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POLITECNICO DI MILANO
PHD SCHOOL OF ENVIRONMENTAL AND INFRASTRUCTURAL
ENGINEERING
DOCTORAL PROGRAMME IN GEOMATICS ENGINEERING
XXVIII CYCLE (2012-2013)
AIRBORNE GRAVITY FIELD MODELLING
by
Ahmed Hamdi Hemida Mahmoud Mansi
December 2015
A DISSERTATION SUBMITTED TO THE PHD SCHOOL OF
ENVIRONMENTAL AND INFRASTRUCTURAL ENGINEERING,
POLITECNICO DI MILANO IN PARTIAL FULFILLMENT OF THE
REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
Supervisor: Dr. Daniele Sampietro
Tutor: Prof. Fernando Sansò
Coordinator: Prof. Alberto Guadagnini
Copyright
The author retains ownership of the copyright of this dissertation. Neither the
dissertation nor substantial extracts from it may be printed or otherwise reproduced
without the author's permission. The author has granted a non-exclusive license
allowing the Library of Politecnico di Milano to reproduce, load, distribute or sell
copies of this dissertation in paper or electronic format.
3
Abstract
Regional gravity eld modelling by means of remove-restore procedure is nowa-
days widely applied in dierent contexts, by geodesists and geophysicists : for in-
stance, it is the most used technique for regional gravimetric geoid determination
and it is used in exploration geophysics to predict grids of gravity anomalies. In
the present work in addition to a review of the basic concepts of the classical
remove−restore, some new algorithms to compute the so called terrain correction
(required to reduce the observed gravitational signal), and to model the stochastic
properties (in terms of covariance function) of the gravitational signal, required to
grid sparse observations have been studied and implemented.
Geodesists and geophysicists have been concerned with the computation of the
vertical attraction due to the topographic masses, the so called Terrain Correction,
for high precision geoid estimation and to isolate the gravitational eect of anomalous
masses in geophysical exploration. The increasing resolution of recently developed
digital terrain models, the increasing number of observation points due to extensive
use of airborne gravimetry in geophysical exploration, and the increasing accuracy
of gravity data introduce major challenges for the terrain correction computation.
Moreover, classical methods such as prism or point masses approximations are indeed
too slow, while Fourier based techniques are usually too approximate for the required
accuracy.
A new hybrid prism and FFT−based software, called GTE, which was thought
explicitly for geophysical applications, was developed in order to compute the terrain
corrections as accurate as prism and as fast as Fourier−based software. GTE does
not only consider the eects of the topography and the bathymetry but also those
due to sedimentary layers and/or to the Earth crust−mantle discontinuity (the so
called Moho). After recalling the main classical algorithms for the computation of
the terrain correction, the basic mathematical theory of the software and its prac-
tical implementation are explained. GTE showed high performances in computing
accurate terrain corrections in a very short time with respect to GRAVSOFT and
Tesseroids.
5
Airborne Gravity Field Modelling
The slowest GTE proler has a superior performance in terms of computational
time to compute the terrain eects on grids with constant heights, sparse points and
on the surface of the provided digital elevation model than both of GRAVSOFT
and Tesseroids. While, the fast proler is able to give an overview with a standard
deviation of the errors below the accuracy of the measurements, roughly in a time
that is at least one order of magnitude less than the time required by the other
software.
A ltering procedure for the raw airborne gravity data based on a Wiener lter
in the frequency domain that allows to exploit the information coming from all the
collected data has been developed and tested too. During this ltering also biases
and systematic errors potentially present in airborne data are corrected by means
of GOCE satellite observations. A remove−like step, removing the low and high
frequencies of the observation, is done in order to reduce the values of the signal to
be ltered, which would be restored afterwards in a restore−like step, after lteringthe data.
The ltering step required almost 7 minutes to lter about 440.000 observations
if the Residual Terrain Correction required to reduce the data is available and about
30 minutes if it has to be computed.
Gridding the ltered data is done via applying a classical least squares colloca-
tion. An innovative idea that allows automatizing the estimation of the covariance
matrix, is done by tting an empirical 2D power spectral density with a series of
Bessel functions of the rst order and zero degree that assures to gain a positive
denite covariance matrix.
Finally, the estimation of the along track ltered noise is estimated through per-
forming a cross−over analysis. The study of the expected noise allows to estimate
a covariance function of the noise itself giving valuable information to be used (in
future works) in the subsequent gridding step. In fact integrating the cross−overanalysis within an iterative procedure of ltering and gridding would result in yield-
ing a better grid estimation and noise prediction of airborne gravimetric data.
All the above algorithms have been implemented in a suite of software modules
developed in C and able to exploit parallel computation and tested on a real airborne
survey. The results of these tests as well as the computational times required are
also reported and discussed.
6
Acknowledgments
Alhamdulillah, all praises to Allah for His blessings in completing this disserta-
tion with his grace.
I would like to express my gratefulness for my tutor, Prof. Fernando Sansò, for
his his continuous inspiration, fruitful discussions, brilliant ideas, advices, and the
support I have received over the last years. I am so lucky to be your last PhD
student.
I wish to express my deepest gratitude to my supervisor, Dr. Daniele Sampietro,
for his help, support, and guidance throughout the course of my Ph.D. program.
His encouragement, discussions, and comments were essential for the completion of
this dissertation. The quality of this dissertation was greatly improved as a result of
the discussions we had and as a result of your thoughtful criticism of the rst draft.
I would like to extend my tribute to the world-class Geomatics team of Politecnico
di Milano, especially Prof. Reguzzoni, Prof. Barzaghi, Prof.ssa Venuti, Dr. Gatti,
and Dr.ssa Capponi. It would be unfair if their eorts had gone unmentioned, thanks
to Ballabio, MG., Besana, L., Camporini, M., Frangi, A., Franzoni, E., Raguzzoni,
E., and Robustelli, P.
Thanks for the reviewers of this research manuscript, Prof. Crespi, M., and Prof.
Sideris, M. G., your eorts are so much appreciated.
Thanks to my parents, the main reason for what I am achieving today. Thanks
to my Mother, Samiha Amin, whose arms are always open. Thanks to my Father,
Hamdi Mansi, whose love for me is evident in everything he does.
Thanks a lot to my beautiful wife, Dr.ssa Neamat Gamal, whose sacricial care
for our little family made it possible to complete this work. Thanks to my little
baby-girl Roqayyah, who lled out our life with happiness and joy and who also
gave us some hard times.
Thanks to my loving and most-caring sister, Abeer H. Mansi, her husband Islam,
and my lovely niece Sandy who were always with me in every moment. Thanks to
my parents-in-law, Gamal Monib and Samia Abdel Wahab and thanks to my family-
in-law, Abd-Allah, Osama, Ahmed, Fatema, and Iman.
7
Airborne Gravity Field Modelling
Graduate studies has been a wonderful experience for me. It has allowed me
to learn, to travel to faraway places, and perhaps most importantly, to make some
lasting friendships. I would like to express appreciation for the support I have
received over the duration of my PhD journey to:
• Past and present members of the Geomatics team of Politecnico di Milano for
encouraging and supporting me;
• Past and present members of the Geomatics team of University of Calgary for
making my stay at University of Calgary a pleasant and joyful experience;
• Victoria Sendureva, Serap Çevirgen, Slobodan Miljatovic, Davide, and Matteo
Bianchi;
• Carla, Gerardo, and Graziella;
• Caroline Minguez-Cunningham for your friendship;
• My best friends: Philip Ghaly, Saber El-Sayed, Ahmed Sayed, Asmaa Sayed,
Osama Saleh, Khalid Hassan, Ahmed Shanawany, Ahmed Said, Mohammed
Said, Mohammed Ali, Okil Mohammed, Mahmoud Serag, Mohammed Reda,
Hani El Kadi, Wael El Sawy, Sayed Salah, Mahmoud El-Sayed, and Ramy
Basta;
• Dina Said, Iliana Tsali, Elaheh Mokhtari, Dimitrios Piretzidis, Babak Amjadi-
parvar, Hani Badawy, Carina and Alex, and Ehab Hamza for your friendship;
• EGEC colleagues: Prof. Magdi Gad, Prof. Mohammed Shokry, Prof. Mostafa
Mossaad, Eng. Mohammed Askar, and Eng. Mona for your advices and
support;
• My Uncles Khalid Amin, Thabit Amin, Mohammed Amin, and Adel Amin,
Adel Abdel Hadi, Anwar Nemr, Fathallah Talab and Ezzat Talab, Ezzat El
Araby, Hesham Abu Stita, Maher Koka, and their families;
• Uncles Abdel Nasir, Nasr, Gamal, Ezzat Tolba, and Aunt Halah and their
families;
• Abdel-Ghani Salah and Anwar Mahmoud and their families.
8
Dedication
This work is dedicated to;
• Allah (SWT) an to the last messenger Mohammed (PBUH),
• My beloved parents who teach me every day by example,
• My amazing wife, Dr.ssa Neamat Gamal,
• My little angle, Roqayyah,
• My dear sister and her family,
• My best friends; Ahmed Sayed, Ahmed Shanawany, and Philip Ghaly,
• All my friends,
• Ahmed Abdel-Aal and Ahmed Salem.
9
Declaration
I declare that this is my original work and my genuine research.
11
Contents
I Abstract 5
List of Figures 17
List of Tables 21
1 Classical Processing of Gravitational Data 23
1.1 Surface Gravity Dataset . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.1.1 Land Gravity Data . . . . . . . . . . . . . . . . . . . . . . . . 24
