:CT)
a- - a- a:C'J
CD 0 -J -I
0 • 0)
LD I - Ld
eJ Lu
crn
(L HAI fro 01
1000 '100 '10 1 FREQUENCY IN HZ
0 CD
Lo ui
ui U) A cr0 = C) a-
0
1000 Ica '10 'i FREQUENCY IN HZ
STATION 01ST. = —0.7lM
1 L1OE
Of f
Ui I cc cT)J I. -
a- a- a: C'J
CD 0 -J — t 0 0)
CD Lu
eD J CD
Ui
gc, ICT)
a-
0, 1600 ice 'io 'i '0.1 b.o
FREQUENCY IN HZ
CD Ui
— Co
Ui (1) cr0 j m(T) a-
0 1000 '106 '10 'I
FREQUENCY IN HZ
152
FREQUENCY IN HZ
FREQU 0.1 0.0
N HZ
STATION 01ST. = 0.1KM
—U)
L11E L11H =
C1) Cl) Lu LU
C') CY,
_ w1IfL CJ c'J
CD CD D D
D D
CD CD
—(0
Lii U ' C) CY) =
a-
C C
1000 '1oo '16 '1 b.i b.o FREQUENCY IN HZ
STATION 01ST. = 0.2KM
I L12E = =
8 . LU
(V)
- a- - a- crc'J a:C'J
CD CD 0 0
-4
C C a)
CD CD
—CO co
LU LU
= Cr) cr = (V)
a- a-
C C
'10 1000 100 '1 b.i o.o FREQUENCY IN HZ
153
-' U, E
E =
(f) Lii Cr- CY,
a-a- - crc'J CD 0 -J
C 0)
L1I4E
STATION 01ST. = 1.4KM
Lt)
L13E
c
ui 1
a-I a c'JJ CD 0 -J
- I
—U) r x
U, Lii
0- a Cr c'J CD 0 -J
to 0,
ui e lD co
CD
I'I1I
Lii
CC O
a-
1000 Ioo lo I 'o.i b.o FREQUENCY IN HZ
0 0)
Lu CD
—Co
Lii Cn I
CY,. a- -
0,
1000 'ioo 10 I a.i b.o FREQUENCY IN HZ
STATION 01ST. = 1.9KM
CD LLI
Lii fIT Cr = C!) a-
1006 '100 'io '1 b.i b.o FREQUENCY IN HZ
—U'
L1L!H r =
I i LLJ
I
CD 0 -J
0
CD I uj otD
Co
ii IIIIla
0J 1000 lao lo 1 '0.1 b.o
FREQUENCY IN HZ
154
U)
I LI5H
m an
[UI CT) .
a— a-
CD 0 —J
-4
-J
= U) LU
& 0 ZC'J
(D 0 -J
Li 5E
STATION 01ST. =
ID
=cY, a-
1006 '1oo '10 '1 b.i b.o FREQUENCY IN HZ
0
MED
LU
=(T) a-
D
1000 '100 '10 '1 FREQUENCY IN Ill
STATION 01ST. = 6.9KM
— li)
E = Cl) LU
a- a- cxc\.1
CD 0 -J -4
L16E
II III I J H!
'j iI" TI
U,.
L16H = U) ui
a-cz" CD 0 —
D
CD
LU RD ecD
Lii U, cED
a-
to
• 0)
CD
i i a1
0J
.0 1000 100 10 i 6.1 FREQUENCY IN HZ
MR
FREQUENCY IN HZ
155
1tuuINLT IN HZ
0 • 0)
cm ui
LU ('
(T)
a-
04_ loot, 100 •10 •1 '0.1
FREQUENCy ' IN HZ 0
STATION GIST. = 8.6KM
LI7E 1
I L17H
i —
LU i cy)
0 .
] c1C'J a. 1 I c.D
iII!I aCJ
I CM
D 0)
RD
Lu Lu
;i
LLJ cn LU
a a-
- ID
I
1000 10 '10 '1 b.i b.o FREQUENCY IN HZ
STATION GIST. = 9.5KM
L18E £ LI,
1 ii L18H
LLJ
I I
cc 4 ' LU
a-I aI a- crc'J1 a- cr i cJ -I
C.DI
0
Li
co wJ CC Wjh1\ 0I I cn 1 a-
to 1000 loo lo '1 a. ii.a
FREQUENCY IN HZ
156
to D
Ui
XC') a- CL
1000 100 10 1 b.i b.o FREQUENCY IN HZ
.0
STATION 01ST. 15.5KM
Lfl
L19E L19H
Ii U) Ui
0- HAD a- EL aCJ
CD o 0
157
Fig 7.4 a & b. Fit of the Glen Loy 2-D model curves to Loch Arkaig data. The
apparent resistivity and phase information are presented for different
polarisations (E & H) at each station. The station distance with respect to the
fault axis is indicated at the top of each set of plots.
158
—U,
AlE E
D .
Ln
A1H x o
ci, Lu —
c'J
cm 0 -j
Cr) I Lu I cccr)1 fi
I' a-I III 1 a- I
cm 01
STATION 01ST. = —9.0KM
D 0)
co
cn
1000 100 10 1 b.i FREQUENCY IN HZ
0 — C)
CD
LU in
0 CT)
a-
0 .0 PE
STATION 0151. = —57KM
El I A2E
El
9I .j . •1 iRI If
I i'fJ c,I LU I Crm 1
a-a- - c'J
—U'
A2H =
FTT-
CL
Lo 0 -J
D —C,
men
1000 100 10 1 0.1 0.0 FREQUENCY IN HZ
0 -J
0 0)
Ld
u-I
a-
.0
FREQUENCY IN HZ
159
D 0)
LLI e lD co
LLJ
CL
STATION 01ST. = —3.4KM
1000 Ica lo 'i b.i FREQUENCY IN HZ
—LI)
A3H =
aCr) • II I Ii
a-a- - crCJ
0 -J
0)
cm
0
III ED
—Co
Iii
CCT) a- II
0
.0 1000 100 '10 1 '0.1 FREQUENCY IN HZ
—U)
E = D .
Cl) L&J CC C!)
a- a-
0 -J
lbi
STATION 01ST. = —1.8KM
L()
x =
Cl') I LJI (T)
CLI
czC'J-1
0 -J
.LI)
U, L&J
CL a- - Q:C\J
CD 0 -j
to
Lu
LuuJ
COD m cn
1000 100 10 1 0.1 FREQUENCY IN HZ
0
ui OED
0)
"-Co
Iii
cr = CT, CL
0
.0 1000 '100 '10 '1 '0.1 FREQUENCY IN HZ
uk
160
Al E
V
D Ei c\J C!) U,
ro CD N
0
'I)
•1-
> U) U) 4, I-
a)
0 E
99
SO m hO
A. CO
10
0
N 0
161
Fig 7.6 a-d Fit of the Glen Carry 2-D response curves to observed data. The
apparent resistivity and phase information are presented for different
polarisations (E & H) at each station. The station position with respect to the
fault axis is indicated at the top of each set of plots.
162
0 0)
CD
Iii U, cr0 = U, a-
0
.0
STATION 01ST. = —15.5KM
G1E L1)
G1H
slip
Ld
I * ui
cy,
Cr c\I
_J D —J
0 0).
Cm LLJ
eco
LLJ 00 cro rv)
0
1000 100 10 1 b.i FAEOUENCY IN HZ
.0
STATION 01ST. = —10.5KM
U, G2E
L()
G2H
- .Iuul. 1111.1
CC CV) I c
Cu
C CD D D -J
0 0 O) ,O)
CD
CD CD
I —Ca
Ui I U, a:0 = C')
0
1600 'lao 10 1 b.i 0.0 1000 loo 'io 1 b.i FREQUENCY IN HZ FREQUENCY IN HZ
.6
163
-fl u,
63E
= T = e •
U, U)
Cr- (n 0-0
CD CD
_j -
STATION 01ST. = —9.0KM
0
Lu
CL
1066 166 16 1 6.1 FREQUENCY IN HZ
D 0)
-I
CD
LLi
= (f)
D
.0 1000 106 io i FREQUENCY IN HZ
.0
STATION 01ST. = —6.5KM
L1) E __Ln.
G 4 H G'4E
• •UI&.. II •
ui cy,
CL a_ Cr c\i
Cr c'J
CD CD 0
0 -4 —I-
0 4_O)
CD
ED eCD
Wi
cc DII 0 = cn.I oI
1060 106 10 1 0.1 0.0 1000 100 10 1 0.1
FREQUENCY IN HZ FREQUENCY IN HZ ME
164
REQUENCI IN HZ
to ,0)
Lu
co LU to = a a-
10
STATION 01ST. = -3.0KM
.Lt)
G5E
tu
E I El
r1 IL II
a-ctc'i
-J —4
.Lfl
G5H E = D
a- - cEC'J
CD 0 -J
-4.
to 0)
CD LU
LU
cED CV)
a-
Wil
D 0)
LU in cr o = CT) a-
.0
FREQUENCY IN HZ
STATION 01ST. = -1.4KM
Lfl El I HE E I
1 Hi
cI,l uJ I a:C° -1
a- a cE
CD
-J -4
—U, x l
I G6E = I EPg ..
BIRO I !JI1 IT
9• 1 LU
CT
0 a- - cE C\J
CD 0 -J
LU Co
CO
0
.0 . 1000 100 10 1 b.i FREQUENCY IN HZ
a-
.0
165
STATION 0151. = 1.7KM
L() 1:1 I G7E
LLJ
El
8. 1 aCfl.I
a-a-ccc\i
CD
-j -4
1 G7H El = I
1 Ui II cr,1
0- 0- cr Cu
CD
-j
D _0)
co
1000 ba 10 1 0.1 0.0 FREQUENCY IN HZ
D 0)
CD RD co
1 cn •1 ~', fe
060 100 10 1 0.1 0.0 FREQUENCY IN HZ
166
7.4 Inverse calculations
7.4.1 Optimal resistivity distribution
Using a 2-D linearized inversion program written by Hill (1987) an optimal
resistivity distribution was derived for the region. The underlying theory of the
inversion algorithm is similar to that discussed in chapters 5 and 6 and can
also be found in Hill (1987). The main feature of the program is that inversion
is by ridge regression or singular value truncation (Jupp and Vozoff 1975, 1977)
techniques. The input models were the two Glen Loy models shown in figs 7.2
a & b. Basically each resistivity unit of an input model is divided into a number
of smaller blocks of the same resistivity and the problem is solved in the least
squares sense. The partial derivatives are computed by determining
numerically the effect due to a finite change in the resistivity of each of the
resistivity blocks. While this approach is less accurate than analytic
differentiation, it was considered adequate for the present study. The result of
this operation is a model with a minimum achievable error.
7.4.2 Errors on the model parameters
Assuming that a model with the minimum achievable error had been found,
we need to estimate the range of values which the parameters may have. The
errors are either determined from the parameter covariance matrix or by a
non-iterative most squares technique (Jackson, 1976). In the least squares
approach the model parameters may be correlated with each other so that the
diagonal components of the covariance matrix may not provide correct
estimates of the individual parameters. Using Hill's program, the most squares
method was applied to the two resultant models. The errors associated with
the model parameters were estimated by setting the threshold residual sum of
squares error (see sec. 5.5.2) at
QT = LS X 1.1
7.1
where QLS is the sum of squared residuals of the optimal solution.
The Jacobian obtained for the final iteration (i.e. evaluated at the optimal
model) was used in the calculations. No iterations were conducted for the
error estimations which, therefore, may have been incorrectly estimated.
Unresolved adjacent parameters (blocks of similar values) were amalgamated
and the corresponding columns of the Jacobian were summed. A new set of
parameters was then calculated and the individual errors re-estimated.
167
2-D GEOELECTRIC MODEL: GLEN LOY TRAVERSE
GGF .
0
10
20
30
I
w ° 40
50
60
70
0) 0)
OHM
rl 150.200 200.300 1500-3000 8000 50000: —30000
100,000
Fig 7.7 Most squares resistivity model for the Glen Loy profile
CHAPTER 8
"Plate tectonics has dominated our thinking for many years and continues to do so while we struggle to fit our geology to the theory and its variations. I wonder what the ruling theory will be in 30 years or more, and how much of present tectonic models will form part of it."
W.H.Poole, 1983.
170
INTERPRETATIONS AND CONCLUSIONS
The interpretive geoelectric models are appraised in the light of the
available geological and geophysical data and 2-13 gravity models are
constructed for the Great Glen region and the Scottish Caledonides. An
integrated interpretation of the various data sets was carried out within the
framework of plate-tectonics to unravel the deep structure of the region. The
possible causes of the observed crustal conductive structures are discussed
based largely on the concept of large-scale convective cells in capped
overpressured systems (Etheridge et al, 1983). Suggestions are made for
further work in this region and for the development of new MT data
interpretation techniques.