1.1.2 Marine Gravity Data . . . . . . . . . . . . . . . . . . . . . . . 25
1.1.3 Airborne Gravity Data . . . . . . . . . . . . . . . . . . . . . . 26
1.1.3.1 Classical Airborne Gravimetry . . . . . . . . . . . . 28
1.1.3.2 Strapdown Airborne Gravimetry . . . . . . . . . . . 28
1.1.4 Satellite data . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
1.2 Preprocessing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
1.2.1 Aircraft Motion . . . . . . . . . . . . . . . . . . . . . . . . . . 31
1.2.2 Eötvös Correction . . . . . . . . . . . . . . . . . . . . . . . . . 32
1.2.3 Vertical Acceleration Correction . . . . . . . . . . . . . . . . . 33
1.2.4 Lever Arm Eect . . . . . . . . . . . . . . . . . . . . . . . . . 35
1.2.5 Low−Pass Filter . . . . . . . . . . . . . . . . . . . . . . . . . 35
1.3 Remove−Compute−Restore . . . . . . . . . . . . . . . . . . . . . . . 37
1.3.1 Global Geopotential Model (GGM) . . . . . . . . . . . . . . . 39
1.3.2 Terrain Correction . . . . . . . . . . . . . . . . . . . . . . . . 41
1.3.2.1 Point−Mass Model . . . . . . . . . . . . . . . . . . . 42
1.3.2.2 Right−Prism Model . . . . . . . . . . . . . . . . . . 42
1.3.2.3 Tesseroid Model . . . . . . . . . . . . . . . . . . . . 45
1.3.2.4 Polyhedral−Body Model . . . . . . . . . . . . . . . . 47
1.3.2.5 Fast Fourier Transform Method . . . . . . . . . . . . 49
1.4 Downward Continuation . . . . . . . . . . . . . . . . . . . . . . . . . 49
13
Airborne Gravity Field Modelling
1.4.1 The Molodensky Concept . . . . . . . . . . . . . . . . . . . . 50
1.4.2 Free−Air Downward Continuation . . . . . . . . . . . . . . . . 51
1.5 Gravity Data Inversion . . . . . . . . . . . . . . . . . . . . . . . . . . 53
1.5.1 Collocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
1.5.1.1 Solution of the Basic Observation Equation . . . . . 54
1.5.1.1.1 Least−Squares Collocation for Non−NoisyData . . . . . . . . . . . . . . . . . . . . . . 55
1.5.1.1.2 Least−Squares Collocation for Noisy Data . 55
1.5.1.2 Covariance Estimation . . . . . . . . . . . . . . . . . 56
1.5.2 The Stokes′s Integral . . . . . . . . . . . . . . . . . . . . . . . 56
1.5.2.1 Planar Approximation of Stokes′s Integral . . . . . . 58
1.5.2.2 Spherical Approximation of Stokes′s Integral . . . . . 58
2 Gravity Terrain Eects 59
2.1 Setting the Stage for GTE . . . . . . . . . . . . . . . . . . . . . . . . 60
2.2 Theory of GTE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
2.2.1 The Planar Approximation . . . . . . . . . . . . . . . . . . . . 62
2.2.1.1 First Order Spherical Correction . . . . . . . . . . . 65
2.2.2 The Spherical Corrections . . . . . . . . . . . . . . . . . . . . 70
2.3 The GTE algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
2.3.1 GTE for The Topography . . . . . . . . . . . . . . . . . . . . 74
2.3.1.1 GTE for a Grid on the DTM Itself . . . . . . . . . . 76
2.3.1.2 GTE for a Grid at a Constant Height . . . . . . . . . 80
2.3.1.3 GTE for Sparse Points . . . . . . . . . . . . . . . . . 81
2.3.2 GTE for The Bathymetry . . . . . . . . . . . . . . . . . . . . 83
2.3.3 GTE for Moho and Sediments . . . . . . . . . . . . . . . . . . 84
2.4 GTE Performances . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
2.4.1 Test 1: TC at a Constant Height . . . . . . . . . . . . . . . . 88
2.4.2 Test 2: TC at the Surface of the DTM . . . . . . . . . . . . . 90
2.4.3 Test 3: TC at the Sparse Points . . . . . . . . . . . . . . . . . 91
2.4.4 Test 4: TC at the Sparse Points of the CarbonNet Project . . 92
2.5 Remarks on GTE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
3 Along-Track Filtering 95
3.1 The Filtering Schema . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
3.1.1 Downsampling of Gravity Data . . . . . . . . . . . . . . . . . 96
3.1.2 Wiener Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
3.1.2.1 The Reference Signal . . . . . . . . . . . . . . . . . . 99
14
CONTENTS
3.1.2.2 The Noisy Observation Signal . . . . . . . . . . . . . 100
3.1.2.3 The Removal-Like Step . . . . . . . . . . . . . . . . 101
3.1.2.3.1 The Reduced Reference Signal . . . . . . . . 102
3.1.2.3.2 The Reduced Noisy Observation Signal . . . 102
3.2 The Filtered Signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
3.2.1 Case-Study 1: Filtering Short Track #1040 (Perpendicular
Direction) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
3.2.2 Case-Study 2: Filtering Long Track #204800 (Reference Di-
rection) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
3.2.3 Case-Study 3: Filtering Full Airborne Gravimetric Survey . . 107
3.2.4 Case-Study 4: Comparison with DTU10 Model Data . . . . . 112
3.3 Remarks on Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
4 Gridding 116
4.1 The Mathematical Arguments . . . . . . . . . . . . . . . . . . . . . . 117
4.1.1 The Formulation of the Least Squares Collocation Solution . . 119
4.1.2 The Estimation of the Covariance Matrix . . . . . . . . . . . . 121
4.1.2.1 Data Reduction . . . . . . . . . . . . . . . . . . . . . 122
4.1.2.2 The Spectral vs. PSD Analysis . . . . . . . . . . . . 123
4.1.2.3 The Covariance Function . . . . . . . . . . . . . . . 123
4.1.2.3.1 The Henkel-Fourier Transformation . . . . . 125
4.1.2.4 The Covariance Matrix . . . . . . . . . . . . . . . . . 127
4.2 The CarbonNet Case-Study . . . . . . . . . . . . . . . . . . . . . . . 128
4.2.1 Comparison between the Dierent Grids . . . . . . . . . . . . 133
4.2.1.1 Comparison 1: 1 Grid Vs. 3 Grids . . . . . . . . . . 134
4.2.1.2 Comparison 2: Downsampling Frequency 1/100 Vs.
1/50 . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
4.2.1.3 Comparison 3: Downsampling Frequency 1/100 Vs.
1/10 . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
4.3 Remarks on Gridding . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
5 The Cross-Over Analysis 142
5.1 Flight Tracks Modelling . . . . . . . . . . . . . . . . . . . . . . . . . 143
5.1.1 Intersection Point Computations . . . . . . . . . . . . . . . . 143
5.1.2 Estimation of the Noise Covariance . . . . . . . . . . . . . . . 145
5.2 Case-Study: The CarbonNet Project . . . . . . . . . . . . . . . . . . 146
5.2.1 The Realization of the Cross-Over Noise . . . . . . . . . . . . 146
5.3 Remarks on the Cross-Over Analysis . . . . . . . . . . . . . . . . . . 148
Ahmed Hamdi Mansi 15
Airborne Gravity Field Modelling
6 Geoid Determination 150
6.1 Case−Study : The CarbonNet Project . . . . . . . . . . . . . . . . . 150
6.1.1 Geoid Comparison . . . . . . . . . . . . . . . . . . . . . . . . 152
7 Discussion and Conclusion 154
8 Recommendations and Future Work 158
9 Appendix A 160
10 Appendix B 162
Bibliography 164
16
List of Figures
1.1 An example of a gravity loop network. . . . . . . . . . . . . . . . . . 25
1.2 The airborne gravity measurement schema. . . . . . . . . . . . . . . . 27
1.3 The aircraft orientation layout. . . . . . . . . . . . . . . . . . . . . . 29
1.4 The Spherical coordinates of the computation point P (r, ϑ, λ) and the
integral point P (r, ϑ, λ). . . . . . . . . . . . . . . . . . . . . . . . . . 43
1.5 Sketch map of the denition of the prism (after Nagy et al. (2000)). . 44
1.6 The tesseroid representation in the spherical coordinates system. . . . 46
1.7 The geometric conventions used in the expression of the gravitational
acceleration at the origin due to a 2D polygon of a constant density ρ. 48
1.8 The 3D polyhedral representation in a 3D coordinates system and the
2D reference frame for a generic face. . . . . . . . . . . . . . . . . . . 50
1.9 The geometry of the planar Bouguer reduction, the terrain correction,
and the free-air correction. . . . . . . . . . . . . . . . . . . . . . . . . 52
2.1 Basic notation and symbols used by GTE. . . . . . . . . . . . . . . . 61
2.2 Geometry of the local sphere and of the tangent plane. . . . . . . . . 62
2.3 The mapping of the topographic body B to the attened B. . . . . . 63
2.4 The mapping of the topographic body B to the attened B. . . . . . 67
2.5 Notation of points and distances in the attened body geometry and
the illustration of the dierent used Cartesian distances. . . . . . . . 71
2.6 The set used to isolate the singularity. . . . . . . . . . . . . . . . . . 78
2.7 The Slicing the topographic body to compute the grid at height H. . 81
2.8 The Spatial interpolation at P . . . . . . . . . . . . . . . . . . . . . . 82
2.9 The geometry of the body composed by Bt (topographic body), Br
(basement with rock density), Bw (basin lled with water); Bw max-
imum depth of Bw , H is the height of the grid above the reference
surface where we want to compute δg. . . . . . . . . . . . . . . . . . 83
2.10 The Digital Terrain Model used for the rst test. . . . . . . . . . . . 87
17
Airborne Gravity Field Modelling
2.11 TC computed with the SLOW prole and its dierences with re-
spect to the gravitational eects computed by means of dierent pro-
les/software. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
2.12 The Digital terrain model used for the fourth test and the black lines
represent the dierent ight tracks followed to acquire the data. . . . 94
3.1 Schematic representation of the ltering procedure. . . . . . . . . . . 96
3.2 The procedure to compute the Reference Signal and the nal ltered
signal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
3.3 The SH coecients of the EIGEN − 6C4 GGM. . . . . . . . . . . . 98
3.4 The degree variances of EIGEN − 6C4 model. . . . . . . . . . . . . 99
3.5 The development of the SH coecients of the model to be removed. . 100
3.6 The observation versus the reduced observation of track #204800. . . 101
3.7 Schema of the ltering software: it computes the ltered signal for
dierent tracks then it computes the nal ltered signal at all the
track points by interpolating the values computed for the dierent
tracks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
3.8 The gravity observations (Signal+Noise) of track #1040. . . . . . . . 103
3.9 The 1D PSD representations of track #1040. . . . . . . . . . . . . . . 104
3.10 The PSD of the RTC signal of track #1040. . . . . . . . . . . . . . . 105
3.11 The 1D PSD function of all the signals over track #1040. . . . . . . . 105
3.12 The Reference and Filtered signals of track #1040. . . . . . . . . . . 106
3.13 The gravity observations (Signal+Noise) of track #204800. . . . . . . 107
3.14 The 1D PSD representations of track #204800. . . . . . . . . . . . . 108
3.15 The 1D PSD function of all the signals over track #204800. . . . . . 108
3.16 The Reference and Filtered signals of track #204800. . . . . . . . . . 109
3.17 The altitude of the ight performed the gravity acquisition of the
CarbonNet project. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
3.18 The gravity observations (Signal + Noise) of the CarbonNet project. 110
3.19 The reference signal of the CarbonNet project. . . . . . . . . . . . . . 111
3.20 The EIGEN − 6C4 (low frequencies) Signal. . . . . . . . . . . . . . 111
3.21 The dV_ELL_RET2012 LEIGEN−6C4max (high frequencies) Signal. . . . . . 112
3.22 The reduced reference signal of the CarbonNet project. . . . . . . . . 113
3.23 The ltered signal of the CarbonNet project. . . . . . . . . . . . . . . 114
3.24 The DTU10 gravity signal computed for the region of the CarbonNet
project. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
4.1 The gridding scheme. . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
18
LIST OF FIGURES
4.2 The spectral estimate of the reduced-ltered signal. . . . . . . . . . . 121
4.3 The 1D PSD representation of the data. . . . . . . . . . . . . . . . . 122
4.4 The graphical representation of Bessel functions of the rst kind. . . . 124
4.5 The 2D spectral estimation of the reduced observations. . . . . . . . . 129
4.6 The 1D empirical Covariance ([red]) and the theoretical Covariance (
[blue]) by tting the empirical Covariance with set of Bessel functions. 130
4.7 The nal gridded data. . . . . . . . . . . . . . . . . . . . . . . . . . . 132
4.8 The prediction error associated with the nal gridded signal. . . . . . 133
4.9 The reduced gridded signal obtained using 6743 observations and 3
LSC solutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
4.10 The prediction error of the reduced gridded signal obtained using 6743
observations and 3 LSC solutions. . . . . . . . . . . . . . . . . . . . . 135
4.11 The dierence of the reduced gridded (3 LSC solutions single LSC
solution). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
4.12 The dierence of the prediction error (3 LSC solutions single LSC
solution). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
4.13 The dierence of the reduced gridded (ωds = 1/100 ωds = 1/50) Hz. 138
4.14 The dierence of the prediction error (ωds = 1/100 ωds = 1/50) Hz. 138
4.15 The dierence of the reduced gridded (ωds = 1/100 ωds = 1/10) Hz. 139
4.16 The dierence of the prediction error (ωds = 1/100 ωds = 1/10) Hz. 140
4.17 The improvements in terms of gravity disturbances are located where
the new data are introduced (i.e., on the border of the gravimetric
campaign and beyond). . . . . . . . . . . . . . . . . . . . . . . . . . . 140
5.1 The graphical explanation of the cross−over of the ight−tracks. . . . 1435.2 The 3D original and modeled ight−tracks projected in the 2D space. 144
5.3 The intersections of all the ight−tracks projected in the 2D space. . 144
5.4 The results of the 12−cycles renement procedure, the [green lines]
represent the actual ight tracks, the [blue lines] represent the 3D
LS estimated lines projected into the 2D space, the [black stars] are
the initial intersection points, the [red stars] are the intermediately
calculated intersection points, the [black circle] is the nal intersection
point. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
5.5 The Empirical covariance function Cνν(d) for the CarbonNet data. . . 146
5.6 The realization of the noise on the CarbonNet tracks (mGal). . . . . 147
5.7 The realization of the noise on the CarbonNet grid (mGal). . . . . . . 148
5.8 The iterative procedure. . . . . . . . . . . . . . . . . . . . . . . . . . 149
Ahmed Hamdi Mansi 19
Airborne Gravity Field Modelling
6.1 The computed CarbonNet−based geoid heights. . . . . . . . . . . . . 151
6.2 The error associated to the estimation of the CarbonNet−based geoidheights. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
6.3 The geoid dierences between the CarbonNet-based geoid and the
ocial AUSGEOID09. . . . . . . . . . . . . . . . . . . . . . . . . . . 152
20
List of Tables
1.1 Summery of acceleration terms . . . . . . . . . . . . . . . . . . . . . 33
1.2 Examples of dierent reference ellipsoids and their geometrical pa-
rameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.1 Number of slices and number of prisms used for each slice to reduce
the FFT singularity for dierent proles. Parameters are reported in
case of computation of topographic and bathymetric eects . . . . . . 85
2.2 The statistics and the computational time on a grid at 3500 m for the
dierent proles and software tested. SLOW prole shows statistics
on the computed signal. For the other rows the statistics refer to the
dierence between each result and the terrain eect computed with
the SLOW prole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
2.3 The statistics and the computational time on a 1000 points for the
dierent proles and software tested. SLOW prole shows statistics
on the computed signal. For the other rows the statistics refer to the
dierence between each result and the terrain eect computed with
the SLOW prole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
2.4 The statistics and the computational time on 404384 points for the dif-
ferent proles and software tested. VERY SLOW prole shows statis-
tics on the computed signal. For the other rows the statistics refer
to the dierence between each result and the terrain eect computed
with the VERY SLOW prole . . . . . . . . . . . . . . . . . . . . . . 93
3.1 The statistics of all the signals aecting track #1040 . . . . . . . . . 104
3.2 The statistics of all the signals aecting track #204800 . . . . . . . . 107
3.3 The statistics of the CarbonNet airborne gravimetric campaign . . . . 113
9.1 Full list of the reference ellipsoids and their geometrical parameters . 161
10.1 Details of dierent GGM combinations. . . . . . . . . . . . . . . . . . 163
21
Chapter 1
Classical Processing of Gravitational
Data
[
HAKP@YË @] [(47)
àñ
ªñ
Ü
Ï A
K @
ð Y
K
AK.
AëA
JJ
K. Z A
Ò
Ë@
ð]
[And the heaven (is also a sign). We have built it with (Our) Hands (i.e.,
Capability) and surely We are indeed extending (it) wide. (47)] [Quran,
Adh−dhariyat]
In this chapter, a detailed discussion will be made about the dierent datasets
available in classical gravity eld modeling for geoid determination (e.g., ground,
shipborne, and aerogravimetric gravity data). The pre−processing schema (e.g.,
the ltering of the raw data), the processing techniques implemented (e.g., the
remove−compute−restore technique) and its dierent stages such as the compu-
tations of the terrain correction, the residual terrain correction, and the downward
continuation will be briey presented too.
1.1 Surface Gravity Dataset
Dierent gravitational data types such as ground, shipborne, airborne and global
gravitational models will be considered and discussed in this dissertation work. Gen-
erally speaking, the Earth′s Gravity eld is a harmonic potential eld (V) and it is
a fundamental geodetic parameter (Heiskanen and Moritz , 1967). The gravity ex-
ploration is used to sense dierent physical properties for the subsurface and to give
an idea about its composition and geological formations. More specically, gravity
surveys exploit the very small changes in gravity eld from a place to another that
are caused by the changes in the densities of the subsurface layers (Rogister et al.,
2007).
23
Airborne Gravity Field Modelling
Up to the late twenties, pendulum was essential instrument for acquiring abso-
lute and relative gravity measurements on land and in most of the oceans of the
world with a specially designed pendulum installed in a submerged submarines.
The resulted anomaly maps were obtained with error as much as 10 mGal (Ven-
ing Meinesz , 1929). Later on, the other gravity measuring instruments such as
sensitive Quartz−spring balances/gravimeters for relative gravity and falling bodies
for absolute gravity measurements showed great performances at laboratory tests
till they have been put to the eld and then dominate the market due to their
continuous−reading, relatively cheaper−operating costs and the promises results of
using them aboard ships and later on board of aircrafts (Dehlinger , 1978). LaCoste
and Romberg are the pioneers of stabilized platform shipborne gravimeters that has
evolved from the early launch on mid−sixties to become the most used gravime-
ter for land, airborne, and shipborne gravity campaigns, the interested reader is
redirected to (LaCoste, 1959a) and (LaCoste and Harrison, 1961) for more details.
1.1.1 Land Gravity Data
The ground gravity acquisition measures the gravity eld using relative or ab-
solute gravimeter. Because of the very weak nature of the gravity forces, the grav-
itational campaign necessitates using highly sensitive gravimeters that have been
classically proven to have a measuring precision of 0.01 mGal or better.
On the one hand, the absolute gravimeter measures the actual value of the grav-
itational acceleration, g, by measuring the speed of a falling mass using a laser beam
with precisions of 0.01 to 0.001 mGal. The usage of absolute gravimeter is highly
expensive, heavy, and bulky. On the other hand, the relative gravimeter measures
the relative changes in g between two locations by using a mass on the end of a
spring that stretches where g is stronger with a precision of 0.01 mGal in about 5
minutes. A relative gravity measurement should be done whenever possible at the
start and/or the end stations thereby the relative gravity measurements get tied,
namely "land ties" and consequently when the land ties are made at the same point,
it could be used to correct the survey values for the drift of the equipment and
other temporal variations (such as tides). If the land ties are spread over dierent
locations therefore they need to be correlated with a worldwide gravity datum by
getting the gravity value at the nearest absolute gravity station that would allow
the survey to be integrated into the regional context. A gravity survey network is a
series of interlocking closed loops of gravity observations. An example of a gravity
loop network is shown in Fig. 1.1.
On one side, permanent gravity stations (Gravity Bench Marks) are equipped
24
CHAPTER 1. CLASSICAL PROCESSING OF GRAVITATIONAL DATA
Figure 1.1. An example of a gravity loop network.
with adequate gravimeters that best serve the purpose of each station. Also, a
permanent GNSS station is a must in order to provide a continuous data about
the position of the station. Traditionally, the permanent gravity stations are tradi-
tionally installed with a permanent GNSS station that belongs to the International
GNSS Network in order to provide simultaneous observations for the coordinates of
the station. Performing auxiliary geometrical connections between the permanent
gravity stations and tide gauges is a common practice. On the other end, while per-
forming gravity campaigns, temporary gravity station is well thought out at each
observatory point of the gravity network and also an accurate leveling/surveying
campaign (e.g., Precise Point Positioning, Long−Time static GPS sessions, . . . etc.)
is performed at each node of the gravity network (Timmen et al., 2006).
1.1.2 Marine Gravity Data
On the early fties, marine gravity surveys have been made in submerged sub-
marines as pendulums were not able to operate reliably on board of ships even
in calm sea states although the operation cost was relatively expensive (Harrison
et al., 1966). LaCoste 1967 introduced the stabilized platforms and the highly
damped−sensors that helped the shipborne gravimetry to dominate the subma-
Ahmed Hamdi Mansi 25
Airborne Gravity Field Modelling
rine gravimetry due to its reduced cost, reliable measurements, and in recent years,
the high accuracy achieved. With the steady advancement of the technology, the
capabilities to mute the acceleration of the ship, and the dierent contributions,
modications, and adaptations made on the original LaCoste & Romberg marine
gravimeter, the current state of marine gravimetry has been achieved (Hildebrand
et al., 1990; Zumberge et al., 1997; Sasagawa et al., 2003).
On the one hand, the great capabilities to sail in a non−rough sea states, the
steady slow−motion of the ship and the installed gravity platform, the technological
tools to average the data over very large intervals, a better gain in the accuracy and
higher resolutions were achieved. In additional the majority of marine gravity data
are collected in conjunction with other expensive survey methods such as seismic sur-
veys, EM surveys, Remotely Operated Vehicles (ROV) projects, multi−beam bathy-
metric surveys, and other hydro-graphic projects that made the cost for collecting
the marine gravity data relatively low. Also acquiring synchronized marine gravity
data with other data typologies such as seismic can be benecial in multi−disciplineenhanced processing, inversion, and interpretation methods. On the other hand,
the stand alone marine gravity campaigns provide the highest quality surveyed lines
because of the optimization processes of choosing the vessel sailing parameters such
as speed and orientation that help yield such optimum data.
1.1.3 Airborne Gravity Data
The spatial resolution of Earth gravity models derived from satellite data is lim-
ited. The only technique available to bridge the gap in spatial resolution between
satellites and ground−based gravimeters is airborne gravimetry, i.e., the measure-
ment of the gravitational eld signal using gravimeters installed on board of air-
crafts. The concept of airborne gravity was proposed more than half a century ago
(Hammer , 1950), while the rst ight was conducted only in late fties, namely
Lundberg′s test (Lundberg , 1957). The implemented system was built upon the
principle of gradiometry and its results were met with great skepticism due to the
inaccurate determination of the aircraft position and velocity (Hammer , 1983). The
key problems for airborne gravimetry at that time were the navigation of the aircraft,
including velocity, elevation space positioning, in−ight accelerations of the aircraft,and the lack of a gravimeter able to work in a dynamic environment (Thompson and
LaCoste, 1960). Just a few years later, the advancement of the marine gravimeters
and the development of navigation systems exploiting Doppler eect were put into
a successful ight in early sixties (Nettleton et al., 1960). The development of the
GPS during the early eighties was a very essential milestone in redesigning and real-
26
CHAPTER 1. CLASSICAL PROCESSING OF GRAVITATIONAL DATA
Figure 1.2. The airborne gravity measurement schema.
izing the present−day airborne gravity system (Schwarz , 1980; Brozena et al., 1988;
Fonberg , 1993).
The early 1960s witnessed a successful attempt to collect gravity data from a
xed−wing aircraft (Gumert , 1998), while the rst successful trial acquiring gravitymeasurements from a helicopter was performed in 1965 (Gumert and Cobb, 1970).
The advantages of a helicopter over a xed−wing aircraft are its capabilities to betterfollow the terrain, the abilities to y at a low altitude that increases the spatial
resolution, and the fact that a helicopter is less aected by turbulent conditions
than most other types of aircraft (Lee et al., 2006). A schematic layout for the
airborne gravity measurements is shown in Fig. 1.2. The classical airborne gravity
system could easily collect the data with 0.5 to 1 mGal accuracy through integrating
the observations made with many dierent sensors and systems installed within one
aerogravimetric platform, such as:
Ahmed Hamdi Mansi 27
Airborne Gravity Field Modelling
1. Gravity sensor system that comprises the airborne gravimeter and the contain-
ing platform. This system helps in computing some corrections to the collected
gravity disturbances;
2. Inertial Navigation System (INS) and Global Positioning System like GPS in
order to provide the optimal real−time navigation data and the coordinates of
the platform as (X, Y, and Z). It allows us to compute an independent solution
for the velocity and non−gravitational acceleration to correct for the Eötvös
and tilt eects;
3. IMU systems to provide data about the orientation of the aircraft in terms of
(pitch, roll, and yaw), see Fig. 1.3;
4. Altitude sensor system in order to provide data about the altitude/height of
the aircraft that would help in computing both Eötvös and vertical acceleration
corrections;
5. Metadata of the acquisitions in terms of;
(a) The lever−arm between the gravimeter and the IMU/INS/GPS systems;
(b) The vertical distance between the odometer and the tie spot;
(c) The gravity value at the tie spot.
The state if the art gravity data could be collected through 2 kinds of airborne
systems. A brief introduction for the sake of completeness will be elaborated within
the following sections.
1.1.3.1 Classical Airborne Gravimetry
The main characteristic of the classical airborne gravimetry systems consists in
having the gravimeter xed on an inertial platform in order to stabilize the sensor
during the data collection phase. The ight can reach a speed of 50 meters/second,
therefore allowing very accurate data to be collected. The spatial resolution of such
classical airborne gravimetric data can straightforwardly reach 1.0 kilometer after
applying a 20 second lter over the raw data. The stabilized airborne platforms have
shown to have long−term stability (Glennie and Schwarz , 1999).
1.1.3.2 Strapdown Airborne Gravimetry
On the other side, the strapdown inertial navigation systems have been developed
to be an alternative to the classical airborne gravimetry (Schwarz et al., 1991). The
28
CHAPTER 1. CLASSICAL PROCESSING OF GRAVITATIONAL DATA
Figure 1.3. The aircraft orientation layout.
strapdown inertial navigation systems do not have a gravimeter on board but it does
have an IMU that is xed on the plane. With certain software, the oscillations and
the acceleration of the aircraft would be computed and then used in order to lter
the collected data. The main advantages of this system over the traditional one are
its smaller size and relatively low cost (Wei and Schwarz , 1998) in additional to it
has shown that it can reach the same level of accuracy (Bruton et al., 2002) and that
the full gravity vector can be obtained (Jekeli , 1994) and that it has the potential
to increase the spatial resolution in the future (Alberts et al., 2008).
1.1.4 Satellite data
CHAMP (CHAllenging Minisatellite Payload) is a small German low-Earth-
orbiting satellite mission for geoscientic and atmospheric research and applications.
CHAMP that operated from July 2000 to September 2010 had generated simultane-
ously high precise gravity and magnetic eld measurements. The CHAMP mission
was the rst big step in gravity satellite missions that opened a new era in global
geopotential research (for more details, see (Reigber et al., 2000, 2002)). Using GPS
satellite−to−satellite tracking and accelerometer data of the CHAMP satellite mis-
sion, a new long−wavelength global gravity eld model, called EIGEN−1S, has beenderived solely from analysis of satellite orbit perturbations (Reigber et al., 2002).
Ahmed Hamdi Mansi 29
Airborne Gravity Field Modelling
The Gravity Recovery and Climate Experiment (GRACE) mission by NASA was
launched in March of 2002. The GRACE mission is accurately mapping variations
in the Earth′s gravity eld with a system of twin satellites that y about 220 km
apart in a polar orbit 500 km above Earth. GRACE maps the Earth′s gravity eld
by making accurate measurements of the distance between the two satellites, using
GPS and a microwave ranging system to provide scientists with an ecient and
cost−eective way to map the Earth′s gravity eld with unprecedented accuracy.
The GRACE follow−on mission scheduled for 2017 will continue the work of moni-
toring the Earth (for more details, see (Adam, 2002; Aguirre-Martinez and Sneeuw ,
2003)). Integrating the data between CHAMP and GRACE has been done to pro-
duce models for the gravity eld of the Earth (Kaban and Reigber , 2005) and these
helped scientists to better understand the mass of the Earth (Kaufmann, 2000).
GOCE is the acronym for the Gravity eld and steady−state Ocean Circulation
Explorer mission. The objective of GOCE was the determination of the station-
ary part of the Earth gravity eld anomalies with 1 mGal accuracy and geoid with
1 to 2 cm with spatial resolution better than 100 km with highest possible accu-
racy (EGG-C , 2010a). GOCE provided completely new information about the mid
frequency range of the gravity eld. GOCE provided a very high precision in the
long−to−medium wavelength part of the gravity eld up to a spherical harmonic
degree of about 250 (Fecher et al., 2011a,b).
The satellite−only Global Gravity Models have major improvements in area
where only a few and less accurate terrestrial measurements are available (Hofmann-
Wellenhof and Moritz , 2005). Many authors tried to sew together various types of
satellite−only data (for instance see, Reguzzoni and Sansò (2012)), to integrate
satellite data and terrestrial data (for more details see, Pavlis et al. (2008)), and
to merge dierent kind of satellite data, terrestrial data, and kinematic orbits, and
satellite laser ranging (SLR) data (consult, Mayer-Gürr et al. (2015)).
1.2 Preprocessing
First, the raw airborne gravity data is corrected for the aircraft motion (vertical
acceleration correction, Eötvös correction, inclination to the horizontal (referred as
tilt) correction and lever arm eect). The vertical acceleration correction is com-
puted to compensate for the high−frequency components added to the observed
gravitational signal due to the vertical motion of the aircraft and due to the vi-
bration of the body and the platform. The application of a low−pass lter or a
high−damping sensor to the gravimeter can result in removing the vertical acceler-
30
CHAPTER 1. CLASSICAL PROCESSING OF GRAVITATIONAL DATA
ation contributions (Dehlinger et al., 1966). On the contrary, the Eötvös correction
is well known as the change of the centrifugal force of the earth rotation due to
measuring the gravity from a moving platform (Glicken, 1962). Moreover, when the
airborne gravity platform is not strictly parallel to the level surface that does not
only aect the gravitational acceleration but also impact on the vertical component
of the horizontal acceleration known as the inclination to the horizontal acceleration
correction (Lu et al., 2014). The lever arm eect happens physically when the Iner-
tial Measurement Unit (IMU) of an INS, the GPS antennas, and the odometers are
not located at the same position. The lever arm is the distance between the sensing
points of the sensors (Seo et al., 2006).
Later on, the manipulation of the airborne gravitational data can be done on two
phases, the former is the so-called the preprocessing phase and the latter is the data
inversion phase. The preprocessing phase consists in many steps such as low−passltering, cross-−over adjustment, and gridding. The low−pass lter is mainly used
to handle the noise and to suppress its high−frequency components. The cross−overanalysis is an essential step to remove the bias and the drift that exist within the
data and to reduce the mist at the locations of crossing ight lines. Although the
aerogravimetric data provided for this research were not delivered as raw data but
as preprocessed dataset with a low−pass lter, this section will be elaborated only
for the sake of completeness.
1.2.1 Aircraft Motion
This section is dedicated to discuss the dynamics of the aircraft motion and how
to handle its motion equation. Bearing in mind that the measured gravitational dis-
turbances by the gravimeter launched on board of an aerogravity platform must be
distinguished from the non−gravitational accelerations. Therefore, the accelerationsmeasured or derived from the GPS data are used in order to produce uncorrelated
gravity measurements. Also, knowing that the gravimeter attached to the plat-
form provides relative measurements of the observed gravity eld, a tie point on the
ground with an absolute gravity value must be generally used upon the take−oin order to correct the raw data observed. However nowadays due to the improve-
ments in the gravity eld modelling from satellite data, the use of absolute ground
gravimeter can be avoided. In the presence work, for instance, as reference eld a
global model containing CHAMP, GRACE and GOCE satellite data will be used.
This will not only improve the low frequency of the retrieved gravitational eld, but
also assure the global consistency of the obtained results. Finally, any instantaneous
deviation from the ideal measuring layout such as instrumentation drift, o−level
Ahmed Hamdi Mansi 31
Airborne Gravity Field Modelling
. . . and/or tilt of the platform must be corrected.
In order to proceed, one must point out the basic observation equation to recover
the gravity disturbances at the ight altitude using a stabilized platform system
(Eq. 1.1):
δg = gm − z + εEotvos + εtilt − gm0 + ga − γh (1.1)
Eq. 1.1: Gravity disturbances at the ight level, where gm is the vertical
acceleration sensed by the gravimeter that is also known as the specic force, z is
the vertical acceleration of the aircraft, εEotvos is the Eötvös correction, εtilt is the
inclination to the horizontal correction, gm0 is the gravimeter reading of the
gravimeter at the stay-still state, ga is the absolute gravity value at the tie point,
and γh is the normal gravity value.
The full acceleration of the moving aircraft in a rotating reference system could
be written implementing the Newton′s law of motion as in Eq. 1.2:
~a = d2~rdt2
+ 2~ω d~rdt
+ d~ωdt× ~r + ~ω × ~ω × ~r (1.2)
Eq. 1.2: The acceleration equation of the aircraft, where t is time, ~r is the vector
from the considered observation point to the axis of rotation of the Earth
perpendicular to the axis itself; ~ω is the angular velocity of the Earth.
Note that the rst term is the acceleration of the aircraft within the considered
coordinate system. The third term is the so−called Euler acceleration (David Scott ,
1957), which mathematically models the acceleration of the coordinate system itself
that equals zero when the assumption of constant rotation rate of the Earth is consid-
ered. The second and the fourth terms represent the Coriolis acceleration (Coriolis ,
1835) and the centrifugal acceleration, respectively, and the vertical contributions of
both terms embrace the Eötvös correction. A summary of main acceleration com-
ponents (Coriolis, Eötvös, and centrifugal accelerations) and their contributions to
the various directions (East, North, and Vertical directions) are reported below in
Table 1.1.
1.2.2 Eötvös Correction
In 1919, this correction was formulated by Eötvös (1953), to compensate for the
aforementioned, Eötvös eect that is the horizontal motion of the aircraft platform
over the Earth′s surface that corresponds to the merged vertical contribution of
Coriolis and centrifugal accelerations. In other words, Eötvös eect can be explained
as the output centripetal acceleration from the motion of a moving platform over a
32
CHAPTER 1. CLASSICAL PROCESSING OF GRAVITATIONAL DATA
Acceleration component X axis (East) Y axis (North) Z axis (Vertical)
Coriolis component 2νNωEarth sinλ 2νEωEarth sinλ 2νEωEarth cosλ
Centrifugal component νNνER
tanλν2E
Rtanλ
ν2E + ν2
N
Rtanλ
Gravitational component g0R2
(R +H)2
Table 1.1. Summery of acceleration terms
curved rotating Earth. As Eötvös correction depends on the speed of the aircraft,
its direction, and the latitude and the altitude of the ight, the resulted correction is
concerned with the steady−state motions of the aircraft (Geyer and Ashwell , 1991).