8.1 Geoelectric structural appraisal
Much of the discussion in this section will draw upon the MT results from
the Glen Loy traverse since it cuts across most of the geological features of
interest in the area of study. The 1-D and 2-D models show common features
of perhaps regional interest. An important feature of these models is the
presence at depth (4-7 Km) of a conductive "dyke-like" body that connects
with the lower crustal layer at about 6-7 Km east of the GGF. Its surface
position corresponds to a major geological feature in the area - the Fort
William Slide, a surface along which at an early stage in the Caledonian
deformation upper crustal rocks became detached from and slid bodily over the
underlying rocks. An outcropping conductive dyke-like body occurs at the
position of the GGF. These two structures do not appear to be connected
together. The zone of thickened crust is asymmetric about the GGF in all the
models but is about 12 Km wide in the 1-0 case and about 18 Km wide in the
2-0 model. The thicknesses of the upper crustal blocks are underestimated by
the 1-D models by about 20%. A deep-lying conductive dyke-like body
appears to be present at about 10-12 Km west of the GGF along the profile
(this body was better constrained by the Glen Garry data) and is connected
with the lower crustal conductor. The lower crust appears to be of the same
electrical character on either side of the fault.
8.1.1 Relationship with previous geophysical and geological observations
In many respects the 2-D geoelectric model (redrawn in fig 8.1a) is in
accord with the results obtained from previous geological and geophysical
studies in the region. Hutton et al (1977) from magnetometer array studies
171
Fig 8.1 Some geophysical results for northern Scotland.
A simplified 2-D geoelectric model for the Great Glen region.
An electrical model for northern Scotland (from Mbipom 1980).
Magnetic anomalies of the Great Glen region (from Smoothed
Aeromagnetic map of Great Britain and Northern Ireland, IGS 1970).
Seismic structure along the LISPB line (after Bamford et al 1978). The
numbers in the figure give P-wave velocity in km/s.
Part of the WINCH profile in the sea of Hebrides (after Hall 1986). The
2-0 MT model is superimposed for comparison.
The SHET deep seismic section across the Walls Boundary Fault (from
McGeary 1987).
172
a. -U
a
0
10
20
30
0. LU
40
50
60
70
2.D GEOELECTRIC MODEL: GLEN LOY TRAVERSE
GGF 46km-44
•/ / i;
lip
g -
- •- - ,-. .. __,,
FPIIãu__ 'T11JT
•.,.I OHM I I I ____ ____ 150.200 200-300 1500-3000 8000 100,000
Maine Thrqst K9Fuit
LEWISIAN CALEDONIAN METAMORPHIC ZONE FORE LAN 0
Lalrg Gricit Glen Grampian Highlands
m m >10000 5000-10000
ElE 50-500 Qm Uncertain
structure
173
7 j35 ,p
4Y h
C:S
C is Ih a 'V +50
UTH Pen
B o + 50— .s4 thy Po
m •
0 L. V
100 H .
d f'.ISt.
2 P iE A N
Z-1 —20 hM, — Ki
bm Hint :
Ligr
: .
a
, +
$ S —
rV
C. M, 0 N 0 C 0 I
R ... h..
$ S
) o.v. ' J ......
a
15 *
00
Lh
A R S E M0a\S
1r o
T if
..U0RDEE'' To.
0 \+i oo"
'S • .'---- ... 0• \
•:. •/4—'.. R ,.•..\
- . . . . •.0.
rl EPPS-
J c3 U r4'ir
—h-_ KjsIols 7t . T1.
—I
cui
j UNO
0 5500
R . Abnood *
P-S
FE HIRE
'-S
E
-c 0.
3 Q)
)A 0 )FJ E
'C 0
lu o_ 0
NW
0
Ei —S V
C 0
10
U V
V I-
e 15
I I I I I
- Dubh A auk / Colonsay Fault
- - '— S_i_I
- } Great Glen -_-------- —
Fault Zcie — 1h-Th
— II
- - I - I —
- J•-------- - - — - -:-- -=7 ---
base 0i the _-_- - crust
- ------
mantle detachment zone
WINCH Sea of the Hebrides 10km
SE
10600 11000 11400 11800
E
M.
GG HBF SUF
on -
- 6S
--
4 4
f
WALLS BOUNDARY FAULT
0 1.0, 2,0 ao140 1 -70
MOHO
• ------ - - --------- - - —
- - —
- - - ;.I-- -- ------- -S.-- -
-:--- -:..---- _- - — - .-: - -_.:- -z.°- ,z-__ -------_ -a-- -. -
- H -10
-S 5- -
-
MO HO? -
15
175
showed that an anomalously conducting zone exists along the Great Glen (the
conductive zone being slightly offset to the east of the fault). Mbipom (1980),
Mbipom and Hutton (1983) and Hutton et al (1981) showed the presence of a
conductor in the Great Glen area and the lower crust was found to be
distinctly conducting while the upper 10-15 Km of the crust in this region was
found to be resistive (5000-10000 lm) with a 10 Km gap at the GGF where an
upper conducting layer connects with the conductive lower crust/upper mantle
(fig 8.1b). The 10 Km gap may at first sight appear over-estimated but when
one considers their station spacing (10-15 Km) and the frequency sounding
bandwidth (5x102 - 300 4Hz) it seems probable that they observed the
combined signatures of the conductive dykes of the Great Glen system.
Kirkwood et al (1981) required models ranging from a line current at 80 Km
depth to a uniform current sheet, 60 Km wide, at 10 Km depth to fit long
period magnetovariational observations near the GGF and favour an
interpretation in terms of a conducting zone in the 20-80 Km depth range for
the fault. However, the structure of the fault was not resolved by these
previous studies that lacked adequate control of the upper crust. The present
study thus provides new information on the resistivity structure across the
Great Glen area.
According to Ahmad (1966,1967) the Great Glen fault is active and the
Strontián and Foyers granite complexes extend to 22Km or greater depth, more
or less as has been shown in the present 2-D geoelectric model. Handa and
Sumitomo (1985) and Meju and Hutton (1987) from studies in Japan and
Scotland respectively have suggested that upper crustal dyke-like zones of low
resistivity may be hallmarks of active fault systems. Dimitropoulos's (1981)
Grampian granitic body with a top at 7Km and bottom at 19Km between the
GGF and Loch Tay fault to the south, - an interpretation rejected by Hipkin and
Hussain (1983) - and the inferences from heat flow measurements (Pugh 1977,
Oxburgh et al 1980) are consistent with the thick (20-30Km) highly resistive
upper crust south-east of the GGF presented here.
The magnetic ridge over the GGF as seen on the Northern Sheet of the
Aeromagnetic anomaly map of Great Britain (fig 8.1c) coincides with the
position of the inferred zone of thickened electrical crust. From three profiles
across this anomaly Hall and Dagley (1970) suggested the presence of a thick
horizontal sheet with a top at 8-15Km and bottom at 15-26Km (i.e. from the
Loch Ness-Moray Firth area to the Foyers granite complex) and bounded in the
northwest by a nearly vertical plane coincident with the GGF. However,
176
* see fig. 1.2r
another profile across this anomaly in the Loch Lochy area was interpreted as
due to a normally magnetized unit bounded by the GGF to the south and
extending from a shallow depth to about 15Km or greater. In either case, the
bottom of this magnetic sheet agrees somewhat with the top of the lower
crustal conductor in the 2-0 model and the GGF is of a vertical disposition.
The seismic refraction model of Bamford et al (1978) is shown in fig 8.1d.
Note that the 20Km refractor may be correlated within the bounds of the
geoelectric model with the top of the lower crustal conductor.
A section of the WINCH deep seismic reflection profile across the offshore
extension of the GGF is shown in fig 8.1e with the Glen Loy 2-D model
superimposed for comparison. Note the correspondence between the base of
the resistive upper crust and the zone of prominent reflectors in the lower
crust. BIRPS' deep seismic reflection profiling in the north of the Shetlands -
the SHET survey (see fig. 1.1) - crosses the Walls Boundary Fault (WBF), a
suggested northern extension of the GGF. The seismic section is shown in fig
8.1f. The results show the WBF as a near-vertical structure which transects
the entire crustal thickness and offsets the continental Moho; the Moho
reflection time was found to be significantly different on either side of the fault
zone implying a small wavelength structure on the Moho directly beneath the
surface position of the WBF (McGeary, 1987). Assuming structural continuity
between the GGF and the WBF, crustal thickening as detected by geoelectrics
may be invoked to explain the local Moho topography in travel-time. Note the
presence of sub-Moho reflectors in the SHET and WINCH (Fig. 8.1e) profiles. It
is now known that beneath the essentially vertical frictional slip horizon of
strike-slip faults is a zone of quasi-plastic shearing (Sibson 1983) which in
some cases may be sub-horizontal decoupling shear zones (Lachenbruch and
Sass, 1980; Prescott and Nur 1981). This may be the case at the typical Moho
depth beneath the Great Glen and the Walls Boundary faults. However, caution
must be exercised when comparing mainland and offshore structures in
Scotland since the exposed metamorphic Caledonides narrow rapidly on the
mainland from north-east to south-west. More so, Hall (1985,1986) suggests
that the Dalradian terrain of SW Scotland occupies an area geophysically
distinct from that of the Grampian Highlands and separated from it by the
Cruachan lineament.
The 2-D geoelectric model can also be shown to accord with the known
geology of the region. The highly resistive upper layer is consistent with a
metamorphosed, highly granitized upper crust. Outcrops of granitic and other
177
crystalline rocks abound in the area. The outcropping conductive dyke at the
position of the GGF may be related to the crushed rocks of the fault zone;
these rocks probably derived in part from the initial sedimentary cover of the
type now represented by the narrow strips of sediments (Old Red Sandstone)
that occur in the fault zone on the south-east side of Loch Lochy (fig. 1.3).
The concealed conductive dykes and the thickened resistive crust re-affirm the
long held geological views of a deeply sheared basement (Watson, 1977) that is
perhaps extensively block-faulted (Johnstone et al, 1969; Tanner et al, 1970;
Watson and Dunning, 1979) thus forming basins within which were deposited
the pre-Caledonian cover-units whose thickness suggest active subsidence
(Johnstone, 1975). The emplacement of the Fort William Slide and the Loch
Quoich Line (Clifford 1957; Roberts and Harris, 1983), two major tectonic
boundaries to the east and west of the GGF, may have been controlled by
these buried shear zones. The GGF is inferred to form splays off the west
coast of Scotland and this feature may possibly be related to the multiple
conductive dyke systems of the geoelectric model. Although the GGF is about
2Km wide at the near surface, it narrows to about 800m at >41(m depth which
agrees with the 500m suggested from surface geology (see IGS Sheet 62E).
8.1.2 Integrated interpretation of MT and Gravity data
Although a large coherent magnetic anomaly is associated with the GGF,
the gravity expression across the faults on land is relatively featureless (broad
negative low) and presents an interpretation problem (Hipkin and Hussain,
1983). Computation of pseudogravity anomalies (Jarvis 1980) failed to resolve
the apparent conflict between gravity and magnetic anomalies over this fault
and Hipkin and Hussain suggest that a plane truncation in the lower crust by
the Curie isotherm may resolve the dilemma.
Since the 2-D MT model appears to be consistent with much of the
geophysical and geological observations on the fault, an integrated
interpretation of MT and gravity data sets was considered a worthwhile
approach to help clarify the nature of the Great Glen gravity anomaly.
The integrated interpretation problem is posed, as follows:' Assuming that
the structural geometry is reasonably well determined by the MT method and
that there is a regional vertical gravity gradient that does not generate any
gravity anomaly, we want to estimate the departures from this background field
which adequately reproduce the observed gravity data. As in the MT case
2-dimensionality is assumed. The gravity data (described by Hipkin and
Hussain 1983) were obtained (with a point spacing of 21(m) from the land
178
A
S
S
G .
D
Fig 8.2 Location map of the gravity and MT profiles
179
a. LA
--8
A £
A £
A
C ci
.; ci
-17
12 1-025 +.025
18. 28k 8k E 24
30 -.01 -C
36 r >80k
42
48 54 'aa observed gravity
60 calculated gravity
A A
7A A / A
A\ I Lt
\o
A
+.025
-.01 >80k
8-90k
—.01
density contrast in g/cc resistivity in ohm-rn
Fig 8.3. An integrated geophysical model for the Great Glen region (a) Bouguer
gravity anomaly of profile GG'.
(b) MT/gravity model for profile GG'
180
gravity data bank for northern Britain (Hipkin and Hussain, 1983) for a profile
approximately parallel to the MT profiles and as clear as possible of known
granites (about 11Km north of Glen Garry). The gravity profile (labelled GO' in
fig 8.2) is coincident with that of Jarvis (1980) and runs from the Moine Thrust
to Loch Ericht fault (194E843N-266E778N, British national grid coordinates in
Km). The Bouguer gravity anomaly of the profile (fig 8.3a) has been described
by Hipkin and Hussain (1983). The data were fitted by the author by 2-D
forward modelling with a Taiwani (Taiwani et al 1959) algorithm (the computer
programs were provided by R.Hipkin and T.Genc).