Because gravity varies with the altitude according to the inverse square law and
the relation with respect to the altitude H can be expressed as shown in Eq. 1.3
(Collinson, 2012):
g = g0R2
(R+H)2 (1.3)
Eq. 1.3: The inverse square law of the gravity value with respect to the altitude H.
Therefore, a lot of eort was done in order to distinguish the ground speed and
the aircraft velocity and to accommodate this Eötvös correction for higher velocities
and higher altitudes choices (where smooth−ight conditions could be obtained) andto demonstrate the large impact of the navigation parameters′ errors on it. Harlan
(1968) expressed mathematically the Eötvös correction as reported in Eq. 1.4 as
follows:
εEotvos = ν2
a
[1− h
a− ε(1− cos2 ϕ(3− 2 sin2 α))
]+ 2ν ωEarth cosϕ sinα (1.4)
Eq. 1.4: The Eötvös correction, where ν = νE + νN with ν is the aircraft speed, νE
and νN the East and the North components of the aircraft speed, a is the
semi-major axis of the reference ellipsoid, h is the altitude of the aircraft, ωEarth is
the angular velocity of the Earth, ϕ and α are the latitude and the azimuth angles
of the aircraft, with ε = ν2
a· sin2 ϕ+ νωEarth.
1.2.3 Vertical Acceleration Correction
When the vertical axis of the platform of the aircraft is deviated and misaligned
from the instantaneous vertical vector, errors due to the horizontal accelerations
contaminate the collected airborne gravitational data (Lu et al., 2014). Aircraft
vertical accelerations for airborne gravimetry have been determined using radar and
Ahmed Hamdi Mansi 33
Airborne Gravity Field Modelling
pressure altimeters (Brozena et al., 1986), laser altimeters (Bower and Halpenny ,
1987), and most recently GPS (Brozena et al., 1988). Because the vertical accelera-
tion due to aircraft motion is inseparable from the gravitational acceleration sensed
by the installed gravimeter, the navigation data must be utilized in order to derive
independent estimate of the vertical acceleration of the platform or to estimate the
o−level angle of the platform horizontal acceleration, in order to obtain a correc-
tion for the tilt eect. Therefore, a continuous observation for the aircraft altitude
is required in order to dierentiate it twice to attain the second derivative of the
height (Kleusberg et al., 1989). After that an essential piece of computation in or-
der to properly quantify the vertical acceleration, z, and therefore to evaluate the
vertical acceleration correction (Meurant , 1987), is to apply a low−pass lter suchas a moving average window (e.g., a 2 kilometers window).
In the following some easy to be implemented formulas found in literature are
reported. In the one hand, Eq. 1.5 is applicable if the horizontal acceleration is
well−computed and corrected for any errors in addition to the availability of the tilt
angle, θ, (Lu et al., 2014).εtilt = g(cos θ − 1) + Ae sin θ(1.5)
Eq. 1.5: The vertical acceleration correction, where θ is the tilt angle between the
platform and the level surfaces and Ae is the horizontal acceleration.
While Eq. 1.6 can perform better if the tilt information is known with respect
to both X and Y axes (Operation manual, MicrogLaCoste).
εtilt = g(1− cos θx. cos θy) (1.6)
Eq. 1.6: The tilt correction, where θx and θy are the tilt angles with respect to X
and Y− axes.
Both exact and approximate corrections are found in Valliant (1992) and they
require precise information about the output of the accelerometer along the cross
and long axis and about the horizontal kinematic accelerations in the East and North
directions. The latter can be derived easily from the navigation data as reported in
Eq. 1.7 and Eq. 1.8.
εtilt =√g2accelerometer + A2 − a2 − ggravimeter
with A2 = (A2X + A2
L) and a2 = (a2E + a2
N)
(1.7)
Eq. 1.7: The exact tilt correction, where AX and AL are the along the cross and
long axis output of the accelerometer, aE and aN are the horizontal acceleration in
34
CHAPTER 1. CLASSICAL PROCESSING OF GRAVITATIONAL DATA
East and North directions derived from the navigation (GPS) data, and ggravimeter
is the gravimeter observed data.
εtilt = A2−a2
2ggravimeter
with A2 = (A2X + A2
L) and a2 = (a2E + a2
N)
(1.8)
Eq. 1.8: The approximate tilt correction, where AX and AL are the along the cross
and long axis output of the accelerometer, aE and aN are the horizontal
acceleration in East and North directions derived from the navigation (GPS) data,
and ggravimeter is the gravimeter observed data.
1.2.4 Lever Arm Eect
Physically, the Inertial Measurement Unit (IMU) of an INS, the GPS antennas,
and the odometers cannot be located at the same position thus generating what is
called the lever arm eect. The result of the lever arm eect is seen in the dierence
between the vertical accelerations computed from the GPS data and those observed
directly by the gravimeter. In order to end−up with accurate navigation data, a
compensation for the eect of lever arm that could be dened as the horizontal
distance between the sensing points of the dierent sensors must be utilized (Seo et
al., 2006). While Olesen (2002) recommend neglecting the lever arm eect for scalar
gravity if it is below 1 meter, De Saint-Jean et al. (2007) advised to accurately model
it for vector gravity. Similar to the lever arms of the GPS antennas−gravimeter,
lever arms of the INS and altimeter instruments must be corrected, if present. All
adjustments are made to the location of the gravimeter, which better be located
close to the center of gravity of the aircraft (Hong et al., 2006).
1.2.5 Low−Pass Filter
Airborne gravity measurements are characterized by its low signal-to-noise ratio
as the measurements are collected in a very dynamic environment. A typical value for
the noise−to−signal ratio of 1000 or higher could be easily found (Schwarz and Li ,
1997). This high noise−to−signal ratio contaminating the raw sensor measurements
makes the extraction of the gravitational disturbances a hard and a challenging task.
Low-pass ltering is an essential processing step that is applied to the acceleration
data in order to separate the high frequency receiver measurement noise from the
low frequency acceleration data. On the one hand, to design the optimum lter
Ahmed Hamdi Mansi 35
Airborne Gravity Field Modelling
for airborne use, we must determine the gravity signal waveband. Therefore, the
optimum lter must imply that:
• It does not aect or distort the low frequency content of the acceleration
signal obtained by airborne gravimetric surveys, as narrowing the transition
band of the lter produce distortion to the characteristics of the low frequency
acceleration signal;
• It lters out the high frequency noise contaminating the gravity observations
(Peters and Brozena, 1995).
The main advantage of using the low−pass lter is the easy design and implemen-
tation of such lter. A secondary advantage of using the low−pass lter is that theband−limited resulted gravitational measurements somehow stabilize the downward
continuation process but in the other hand, it will generate in a smooth geoid.
Traditional lters used in airborne gravimetry are the 6 x 20−s resistor-capacitor(RC) lter and the 300−s Gaussian lter, heavily attenuate the waveband of the
gravity signal and they are much more suitable to be used in marine gravimetric
surveys. While, the concept of model−based ltering has been proposed by Ham-
mada and Schwarz (1997). Childers et al. (1999) studied a low−pass lter that
involves identifying the waveband of the gravity signal based upon the survey pa-
rameters and an iterative approach in implemented in order to design the lter
which is repeatedly tested and modied to yield the optimum results. While Al-
berts et al. (2007a) studied how to replace the concept of low−pass ltering by a
frequency dependent data weighting to handle the strong colored noise contained
within the raw data. The ideal low−pass lter can be represented mathematically
(Eq. 1.9) as a transfer function, H(ω), transforming the signal up to the cut−ofrequency, ωcut−off asfollows :
H(ω) =
1 if 0 < ω ≤ ωcut−off
0 if ωcut−off < ω ≤ ∞(1.9)
Eq. 1.9: The mathematical model of the low−pass lter, where H(ω) is the
transfer function, ω is frequency of the gravimetric signal, and ωcut−off is the
cut−o frequency.
The airborne gravity data used within this dissertation are characterized by being
bandwidth−limited data because of the usage of low−pass lters while collecting thedata in addition to the limitation of the aerogravimetry area coverage. In any case to
lter the raw gravitational acceleration we propose in this work the use of a Wiener
36
CHAPTER 1. CLASSICAL PROCESSING OF GRAVITATIONAL DATA
lter in the frequency domain. In particular the lter is obtaining by studying end
exploiting the spatial correlation of the gravitational eld along each airplane track.
Further details on this approach will be discussed in Chapter 3.
1.3 Remove−Compute−Restore
The classical Remove−Compute−Restore (RCR) technique is the most adopted
and applied technique for regional gravimetric geoid determination (Schwarz et al.,
1990). The RCR is composed by three essential steps; the rst step namely, Remove,
targets the removal of the long−wavelength contributions utilizing the maximum
degree or a truncated spherical harmonics expansion of a certain global geopotential
model (GGM) exploiting either a satellite−only GGM or a combined GGM (Abbak
et al., 2012). In addition to, the removal of the short−to−medium wavelength
contributions of the topography existing above the geoid, this computation is called
terrain correction (TC). The last piece of the removal step is to compute the residual
terrain correction (RTC) and remove it from the observed signal in order to suppress
the short−wavelength contributions. The data model can be explained as reported
in Eq. 1.10:
∆gred = ∆g −∆gGGM −∆gTC −∆gRTC (1.10)
Eq. 1.10: The gravimetric measurement model, where ∆g is the low-pass ltered
gravimetric signal, ∆gGGM is the gravimetric signal imprints of the
long−wavelength contributions computed from the global geopotential model,
∆gTC is the terrain correction gravimetric signal representing the
short−to−medium wavelength contributions, and ∆gRTC is the residual terrain
correction gravimetric signal.
Secondly, the Compute step, where the reduced signal of the gravity anomalous
would be processed in order to compute the geodetic functional of interest, namely
the geoid undulation. On the one hand, a grid of the reduced signal ∆gred is essential
in order to apply the 1D or the 2D Fast Fourier Transformation (FFT) methods to
evaluate the Stokes′ integral (Stokes , 1849), while evaluating the Stokes′ integral
using the Least Squares Collocation (LSC) does not invoke having gridded data.
The geoid undulation N is computed through the implantation of Eq. 1.11 that
represents the Stokes′ integral (i.e., the most important formula in physical geodesy),
the solution of the boundary value problem in the potential theory permitting the
determination of the geoid undulation from gravimetric data (Heiskanen and Moritz ,
Ahmed Hamdi Mansi 37
Airborne Gravity Field Modelling
1967) as follows:
N∆gred = R4πγ
∫ ∫(∆gred + gMolodensky) S(ψ) dσ (1.11)
Eq. 1.11: The Stokes′ integral, where R is the mean Earth radius, γ is the normal
gravity on the reference ellipsoid, ∆gred is the reduced gravity anomaly signal,
gMolodensky is the rst term in Molodensky expansion, ψ is the geocentric angel, dσ
is an innitesimal element on the unit sphere, and S(ψ) is the original Stokes
function.
If the RTC is subtracted to obtain the reduced signal then the Molodensky ex-
pansion gMolodensky would be insignicant (Forsberg and Sideris , 1989) and therefore
could be ignored (Schwarz et al., 1990). The Stokes′ function can be computed by
using Eq. 1.12 expressed in terms of a series of Legendre polynomials (Snow , 1952)
over the sphere σ.
S(ψ) =∞∑n=2
2n+1n−1
Pn(cosψ) (1.12)
Eq. 1.12: The Stokes′ function, where n is the spherical harmonic degree, and
Pn(cosψ) is the series of Legendre polynomial.
The classical Remove−Compute−Restore (RCR) technique would be applied,
consequently the integration domain would be spatially restricted because of the
lack of coverage and the limited availability of the terrestrial gravimetric data over
the whole Earth surface.
The main emphasis of the last step, namely called the Restore is to recover the
eects of the removed GGM gravitations signal (∆gGGM) and the TC signal (∆gTC)
and the RTC signal in terms of terms of geoid undulation as seen in Eq. 1.13.
N = NGGM +N∆gred +Nindirect (1.13)
Eq. 1.13: The geoid undulation, where NGGM geoid undulation contribution of the
global geopotential model, N∆gred is the residual geoid undulation computed by
band−pass ltered ad reduced gravity measurements, and Nindirect is the indirect
eects of the terrain and topography on the geoid height.
The following subsections will be dedicated to the discussion of the remove and
restore computations of the GGM, TC, and RTC.
38
CHAPTER 1. CLASSICAL PROCESSING OF GRAVITATIONAL DATA
1.3.1 Global Geopotential Model (GGM)
Simply, a global geopotential model could be dened as the mathematical ap-
proximation of the gravity potential eld of an attracting body, the Earth for our
geodetic applications. The GGM consists in a set of numerical coecients of a
spherical harmonic expansion truncated up to a maximum degree (Lmax), the statis-
tics of the error associated to these coecients (error covariance matrix) and the
mathematical expressions and algorithms that permit:
• The rigorous and ecient computation of the numerical values of any func-
tional of the potential eld such as geoid undulation, gravity anomalies, de-
ection of the vertical, second order gradients of the potential at any arbitrary
point on or above the surface of the Earth;
• The evaluation of the error propagation such as the expected errors of the
computed functionals by propagating the errors of the GGM parameters.
All these computations must be done in a consistent manner, which means that
the GGM must preserve the dierential and integral relationships between the var-
ious functionals. It is also characterized by fullling the constraining conditions of
the potential theory and strictly follows their corresponding physics concepts such
as representing a harmonic potential eld outside the attracting mass that vanishes
at innity.
The signal of the GGM can be thought as the eect of the normal ellipsoid and
the topographic eect as explained in Eq. 1.14. The rst term on the right hand side
represents the terrain and topographic eects that would be covered and explained
in details on subsection 1.3.2. On the other hand, the second term that represents
the gravitational eect of what is called equipotential ellipsoid of revolution. One
particular ellipsoid of revolution, called the "normal Earth", is the one having the
same angular velocity and the same mass as the actual Earth, the potential U0 on
the ellipsoid surface equal to the potential W0 on the geoid, and it center of mass is
coincident with the center of mass of the Earth (Li and Götze, 2001).
∆gGGM = ∆gTC + γEllipsoid (1.14)
Eq. 1.14: The GGM signal, as γEllipsoid = γEllipsoid(P ) + δghP is the gravitational
signal of the reference ellipsoid that contains long−wavelength contributions at
point P and δghP is the corresponding height correction.
Due to the advancement of geodesy and the consequent improvements to the
dening parameters of the reference ellipsoid (for instance see Table 1.2, and for the
Ahmed Hamdi Mansi 39
Airborne Gravity Field Modelling
Ellipsoid name Semi−major axis (a) Reciprocal of attening (1/f)
Airy 1830 6377563.396 299.324964600
Helmert 1906 6378200.000 298.300000000International 1924 6378388.000 297.000000000Australian National 6378160.000 298.250000000
GRS 1967 6378160.000 298.247167427
GRS 1980 6378137.000 298.257222101WGS 1984 6378137.000 298.257223563
Table 1.2. Examples of dierent reference ellipsoids and their geometrical param-eters
full list see Table 1.2 in Appendix A), there is a great impact on the computation
of ∆gEllipsoid at an arbitrary point (P). The formula by Moritz (1980a), reported in
Eq. 1.15, is the most common formula and the worldwide used one.
γEllipsoidP = γ0(1 + a1 sin2 φp + a2 sin2 2φp) (1.15)
Eq. 1.15: The reference ellipsoid signal, where γ0 = 978032.7 mGal,
a1 = 0.0053024, a2 = −0.0000058, and φp is the geodetic latitude of point P .
Consequently to these complicated computations, the ultimate goal for geodesists
has been formulate a unique, general purposed GGM that could perform dierent
and diverse applications in an optimum way and to be able to ease the computa-
tions of all the gravitation functionals. From one point of view, this optimal GGM
has facilitated the complicated computations but from the other point of view, it
has created a new challenge to compute the RTC that coincides with the maximum
order/degree used within the processing of the GGM. Currently GGMs are repre-
sented as a spherical harmonic series truncated up to a maximum degree Lmax while
the formulas implemented to compute the various functionals are found in literature
and they could be seen as a reection of the relationship between the spatial and
spectral domains of the computed geopotential component as seen in Eq. 1.16.
V (r, ϑP , λP ) = GMR
Lmax∑l=2
l∑m=0
(Rr)l+1Vn,mYn,m(ϑ, λ) (1.16)
Eq. 1.16: The implicit representation of the gravitational potential in terms of
Spherical Harmonics.
Eq. 1.16 could be further elaborated and represented in terms of Legendre poly-
nomials as reported in Eq. 1.17 (as in Torge (1989); Dragomir et al. (1982)) and the
40
CHAPTER 1. CLASSICAL PROCESSING OF GRAVITATIONAL DATA
evaluation of this equation shows that a smoothening eect hits the signal and there-
fore it loses the high frequency components and gradually damps with the height
and would vanish at innity.
V (r, ϑP , λP ) = GMR
Lmax∑l=2
l∑m=0
(Rr)l+1[Clm cosmλP + Slm sinmλP ]Plm cos(ϑP ) (1.17)
Eq. 1.17: The explicit representation of the gravitational potential in terms of
Spherical Harmonics.
1.3.2 Terrain Correction
The raw airborne gravimetric measurements are characterized by a huge variation
that could be up to 5000 mGal, while the resulted low−pass ltered signal varies
only of some tenth of mGals. A further smoothening eect for the gravitational
measurements at the ight altitude is performed by applying the terrain correction
that is an essential step in geoid computation (Nahavandchi , 2000). Very often, the
topographic and bathymetric gravitational eects are the main sources for the local
gravity variations (MacQueen and Harrison, 1997). However, due to the existence of
the topography outside the geoid, the terrain correction must be applied to fulll the
theoretical requirement which mandates that the disturbing potential is harmonic
outside the geoid accordingly the existence of no masses outside the geoid. The
removal of the eect of the topography would increase the applicability of the Stokes′
formula and consequently enhance the geoid determination (Sun, 2002).
The methods which are widely implemented for the computation of the TC are
dependent of the typology of the data, the direct integrations of the TC is preferred
for point−wise computations(Vannes , 2011) while the FFT is in general an ideal
method for grid−wise computation (Schwarz et al., 1990) and it also requires less
time comparing to the direct integral method.
In order to evaluate such correction there is an urgent need for densely sampled
DTM (Tsoulis , 2001) (Tsoulis, 2001). To be able to implement such procedure,
in case no detailed DTM model are available in the area, both land topography
values by SRTM (Farr et al., 2007) (1 arc−second of about 30 meters of spatial
resolution) and oceans bathymetry values by ETOPO1 (Amante and Eakins , 2009)
(1 arc−minute grid cell resolution of about 1.8 km) could merged by a procedure
of Kriging to build an adequate DTM model assuring at least a 50 km extension in
every direction around the studies area.
Ahmed Hamdi Mansi 41
Airborne Gravity Field Modelling
1.3.2.1 Point−Mass Model
The direct integration is based on the Newtonian volume integral. In the point
mass model in order to compute the gravitational potential V at of attracting mass,
the attracting body is condensed and represented as a set of point−masses each
located at specic pointP (x, y, z). The eect of each point mass at any arbitrary
computation point P (xP , yP , zP ) is reported in Eq. 1.18 in its nal form in Cartesian
coordinate system.
V (xP , yP , zP ) = G
∫∫∫v
= ρ(x,y,z)√(x−xP )2+(y−yP )2+(z−zP )2
dxdydz (1.18)
Eq. 1.18: Newton′s volume integral for gravitational potential evaluation in the
Cartesian coordinate system, where G = 6.67 · 10−11 is Newton′s gravitational
constant m3 · kg−1 · s−2 and v is the volume of the attracting mass.
Eq. 1.18 could be extended from the Cartesian to the spherical coordinate system
(see Fig. 1.4) in order to compute the eect of the innitesimal point−mass located
at P (r, ϑ, λ) at the computation point P (r, ϑ, λ) in terms of potential value as in
Eq. 1.19.
V (r, ϑ, λ) = G
∫ 2π
λ=0
∫ 2π
ϑ
∫ rmax
r=0
ρ(r, ϑ, λ)√r2 + r2 − 2rr[cosψ]
r2 sin ϑdrdϑdλ (1.19)
Eq. 1.19: Newton′s volume integral for gravitational potential evaluation in
Spherical coordinate system, where cosψ = cosϑ cos ϑ+ sinϑ sin ϑ cos (λ− λ).
Finally, the total eect of the body is computed by summing up all the eects
of all the innitesimal point−masses.
1.3.2.2 Right−Prism Model
In case of the availability of topographic data which could be discretized as
columns of attracting masses above or below the geoid, a closed−form solution has
been developed to compute the gravitational potential and its derivatives up to the
third order of a right−prism ((Nagy et al., 2000; Wang et al., 2003; Nagy et al.,
2002; Han and Shen, 2010)), then a new expression of the gravitational potential
and its derivatives were elaborated by D′Urso (2012). The rectangular−prism rep-
resentation (see Fig. 1.5), is a rigorous and useful model for numerical integration
of Eq. 1.18 that can be rewritten as reported in Eq. 1.20:
42
CHAPTER 1. CLASSICAL PROCESSING OF GRAVITATIONAL DATA
Figure 1.4. The Spherical coordinates of the computation point P (r, ϑ, λ) and theintegral point P (r, ϑ, λ).
Ahmed Hamdi Mansi 43
Airborne Gravity Field Modelling
Figure 1.5. Sketch map of the denition of the prism (after Nagy et al. (2000)).
V (r, ϑP , λP ) = Gρ|||xyln(z + r) + yzln(x+ r) + zxln(y + r)− x2
2tan−1 yz
xr
−y2
2tan−1 xz
yr− z2
2tan−1 xy
zr|x2x1|y2y1|z2z1
(1.20)
Eq. 1.20: Newton′s volume integral for gravitational potential evaluation in (planar
approximation) Cartesian coordinates, where the prism is bounded by planes
parallel to the coordinate planes dened by coordinates X1, X2, Y1, Y2, Z1, and Z2
and
x1 = X1− xP , x2 = X2− xP , y1 = Y1− yP , y2 = Y2− yP , z1 = Z1− zP , z2 = Z2− zP .
The simple discretization of the terrain shape in term of prisms will consequently
make the evaluation of the integral reported in Eq. 1.20 as sums. Also, it is possi-
ble, to produce a better discretization that accounts for the spherical or ellipsoidal
shape of the reference surface and use accordingly spherical/ellipsoidal prisms (Heck
and Seitz , 2007). It should be pointed out that the numerical implementation of
prism formulas is a time−consuming procedure and it demands advanced computer
resources, especially when dense DTMs are used. Therefore, in practice, when the
studied area is relatively large, for any arbitrary computation point P (xP , yP , zP )
44
CHAPTER 1. CLASSICAL PROCESSING OF GRAVITATIONAL DATA
orP (r, ϑ, λ) , we do not need to consider prisms everywhere, those which are very far
from P and presumably produce an insignicant contribution that can in general be
computed with approximated formulas or simple models such as a point−mass thus
reducing the computational time without decreasing the accuracy (Forsberg , 2008).
In this dissertation, the theory and the implementation of a new technique will be
explained in chapter 2. Basically the classical prism−based software uses a detailed
grid around the computation point and a coarser grid for the remaining test area,
while the new technique is a hybrid model, which simultaneously implements both
the prism and FFT models.
1.3.2.3 Tesseroid Model
The tesseroid model exploits a similar geometric element to the right−prismmodel (see, section 1.3.2.2) theoretically acknowledged as the spherical prism but
world−wide known as Tesseroid model (Smith et al., 2001), whose is projected and
manipulated within the spherical coordinates system, as shown in Fig. 1.6. The
tesseroid element is bounded by 2 meridians λ1 and λ2 (whereλ1 < λ2), 2 parallels
ϕ1 and ϕ2 (whereϕ1 < ϕ2), and 2 spheres of radii r1 and r2 (wherer1 < r2) (Uieda
et al., 2011).
Through implementing the aforementioned shape parameters as integral limits,
the gravitational potential of the tesseroid model can be computed using Eq. 1.21
(Heiskanen and Moritz , 1967).
V (r, ϕ, λ) = Gρ
∫ λ2
λ1
∫ ϕ2
ϕ1
∫ r2
r1
1√r2+r2−2rr cosψ
r2 cos ϕdrdϕdλ (1.21)
Eq. 1.21: The gravitational potential of the tesseroid model in Spherical
coordinates.
On the one hand, Grombein et al. (2013) elaborated the formulas to compute the
gravitational attraction for the X, Y, and Z directions as reported in Eq. 1.22.
gx(r, ϕ, λ) = Gρ
∫ λ2
λ1
∫ ϕ2
ϕ1
∫ r2
r1
r(cosϕ sin ϕ−sinϕ cos ϕ cos(λ−λ))
(r2+r2−2rr cosψ)3/2 r2 cos ϕdrdϕdλ
gy(r, ϕ, λ) = Gρ
∫ λ2
λ1
∫ ϕ2
ϕ1
∫ r2
r1
r cos ϕ sin(λ−λ)
(r2+r2−2rr cosψ)3/2 r2 cos ϕdrdϕdλ
gz(r, ϕ, λ) = Gρ
∫ λ2
λ1
∫ ϕ2
ϕ1
∫ r2
r1
r cosψ−r(r2+r2−2rr cosψ)3/2 r
2 cos ϕdrdϕdλ
(1.22)
Ahmed Hamdi Mansi 45
Airborne Gravity Field Modelling
Figure 1.6. The tesseroid representation in the spherical coordinates system.
46
CHAPTER 1. CLASSICAL PROCESSING OF GRAVITATIONAL DATA
Eq. 1.22: The gravitational attraction of the tesseroid model in X, Y, and Z
directions in Spherical coordinates, where
cosψ = sinϕ sin ϕ+ cosϕ cos ϕ cos(λ− λ).
Also, they elaborated the dierent formulas to compute the dierent gravity
gradients that can be summarized in a general formula as in Eq. 1.23.
gαβ(r, ϕ, λ) = Gρ
∫ λ2
λ1
∫ ϕ2
ϕ1
∫ r2
r1
Iαβ(r, ϕ, λ)drdϕdλ As α, β ∈ x, y, and z (1.23)
Eq. 1.23: The general formula to compute the dierent gravity gradients in
Spherical coordinates, where
Iαβ(r, ϕ, λ) = (3∆α∆β
(r2−r2−2rr cosψ)5/2
δαβ(r2−r2−2rr cosψ)3/2 )r2 cos ϕ with
∆x = r(cosϕ sin ϕ− sinϕ cos ϕ cos(λ− λ)), ∆y = r cos ϕ sin(λ, and
∆z = r cosψ − r.
On the other hand, Asgharzadeh et al. (2007) implemented the Gauss−LegendreQuadrature rule in order to evaluate the various gravitational attraction and gravity
gradients signals.
1.3.2.4 Polyhedral−Body Model
In a very simple denition, polyhedral body model is a body that is solely de-
scribed using the coordinates of the vertices of the relevant faces (D′Urso, 2014). In
the beginning of this line of thoughts, Hubbert (1948) studied the gravitational at-
traction of a 2D body and found that its gravitational attraction can be expressed in
terms of transforming the surface and volume integrals into a line integral around its
periphery (Won and Bevis , 1987). Few years later, Talwani et al. (1959) presented
a methodology to compute the gravitational attraction signal for any arbitrary 2D
n−sided polygon (for instance, see Fig. 1.7) by breaking the line integral up into n
contributions, each associated with a side of the polygon as reported in Eq. 1.24.
∆gx = 2Gρn∑i=1
Xi
∆gz = 2Gρn∑i=1
Zi
(1.24)
Eq. 1.24: The horizontal and vertical components of the gravity anomaly of a 2D
n−sided polygon.
Then, the 2D polygon was used as a base to represent a 3D body ((Talwani and
Ewing , 1960; Collette, 1965; Takin and Talwani , 1966)) that was generalized for a
Ahmed Hamdi Mansi 47
Airborne Gravity Field Modelling
Figure 1.7. The geometric conventions used in the expression of the gravitationalacceleration at the origin due to a 2D polygon of a constant density ρ.
3D polyhedral allowing for a more ecient modelling of real complex bodies (see
Fig. 1.8) ((Paul , 1974; Barnett , 1976)).
On the one hand, the Polyhedral model, which is characterized with a homoge-
neous density distribution drew the attention of a great number researches, see for
instance ((Okabe, 1979; Götze and Lahmeyer , 1988; Petrovi¢, 1996)). On the other
hand, Artemjev et al. (1994) studied the polyhedral model with a linearly increasing
density, while, Pohánka (1998) studied polyhedral with linearly varying density.
D′Urso and Russo (2002) developed a new algorithm to evaluate the gravita-
tional acceleration for a point−in a 2D polygon, then D′Urso (2013a) developed
the formulas needed to compute the gravitational potential and its derivatives for a
point belongs to the interior face of the right−prism that was extended to the com-
putation of the gravity eect of polyhedrals with linearly varying density (D′Urso,
2013b).
For instance, the gravity potential V at the observation point P from an arbitrary
3D polyhedral in a Cartesian reference frame starting from the Newton′s volume
integral reported in Eq. 1.18 can be simplied by applying the approach presented
by Petrovi¢ (1996) that transforms the triple (3D) integral into a summation of 1−Dintegrals (Hamayun et al., 2009) as reported in (D′Urso, 2014) as follows:
V (xP , yP , zP ) = Gρ
NF∑i=1
di IFi − |di|αi (1.25)
Eq. 1.25: The gravity potential at any observation point P from an arbitrary 3D
48
CHAPTER 1. CLASSICAL PROCESSING OF GRAVITATIONAL DATA
polyhedral in a Cartesian reference frame, where NF is the number of faces
belonging to the boundary and di is the signed distance between P and Fi.
1.3.2.5 Fast Fourier Transform Method
Fast Fourier transform (FFT) technique is one of the most ecient tools for
treating large amounts of height data, although special attention should be paid to
the problems arising from the numerical evaluation of such integrals (for instance,
see (Forsberg , 1984; Sideris , 1984, 1985). With a single elaboration on Eq. 1.18,
the rigorous terrain correction could be seen as a convolution integral as reported in
Eq. 1.26. The availability of digital models of topography and bathymetry in forms
of regular grids and the convolution integrals can be eciently evaluated by means
of 2D FFT (Li , 1993) and 3D FFT (Peng , 1994).