The resultant 20 MT/Gravity model is shown in fig 8.3b. This model was
obtained only after a few forward calculations. Note the good fit between the
observed gravity data and the calculated responses of the MT/Gravity model
(fig 8.3a). The regional MT model incorporates recent geoelectric results (Hill,
1987) from the Moine Thrust area (described in subsection 8.2.2). The zones of
thickened electrical crust coincide with the broad gravity lows. These zones
are characterised by smaller density contrasts (- -0.01g/cc) than the
surrounding regions (- +0.025 g/cc). Thus an interesting observation in the
Great Glen area is that the magnetic field values (>150nT) are high where the
gravity field values are low and the electrically resistive crust is thickest.
It cannot be emphasized enough that the 2-D geoelectric model appears to
be consistent with the Great Glen gravity data and that crustal thickening (as
detected by geoelectrics) can be invoked to explain some of the major gravity
anomalies in the Scottish Highlands. The author has no doubt that the 2D
geoelectric model shows significant (hitherto undiscovered) features compatible :
with much of the geophysical and geological observations in the Great Glen
region. On these grounds, it is reasonable to propose a close relation between
the complex crustal electrical structure and regional tectonics.
8.2 Deep geology of northern Scotland
As the Great Glen forms a natural boundary between the Northern and
Grampian Highlands it is also appropriate that the results of this study be
interpreted in terms of the deep structure of the Scottish Highlands as a whole.
Previous geological and geophysical results from the areas adjacent to the
Great Glen area are now considered and reconciled with the present
geoelectric model into a feasible tectonic framework. Firstly, however, a
synopsis is presented of some outstanding geological problems in the region
which any tectonic model should try to explain.
181
8.2.1 Gross geological overview: problematic observations
Current hypotheses suggest that most of the Moine rocks north of the GGF
as well as some areas immediately to the southeast of the Great Glen, have
suffered pre-Caledonian - Grenville (1000 Ma) and Morarian (720 Ma)-
deformation and metamorphism and have merely been recrystallised during the
Caledonian orogeny. The areal extent of these orogenies remains unclear
(Johnson et al 1979) and their influence on the metamorphic zonation in these
areas is still uncertain (Fettes 1983). These problems are compounded further
by the discovery of only Caledonian deformations in the Moines of Sutherland
by Soper and Wilkinson (1975).
Detached slices and wedges of Lewisian rocks are interleaved with oine
rocks towards the boundary between the Morar and Glenfinnan Divisions
forming a laterally extensive (250Km) NNE trending zone (Tanner et al 1970).
Tanner and coworkers suggested that these may have been derived from a
basement rise dividing the Moinian depositional basin into two troughs.
Johnstone et al (1979), adopting the assumptions of Tanner and coworkers,
raised the possibility that whereas the western trough was filled largely by
fluviatile or deltaic sediments of westerly provenance, the easterly trough
received detritus from different sources. The location and areal extent of these
depocentres are at best conjectural. Also, Watson (1984) argues that about
25-35 Km of crystalline material was eroded from the Highlands forming
sediments 1.5 times this volume and that most of this volume should have
found its way into Ordovician and Silurian basins in or at the margins of the
North Atlantic continent. Such basins have not yet been identified.
Garson and Plant (1973) suggested a subduction related Morarian event
with a suture coinciding with the GGF but Johnson (1975) postulates that the
suture is hidden by the Caledonian (Glenelg and Ross-shire) nappes (see also
Stewart 1982, p.418) and argued for a continental collision tectonics for the
Morarian episode, with the Moines deposited on one continent (the Baltic
Shield or microcontinent) and the Stoer and Torridon groups on another (the
Laurentian). These hypotheses need considerable testing as microplate
structure of this part of the Caledonides has not been identified.
Four regional deformation episodes (D1-D4) are recognized in the
Glenfinnan and Loch Eil divisions of the Moines and Roberts and Harris (1983)
suggested that the D1-D2 structures are of pre-Caledonian age. Structural
studies in the Glen Garry area (Holdsworth and Roberts 1984) showed a N-S
transport direciton during D2 which is markedly different from the Caledonoid
182
WNW-directed movements. The significance of this N-S trend is not known.
The relationship between these northern Moines and those to the south of the
GGF is still debatable (see Johnson et al 1979; Piasecki 1980; Piasecki et al
1981; Baird 1982; Roberts and Harris 1983).
The origin and nature of the GGF as well as the translations on it is also
still controversial. Considerable strike-slip movements have been suggested
from regional studies by several workers (see section 1.3). However, evidence
for some brittle or ductile deformation of the wall rocks indicative of
large-scale strike-slip movement is missing; near the GGF, Parson (1979)
detected small-scale structures indicating NW-SE compression and no
structures indicative of lateral NE-SW displacement were observed.
The enormously thick Dalradian rocks of the Grampian Highlands are
traditionally believed to have accumulated in a Dalradian basin whose exact
position is a matter of conjecture. The nature of the basin (whether
continental or oceanic) is not fully resolved (e.g. Phillips et al 1976; Graham and
Bradbury 1981). The high pressure to which much of the Dalradian fold-mass
was subjected is thought to be indicative of extreme thickening and shortening
of the weak basin fill while the anomalous low pressure Buchan region of NE
Scotland (which is permeated by large volumes of basic magma) represents an
area of least thickening (Watson 1984). However, no satisfactory mechanism
has been found as yet to achieve the required crustal thickening (see Watson
1984, p.198). The southwest Dalradian basaltic activity was also thought to
have occurred along a NE-SW trending (Caledonoid) rift related to the
presumed opening of the lapetus Ocean to the southeast (e.g. Graham 1976;
Graham and Bradbury 1981; Plant et al 1984) but Fettes and Plant (1985), Fettes
et al (1986) and Graham (1986) present radically different interpretations based
largely on trans-Caledonoid deep structures. However, no direct, and by all
means compelling, geophysical evidence for the deep structure of the region
has so far been provided.
Along the southern margin of the Dalradian and largely in faulted contact
are a series of exotic rocks (black phyllites, cherts, grits, amphibolites, spilitic
lavas and serpentinites) known as the Highland Border series. Henderson and
Robertson (1982) interpreted these rocks as tectonically emplaced ophiolites
(see also Garson and Plant 1973) derived from an ocean basin to the north of
the Highland Boundary (in the Central Highlands or farther north) but there
remains the chief difficulty of identifying where the respective ocean basin
developed. According to Henderson and Robertson, the position of the
183
postulated basin bears strongly on the problem of 'root-zone' to the Tay nappe
and related structures which are outstanding geological puzzles.
Although the isotope systematics of some Caledonian granites indicate a
dominance of crustal provenance (Hamilton et al 1980) and a change in the age
of the basement across the Grampian Highlands (Halliday 1984), the Caledonian
granites are not easily related to active subduction and there is no obvious
mechanism for their generation (Halliday 1984).
Multi-disciplinary evidence of the deep structure across the Scottish
Highlands will now be presented that will test some of these geological
hypotheses and hopefully facilitate a clearer assessment of the above
problems.
8.2.2 Evidence of deep structure
A recent short period (0.1 - 800 Hz) MT study of the Moine Thrust region
(Hill, 1987) (profile CC' in fig 8.2) revealed an anomalous resistivity distribution
to the west of the Lairg area. Although the data are sparse with poor control
on the precise location of lateral variations (5Km average station spacing), most
of the measurements were made between Loch Oykel and Lairg, that is in the
region of the inferred Moho offset of Faber and Bamford (1981) and the
effective orogenic front of Watson and Dunning (1979). Hill derived a 2-D
model which showed the presence of a 10-20 Km wide zone of thickened
(>40Km) resistive (>80000 m) crust at the centre of the profile, bordered by a
normal 16-20Km thick resistive (8000 czm) upper crust. The lower crust was
found to be more conductive (200 - 1000 2m) everywhere. Mbipom (1980)
indicated a thickening of the resistive upper crust and a thinning of the
underlying conductive layer in the region of the Dalradian outcrop (see fig
8.lb).
On the basis of the interesting results obtained from the Great Glen
MT/Gravity modelling (fig 8.3a&b), it seemed worthwhile to fit the gravity data
along an extended profile (HH' in fig 8.2) running from the Lewisian Foreland
(164E870N) to the Midland Valley (335E720N) and parallel to the WINCH seismic
reflection profile (Brewer et al 1983). The objective was to shed some light on
the deep structure of the Scottish Highlands and to see if some of the features
seen offshore could be detected on the mainland using techniques other than
seismics. The Bouguer gravity anomaly for this profile is as described by
Dimitropoulos (1981). A good agreement between the calculated and observed
regional gravity data was obtained by the author as shown in fig 8.4a for the
model of fig 8.4b. An interesting feature of this resistivity-density structural
184
E 1
>' a
0
0
>'-
a I-
0-3
— 3 -c
0 41, 04
4
1
AA a
A
)
2 AA
6
j
LU
D <22kcn
)
6 I+.oe t,025-8k 015 -.011 .03 H _.o;cc
+.025 !cc 2
1'lk 8-80k
B . ______________ +.043g/c CL p.025 '•025
8k >80k 4
.02 >80k Tj
2-3k
(. observed gravity density contrast: g/cc
-.. calculated 9ravity; b resistivity : ohm.m
Fig 8.4. An integrated geophysical model for northern Scotland.
(a) Bouguer gravity anomaly along profile HH
(B) MT/gravity model for profile HH'
185
Moine Thrust extcnsion(?) - Colonsay Basin Highland Boundary Fault
Minch Basin Stanton Trough L•.___.___ Great Glen Fault Zone J Rathlin Island Southein Uplands Fault
Soiway Basin
10 I
:
Flannan Thti Flnnan Thrust(?) Sub-MOHO reflector lapetus Suture(?1 Reflective Lower Crust MOHO
12
km
50km -,
Fig 8.5. A comparison of MT/Gravity and seismic (after Brewer et al, 1983)
structure across the Scottish Caledonides
CO 0,
model is the need for another zone of thickened crust (-.42Km) in the region of
the Dalradian outcrop. The model thus confirm and refine Mbipom's model for
the Dalradian region. This alternative interpretation may possibly shed some
light on some of the on-land problematical geological phenomena and the
unexplained features on the WINCH seismic section (assuming lateral continuity
of structure offshore).
The regional MT/Gravity model of fig 8.4 is shown again in fig 8.5 together
with the WINCH seismic section (on the same horizontal scale). Note the
correspondence between the zones of thickened (-.421(m) crust in the
MT/Gravity model and the regions of truncated seismic reflections that extend
from the near-surface to sub-moho depths (-42Km). It can hardly be
coincidental that almost identical structures are revealed across the Caledonian
orogen by the three geophysical methods since each measures a different rock
property. Because each of these methods is most effective when the contrast
in the observed parameter is greatest and the contrasts may occur at different
depths for each method, this structural concordance is seen as evidence of
strong lateral (NW-SE) differences in the deep structure of the Scottish
Highlands. No such correspondence has been reported elsewhere to the
author's knowledge. It is perhaps noteworthy that Bamford et al (1978)
ascribed the cause of the uncertainty in LISPB interpretation in the Grampian
Highlands (see fi.g 8.1d) to rapid deepening to the south of the GGF and north
of the Highland Boundary Fault. This inference is consistent with the present
interpretation and the MT/Gravity model perhaps explains the seismically
detected Moho offset (Bamford et al 1978; Faber and Bamford 1981).
The approximate locations of the inferred zones of thickened crust are
shown in fig 8.6a. Note the southward increase in the size of these crustal
anomalies.
Further geophysical evidence of the deep structure of the region comes
from the magnetometer arrays study of Hutton et al (1977) the results of which
are shown for a period of 700 sec in fig 8.6b. In this figure, notice that the
hypothetical-event Z (vertical magnetic field component) zero contours (labelled
B and C) seem to correlate with the Great Glen and Dairadian anomalous zones
respectively. Hutton et al (1977) drew attention to the relation between their
data and the results of Garson and Plant (1973) and suggested that these
anomalous magnetic variation features may also be related to ancient plate
boundaries. It is also noteworthy that the areas of recent deep-focus (221(m)
earthquakes (BGS data), Kintail (10-08-74), Fort William (13-11-72) and Oban
187
Fig 8.6. Comparison of the MT/Gravity model and previous models for northern
Scotland.
Location map of the MT/gravity inferred zones of crustal thickening.
Magnetic variation anomaly map of Scotland (after Hutton et al 1977).
The Z contour interval is 0.2nT and the zero Z contours are represented by the
dotted lines.
(C) Distribution of belts of alpine type ultramafic rocks in Scotland (from
Garson and Plant 1973).
Map showing the distribution of wedges and slices of Lewisian rocks in
the Moine rocks of Scotland (from Johnstone 1975).
Distribution of late Caledonian granites and associated rocks in Scotland
(from Watson 1984).
Distribution of late Caledonian granites and the location of the
Mid-Grampian Line (after Halliday 1984). The MT/gravity profile HH' and the
position of the Dalradian crustal anomaly are indicated.
188
t U L1P
PILO
Aft
Okm / Unst
4 Zone of discontinuous outcrops F.$Igr
4 &$ of alpine-type ultramafic rocks
cJSit. of ancient oceanic tr.nch
ç and direction of lithosphunc sub- dctson
Scal. 6
ii
/ c a
'WI
• ....A ...
I I F ,y.cc Phoy
S verness
mtly
I dor
4 • •
Colon
I -?•'•3
Cf,) • ,' i —063
........s......
189
190
+ MO/NE CLUANIES THRUST .13(1)
RAIAGAI N -133(l) \ \ \..—
GREAT GLEN FAULT -.
BEN NEVIS C)
-76(1)
STRONTIAN
-63- .06 (6) 7
C TI yE -48(4)
ROSS OF MULL
-97(I) KILMELFORD -5e-.04(1I)
DISTINK HORN +1 3 (1)
Late Cal.000aan gran$teS and .SsOC.Sl,d PIStOnS
S tysolte
Post. CaIsdOttIen
S Sc
Anomalous tranIverse r ZOtt• marked by gravity gr.d..nt. i5\
pprfl,tO 0 ,0.1
U I Loch Shin n. i i Cruachen Un. ( ) 0 ' c' ç3
C Catrngovm granIte
rIG Gl.ncoe ring con.olev - 0 50 radOn,
a
4. II e -
\\\\V
k
BONAR BRIDGE m
0 50 100
ISO -68(l)
HELMSDALE
50 100
mIles
-$6- - 5- 0(6)
FINOHORN -106(1)
FOYERS
HILL OF FARE -18(1)
ABERDEEN I 0 127(1)
HIGHLAND BOUNDARY FAULT
LOCHNAGAR - 27(%)
SIRATH OSSIAN -63 (I) MID GRAMPIAN LINE
GARABAL HILL -37(1)
•1
- CHEVIOT
4 2(1)
SOUTHERN CRIFFEL L
UPLANDS FAULT -04 (3)
DOOM HAP 0(1)
FLEET
ESKDAIE IAPETUS -51(1)
KEY: CNd 1 (N) SUTURE f
191
(29-09-86) lie within the inferred zones of crustal thickening.
Much geological evidence can be shown to support the above geophysical
inferences. The NNE trending belts of alpine type ultramafic rocks in the region
discussed by Garson and Plant are presented in fig 8.6c. They agree in
geographical location not only with the magnetic variation anomalies but also
with the three anomalously thick crustal zones discussed in this thesis
especially in the Great Glen and Dalradian areas. The northernmost zone
correlates with the laterally extensive NNE trending belt of Lewisian rocks in
the northern Moines (fig 8.6d). Also, a remarkably good agreement is seen in
the northwest corner of Sutherland and in the Southwest Highlands (Oban area)
between the Z zero contours (8.6b) and the anomalous trans-Caledonoid zones
marked by gravity gradients and appinite pipes (fig 8.6e) namely the Loch Shin
line (Watson, 1984) and the Cruachan lineament (Watson 1984; Plant et al 1984;
Fettes et al 1986; Graham 1986; Hall 1986). The appinite suite of rocks is
regarded as being derived from mantle sources and the Cruachan lineament
has been shown by the above workers to be a deep basement feature
separating crustal blocks of contrasting physicochemical characteristics. The
MT/Gravity model also shows some correspondence with the thick-skinned or
autochthonous model for the Moine thrust area suggested by Watson and
Dunning (1979) and Stewart (1982).
Isotopic evidence can also be shown to support the above geophysical
inferences. From the distribution of inherited zircons (<1000 Myr old) Pidgeon
and Aftalion (1978) proposed a major Proterozoic crustal component in granites
north of the Highland Boundary fault, and isotope systematics (Hamilton et al
1980) of some of the Caledonian granites (such as Strichen, Ben Vuirich, Ben
Nevis, Etive and Foyers intrusions and Strontian biotite granite) also indicate a
dominance of crustal provenance. Recent isotopic data (Halliday, 1984)
confirmed and refined these previous models of the basement under northern
Britain. It was shown (Halliday 1984) that although a major mantle component
is important, the lower and middle continental crust have affected the isotopic
compositions of the late Caledonian granites to the extent that a change in the
age of the basement is recognizable across the Grampian Highlands. Halliday
placed the basement boundary through the middle of the Grampian Highlands
and called it the Mid-Grampian Line (fig 8.6f). It is significant that the
Mid-Grampian Line correlates with the northern margin of the inferred zone of
thickened crust and Hutton et al's (1977) magnetic variation anomaly in the
Dalradian region.
192
As mooted previously, the appearance of linear features in the same
geographical location plus quantitative estimation of source depth in different
data sets suggests the presence of strong lateral differences in structure
across the region and a deep-seated origin of the structures. Thus the
available geophysical and geological data form a coherent body of evidence for
the presence of deep Caledonoid and non-Caledonoid structures in the region.
The association of alpine type belts with zones of thickened crust is significant
as it seems to suggest that alpine type tectonic processes may have been
operative in this region.
The origin and emplacement of alpine type ultramafic rocks have been
discussed by Gresens (1970) and Chidester and Chady (1972) among others.
They are believed to originate in rifts in ocean-floors or in the mantle
underneath continental crusts. In continental environments they are emplaced
tectonically through the crust into the overlying thick pile of basin sediments.
However, Fyfe (1967, p.47) suggested that dunites and peridotites can also
result from in situ differentiation of mafic or ultramafic magma. Garson and
Plant (1973) have interpreted the suites of ultramafic rocks in Scotland as
ophiolites implying oceanic conditions and subduction.
Areas of soda metasomatism characterised by crocidolite, probably of early
Devonian age, occur in the Inverness region in association with breccia pipes
and the chemical changes have been tentatively related by Deans, Garson and
Coats (1971) to saline diapiric intrusions or to deep-seated carbonatite intruded
in the plane of the GGF. Evaporite deposits often form, given certain
limitations, in the early rifting stage of the fragmentation of continents and of
the formation of proto-oceans (Windley 1977, p.215), and carbonatites, which
result sometimes from partial fusion of mantle peridotites (Wylie and Huang
1975), are also critical components of rift magmatism (Windley 1977).
Whichever interpretation is preferred, however, it is evident that the attendant
processes are deep-seated and probably involve megafractures. It can be no
accident that the aegirine granulites of Glen Lui (with blue amphiboles),
situated near the Loch Tay fault, resemble some of the rocks from the Great
Glen (Deans et al 1971). The question thus arises, 'can we reconcile the above
observations into one feasible structural framework ?' This is attempted in the
next section.
193
8.2.3 Regional synthesis: an alternative tectonic view
Although it is generally believed that the Northern and much of the
Grampian Highlands are underlain by Lewisian-type material (Bamford et al
1978; Watson and Dunning 1979; Watson 1984), the MT/Gravity model
presented here shows discontinuity in structure across the Scottish
Caledonides. Also, a comparison (Hamilton et al 1980) of the Sm-Nd
systematics of the Lewisian gneisses with those of some early Caledonian
granites shows that these granites are not dominated by a Lewisian
component. Thus the simplest interpretation of the MT/Gravity model (fig 8.4b)
is that the zones of thickened upper crust demarcate difterent terranes. These
terranes were probably continental fragments with attached crust and upper
mantle as is the case in the accreted terranes in the northern part of the
Phillipines (Karig, 1983). They may have been separated by epherheral basins
some of which were probably oceanic or quasi-oceanic and individually filled;
these basins could have formed in rift systems in the underlying crust and may
have developed as part of the early Palaeozoic continental shelf (or shelves) as
envisaged by Graham (1976), Graham and Bradbury (1981) and Plant et al (1984)
for the Dalradian. These basins were closed in space and time by convergence
leading to continental collision and expulsion of the basin-fill sediments as
major nappes. Oblique movements as envisaged by Badham (1982) for the
Hercynides were probably important in the development of the present crustal
architecture. It follows from this terrane accretion interpretation that the
geophysically inferred areas of thickened crust are suture zones. This
hypothesis needs considerable testing.
This model is consistent with Hutton et al's (1977) plate-boundary
interpretation for the Great Glen and Dalradian regions. It is also consistent
with Halliday's (1984) proposal - rejected by Watson et al (1984) - that a crust,
younger than the basement to the north of the Mid-Grampian Line, was thrust
under the south side of the Grampian Highlands. This would seem to suggest
that the suture (concealed) in the Dairadian region lies near the Mid-Grampian
Line and Hutton, et al's magnetic variation anomaly. A suture in the
Kintail-Oykel region would support Johnson's (1975) and Stewart's (1982)
proposal of a concealed Morarian suture in NW Scotland. Garson and Plant
(1973) suggested the Great Glen as a Morarian suture and the Dalradian belt of
alpine type ultramafics was interpreted as another suture. This is in remarkable
agreement with the geophysical evidence presented here. However,
considering the effective phase information (fig 6.12),and Hutton et al's (1977)
194
magnetic variation anomaly, it is also possible that the Great Glen suture is
probably hidden by the Caledonian Fort William slide. It probably corresponds
to the concealed conductive dyke observed in this study to the east of the
GGF. Another plausible explanation is offered below for the Great Glen fault.
Areas of fenite-type soda metasomatism occur in the Great Glen region
(from south of Foyers to north of Inverness) in association with breccia bodies
(Deans et al 1971). These workers ascribed the chemical changes of the
country rocks to a deep-seated carbonatite intruded in the plane of the GGF.
The close association between continental rifting and alkaline magmatism is
well known and rift magmatism may be associated with abnormal enrichment
of volatiles (and alkalis) which may lead to highly explosive and fragmental
volcanism (see Windley 1977, p.220). Also, peralkaline (soda-rich) subvolcanic
ring complexes (whose rocks include carbonatites and fenites formed by in situ
alkali metasomatism of the country rocks) are associated with rift valleys
(Windley 1977, p.222). Thus the Great Glen is probably a fossil rift system.
The rifting is probably of late Silurian or early Devonian age if it can be dated
by the soda metasomatic event (Deans et al 1971). The breccia bodies in the
Great Glen probably suggest explosive events and may be fossil earthquakes,
affected by later strike-slip movements. The fenite-type rocks probably
represent exhumed subvolcanic ring complexes. If these contentions are
correct, then we can speculate that the apparent absence of fenite-type rocks
around Strontian and between Strontian and Foyers (Deans et al 1971) is
suggestive of a southwestward deepening magma chamber or may be a
reflection of the deeper erosion in the northern Great Glen region.
The above hypothesis may be used to shed some light on the aeromagnetic
anomaly of the Great Glen region. It is possible that the linear magnetic
anomaly in this region arises from magmatic rocks associated with the
proposed rifting. From three profiles across the magnetic ridge over the GGF,
Hall and Dagley (1970) suggested the presence of a thick horizontal sheet with
a top at 8 to 15 Km and bottom at 15 to 26 Km from the Loch Ness - Moray
511-6-01— '2 Firth area to the"Foyers granite complex and bounded in the northwest by the
GGF. This implies a southward dip for the magnetic sheet and accords with
the author's suggestion that subvolcanic complexes probably lie deep further
south of Foyers. However, another profile across this magnetic anomaly in the
Loch Lochy area was interpreted by Hall and Dagley as due to a normally
magnetized unit bounded by the GGF to the south and extending from a
shallow depth to about 151(m or greater. This would imply a restricted
195
occurrence of this unit and an initial southeastward underthrusting followed by
sinistral strike-slip movement along the fault axis. However, dykes and sheets
of ultrabasic to acidic composition occur to the west and southwest of Loch
Lochy and the magnetic unit is probably related to these intrusive bodies.
196
F S
Fig 8.7. The lid of the continental collision zone (orogen) and its frontal
wedges model (after Laubscher 1983). Essential points are (a) two subducted
slabs with intervening funnel of detached middle and lower crustal material
that may be detached (as slices and wedges) and transformed into ophiolite
and (b) the surficial masses are detached and transformed into recumbent folds
and nappes. Lateral displacement between the two slabs may be
accommodated along a fault (F).
197
The structural model presented above appears to the author to form a basis for
other speculations. In the Northern Highlands a complex geological history has
been suggested from geological evidence, and isotopic evidence (see Powell et
al 1983, for a bibliography) supports two or more cycles of orogenic activity.
However, no. internal unconformities that might relate the polyphasal nature of
both the deformation and regional metamorphism to the isotopic data (see
Powell et al 1983, p.813) have yet been recognised within the Moine rocks in
this region. It appears that all the major lithological divisions of at least the
southwestern Moine share the same deformational and metamorphic history
(Baird 1982; Roberts and Harris 1983). The Kintail-Oykel crustal anomaly may
therefore provide the important missing link in the complex history of NW
Scotland. The complex structural pattern of the Scottish Caledonides may be
explained by the hypothesis of episodic and local collision and expulsion of
basin-fill sediments as major nappes. Collision tectonics as they relate to
nappe and recumbent fold formation can easily be demonstrated in the light of
the "orogenic lid and its frontal wedges' model developed by Laubscher (1983)
for the central Alps. The basic concepts are illustrated in fig 8.7.