Vz(xP , yP , zP ) = G
∫∫∫v
ρ(x,y,z)(z−zP )
[(x−xP )2+(y−yP )2+(z−zP )2]3/2dxdydz (1.26)
Eq. 1.26: The gravity potential at any observation point P from an arbitrary 3D
polyhedral in a Cartesian reference frame.
The 3D FFT method is favorable over the 2D FFT method because it is un-
aected by terrain inclination and accordingly it avoids the numerical diculties
present in the 2D FFT method. The other advantage is that it can handle varying
density in the Z direction, which is not possible with the 2D FFT method (Peng et
al., 1995). Finally, we must point out that the main drawbacks of FFT−based tech-
niques are that they necessitate the input to be in as a grid and they also demand
much more computer memory.
1.4 Downward Continuation
Downward continuation is often used in gravimetric data processing, especially
in airborne gravimetry in order to transfer the gravity anomalies observed at the
ight altitude,h = zP , to estimate the gravity anomalies at surface of a reference
surface, namely the geoid, h = 0 for geodetic purposes. Undoubtedly, the downward
continued gravity anomalies are not the original gravity anomalies, which could
be sensed inside the Earth. In simple words, the downward continuation gives a
ctitious gravity anomaly on the ellipsoid that generates a disturbing potential on
and outside the surface of the Earth that coincides with the original disturbing
potential T on and outside the Earth (Wang , 1988).
Ahmed Hamdi Mansi 49
Airborne Gravity Field Modelling
Figure 1.8. The 3D polyhedral representation in a 3D coordinates system and the2D reference frame for a generic face.
Moritz (1980a) suggested that the free−air anomalies be continued to the ref-
erence surface, which could be chosen to be the ellipsoid and use this method to
analytically continue the gravity anomalies down to the ellipsoid. While, Bjerham-
mar (1964) proposed performed an iterative numerical solution of the Poisson′s
integral to continue the free−air anomalies downward to a sphere embedded inside
the Earth, which was fullled using the discrete technique and matrix formulas, for
more information see (Wang , 1987). In the sequel, we will discuss Molodensky′s
concept and the free−air for downward continuation.
1.4.1 The Molodensky Concept
The Molodensky concept is basically that the knowledge of gravity eld outside
the masses can be exclusively fullled solely using the gravity data collected on the
surface (Eq. 1.27). From the modern mathematical point of view this is an early
50
CHAPTER 1. CLASSICAL PROCESSING OF GRAVITATIONAL DATA
formulation of a so called free boundary, boundary value problem.
∆g(xP , yP , zP ) = 12π
∫∫E
∆g(x, y, 0) zP[(x−xP )2+(y−yP )2+(z−zP )2]3/2
dxdy
= ∆g(xP , yP , 0) ∗ lu(xP , yP , zP )
(1.27)
Eq. 1.27: Molodensky concept; the gravity anomaly computed outside the masses
expressed in terms of gravity anomaly observed on the surface of the mass, which
is seen as convolution, where the upper continuation kernel
lu(x, y, zP ) = zP2π(x2+y2+z2
P )3/2 .
Eq. 1.28 could be reversed to obtain a formula for the downward continuation
and its kernel `d using the analytical denition of the upward kernel (Eq. 1.29).
∆g(xP , yP , zP ) = F−1F∆g(xP , yP , 0)Flu(xP , yP , zP ) (1.28)
Eq. 1.28: Molodensky concept; the evaluation of the convolution.
Flu(xP , yP , zP ) = lu(u, v, zp) = e−2π(u2+v2)1/2
= e−2πzpq (1.29)
Eq. 1.29: The analytical denition of the upward continuation.
As expected, Eq. 1.30 illustrates that on the one side, the upward continuation
attenuates the high frequencies of the gravity eld, while on the other side, the
downward continuation amplies the high frequencies and the noise that contaminate
the data, and therefore a proper lter could be utilized to stabilize the solution.
∆g(xP , yP , 0) = F−1F∆g(xp,yP ,zP )Flu(xP ,yP ,zP )
= F−1F∆g(xP , yP , zP )Fld(xP , yP , zP )(1.30)
Eq. 1.30: Molodensky concept; the evaluation of the convolution, where the upper
continuation kernel Flu(xP , yP , zP ) = 1(lu(u,v,zP )
= e2π(u2+v2)1/2= e2πzpq.
1.4.2 Free−Air Downward Continuation
After applying the terrain reduction, a correction for the height (Eq. 1.33) that
is used to transfer the gravity measurements from the observation point P located
on/above the surface of the Earth to point P0 on the reference surface, namely the
geoid, as shown in Fig. 1.9. This height correction is known as the free−air correction
Ahmed Hamdi Mansi 51
Airborne Gravity Field Modelling
Figure 1.9. The geometry of the planar Bouguer reduction, the terrain correction,and the free-air correction.
that ignores the masses between the Earth′s surface and the geoid, as the terrain
reduction removed the full eects of the topographic masses. The gravity change
seen by the free−air correction is given by the actual gravity gradient (Eq. 1.31).
δhhp = − ∂g∂HH∗ (1.31)
Eq. 1.31: The free−air reduction in terms of actual gravity gradient.
As the evaluation of the free−air is essential in order to completely evaluate the
∆gEllipsoid, and due to the fact that the normal height, H∗, is not often available in
practice, therefore the actual gravity gradient is replaced with the normal gravity
gradient (Eq. 1.32) using the Orthometric height, HP . This approximation however,
introduces non−negligible systematic errors in the analysis of surface gravimetric
data (Pavlis , 1988).
δhhp = − ∂g∂HHP (1.32)
Eq. 1.32: The free−air reduction in terms of normal gradient of gravity.
Moritz (1980a) exploited the precise gravity measurements across the globe done
by precise absolute and relative gravimeters and reported Eq. 1.33, which estimates
the change of theoretical gravity values with latitude computed on the surface of the
ellipsoid due to the reference ellipsoid.
δghp = −(b1 + b2 sin2 φ)hP + c1h2P (1.33)
52
CHAPTER 1. CLASSICAL PROCESSING OF GRAVITATIONAL DATA
Eq. 1.33: The height correction applied to the reference ellipsoid signal, where
b1 = 0.3087691, b2 = −0.0004398, c1 = 7.2125 · 10−8, and hP is the ellipsoidal height
of point P .
By ignoring the second−order term and using φP = 45 the height correction
is approximated to Eq. 1.34. Generally, the name "free−air" correction has re-
placed the height correction, therefore it has been thought to be associated with
the elevation H, not the ellipsoid height h. In geodesy, the "free−air" correction
was interpreted misleadingly as a reduction to the geoid of gravity observed on the
topographic surface (Nettleton, 1976).
δghp = 0.03086hP (1.34)
Eq. 1.34: The approximated height "free−air" correction.
Although the free−air correction could be expressed employing closed−analyticalformulas for every term, yet for points up to topographic altitude above the reference
ellipsoid, it is preferred to use the approximate formula reported in Eq. 1.34 over
the rigorous formula stated in Eq. 1.33.
1.5 Gravity Data Inversion
Methods that have been used to process, invert, and interpret the airborne grav-
ity data are integral methods, least−squares collocation, and sequential multi-pole
analysis. The main drawbacks of the Integral methods are their urge to collect the
data in a much larger area than for which the gravity functionals are to be evaluated
and the border eect that results from having no data outside the area of interest.
On the other side, Least−squares collocation suers much less from these errors and
can yield accurate results, provided that the auto−covariance function gives a good
representation of data in− and outside the area. However, the main disadvantage
of least−squares collocation is being numerically less ecient because it requires
to use equivalent number of base functions to the number of observations therefore
it demands high eort to establish and solve this large number of equations (Kaas
et al., 2013). Several authors have compared the performance of the approaches,
especially in terms of geoid height errors. Alberts and Klees (2004) investigated
the accuracy of the integral methods and the least−squares collocation and con-
cluded that the least−squares collocation performed slightly better. Marchenko et
al. (2001); Klees et al. (2005) stated that the sequential multi-pole analysis approach
Ahmed Hamdi Mansi 53
Airborne Gravity Field Modelling
and least−squares collocation produce comparable results. Therefore, it can be con-
cluded that these dierent approaches yield similar results in terms of geoid height
and gravity disturbance errors with similar accuracies but with variant computa-
tional and numerical complexity.
1.5.1 Collocation
Least−squares collocation (LSC) is the most used stochastic model to perform
an optimal linear estimation for gravity modelling. LSC is often used for the down-
ward continuation of airborne gravity data (Forsberg and Kenyon, 1994) and the
computations of dierent gravity functionals at ground level, for more details, con-
sult, (Marchenko et al., 2001; Forsberg , 2002). LSC is also used to sew together
dierent heterogeneous gravity data (Gilardoni et al., 2013). It is based on ideas
in the elds of least−squares estimation, approximation theory, functional analysis,
potential theory, and inverse problems (for more details, see
Table 10.1 in Appendix B gives a detailed insights about the dierent elements
required to perform a collocation estimation. The LSC solution has many advan-
tages:
• The LSC solution is stable for the generally ill−posed problem of gravity eld
determination;
• The solution is independent of the number of signal parameters to be esti-
mated;
• The solution is invariant to linear transformations of the data and results;
• The result is optimal with respect to the covariance function used.
For the dierent theoretical variants and the numerous interpretations developed
over the years, see (Moritz , 1978; Sansò and Tscherning , 1980; Moritz and Sansò,
1980; Kotsakis , 2000). If the data are given on grids, the multi−input multi−output(MIMO) Wiener lter is a fast and equivalent alternative to LSC (Bendat and Pier-
sol , 1986; Vassiliou, 1986; Schwarz et al., 1990; Bendat and Piersol , 1993; Sideris ,
1996; Li and Sideris , 1997; Sansò and Sideris , 1997; Andritsanos et al., 2001).
1.5.1.1 Solution of the Basic Observation Equation
In the Sequel we will highlight the dierent solutions of the LSC with and without
the presence of the contaminating noise.
54
CHAPTER 1. CLASSICAL PROCESSING OF GRAVITATIONAL DATA
1.5.1.1.1 Least−Squares Collocation for Non−Noisy Data
The basic observation equation for a least−squares collocation is reported in
Eq. 1.35.yi = Li(x), or y = L(x) i = 1, · · · , n in case of no noise
yi = Li(x) + νi, or y = L(x) + ν i = 1, · · · , n in case of noise(1.35)
Eq. 1.35: The observation equation for the Least−Square Collocation, where yi is avector of n residual observations reduced by the eects discussed within this
chapter, νi is the observational errors, Li is a vector of linear functionals
associating T with the observations, and x is the approximation to T which will be
determined.
The development of collocation without errors, νi is null, is called the "exact
collocation" with innite number of compatible solutions, x, and among them the
smoothest, x, by minimizing the norm (as in Eq. 1.36), is obtained. x is the orthog-
onal projection of x onto a subspace of the Hilbert space, H.
x = Argmin||x||,〈K(L, ·)〉x = y,
where ||x|| ≥ ||x||, isx = (LK)T (L(LK)T )−1y
= Cxy(Cyy)−1y
(1.36)
Eq. 1.36: The problem of minimizing the norm to obtain the smoothest solution by
the collocation.
The magnitude of the error of the exact collocation can be computed by imple-
menting Eq. 1.37.
εtot(x) = ||x− x|| (1.37)
Eq. 1.37: The magnitude of the error of the exact collocation.
Where the kernel function K(P,Q) is identied by the covariance function of the
disturbing potential, T .
1.5.1.1.2 Least−Squares Collocation for Noisy Data
In the case of the existence of the noise, the solution of the basic observation
equation will be achieve using the probabilistic LSC approach, where the unknown
disturbing potential is modeled as a zero−mean stochastic process and the available
Ahmed Hamdi Mansi 55
Airborne Gravity Field Modelling
observations are considered as zero−mean random variables (Moritz , 1962). The op-
timal solution,x is obtained by minimizing the mean square estimation error (MSE)
and should be an unbiased estimator, using the Wiener−Kolmogorov principle, as
in Eq. 1.38.
E[y − y]2 = minλ (1.38)
Eq. 1.38: The Wiener−Kolmogorov principle solution by the collocation, where λ
is the combination weights.
The main drawback of this probabilistic LSC is that the gravity eld is not a
stochastic phenomenon, since repetitive gravity measurements should always provide
the same result (excluding time−dependent variations and measurement errors).
1.5.1.2 Covariance Estimation
The quality of the inversion of the gravimetric data highly depends on the co-
variance function (Knudsen, 1987). As seen in Eq. 1.36 and Eq. 1.39, the covariance
counts for the noise if exist. If the noise is zero, the solution will agree exactly with
the observations (see Tscherning (1985)).
x = Cxy(Cyy + Cνν)−1y (1.39)
Eq. 1.39: The solution of minimizing the mean square estimation error by the
stochastic collocation.
The rst step is to estimate the empirical covariance of the residual data. This
task could be easily fullled using the global covariance function implemented in
LSC is simply a triple integral as Eq. 1.40.
COV (P,Q) = 18π2
∫ 2π
0
∫ π/2
−π/2
∫ 2π
0
T (P )T (Q)dα cosϕdϕdλ (1.40)
Eq. 1.40: The global covariance function used in LS collocation, where α is the
azimuth between P and Q, ϕ and λ are the coordinated of P while Q has a xed
spherical distance from P .
Then, for the second step, we must choose one of the well−known covariance
functions that best t this empirical covariance computed from the data.
1.5.2 The Stokes′s Integral
The Strokes′s integral is the solution for a geodetic boundary value problem using
the geoid as the boundary surface and exploiting the reduced gravity anomalies
56
CHAPTER 1. CLASSICAL PROCESSING OF GRAVITATIONAL DATA
∆g collected over this boundary surface in order to dene a potential, which is
characterized being harmonic outside the masses (Stokes , 1849). The evaluation of
such Stokes′s integral mandates the existence of no masses outside the geoid, and this
condition has been fullled by the various reductions made to the collected gravity
data (terrain reduction, free−air reduction . . . ). The classical Stokes boundary valueproblem determines a solution for the problem, explained in Eq. 1.41, by nding the
disturbing potential that satises Laplace′s equation reported on Eq. 1.42.∆T = 0 in Ω
−∂T∂r− 2
rT = ∆g(P ) on S
T → 0 when r →∞(1.41)
Eq. 1.41: The Stokes problem, where Ω is the space exterior to the geoid, S is the
surface of the geoid, and r is the radius of the reference sphere of point P .
∆T = ∂2T∂x2 + ∂2T
∂y2 + ∂2T∂z2 = 0 (1.42)
Eq. 1.42: The Laplace′s equation.
The solution of this problem is given by Stokes′s integral in Eq. 1.43.
T = R4π
∫∫S
∆gS(ψ)dσ = ∆S(∆g) (1.43)
Eq. 1.43: The solution of the Stokes′s boundary value problem, where
S(ψ) = 1sin (ψ/2)
− 6 sin (ψ/2) + 1− 5 cosψ − 3 cosψ ln (sin (ψ/2) + sin2(ψ/2),
sin2(ψ/2) = sin2(ϑ−ϑ2
) + sin2(λ−λ2
) cosϑ cos ϑ , ψ is the spherical distance between
the data point P (r, ϑ, λ) and the computation point P (r, ϑ, λ), and σ stands for
the integration area.
At this point, the one can easily exploit Brun′s equation (Eq. 1.44) in order to
compute the functional of interest, which is the geoid undulation, N , for our purpose.
N = Tγ
(1.44)
Eq. 1.44: The Brun′s equation.
The integration area, theoretically, should cover the whole Earth, but due to the
limitation in the area coverage and the point density of the gravity measurements,
the Stokes′s integral can be limitedly evaluated within limited local/regional areas.
Ahmed Hamdi Mansi 57
Airborne Gravity Field Modelling
Accordingly, these limitations restrict and limit the minimum and maximum re-
solvable wavelength of the computed geoid. For more explanations and a complete
discussion about the dierent approximations and implementation of the Stokes′s
Integral and the detailed computation of the geoid undulation, please see ((Heiska-
nen and Moritz , 1967; Sideris and Tziavos , 1988; Schwarz et al., 1990; Haagmans
et al., 1993)).
1.5.2.1 Planar Approximation of Stokes′s Integral
For the sake of consistency and self−contained, we will report the dierent forms
for the geoid undulation, N, written as a convolution solvable by methods of fast
Fourier transform. The rst case (Eq. 1.45) expresses the planar approximation
of the Stokes′s integral (for more details, see (Kearsley et al., 1985)). The planar
approximation formula is valid in the vicinity of the computation point that has the
extension of the area of local data lower than several hundreds of kilometers in each
direction in order to avoid long−wavelength errors (Jordan, 1978) (Jordan, 1978).
N(xP , yP ) = 12πγ
∫∫S
∆g(x,y)√(x−xP )2+(y−yP )2
dxdy
= 1γ∆g(xP , yP ) ∗ lN(xP , yP )
(1.45)
Eq. 1.45: The planar approximation of the Stokes′s integral as a convolution.
1.5.2.2 Spherical Approximation of Stokes′s Integral
A spherical approximation of the Stokes′s integral has been found under the
boundary condition, which neglects the relative error of the attening of the reference
ellipsoid (Eq. 1.46).∂T∂r
+ 2rT + ∆g(P ) = 0 (1.46)
Eq. 1.46: The attening of the reference ellipsoid.
The spherical approximation of the Stokes′s integral for any arbitrary point
P (rP , ϑP , λP ) located on the reference surface can be written explicitly as in Eq. 1.47.
N(ϑP , λP ) = R4πγ
∫∫S
∆g(ϑ, λ)S(ϑP , λP , ϑ, λ) cos(ϑ)dϑdλ (1.47)
Eq. 1.47: The planar approximation of the Stokes′s integral as a convolution.
58
Chapter 2
Gravity Terrain Eects
[ÉjJË @
èPñ] [(15)
àð
Y
JîE
Ѻ
ʪ
Ë C
J.
ð @ PA
îE
@ð
Ñ
ºK.
YJÖ
ß
à
@ ú
æ@ð P
P
B@ ú
¯
ù®Ë
@ð]
[And He has axed into the Earth Mountains standing rm, lest it should shake
with you, and rivers and roads, that you may guide yourselves. (15)] [Quran,
An−nahl]
The computation of the gravitational eects of the topographic masses, the
masses distributed in the volume of the so called topography, namely those that
are located between the geoid or the Earth ellipsoid and the actual topographic
surface of the Earth, is known as the Terrain Correction. Recalling from section
1.3.2, the terrain correction is a very crucial step in geodetic and geophysical ap-
plications, especially for the purpose of this research, which is to develop a high
precision estimation for the geoid on a local/regional scale (i.e. diameter smaller
than 200 km). The increasing resolution of recently developed digital terrain mod-
els, the increasing number of observation points due to extensive use of airborne
gravimetry in geophysical exploration and the increasing accuracy of gravity data
represents nowadays major issues for the terrain correction computation.
Classical techniques that exploit the prism or the point−mass approximation
models are indeed too slow while the other techniques based on FFT methods are
usually too approximate for the required accuracy. New software, named Gravity
Terrain Eects (GTE), is developed in order to perform fast computations of the
terrain corrections with high accuracy. GTE has been thought expressly for geophys-
ical applications allowing the computation not only of the eect of topographic and
bathymetric masses but also those due to sedimentary layers and/or to the Earth
crust−mantle discontinuity (the so called Moho). In any case, since the main topic
of the present research is on physical geodesy we will concentrate here only on the
computation of the topographic layer leaving the interested reader to the algorithms
59
Airborne Gravity Field Modelling
described in Sampietro et al. (2015).
The following sections are dedicated for the explanation of the theory that helped
developing the GTE. Section 2.1 explains the motivations led to the development
of the GTE software, while Section 2.2 is dedicated to explain the mathematical
formulations of both the planar approximation formulas (adopted by the GTE) and
the spherical correction terms. Section 2.3 explains the dierent cases where GTE
can be used and how the equations are modied and tuned in order to consider the
dierent cases, which geophysicists and geodesist face in reality. Section 2.4 is dedi-
cated to the numerical tests performed to compare the results of the GTE adopting
dierent techniques/proles with the results obtained by other commonly used soft-
ware for geodetic and geophysical applications such as GRAVSOFT (Forsberg , 2003)
and Tesseroids (Uieda et al., 2011).
2.1 Setting the Stage for GTE
The Gravity Terrain Correction (GTE) is basically an implementation of classical
prisms and FFT methods improved and combined in order to maximize the accuracy
of the results minimizing the computational time. GTE exploits the velocity of FFT
techniques to compute the gravitational eect at any given altitude, not only at the
surface of the topography as other techniques, thus allowing for accurate and fast
terrain corrections of airborne data. Finally, GTE is expressly thought for geophys-
ical applications allowing not only the computation of the eects of the oceanic and
topographic masses but also those due to sediments and Moho undulation.
Gravity Terrain Correction (GTE) is sought to compute the value of the terrain
correction to calculate the gravitational potential of the topography, Tt, and/or its
gradients, mainly the vertical components, known as the gravity disturbance δg, as
in Eq. 2.1.
δg = δgt = −νTt (2.1)
Eq. 2.1: The gravitational disturbance as the vertical component of the gradient of
the potential eld.
Eq. 2.1 can be evaluated using the Newtonian volume integral (Eq. 1.18, Eq. 1.19,
and Eq. 1.20) to forward model the eect of the masses between the surface of the
topography and the reference ellipsoid of geoid, as in Eq. 1.26.
These values are computed at point P , which is located either on the surface
of the topography (e.g. terrestrial and/or shipborne gravimetry), Pot, or in the air
(e.g. airborne gravimetry), Pof . Fig. 2.1 represents the generic prole used for
60
CHAPTER 2. GRAVITY TERRAIN EFFECTS
Figure 2.1. Basic notation and symbols used by GTE.
the evaluation of the various corrections, where the notation is also depicted and
summarized as follows:
• Pot is a computation point located on the terrain;
• Pof is a computation point located on the ight track;
• St is the exterior topography surface;
• Sw is the surface of the sea oor;
• SM0 is the surface of the Moho;
• ht is the topographic height;
• hw is the bathymetry depth;
• hM0 is the depth of the Moho;
• ρr ∼ 2670 kgm3 is the average density of the topography, crystalline crust;
• ρw ∼ 1030 kgm3 is the average density of the oceans′ saline water;
• ρs ∼ 2200 kgm3 is the average density of the sediments;
• ρm ∼ 3300 kgm3 is the average density of the upper mantle;
• h is the arbitrary height of the computation point, P ;
• H is the case of having a constant height of the gird.
Ahmed Hamdi Mansi 61
Airborne Gravity Field Modelling
Figure 2.2. Geometry of the local sphere and of the tangent plane.
As the goal of this research is to develop a high precision local estimation of
the geoid, therefore our areas of interest are bounded to be within the limits where
the planar approximation principles and formulas are feasibly usable and applicable,
namely considering ν as a eld of parallel unit vectors.
2.2 Theory of GTE
This section will give a detailed review for the arguments leading to the use
of a planar approximation for Tt and δgt a local area, dened as one which can
be inscribed in a cap of 100 : 200 km diameter, providing an explicit expression
for the largest part of the dierence between terrain correction in spherical and
planar approximations. Within literature, two dierent solutions to the problem
of spherical approximation by means of FFT techniques have been given by Strang
van Hees (1990) and more recently in Sampietro et al. (2007) in which a numerical
trick is applied to TCLight terrain correction software (Biagi and Sansò (2000)) to
extend the solution also to an spherical approximation.
2.2.1 The Planar Approximation
Starting from Fig. 2.2, where a tangential local sphere to the ellipsoid at point P ,
the central point of the studied area, is considered. This local sphere is characterized
62
CHAPTER 2. GRAVITY TERRAIN EFFECTS
by a center O and a radius R equivalent to the Gaussian radius of the ellipsoid
at point P , where the vector ~OP , lies along the ellipsoid normal direction. The
vector ~rP , with modulus rP is the position vector of point P and the center O, where
eP = ~rPrP.
From Fig. 2.3, we can develop and use the notations of Eq. 2.2. The notations
∆hPQ and ∆h will be used synonymously, eZ is the vector where the axis Z goes
along with νP ,eOP is the unit vector along the projection of ~rP on the tangent plane,
where sPQ and its synonym s represent the projection of `PQ on the tangent plane
of the local sphere drawn at point P and αPQ and its synonym α is the angle in the
tangent plane eOP eOQ.
Figure 2.3. The mapping of the topographic body B to the attened B.
hP = rP −RεP = hP
R
rP = RεP +R
∆hPQ = hP − hQlPQ = | ~rP − ~rQ|
ψP = eP ez
(2.2)
Eq. 2.2: The mapping notations.
Starting from the well−known identity of the distance between two vectors in theSpherical Coordinate System expressed in a very compact way in Eq. 2.3, Eq. 1.11 to
develop a generic denition for eP as in Eq. 2.4, recalling the dot product between
Ahmed Hamdi Mansi 63
Airborne Gravity Field Modelling
the two vectorseP and eQ as expressed in Eq. 2.5, and the local area of ψ ∼= 1,
therefore we can use the approximations expressed in Eq. 2.6.
l2PQ = | ~rP − ~rQ|2 = r2P + r2
Q − 2rP rQeP .eQ
= (rP − rQ)2 + 2rP rQ(1− eP .eQ)
(2.3)
Eq. 2.3: The distance between 2 vectors in the Spherical Coordinate System.
eP = sinψP eOP + cosψP ez (2.4)
Eq. 2.4: Resolving a vector into 2 perpendicular components using the triangle trig
relationships.
eP .eQ = sinψP sinψQ cosαPQeOP + cosψP cosψQez (2.5)
Eq. 2.5: The dot product of two vectors.
Considering that the gravitational terrain eects can aggregate up few hundreds
of mGal, we shall consider approximations with terms of a relative order of 10−4 up
to 10−3, certainly paying no attention to terms of orders below 10−4. Hence, we will
use only the term ψ2 ∼= 3.1 · 10−4 that is about our lower acceptance limit, which
could be neglected while evaluating 12ψ2 ∼= 3.1 ·10−4 with a relative error of 1.5 ·10−4.
We will neglect the higher powers of ψ such as ψ3 ∼= 5.3 ·10−6 and ψ4 ∼= 9.3 ·10−8. A
note must be taken about cases of a very rugged topography with high mountains,
we have at most |ξ| and |∆| of order 10−3, although it is clear that in such areas we
expect the planar approximation to have a poor performance.When ψ ∼= 1 ∼= 1.7 · 10−2(rad)
sinψ ∼= ψ
cosψ = 1− 12ψ2
(2.6)
Eq. 2.6: The approximations due to using local area caps of 2× 2 degrees .
For instance, we can elaborate the term 2rP rQ(1− eP · eQ) of Eq. 2.3, exploiting
the mapping notations, of Eq. 2.2, to obtain Eq. 2.7, where sP = R sinψP ∼= RψP ,
64
CHAPTER 2. GRAVITY TERRAIN EFFECTS
and sQ = R sinψQ ∼= RψQ.
2rprQ(1− eP · eQ) ∼= 2 · (RεP +R) · (RεQ +R) · (12ψ2p + 1
2ψ2Q − ψPψQ cosαPQ)
∼= (1 + εP ) ·R · (1 + εQ) ·R · (12ψ2p + 1
2ψ2Q − ψPψQ cosαPQ)
∼= 2 · (1 + εP )(1 + εQ) · (12ψ2pR
2 + 12ψ2QR
2 − ψPψQ cosαPQR2)
∼= 2 · (1 + εp + εQ + εpεQ) · (12ψ2pR
2 + 12ψ2QR
2 − ψPψQ cosαPQR2)
∼= 2 · (1 + εp + εQ) · (12s2P + 1
2s2Q − sP sQ cosαPQ)
∼= (1 + εp + εQ) · (s2P + s2
Q − 2sP sQ cosαPQ)
∼= (1 + εp + εQ) · (sP − sQ)2
∼= (1 + εp + εQ) · s2PQ
(2.7)
Eq. 2.7: The simplication of 2rP rQ(1− eP · eQ), where sP eOP and sQeOQ are
(almost) the projection of point P and Q on the tangent plane, respectively.
2.2.1.1 First Order Spherical Correction
For the next passages, we will use εPQ = εP + εQ and `2PQ as reported in Eq. 2.3.
From Fig. 2.4 the value of L2PQ can be computed from Eq. 2.8.
LPQ =√
(hp − hQ)2 +D2PQ
∼=√
∆h2PQ + S2
PQ
(2.8)
Eq. 2.8: The Cartesian distance, LPQ, between two points mapped from P and Q,
where hP and hQ are considered as the component in z direction.
Furthermore, starting from Eq. 2.3, substituting the mapping notations, of Eq. 2.2,
and using the planar approximation, Eq. 2.6, a simplied expression for `PQ could
Ahmed Hamdi Mansi 65
Airborne Gravity Field Modelling
be reached as expressed in Eq. 2.9.
lPQ = [r2P + r2
Q − 2rP rQ cosψPQ]12
= [(rP − rQ)2 + 2rP rQ(1− cosψPQ)]12
= [(hP +R− hQ −R)2 + 2rP rQ(1− (1− 12ψ2PQ))]
12
= [(hP − hQ)2 + 2rP rQ(12ψ2PQ)]
12
∼= [(∆hPQ)2 + rP rQ(D2PQ
R2 )]12
∼= [(∆hPQ)2 + rPR
rQR
(D2PQ)]
12
∼= [(∆hPQ)2 + (hP+R)R
(hQ+R)
R(D2
PQ)]12
∼= [(∆hPQ)2 + (1 + hPQ
)(1 +hQR
)(D2PQ)]
12 → [(∆hPQ)2 + (D2
PQ)]12 ∼= LPQ (seeEq. 2.8)
∼= [(∆hPQ)2 + (1 + hPR
+hQR
+ hPR
hQR
)s2PQ]
12
→ replacing D(planar) with s(sphere) ∼= [(∆hPQ)2 +D2PQ + (
hP+hQR
)s2PQ]
12
∼= [L2PQ + (
hP+hQR
)s2PQ]
12 ∼= [L2
PQ(1 + (hP+hQ
R)s2PQL2PQ
)]12
∼= LPQ[1 + (hP+hQ
R)D2PQ
L2PQ
]12 ∼= LPQ[1 + (hP
R)s2PQL2PQ
+ (hQR
)s2PQL2PQ
]12
∼= LPQ[1 + εP + εQ]12
∼= LPQ[1 + εPQ]12
(2.9)
Eq. 2.9: The simplied version of the spherical distance, `PQ, where O(ε) ≤ 10−3.