Northwest of the GGF the structural vergence is in general NW but
structures with eastward vergence occur in the Morarian alpine-type belt of
Johnson (1975). The root zone of the Morarian antiforms is exposed in the
Glenelg region where the western part of the zone has been truncated by the
Moine thrust, and in Sutherland it is possible that the root zone lies wholly or
in part beneath the Moine thrust (Johnson 1975). The Morarian deformation
obviously pre-dates the formation of the Moine thrust and the suggested root
zone seems to correlate with the Kintail-Oykel anomaly. The Great Glen region
appears to be the root zone of the later Moine thrust. This view is supported
by Coward (1980) and by the recent estimates of the displacement in the Moine
thrust zone by Butler (in Coward 1983) which suggest that the Cambrian strata
can be restored to some 60Km east of the present trace of the Moine thrust.
The upright to NW verging Sgurr Beag slide may also be rooted in the Great
Glen region. This is supported by Coward (1983), who considered the slide to
have a sigmoidal form branching off a floor thrust in the lower crust and
estimated the footwall ramp to be some 60Km east of the Moine thrust outcrop
and east of the outcrop of the Sgurr Beag slide.
Southeast of the GGF the structures are dominated by large fold nappes
which face the northwest near the Great Glen but face southeast near the
Highland Boundary fault. According to Thomas (1979) the structures (Fl
198
nappes) appear to diverge from an upward facing axial zone in the Central
Highlands but Shackleton (1979) pointed out that it is hard to see how such
enormous volumes of nappes could be ejected, fountain-wise, from this steep
belt. The author agrees with the interpretation of Thomas (1979) in the light of
Laubscher's (1983) model but suggests that the 'root zone' of these nappes
probably lies in the region of the proposed Dalradian suture.
8.2.4 Tectonic summary
The tectonic setting for the Scottish Caledonides involves compressive as
well as extensional movements.
The geophysical models presented and discussed here evoke a picture of a
basement that is divided into blocks. The orogen is viewed therefore as a
mosaic of accreted teçranes or microplates. It is noteworthy that Coward
(1983, p.808) pointed out that the Scottish Caledonides have many similarities
to the North American Cordillerra (see Coney et al 1980) and may be a mixture
of allochthonous or suspect terranes, and relics of an auth 1ochthonous foreland.
It is remarked here that Peddy and Keen (1987) sugges4ed that the Flannan
reflector seen on the DRUM seismic section (McGeary and Warner 1985) may
be a fossil subduction zone.
The foregoing analysis of the Caledonian orogen is crude but draws on the
increasingly popular concept of suspect terranes of orogenic belts (Coney et al
1980; Williams and Hatcher 1983), the strike-slip orogen hypothesis of Badham
(1982), the transpression model of Saleeby (1981) involving a combination of
strike-slip and convergence and Laubscher's (1983) model for alpine-type
collision. The structural model presented here provides a basis for informed
speculations by other authors as well as a fresh approach to Caledonian
syntheses. An interpretation of the deep geology of northern Scotland in the
light of geophysical data is given in fig 8.8. This model shows some
similarities with that proposed by Yardley et al (1982) for the southern part of
the Scottish Caledonides and with Garson and Plant's (1973) model for the
Scottish Highlands.
199
tJ 0 0
SPECULATIVE TECTONIC MODEL
GGF
OIT P.1-Ill
7 I
J r
• ,,
FT J I rj
111*11
C;11
-\
-60
C>
DRUM. MOIST & WINCH Prosiles
HILL (1 987) __________ MEJU(MT) MEJU(Grcwity) & MBIPOM(1980)
FT- FLANNAN THRUST? MTh-NOINE THRUST FWS-FORT WILLIAM SLIDE OIl-OUTER ISLES THRUST?
HOE-HIGHLAND BOUNDARY FAULT
u
LEWISIAN CRUST -v
OROGENIC LID &WEDGES (mostlyCaledoflian older
MIDLANDIAN (TRANSPRESSIVE?
DALRADIAN TERRANE
metamorphics) = BELT BASEMENT
'I I IT MOINIAN
r
I UPPER H i II TERRANE MANTLE I I ii BASEMENT
I-W5
at
8.3 Interpretation of high electrical earth conductivities
The causes of high rock electrical conductivities have been discussed
extensively in the literature (e.g. Brace 1971; Shankland and Ander 1983; Haak
and Hutton 1986). For physical reasons conductivity is expected to increase
generally with depth. It depends on porosity, pressure, temperature and other
parameters. As regards the lower crust/upper mantle layer in northern
Scotland, it can only be said that an impressive array of factors may be
responsible for its conductive nature, e.g. petrology, presence of fluids in
interconnected spaces (electrolytic conduction), pressure, solid conduction
through the bulk silicate material itself due to increased temperature, and
partial melt. Partial melting seems unlikely and solid conduction through
hydrated silicate minerals may be possible but is unattractive. Petrological
factors may be called upon for geological reasons but this can be misleading
as the presence of water affects a variety of petrological materials (gabbro,
tonalite, granite) in similar ways (Lambert and Wylie, 1970).
When crustal blocks collide, they bore into each other forming "wedge into
split-apart" (Laubscher, 1983) structures since the respective originally
continuous sequences are forced apart (as during sill emplacement), with the
uplifted splits being bounded by basal detachment surfaces. The basement and
cover responses to deformation will be different resulting in a contrast in style
and fabrics. This mechanism may well account for the complex wedge-type
structures seen on the MOIST, WINCH and SHET profiles. This implies that
some of the seismic reflective zones may not be zones of strong lithological
contrast but are merely structurally developed zones within the basement. The
sheared nature of the deep crust as suggested from geological studies (Tanner
et al 1970; Watson 1977; Coward 1983; Soper and Anderton 1984) and
confirmed by recent seismic reflection profiling (Smythe et al 1982; Brewer et
al 1983; Brewer and Smythe 1984) and the present study make the presence of
fluids in connected cracks at deep crustal levels an attractive possibility.
Moreso, the presence of stress-aligned liquid-filled microcracks pervading at
least the top 10 to 201(m of the earth's crust has been recognized by Crampin
(1985) among others from shear-wave splitting in various geotectonic
environments and may. be a plausible explanation for the steep conductive
zones found in the Great Glen region. Significantly, the Great Glen fault is still
active and frictional heating with perhaps subterranean contributions may be
another factor contributing to the high conductivity of this structure. However,
it may be unsafe to ascribe the high conductivity of this structure to an exotic
201
geothermal reservoir such as a deep-lying congealed carbonatite magma in the
plane of the GGF (Deans et al 1971): a more practical hypothesis is preferred.
It has been demonstrated by Etheridge and Cooper (1981) among others
that metamorphic fluids may be focused or channelled through high
permeability pathways that are structurally or lithologically determined. It is
also known from petroleum geology that the presence of an impermeable cap,
say a shale layer, within a thick sequence of water-saturated sediments can
lead to the development of anomalous pressures underneath the cap rock.
Etheridge et al (1983) used the concept of large-scale convective cells in
capped over-pressured systems to demonstrate that large-scale fluid
circulation is possible in regional metamorphic belts. The basic concept is
illustrated in fig 8.9. In this figure, we see that fluids are focused by
deep-rooted shear zones. There is limited circulation for anatectic regions but
circulation becomes increasingly active at upper parts of the metamorphic pile.
A relatively impermeable cap separates the overpressured metamorphic system
from upper crustal hydrostatic systems preventing exchange between the two
hydro-systems. Thus the whole pile is deformed with the upper crustal open
system losing its fluid and electrolyte conduction capabilities while large
amounts of fluids are trapped at its base (Etheridge et al 1983).
The above concept may be used to explain the observed conductivity
distribution in northern Scotland. It also allows a great deal of flexibility in the
petrological or physical characterization of crustal conductive structures. The
necessary conditions for its operation are rife and the preponderance of deep
structural discontinuities in this region ensures that the basal fluid system is in
circulation. For a steep shear, large-scale fluid circulation is possible along the
higher permeability crushed zones. This probably explains the conductive
nature of the system of dykes in the Great Glen area. The original cap rocks
have become severely eroded or lost so that the Great Glen fault zone has
become open to the meteoric system. This concept can also be used to throw
light on the distribution of metamorphic zones in the region . For a normal (or
thrust) fault, the emplacement of a slab of hot, deeper lying material over cool,
water-saturated crust may initiate a convective circulation in the footwall block,
with some fluid penetration into, and retrogression of, the hanging wall
(Etheridge et al 1983). In other words, thrusting of rocks undergoing
/ metamorphism over cold foreland rocks would resul?in retrogression in the
overlying slab and a prograde metamorphism of the footwall rocks. If these
contentions are correct, then we expect a significant amount of fluids at the
202
base of the resistive upper crust or in the lower crust in general and especially
along the numerous detachment surfaces. This fluid system, together with the
contributions from the connected steep shears, may have influenced the
petrological, electrical and seismic character of the lower crust to a large
extent.
"".JMCe!Znc Fluid 2-1Ok'
Lation
LESS PEI1EABLE CAP Precipitation in Pores and Fractures
upper parts of
metamorpliic ()() k—TZone~— ctive ulation
pile Less active
t , . Icircutation
T Anafec tic FFtIe L.-i circu -'--. I region
-------
MANTLE HEAT SOURCE
II
Fig 8.9 Schematic model of fluid circulation systems in regional metamorphic
belt (after Etheridge et al 1983).
8.4 Discussions
As shown in the discussion which follows the main objectives of this
project have been realized. The deep electrical structure of the Great Glen
Fault has been clarified and structural problems encountered by seismics,
gravimetrics and geology have been overcome by this geoelectric study. All
available useful geophysical and geological data have been integrated into a
coherent structural framework to illuminate the previously- ellusive deep
structure of the Scottish Caledonides. Some light has also been shed on the
possible nature of movements on the GGF.
While the MT technique has been successfully used in this study, it must be
noted that the data acquisition were hampered by numerous factors of which
cultural noise and terrain conditions are the most problematical; resistive
203
mountainous terrain propagates electrical noise over great distances and the
noise degraded low frequency data limited model resolution to some extent.
The use of non-polarizing electrodes for low frequency measurements and of a
simple delay-line filter is strongly recommended.
The effect of the surrounding seas ('coast effect') has been discounted in
this study on the grounds of the numerical modelling studies by Mbipom
(1980). At the geomagnetic latitude of Scotland, source fields associated with
polar electrojet or substorm activity have complicated morphologies such that
the recognition of internal anomalies may be difficult. However, the work of
Hutton et al (1977) showed that source field effects are not very significant at
periods shorter than 1000 sec. Recent geological studies in the Dairadian
region (Fettes et al 1986) show considerable along-strike variations in
structures, therefore the 2-D geoelectric model should be regarded as the best
approximation to the more realistic 3-D earth structure.
This study shows that the MT technique is suited to the detection or
reconstruction of tectonic processes in orogenic belts. However, it should be
emphasized that this was made possible only by use of a dense network of
observational sites and appropriate sounding frequencies; short-wavelength
anomalies were thus more accurately resolved. This study also shows that the
effective 1-D resistivity and phase sections can provide very useful structural
information in areas that show strong 2-13 features and also stresses the need
to use a priori information to resolve non-uniqueness in MT inversions. But
the author argues that there is no case for forcing geoelectric models into
conformity with say, seismic models unless it is justified by the MT data: the
MT technique is as useful and independent as any other technique for
structural mapping and its value in resolving structural disputes is evident in
the above multi-disciplinary synthesis. Each geophysical technique is beset
with particular problems and so efforts should be directed towards improving
current geoelectric interpretation methods as attempted in this study. The
author wonders why attention is directed so strongly to use of seismics when
as shown in this study MT and the simpler Gravity techniques can help solve
major crustal problems and are substantially less expensive.
8.5 Conclusions
Deep magnetotelluric profiling in the Great Glen region, together with
information derived from the adjacent regions, has provided new insights into
the architecture of the crust beneath, the deformed Scottish mountain chains.
Not only has it been shown that anomalous Caledonoid (NE-SW) trending
204
zones of thickened crust exist in the alpine-type belts (see Johnson 1975;
Carson and Plant 1973) of Scotland but also that continental collision tectonics
has been a common ingredient in the evolution of the Scottish mountain belts.
Light has been thrown on the possible microplate or accreted terrane structure
of the Scottish Caledonides.
The following main conclusions are drawn from this study:
The invariant 1-0 MT sections are useful structural mapping tools but the
depths to interfaces are underestimated in this area of complex geology.
The Great Glen study area is shown to be 2-dimensional. The 2-0 model
derived from the study is consistent with the results of numerous
geophysical and geological studies of the GGF. It is thus the first
unequivocal model for the Great Glen region.
The upper crust in this region is very resistive but with different values
on either side of the fault. Its thickness varies laterally from about 201(m
at the traverse extremes to about 40Km near the fault zone. The zone of
thickened upper crust is about 181(m wide and is asymmetric about the
main fault axis. The lower crust is of low resistivity (150-400 2m) which
agrees with past observations in the region.
The basic structure of the GGF is an outcropping low resistivity (150-200
2m) dyke-like body, about 1Km wide, and transects the whole of the
upper crust in a vertical fashion but may be sub-horizontal at lower
crustal/upper mantle depths. Comparable structures are concealed
underneath the surface position of the Fort William slide and east of the
Loch Quoich Line at about 4-71(m depth; these. are interpreted as shear
zones and may have developed in response to compression during
microplate collision. The Great Glen is interpreted as a fossil rift system
or contact zone (suture) between two accreted geoelectrically different
terranes.
The interpretation of MT data sets from 3 profiles with dense networks of
observational sites has ensured good resolution of possible
short-wavelength anomalies in the Great Glen region and an integrated
interpretation of MT and gravity data sets has clarified the previously
problematical gravity anomaly.
The crustal architecture of the Scottish Caledonides is attributed to rifting
and collision tectonics. The coherent body of geological and geophysical
evidence presented above goes some way in solving some of the
problems of the deep structure of the Caledonides. Furthermore, the
205
proposed model provides us with a reasonably detailed picture to account
for the observed deformation (alpine type structures and strike-slip faults)
and metamorphism (considerable amount of fluid transport is envisaged in
the conductive structures). However some of these hypotheses need
considerable testing.
The value of magnétotellurics in resolving geological and other
geophysical disputes is evident. Each of the inferred collision zones
apparently has a distinctive geoelectric structure by which it may be
identified. Gravity provides a useful constraint; It is suggested that the
Great Glen model be used as a 'pathfinder' for any other undiscovered
Scottish microcontinental collision zone(s). A corollary that follows, but
needs further testing is that the electrical anomaly across the Southern
Uplands Fault (Sule 1985) is suggestive of a collision zone in that region.
If this proves to be true, then this combined MT/Gravity approach may be
of equal value in resolving structural problems in other mountain belts.
MT has summarily been shown to be a very powerful and indispensable
geological mapping tool in the deformed (?Caledonian) mountain chains of
northern Scotland.
Above all, this study has shown that, in the words of Hatcher Jr., Williams
and Zietz (1983, p1), "capability of integrating the multitude of data from
geologic and geophysical investigations in orogenic belts is essential to
arriving at solutions to some of the great problems related to the
development of mountain chains".
8.6 Recommendations and plans for future research
The history and deep structure of the crust in Scotland can only be well
understood if the nature and extent of translations along the major crustal
dislocations are fully ascertained. While this study has thrown some light
on the nature of the Great Glen Fault, the extent of translations along it is
still uncertain. The nature and extent of movement along the various
geotectonic boundaries in Scotland should be regarded as research
problems of high priority.
The proposed zone of crustal thickening north of the Highland Boundary
Fault should be investigated by the Magnetotelluric method. The
determination of other zones of crustal thickening will improve our
understanding of the past history of the region.
Development of improved MT interpretation methods should also be
206
regarded as projects of high research priority in view of the depth of
information that can be obtained from MT studies under ideal conditions.
New, accurate and cost-effective 2-D and 3-D interpretation methods that
can cope with complex earth problems need to be developed. Along this
line, it is planned that the applicability of the method of summary
representation (Polozhii decomposition) (Polozhii 1965; Tarlowski et al
1984) in EM modelling should be investigated. A simple 2-D data
transformation (see Coen et al 1983) is feasible and would cut down on
computing costs. Also, the amount of unambiguous information produced
by 2-D geoelectric modelling can be greatly increased by employing
suitable a priori data as shown in the 1-D case. As a potential
improvement of 1-D geoelectric interpretation method, the possibility of
joint inversion of MT and Transient electromagnetic and/or other
geoelectric data will be examined in the near future.
The importance of improved MT observational techniques on land cannot
be over-emphasized. Efforts should be geared towards improving the
data acquisition method in order to reduce the ambiguity which arises
during data interpretation.
For crustal structural studies an integrated interpretation of various
geophysical and geological data sets is highly recommended. MT and
Gravity (or MT and seismics) should be regarded as essential
combinations.
A joint inversion technique for gravity, magnetics and/or MT data is
currently under study. An empirical relationship between gravity and MT
anomalies in Scotland is also being sought as this will aid MT/gravity
interpretation. It is hoped that the integrated interpretation aproach will
be extended to the southern Caledonides of Britain in the near future.
207
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224
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226
APPENDIX I
Glen Garry field results The MT results obtained at 7 sites along the Glen Garry profile (G1-7) are
presented here.
For each site, the results shown are : the major and minor apparent
resistivity and phase, coherence, the number of estimates averaged, skew and
the azimuths of the major impedance.
227
0
C
It,
'JO 'J O
'J O
t
0
UI C hi -. I- a:
8- C U) C hi - I.- a
ui 03
C
U,
C
'Ii
UI
C
C
C UI hi
a In Iii a z
8-
'.4 a:
C C
C C
C
C
SITE. : 553A TELLURIC
CARTRIDGE , 55l 5531R 5532R 5533R
BAND , 1 2 3 14
COMPONENTS . 5 5 5 5
SAMPLES/WINDOW 2 .50 258 258 256
NUMBER WINDOWS. 110 lie 110 27
SAMPLE RATE HZ , 2818 258 32 11
PLOT HPF t 8'.L80 6.00 0.50 0.05
PLOT LPF , S'j'LOT 68.00 15.86 6.85
FREQS/OECADE . 8 8 8 8
FREQS/BRND , 8 11 16
WIN COHERENCY 8 ."5 0.86 0.86 6.88
REJECTION LOOPS. 1 2 2 2
C
C C C C
t
MINOR
!j H
I
I080. 160.00 10.00 1.06 0.18 0.0
FREQUENCY IN HZ
u•. p fi
-w
-- I
IT
1608.86 168.80 10.88 1.60 8.10 0.01
FREQUENCY IN HZ
228
'Ii 'U
13 'U
2
cc
2 a-
us
uj K.
Ic mi 03 z z
2 w 14 a,
MD ma
.Y?Ey Tjj
II
9jJIjfl_
NIKON
II
1058.55 255.88 18.88 1.88 8.18 0.82 FREQUENCY IN HZ
SITE : 574R TELLURIC CRRTNIOS! , 57'L6fl 571410 57112R 57!In ORNU 1 2 3 Ii COMPONENTS • 1 14 14 4 SRNPLES/NINUOII e 259 256 256 255 NUMBER NINDOMS, F5 36 66 55 SPNPLE RRTE HZ . 254! 258 32 '4 PLOT HPV a 54.66 6.60 0.50 8.55 PLOT LP? i 515.55 68.66 16.06 6.55 FREQS/OECRDE a 8 8 0 9 FREQS/BRND a 5 8 11 15 KIN COHERENCY a 6.°5 6.90 0.80 5.90 REJECTION LOOPS, 2 2 2 2
i III'
I • RW U
U I I
I a
JII 1 11 1 11" ill ii J __
• II J 2
I
1585.85 206.85 15.85 1.58 8.18 8.01 FREQUENCY IN HZ
229
a C Cl) 0 C-
C- C (/) C Cli -
cc lc
U-0
z
0
'U le Ci,
C C
(l)C 'U Cli
UI 'U a 0 z
a
I-
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a a
U, N
Cl,
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cx
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X , a- -
a a o MAJOR 0 01 I I
02 T IIj
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Cl) o
W a IT II
fI1hi II LI a
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a
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2
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H Ii!HffiIJJJ II I I C-
cr a. a-
C
mmmmm
•iurniiiiu_____
IN _
SITE 550R TELLUFUC
CARTRIDGE i 55,30A 55018 5502R BAND 1 2 3 COMPONENTS i 5 5 5 SAMPLES/WINDOW , 25 256 256 NUMBER WINDOHS 72 I&e 48 SAMPLE RATE HZ $ 2048 256 32 PLOT HPF s 64.0 6.00 0.50 PLOT LPF i 500.05 60.00 10.00 FREOS/DECADE s 8 8 8 FREOS/BAND i 6 8 11 MIN COHERENCY 0.aO 0.80 0.80 REJECTION LOOPS, 2 2 2
•
S
- '. S
IT T 1111 11111 I___
----.. jI
1000.00 100.00 18.00 1.00 0.10 0.01 FREQUENCY IN HZ
230
SITE : 551R TELLURIC
CRftTRIDG€ 551cif 5511R 5512R 5511C
BRNO 1 2 3 1 COMPONENTS . 1 5 5 5 SAMPLES/HINOOI4 i 26b 258 256 256 NUM8ER WINDOWS. 113 48 48 72
SRMPLE RATE HZ ; 284d 256 32 it PLOT HPF a €1.06 6.00 0.50 0.65
PLOT LPP * 506.06 68.00 10.00 0.90 FREOS/DECADE , 8 8 8 6 FREGS/BAND ; 6 8 11 11
NIH COHERENCI a 0.ao 0.88 0.80 8.80 REJECTION LOOPS. 2 2 2 2
P4AJ Oft
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0 a.
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cc
to
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5 a
C
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1000.00 100.00 10.00 1.00 0.10 0.01
FREQUENCY IN HZ
TT
1000.00 100.00 18.00 1.00 0.18 0.01
FRQUENCT IN HZ
231
z 0
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0,0
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U a -
a
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- 1686.68 155.55 18.55 1.06 5.15 5.51 FREQUENCr IN HZ
uj cc
U, -
U- a
Ic Lu
0
0
U,
a, 0 0
tu U, a: 0 Ui 0
cc
2
0
I-.
MAJOR
!
MT NOR
155.55 10.55 1.88 5.15 0.01 FREQUENCY IN HZ
SITE : 552R TELLUflIC CRRTRIDOE 52.5fl 5521fl 5522A 5523fl
BAND s 1 2 3 1
COMPONENTS • 5 5 5 5 SRMPLES/NINDOH 25 258 258 258
NUMBER MINDONSi I8 l8 31 SAMPLE RATE HZ , 2518 256 32 t
PLOT HPF , 81.53 8.65 8.50 6.65
PLOT LPF , 555.55 68.88 10.86 1.55
FREQS/DECRDE 8 8 8 8 PREQSIBAHD . 5 8 11 11 MIN COHERENCY . 5•a5 0.86 6.86 6.98
REJECTION LOOPSi i 2 2 2
232
88
U rn
5
U 55
S S
S
S S
S S
I i f
I
• J JB IIIIj I I I I
1666.60 160.00 10.00 1.60 0.10 6.01 FREQUENCY IN HZ
U) C Ui -. I—
I— U) U) —
U- 0
ui z z
C
U,
C
Ui
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U) U.'
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z
= C
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5- 5-
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U, Ui
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NRJOR
• •s ..*_, I ;
11 •1 TI
MINOR I
TZ I !' I
MMMMM I.' muu____
IINUI1I
_I- FREQUENCY IN HZ
SITE : 555A TELLURIC
CRRTRIOGE , 55509 5551R 5552R 5553P BRND i 1 2 3 'I COMPONENTS s 5 5 5 5 SRMPLES/NINOOW , 251 256 258 256 NUMBER WINUOW5i 48 46 48 23 SRMPLE RPTE H? , 2648 256 32 4 PLOT HPF , 64.66 6.00 6.56 6.05 PLOT LPF , 566.66 B'1.00 8.86 1.00 FREOS/DECROE , 6 B 8 8 FREOS/BRNO , 6 7 10 11 NIH COHERENCY , 8.68 0.80 886 8.86 REJECTION LOOPS, 2 2 2 2
233
I-
B,
a) Id
I- B,
Ui!
cc 0. 0. cc
C
B, B,
B, N
U, ILl w at (3 B, (Li 0
cr B,
0.-
C
I I!.
• B! B
B
B B
p
JJi iiJ L_ __
B,
cc
I- a, Ui —
II- 0
cc
B,
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a) B,
B,
B, U,. (U Ui (3 B, ILl
cr
0 z
B, x I-
I KRJOH
B, -. 0
I- I 1 T I
LLI
tu
& & cr
B,
B, B,
B, I-
a, Id B,
cc 0 B, IIJ 0
us
B,
C
I MINOR
z
I •
- B! I
TjI I
-
-I,,-
SITE : 556R TELLURIC CRATRIDGE , 556q 5561R 5562R BRND 1 2 3 COMPONENTS i 5 5 II SRNPLES/WINOON i 25P 256 256 NUMBER WINDOWS, 145 18 19 SRNPIE RRTE H! . 2618 256 32 PLOT NPF e 61.68 6.86 6.58 PLOT LPF , 563.86 68.88 10.06 FREQS/DECRDE , P 8 8 FREQS/BRNO i 8 8 11 HIM COHERENCY . 8.B6 0.80 0.80 REJECTION lOOPS, 2 2 2
1650.68 108.88 16.00 1.68 0.10 6.61 FREQUENCY IN HZ
FREQUENCY IN H?
234
U-Ta
Loch Arkaig field results
The MT results obtained at 5 sites along the Loch Arkaig profile (A1-5) are
presented here.
For each site, the results shown are : the major and minor apparent
resistivity and phase, coherence, the number of estimates averaged, skew and
the azimuths of the major impedance.
235
IE
SITE : 573R TELLURIC
CARTR!DOE s 578877 5731R 573277 573377
577*70 1 2 3 'a COMPONENTS , 'i 'a 'a 'a SRMPLES/NIN0ON 2rp 258 258 258 NUMBER WINDOWS, 57 58 Be al SRNP1.E RRTE HZ , 2879 258 32 'a PLOT HPF i 51.88 6.68 5.58 0.05 PLOT LPF s 585.88 68.86 18.86 0.88 FREQS/DECROE • 8 9 9 8 FREUS/BRND , 5 8 ii 18 WIN COHERENCY , 5.'8 8.86 8.80 5.88 REJECTION LOOPS, 2 2 2 2
MINOR
D
I-
*
ii: z 0.-
1806.50 ISLeS ILOS 1.56 REOUENCY IN HZ
I 4 .ls
_L ....
L' Aii[1i:iL:2J 0 w g
f 5 1 ---•-- --
*000.00 105.00 15.50 1.06 5.15 PREQUENCY IN HZ
cc
hi -
U.
ui z
236
g
N'
U, uj
0 N, U
0
cc = N, & —
N,
*600.60 105.50 15.00 1.00 0.10 0.5* FREQUENCY IN HZ
= 0 3- 3-
to
0) w
UA ic CL
CE
ui uj cc uj
I-
N,
N,
= in IL —
I RI m
U its
Tl
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Ti
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±, .-.
,000.00 105.50 10.00 1.56 5.10 0.51 FREQUENCY IN HZ
U.
us
S
N, N,
*1* U,
N, N,
uJ Ui W N, UJ U 0 z
N, = I-
F?
MRJOR
ftf hiiif
___iij'II HIIIJ II
MINOR
I I I I = 0
IT
TI jhhifJ
fn II tu cc cc
CL
SITE : 572R TELLURIC
CRRTR100E . 5723H 5721R 5722R 5723R BRNO 1 2 3 'I COMPONENTS . 1 4 '1 14 SRNPLES/WINOOH . 25 4s 256 258 255 NUMBER WINOONS, 63 60 60 55 SRMPLE RRTE HZ t 2019 258 32 i PLOT HPF . 61.58 6.00 6.56 8.05 PLOT LPF , 553.55 68.00 16.66 6. FREQS/DECRDE * 8 8 8 8 PREQS/BRNO * 6 8 11 18 NIN COHERENCY , 3.3 6.80 6.80 0.85 REJECTION LOOPS. 2 2 2 2
/ 237
I
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FREQUENCY IN HZ
C 0
U) 0 Ui - I.-
1- 0 U) C Ui -
U- 0
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"I cr
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F•I! f'f If II _____ _____
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SITE 571R TELLURIC CARTRIDGE , 571 l 5711P 5712A 571'fl BAND I 2 3 COMPONENTS s 5 5 4 SAMPLES/WINDOW t Es" 258 258 25 NUMBER WINDOWS, El 1B '*8 5 SAMPLE RATE HZ u 2pllp 258 32 'I PLOT HPF , 6.88 0.50 0.85 PLOT LPF 68.00 10.60 0.0 FREOS/OECADE e 8 8 8 FREOS/BAND r B II 10 HIM COHERENCY , 'i 0.80 8.80 S.P8 REJECTION LOOPS. 2 2 7
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In
cn
z 0
238
x a
I-.
LS ic
Lu cr ft. a-
0
0 N
I: ui cr
0 & 0
cc
to Iii —
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ui a,
C
N,
0
z Ui
a, C
C
Ui Ui
a o III a
z
cc
0 = I-
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go tno
8 -j!
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-jiIT
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1f r+f4!
MINOR
Ei- I I = a
,= p z i Ui I-
cc a- i
cc
IJ•t11I1I -ii
Ui
a Ui a -
Ui a, =0 a- —
C 1565.80
185.60 15.55 1.OD
FREQUENCY IN HZ
SITE : 570R TELLURIC
CRRTRIOCE , 5700 5761R 5762R 5703R
BRNO i 1 2 3 1
COMPONENTS . 5 5 5 5
SRMPLES/WINOOW • 25p 256 258 25
NUMBER WINDOWS. 32 24 '18 13
SRMPLE RRTE HZ , 2318 256 32 I
PLOT HPF , 6!L00 6.60 0.56 0.05
PLOT LPF . 5o.33 68.00 10.00 0.83
FREQS/DECRUE . 0 8 8 0
FREQS/BRHD s 5 8 11 13
NIH COHERENCY O.Fo 6.80 0.80 3.P3
REJECTION LOOPS. 2 2 2 2
ft p ft S B S
ft 0 ft •
B B to ft ft rn_I- ft
ft -ft-rn ft!I ft
ft ft
ft ft
_1li
Lu _
-
1605.65 165.60 18.60 1.55 0.10 8.6
FREQUENCY IN HZ
239
z a
uj cc
I-
A. 0. cc
ui
0
U,
a 'U a
cr
I
SITE : 557R TELLU9IC CRRTRIOOE I 556R 5571R 55729
1 2 3 COMPONENTS . 5 5 5 SAMPLE 3/WINOON I 25R 258 258 NUMBER NINOO1IS1 '8 48 28 SAMPLE RATE HE . 2648 258 32 PLOT HP? s 64.86 6.60 6.56 PLOT LP? , 666.08 88.88 18.00 FREQ$/OECROE • 16 10 16 FRESS/BAND $ 8 16 13 PUN COHERENCY . 0.P0 0.80 6.86 REJECTION LOOPS, 2 2 2
T
11ILII _1IIFI Ii
iLT_r!+1T Till 1055.50 150.88 15.00 1.05 0.10 6.I
FREQUENCY IN HZ
IF
r cm
ui
z
0
0 a)
'U
U,
a) a)
O R 'U uJ a (U a z
a)
I-
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NO_Iflq
KINOR
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cc
IU
a 'U a uJ a)
0
0
£005.05 105.85 15.85 1.55 5.18 5.01 FREQUENCY IN HZ
240
APPENDIX III
Glen by field results The MT results obtained at 19 sites along the Glen Loy profile (1-1-19) are
presented here.
For each site, the results shown are : the major and minor apparent
resistivity and phase, coherence, the number of estimates averaged, skew and
azimuths of the major impedance.
241
SITE : 607 R
CRRTRIOGE i 6070A 6071A 6072A 60738 BAND , 0 1 2 3 COMPONENTS • 4 4 4 4 SAMPLES/WINDOW • 256 256 256 256 NUMBER WINDOWS. 211 211 24 24 SAMPLE RATE Hi • 512 611 8 1 PLOT HPF • 20.00 2.00 0.20 0.01 PLOT LPF 200.00 214.00 3.00 0.30 FREQS/OECROE • 8 8 8 8 FREOS/BAND • 7 9 10 11 KIN COHERENCY i 0.80 0.80 ().Be 0.80 REJECTION LOOPS. 2 2 2 2
mBa
a Rc
a a
• aa a a aaoa!
a a
a
R ill
If
I PipJ
1880.08 108.08 18.00 - 1.06 0.10 0.81 FREQUENCY IN HZ
Cl, - Li I- a:
I- C U) C Li - U-
C
U)
C
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rw
z C
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a: C a,
MAJOR
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mommmm -I
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- F—F - 1838.120 100.08 10.08 1.00 0.18 0.01
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SITE 560R TELLURIC
CARTRIDGE 5880A 5801R 5802A 5803A
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I I I I r
C
0!
I- I—
—n I L
IIf Li I
I.- z 0 WE
cr 0 a-
C
M wu-i-
cn cc - -
0 - -- -- -- . -. • MI v.o 1800.00 iug.ua & .... FREQUENCY IN HZ
SITE 60 11A TELLURIC
CARTRIDGE 640A 6041R 60'12A 6043R
bAND 1 2 3
COMPONENTS q 256
'I 256 256 256
SAMPLES/WINDOW • 60 '12 32
NUMBER WINOOWSi 38 6 SAMPLE RATE HZ i 512
8.00 6'1 2.80 0.16 8.02
PLOT I4PF , PLOT LPF i 128.00 8.00 1.00 0.10
FREOS/DECADE a 10 10 10 10
FREOS/BAND a 10 6 10 8
MIN COHERENCY i 8.70 8.70 0.70 0.70
REJECTION LOOPS, 2 2 2 2
259
a, ui
lu cc
I-
a- a-
*
a, hi
cc Iii
a- *
I III -
0
ty a, x
V
Ui
a, V
55* U, Iii
43 Ui 0
cc
z V
I-
*
_fT-1 fHfij
-t.---.Th.-
-fill
MINOR
10 - - .--- --- I I
Ii huT IIIIjj
IL
I: ui
155.58 15.55
FREQUENCY IN HZ
SITE : 566A TELLURIC CR1118100! , 58800 5881R 588211 5883R
5111(0 e 1 2 5
8 '1
COnPD1(EN15 . SRNPLES,HINOON ,
5 255 258 258 258
HUNSEPI MINOONS. (18 (18 80 65
SRflPt.E 8417! HE . 250 6U.06
258 8.08
32 0.58
'1 6.85
P1-01 HP? • PLOT LPF 588.50 6U.00 8.00 1.06
PR! (13/Of CR0! , 8 5 8 8
FRE23/B11MO s NIH COHERENCY ,
8 0.80
7 0.80
10 0.80
11 0.80
REJECT 1011 LOOPIs 2 2 2 2
..'...;.• •r. 55
( •
•
u' a •
.l
-
1iJTflIh1 1hIMh!LI I --1 -
- 1658.58 158.58 15.55 1.55 5.15 U.0
FREQUENCY IN HZ V
ISOLSS
Wff
0) 0 U - I-
I- a,
U-0 dc uj a,
x
0
WI
WI
0) WI
0 0 WI 1.4
CDVI UJ WI 0
CC
z 0
I-
0 WI
a
a
E
r a o
1- I-
>.
n9 U, 1.4
I-z 0 IAJ CC
cr. IL a.
MAJOR
•1.1 •..
'I I
0 O19
U, 11 "1 JI T'PiIiI 1.4
w 0
La
CE I a- -
C C C NIOR
I- I-
- C jil
I
11011 U, U CC
I-
U -. CC II a- a-
2
WI I- I
a, --TT 1! 1i Wa I_114I
WI iitiii ijj J
11111 WI 11111 uj
0
tn CC
4.-
1000.60 100.00 10.06 1.00 0.10 0.01
FflEQUENCT IN HZ
SITE : 610A TELLURIC
CARTRIDCE t 6100A BlOIR 6102R 5103fl BAND 1 2 3 1 COMPONENTS , 5 5 5 5 SAMPLES/WINDQII * 256 256 256 256 NUMBER KINOOWSi 2'4 211 214 21 SAMPLE RATE HZ . 512 614 8 I PLOT HPF s 8.00 2.00 0.10 0.02 PLOT LPF i 120.00 8.00 1.00 0.10 FREQS/OECROE , 10 10 to 10 FREOS/BANO , 10 6 10 0 NIN COHERENCT s 0.70 0.70 0.70 0.70 REJECTION LOOPSI 2 2 2 2
MENEM U II __
-t .II
momillmn
0""lliliffilm mmmmmm mmmmm mmmm'm mmmmis 1000.00 100.00 10.00 1.80 0.10 0.01
FREQUENCY IN HZ
261
APPENDIX IV
Individual site 1-13 models The individual station 1-13 inversion results are presented here, going from
the west to the east, for the Glen Garry (G1-G7), Loch Arkaig (A1-A4) and Glen
Loy (1-1-1-19) profiles. Only the most squares and in some cases the Occam
and Niblett-Bostick models are shown. Irreducible bias effects limited model
resolution to a large extent for the Loch Arkaig stations. Consequently, station
557 (A5) has been deleted from the interpretation process. Notice in the Glen
Loy models that the sites to the east of the surface position of the Fort William
slide (1-17, L18, L19) show a somewhat different deep structural behaviour from
the rest of the profile.
262
a, Ui C U) C
U)
a, C =
-
108.50 18.00 1.00 0.10 0.81 FREQUENCY IN HZ
x I- a- U) a
01 910 9920
02 9959 11027
10 MOST-SQUARES SM MODELS FOR SITE 61 EFFECTIVE CHISQ22. 80 NFREQ25 3 LAYERS AHP+PHASE FIT
RESISTIVITY OHMM
E
I- 8 tJ a
01 857 357:
32 12181 2339
O.00 1.00 0.10 0.01 FREQUENCY IN HZ
10 MOST-SQUARES SM MODELS FOR SITE 62 EFFECTIVE CHISQ27.36 NFREQ30 3 LAYERS AMP+PHRSE FIT
RESISTIVITY OHM.H
263
10 100
flCalOuIYIul unn.n
1000 10000
- _
0
I-0 Ui
1000.00 100.00 10.00 1.00 0.10 0.01 FREQUENCY IN HZ
OCCAM MODEL FOR SITE 550A EFFECTIVE CHISQo22.66 NFIiEQo23 It LAYERS AMP FIT
8 8 8 16
RESISTIVITY OHH.M 100 1000 00006
1000.00 000.00 00.00 1.00 0.00 0.01 FREQUENCY IN HZ
00000
0200
Dl 756
5110
02 5705
1289
10 MOST-SQUARES SM MODELS FOR SITE 63 EFFECTIVE CHISQ=20. 98 NFREQ=23 3 LAYERS RMP+PHRSE FIT
264
r
F; C
= Ii,
1000.00
=
a. a
Dl 208
02 0230
188.00 10.00 1.00 0.10 0.01 FREQUENCY IN HZ
=
4.LlJ
al
100.00 10.00 1.00 0.18 0.01 FREQUENCY IN HZ
U' hi C
hi 0
U' C = in —
1000.08
E
= - a- Lii a
10 MOST-SQUARES SM MODELS F-OR SITE Gil EFFECTIVE CHISQ=47. 09 NFREQ=33 3 LAYERS RMP+PHASE FIT
RESISTIVITY OHM.M
OCCAM MODEL FOR SITE 551R EFFECTIVE CHISQ-17.08 NFREQC33 16 LAYERS AMP FIT
RESISTIVITY OHM.M
265
- C
0, (Si =
C
C-
Ui Ci =
C r
1000.00 teo.00 10.00 1.60 FREQUENCY IN HO
x
a- Ui 0
S
-
01 5056
18
02 15036
5.10 0.01
10 MOST-SQUARES SM MODELS FOR SITE 65 EFFECTIVE
CHISQ=29. 18 HFREQ32 3 LAYERS AHP+PHASE FIT
RESISTIVITY OHM.II
r
:
U, lU 0 Ui
t&J S C
r in
a- Ui
:5
DI 1251 IS
15
02 15058 3
2
155.5 165.50 10.00 1.00
FREQUENCY IN HZ
10 MOST-SQUARES SM MODELS FOR SITE 66 EFFECTIVE
CHISQn24.62 NFREQ27 3 LAYERS AMF+PHASE FIT
RESISTIVITY OHM.K
266
F
rf 1000.05
___
155.50 10.50 1.00 0.15 0.01
FREQUENCT IN II!
OCCRM MODEL FOR SITE 67 EFFECTIVE CHISQ11.92 NFREO2'I 12 LAYERS AMP FIT
RESISTIVITY OHM.K IS -" I 100 1000
F
I J4frf 1ii f__
1000.00 100.00 10.00 1.00 0.10 0.01 FREQUENCY IN HZ
60000
I 2166
DI 602 IS
10157
02 12163 20!
5606
10 MOST-SOURRES SM MODELS FOR SITE 67 EFFECTIVE CHISO=26.'10 NFREQ=24 3 LAYERS AMP+FHRSE FIT
RESISTIVITY OHH.H
267
D
r
C
C
C = 0.
i55.05 250.05 18.0 5.00 0.20 0.55 FREQUENCY IN HZ
00500
'see
01 1156 5
15605
02 52656 2:
757
= = C
0,
C
C
0
-
5000.00 105.00 55.00 5.50 FREQUENCY IN HZ
01 066
52 6335 171
5.50 0.05
10 MOST-SQURRES SM MODELS FOR SITE Al EFFECTIVE
CHISQ=73. 10 NFREQ=3L1 3 LAYERS RMPPHASE FIT
RESISTIVITY OHM.H
10 MOST-SQUAREs SM MODELS FOR SITE fl2 EFFECTIVE CHISQ34. 10 NFAEQ-31 3 LAYERS ANP+FHASE FIT
RESISTIVITY OHM.H
268
r C C
- C 0. a-
r--- ---- -
1600.06 100.60 10.00 1.00 FREQUENCY IN HZ
ID MOST-SQUPHES SM MODELS FOR SITE All EFFECTIVE
CHISQI7. 29 NFREQ32 3 LAYERS AMF+PI-IASE FIT
RESISTIVITY OHM.M
U,
E
9.60 0.21
10 MOST-SQUARES SM MODELS FOR SITE A3 EFFECTIVE
CHISO=t1O. 71 NFREO=28 3 LAYERS RMP+PHRSE FIT
RESISTIVITY OHM.M 18229
III 1128 1563
i
01 735 6936
1: 1
I- I a_ .1 Ui 112 2655 3521 I
02 0588 16555
Ui U, C
O52.C2 199.20 10.99 1.60
FREQUENCY IN HZ 0.10 0.01
53 270 855
2* 788 1831
02 602*8 2289
269
ID MOST-SQUARES SM MODEL FOR SITE Li EFFECTIVE CHISO=85.06 NFREQ=32 3 LAYERS RMPPHASE FIT
RESISTIVITY OHM.M
r
a
a r &
a
1880.85 188.80 10.85 1.00 0.10 0.81
FREQUENCY IN lIZ
OCCAM MODEL FOR SITE Li EFFECTIVE CHISQ=31.43 NFREQ32 16 LAYERS AMP FIT
RESISTIVITY OHM.M
ui
=0
I a-
= a I- a-
ao.8g 10.00 0.00 0.10 0.80 FREQUENC1' IN HZ
80000
2882
II 033 2310
0 7850
02 03062 1711
112
270
CL LU
7
*6
02 10322 1
62
10 MOST-S0UAFE5 SM MODEL FOF SITE L2 EFFECTIVE CI1ISQ68.99 NFREO39 3 LAYERS AMP+PHASE FIT
RESISTIVITY 01*1.11
1006.00 100.00 10.00 1.00 0.16 0.0*
FREQUENCY IN HZ
10 MOST-SQUARES SM MODEL FOR SITE L3 EFFECTIVE CHISQ=108.16 NFREO=31 4 LAYERS RMP+PHRSE FIT
RESISTIVITY 01111.11
I C
I-
LLJ
AJ
IT
*000.00 100.00 *0.00 1.00 0.10 0.01
FREQUENCT IN HZ
5
Dl *7*1
0
02 *655*1
6
03 27266
271
100.00 10.00 1.00 0.10 0.01 FREQUENCY IN HZ
U,
0
-
1000.00
000RM MODEL FOR SITE L5 CHISO=33.00 17 LAYERS
a- a lU
Q
EFFECTIVE NFREQ=3q AMP FIT
RESISTIVITY OHM.M
200.50 10.00 1.00 5.10 5.01 FREQUENCY IN HZ
Lu - ILL C
ILL 0
C r ,
1050.00
10 MOST-SOURRES SM MODEL FOR SITE LII EFFECTIVE CHISQ=73.94 NFHEQ32 q LAYERS RMPPI1ASE FIT
00200
2767
Dl 611
7659
02 28365
12611
2115953 52716
272
OCCRM MODEL FOR SITE LB EFFECTIVE CHISU33. 96 N.FREQ=311 17 LRTERS R1IP FIT
RESISTIVITY OHM.M
___•__
-1-- 100.00 iø.ce 1.00 o.o FREQUENCY IN Hi
1D MD5T-5UHHES HR MODEL FOR SITE LB EFFECTIVE CHIS059. 39 NFREQ34 ' LRYEAS RHF+PHRSE FIT
RESISTIVITY OHtI.M
r C
02
U,
=
-4- I- 0. Ui
I6U5. I66.00 10.0 4.00 0.10 0.04 —I-- FREQUENCY IN HZ
Dl I0
02 32306
03 50174
273
MOST-SQUARES SM MODELS FOR SITE L7 EFFECTIVE
CHISQ=24. 62 NFAEQ27 3 LAYERS RMPPHASE FIT
RESISTIVITY OHM.H
r
3 jjJJL
0
TJO
Ui
10.e6 10.00 1.00 0.10 0.01 FOEQUENCY IN HZ
OCCAM MODEL FOR SITE 602A EFFECTIVE LB
CHISQ32. 64 NFREQ-33 16 LAYERS AMP FIT
o RESISTIVITY OHM.M
r
I
a. uj
TTtllf
1000.00 100.00 10.00 1.00 0.10 0.01
FREOUENCT IN HZ
Dl 1702 20
I0
02 32250 00
274
C
C
a-
0
C r a. -.
1000.00 100.00 10.00 1.00 0.10 0.01 FREQUENCY IN HZ
00000
3903
I- a-
0
01 2350
11235
02 00732 1 2317
10 MOST-SQIJAHES SM MODEL FOR SITE L9 EFFECTIVE CHISQ23.71 NFREQ26 3 LAYERS AMP+PHRSE FIT
RESISTIVITY OHM.M
10 MOST-SQURRES SM MOOEL FOR SITE 558B EFFECTIVE 19
CHISQS. 83 NFREQ26 3 LAYERS PHASE FIT
RESISTIVITY OHK.H
C
_JI I_
C
C 0
C C a. a- -
C
0
a
1000.00 100.00 10.00 1.00 0.10 0.0* FREQUENCY IN HZ
E
a- Ui 0
00000
1005
01 *003 01
3032
-
02 33800 80 *220
275
DI 772 ISIS
22 15238 08737
r
U, IJ 0
0
UJ
= U,
loss. 60 *60.00 10.06 1.00 6.10 6.61 FREQUENCY IN HZ
= I—a-'U 0
l620.00
100.60 16.60 1.66 6.16 0.61 FREQUENCY IN HZ
= 0
DI 787 1530
I-a-1J 0
II
02 10582 12332
10 MOST—SOUcIRES SM MODEL FOR SITE LiC EFFECTIVE CHISQ=27.36 NFREQ=35 3 LAYERS RMP+FHASE FIT
RESISTIVITY OHH.M
ID MOST—SOURRES SM MODELS FOR SITE Lii EFFECTIVE CHI5QU44.00 NFREQ=40 3 LAYERS AMPPHRSE FIT
RESISTIVITY OHM.H
276
a: = a -
111311.110 110.0e 10.0.
FREQUENC'T IN Ill
t
67 C.
U, Ia C
C
C
Iii S 0
Ia - U, C = C 0.
I030.00
x
I- a- Ia 0
01 S3
17
32 29711
100.00 10.00 140 11.10 0.111 FREQUENCY IN HZ
10 MOST-SQUARES SM MODELS FOR SITE L12 EFFECTIVE c111SQ35. 26 NFREQa32
3 LAYERS AHF+PHASE FIT
RESISTIVITY OIIH.M
r
tO
LIL1 rn 11 rn --- Iti1W1IIILllI LU iiiIN!II
Dl 9110 11
15
32 lOSS 311
10 MOST-SQUARES SM MODELS FOR SITE L13 EFFECTIVE CHISQ55.37 NFREQa28
3 LAYERS RMP+PHASE FIT
RESISTIVITY OHH.M
277
=
0,2
C
C
On UI C Isi C
C
Or = 0
1000.02 100.00 10.00 1.00
FREQUENCY IN HZ
E
E
I- 0 UI C
at 1073 01
05
02 01025 07
S
10 MOST-SQUARES SM MODEL FOR SITE L14R EFFECTIVE CHISQ=10. 26 NFREQ=26 3 LAYERS AKF+PHASE FIT
RESISTIVITY OHM.M 00000
I 2150
01 2023 3333
2 5602
02 *2505 1760
617
10 MOST-SQUARES SM MODEL FOR SITE LiS EFFECTIVE CHISQ-12.77 NFREQ-lq 3 LAYERS RHF+PHASE FIT
CTYTVTTY flWIl M
1000.00 100.00 10.00 1.00 0.10 0.01 FREQUENCY IN HZ
278
E
= VI 0 -
U,
C
C 0 0 C
C
0
C = VI -
IQVO, DD 100.00 26.06 2.00 6.10 6.61 FREQUENCY IN HZ
02 1305 30 15
02 9691 -92
C
*56.66 20.65 2.00 6.10 6.51 FREQUENCY IN HZ
C
Iii S 0
Hi C In C = VI
50
= a- Hi
02 699 27
it
02 7377 021
03 09323 251
10 MOST-SQUARES SM MODELS FOR SITE L16 EFFECTIVE CIiISQ=L16. 33 NFRE0=29 3 LAYERS AMP+PIIASE FIT
RESISTIVITY OHM.M
10 MOST-SQUARES SM MODEL FOR SITE L17 EFFECTIVE CI1ISQ73.8 NFREQ-33 'I LAYERS RMF+PHASE FIT
RESISTIVITY OHM.Pt
279
= 0. w 0
0.90 0.09
r
0
—
9000.00 900.00 90.00 9.00
FREQUENCY IN II?
09 922 329
3'
22 flU II!
2
03 29670 62
9— a- Ui 0
0.90 0.09
r
U,
C o , 0
C = In
9022.00 902.00 90.00 9.00 FREQUENCY IN HZ
01 IOU 22
95
02 7999 09
03 111870 29
10 MOST-SQUARES AR MODELS FOR SITE LiB INVARIANT
CHISQ28. 27 NFREQ31 '1 LAYERS AMP+PHASE FIT
RESISTIVITY OHM.M
10 MOST-SQUARES SM MODELS FOR SITE L19 EFFECTIVE
CI1ISQ27. 36 NFREQ30 4LAYERS RMP+PIIRSE FIT
RESISTIVITY OHK.N
280
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