By elaborating the results obtained in Eq. 2.7, Eq. 2.8, and Eq. 2.9, we get a
formula to evaluate 1`PQ
(Eq. 2.10) that will be exploited later in the formulation
66
CHAPTER 2. GRAVITY TERRAIN EFFECTS
Figure 2.4. The mapping of the topographic body B to the attened B.
and the evaluation of the gravity potential.
1lPQ
= [(rP − rQ)2 + (1 + εP + εQ) · s2PQ]−
12
= [(∆hPQ)2 + s2PQ + (εP + εQ) · s2
PQ]−12
= [((∆hPQ)2 + s2PQ) + (εPQ) · s2
PQ]−12
= [L2PQ + (εPQ) · s2
PQ]−12
= [L2PQ(1 + εPQ
s2PQL2PQ
)]−12
= 1LPQ
[(1 + εPQs2PQL2PQ
)]−12
as 1√1+x
= 1− 12x+ 3
8x2 − 5
16x3 + 35
128x4 − 63
256x5 + · · ·
∼= 1LPQ
(1− 12εPQ
s2PQL2PQ
)
∼= 1LPQ− 1
2εPQ
s2PQL2PQ
(2.10)
Eq. 2.10: The simplied version of the spherical distance, 1`PQ
.
Ahmed Hamdi Mansi 67
Airborne Gravity Field Modelling
We have to analyze the dierent area elements such as, the spherical area element,
the area element of the tangential plane, and any other shape that we might use
within the formulation of our integrals. On the one hand, the spherical area element
can be represented as reported in Eq. 2.11 and Eq. 2.12.
r2Qdσ = (rQ · rQ)dσ
= (hQ +R)(hQ +R)dσ
= R(1 +hQR
)R(1 +hQR
)dσ
= R2(1 +hQR
)2dσ
∼= R2(1 + εQ)2dσ
∼= R2(1 + 2εQ)dσ
(2.11)
Eq. 2.11: The precise formulation of the area element in the spherical polar system.
r2Qdσ = (rP · rQ)dσ
= (hP +R)(hQ +R)dσ
= R(1 + hPR
)R(1 +hQR
)dσ
= R2(1 +hP+hQ
R)dσ
∼= R2(1 + εPQ)dσ
(2.12)
Eq. 2.12: The approximate formulation of the area element in the spherical polar
system.
On the other hand, the expression for the area element, d2x, of the tangential
plane (reported in Eq. 2.13) that expresses the planar area element in terms of the
elements of the spherical coordinate system can be reversed to derive an explicit
expression for the area element, R2dσ as reported in Eq. 2.14.
d2x = R2dσ cosψ
∼= R2dσ − 12ψ2R2dσ
(2.13)
68
CHAPTER 2. GRAVITY TERRAIN EFFECTS
Eq. 2.13: The area element of the tangent plane.
R2dσ ∼= d2x+ 12
s2PQR2 d2x (2.14)
Eq. 2.14: The area element R2dσ.
Now, letting G be the Newton constant, ρ the density of the attened body,
which is assumed to be a constant value within the body, and by using µ = G · ρwhere drQ = dhQ, we can write a formula for the gravity potential as follows;
T (P ) = G∫dσ∫ R0+hQR0
ρ(Q)·r2Q
lPQdrQ
= Gρ∫dσ∫ R0+hQR0
r2Q
lPQdrQ
= µ∫dσ∫ R0+hQR0
R2(1+2εQ)
lPQdrQ
= µ∫d2x
∫ R0+hQR0
(1+12
s2QR2 )(1+2εQ)
lPQdrQ
= µ∫d2x
∫ R0+hQR0
(1+2εQ+12
s2QR2 +εQ
s2QR2 )
lPQdrQ and by ignoring [εQ
s2QR2 ]
= µ∫d2x
∫ R0+hQR0
(1+2εQ+12
s2QR2 )
lPQdrQ
= µ∫d2x
∫ HQ0
(1 + 2εQ + 12
s2QR2 )( 1
LPQ− 1
2εPQ
s2PQL3PQ
)dhQ
= µ∫d2x
∫ HQ0
( 1LPQ− 1
2
εPQs2PQ
L3PQ
+ 2εQLPQ− εQεPQs
2PQ
L2PQ
+ 12
s2QLPQR2 − 1
4
εPQs4Q
L3PQR
2 )dhQ
∼= µ∫d2x
∫ HQ0
( 1LPQ− 1
2
εPQs2PQ
L3PQ
+ 2εQLPQ
+ 12
s2QLPQR2 )dhQ
→ ignoring (εPQs2PQL3PQ, 1
4
εPQs4Q
L3PQR
2 )
∼= µ∫d2x
∫ HQ0
( 1LPQ
)dhQ + µ∫d2x
∫ HQ0
(2εQLPQ
+ 12
s2QLPQR2 − 1
2
εPQs2PQ
L3PQ
)dhQ
∼= T Pt (P ) + T SCt (P )
(2.15)
Ahmed Hamdi Mansi 69
Airborne Gravity Field Modelling
Eq. 2.15: The gravity potential, T , where T Pt is donated for the planar
approximation of the gravity potential, Tt and TSCt is the spherical correction term.
2.2.2 The Spherical Corrections
At this point, as seen in Eq. 2.15, we can split the computations of the potential
eld into two parts, the rst is the eect of the planar approximation, and the latter
is the spherical correction term. Similarly, we can do the same for the gravitational
eect, where the planar terrain correction can be computed as follows:
δgPt (P ) = − ∂∂hQ
µ
∫d2x
∫ HQ
0
( 1LPQ
)dhQ
= µ∫d2x 1
LPQ− 1
LPQ0
(2.16)
Eq. 2.16: The planar terrain correction, The planar terrain correction, δgPt .
Where the planar integral in Eq. 2.16 must be extended to the actual base, D, of
the topographic body as illustrated in Fig. 2.5. Also, Fig. 2.5 explains the dierent
notations and the distances in the attened body geometry, which helps us compute
the existing integrals within the dierent terrain correction formulas, where LPQ
can be commuted from Eq. 2.8 or Eq. 2.17 therefore LPQ0 can be computed from
Eq. 2.18.
LPQ ∼= [(hP −HQ)2 + (ξP − ξQ)2]12 (2.17)
Eq. 2.17: The value of LPQ, the distance between P and the point Q on the
topography (hQ = HQ).
LPQ0 =√h2P +D2
PQ
∼= [(hP )2 + (ξP − ξQ)2]12
(2.18)
Eq. 2.18: The value of LPQ0 , the distance between P and the point Q0, the
projection point of Q on the tangential plane (hQ = 0).
70
CHAPTER 2. GRAVITY TERRAIN EFFECTS
(a) The notation in Spherical coordinate system
(b) The notation in planar approximation
Figure 2.5. Notation of points and distances in the attened body geometry andthe illustration of the dierent used Cartesian distances.
Ahmed Hamdi Mansi 71
Airborne Gravity Field Modelling
As for the spherical corrections of the terrain correction, δgSCt , the following
approximated expressions hold:
δgSCt (P ) = − ∂∂hQ
µ∫d2x
∫ HQ0
(2εQLPQ
+ 12
s2QLPQR2 − 1
2
εPQs2PQ
L3PQ
)dhQ
= 2µ ∂∂hQ
∫d2x
∫ HQ0
εQ1
LPQ+ µ
2∂
∂hQ
∫d2x
s2QR2
∫ HQ0
1LPQ
dhQ
− ∂∂hQ
µ2
∫d2x
∫ HQ0
εPQs2PQ( 1
L3PQ
)dhQ
= 2µ∫d2x
∫ HQ0
dhQεQ∂
∂hQ
1LPQ
+ µ2
∫d2x
s2QR2
∫ HQ0
∂∂hQ
1LPQ
dhQ
−µ2
∫d2x
∫ HQ0
s2PQL3PQ
∂∂hQ
(εPQ)dhQ
−µ2
∫d2x
∫ HQ0
εPQs2PQ
∂∂hQ
( 1L3PQ
)dhQ
= 2µ∫d2x
∫ HQ0
dhQεQ∂
∂hQ
1LPQ
+ µ2
∫d2x
s2QR2
∫ HQ0
∂∂hQ
1LPQ
dhQ
− µ2R
∫d2x
∫ HQ0
s2PQL3PQdhQ − µ
2
∫d2x
∫ HQ0
εPQs2PQ
∂∂hQ
( 1L3PQ
)dhQ
≡ I1 + I2 − I3 − I4
(2.19)
Eq. 2.19: The spherical corrections of the terrain correction, δgSCt .
Where, the expression for I1, I2, I3, and I4 are reported in Eq. 2.20, Eq. 2.21,
Eq. 2.22, and Eq. 2.23, respectively.
I1 = 2µ
∫d2x
∫ HQ
0
dhQεQ∂
∂hQ
1LPQ
= 2µ
∫d2x
HQR
1LPQ− 2µ
R
∫d2x
∫ HQ
0
dhQ1
LPQ
∼= 2µ
∫d2x
HQR
( 1LPQ− 1
LPQ0)
(2.20)
Eq. 2.20: The evaluation of I1.
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CHAPTER 2. GRAVITY TERRAIN EFFECTS
I2 = µ2
∫d2x
s2QR2
∫ HQ
0
∂∂hQ
1LPQ
dhQ
∼= µ
∫d2x
s2Q2R2 ( 1
LPQ− 1
LPQ0)
(2.21)
Eq. 2.21: The evaluation of I2.
I3 = µ2R
∫d2x
∫ HQ
0
s2PQL3PQdhQ
∼= µ2
∫d2x
HQR
s2PQL3PQ0
(2.22)
Eq. 2.22: The evaluation of I3.
I4 = µ2
∫d2x
∫ HQ
0
εPQs2PQ
∂∂hQ
( 1L3PQ
)dhQ
= µ2
∫d2x
∫ HQ
0
s2PQ(hP
R− HQ
R) 1L3PQ
−µ2
∫d2x
∫ HQ
0
s2PQ
hPR
1L3PQ0
− µ2R
∫d2x
∫ HQ
0
s2PQL3PQdhQ
∼= µ2
∫d2xs
2PQ
hPR
( 1L3PQ− 1
L3PQ0
) + µ2
∫d2xs
2PQ
HQR
( 1L3PQ− 1
L3PQ0
)
(2.23)
Eq. 2.23: The evaluation of I4.
In order to conclude the discussion of the theory of GTE, several notes must be
taken into consideration while evaluating or using these integrals. For the above
expressions whilesPQLPQ≤ 1, we computed rough estimate for the order of magnitude
for I1, I2, I3, and I4 (Eq. 2.25) computed under extreme conditions reported in
Eq. 2.24. εP ∼ 10−3
εQ ∼ 10−3
(SQLPQ
)2 ∼ 3 · 10−4δgPt
· · · , etc.
(2.24)
Ahmed Hamdi Mansi 73
Airborne Gravity Field Modelling
Eq. 2.24: The numerical representation of the extreme conditions used to compute
the order of magnitude forI1, I2, I3, and I4.
Indeed restricting the area e.g. to a 100 km diameter, I2 becomes irrelevant and
the other integrals are reduced considering that R can be brought to a mean height
and it is only in particular area that the one can have 6 km of height dierence in a
100 km horizontal distance. Hereafter, we will consider that the topographic body
is in terms of Cartesian coordinate system and its attraction is given as described
in Eq. 2.16 and Eq. 2.25. O(I1) ∼ 2 · 10−3δgPt
O(I2) ∼ 3 · 10−4δgPt
O(I3) ∼ 12· 10−3δgPt
O(I4) ∼ 1 · 10−3δgPt
(2.25)
Eq. 2.25: The estimates of the order of magnitudes of I1, I2, I3, and I4 compared
to the planar terrain corrections.
2.3 The GTE algorithms
Two main hypotheses are considered within the mathematical evaluation of the
terrain correction eects (Eq. 2.16 and Eq. 2.19), the former is that the density is
considered a constant value across the whole body while the latter is that the body
itself is constituted by prisms. Consequently, we have the input data represented as
a regular grid on the (x, y) plane with cells of size ∆x and ∆y, which is a common
and typical practice when the terrain data is given in the form of a digital terrain
model (DTM). The centers of the cells are thought as sampling points of the digital
terrain and throughout the cell itself the terrain is assumed to have a constant
height. Similar is the situation when we have to deal with bathymetry, the main
and only dierence with the aforementioned case being that now h = −H ≤ 0, while
for topography we have h = H ≥ 0. For a body not totally below the (x, y) plane,
like in the case of sediments, one can modulate the calculation by decomposing the
eects of the upper surface of the body and the lower surface.
2.3.1 GTE for The Topography
Considering that the density is included in µ = G · ρ, namely the multiplicative
constant, it is clear that the algorithm is basically only one but adapted to dierent
circumstances. So we shall concentrate on the topographic case, leaving to a few
74
CHAPTER 2. GRAVITY TERRAIN EFFECTS
remarks the application to other cases. Since the formula for the attraction of a single
prism is explicitly known (Nagy et al., 2000) and can be split into a contribution
δg+t of the upper face and a contribution δg−t of the lower face, in principle we can
exactly compute as well δgt at any point P outside the body by summing up the
eects of each prism as reported in Eq. 2.26.
δgt(P ) =∑(j,k)
[δg+jk − δg
−jk] (2.26)
Eq. 2.26: The terrain eect as a sum of prisms′ eects, as (j, k) are the grid
indexes.
We do not report here the explicit expression for δg±jk that highly depend on the
shape of the prism and the relative position of point P with respect to the center of
the face (i.e., upper face (+) or lower face (−)), the interested reader can read for
instance (MacMillan, 1930). This approach is classical and has been implemented in
so many algorithms to provide an "exact" solution, given our planar approximation
hypotheses.
On the one hand, GTE implements the same algorithm, especially to compute the
terrain eects, δgt, for the case of sparse points. On the other hand, this approach
can be time−consuming and inecient especially in the case of grids with up to
106 nodes and the computational points of the same order of magnitude due to
the necessity to compute several times the logarithms and arctangents present in
the prism formula. Tsoulis (1999) noticed that the bases of the prisms can be
conglomerated in order to reduce the number of computations to one only, therefore
the computational time was divided by a factor of two. This is helpful is helping
but yet it does not solve the problem.
Sideris (1984) reported an innovative solution for the problem implementing the
idea of applying a FFT approach. However the numerical advantage of the Fourier
approach is very large in terms of velocity specically when we have to compute
convolutions and when the discretized form is referring to grids with size equal to
a power of 2 (or using a mixed radix algorithm to a power of 2 by a power of
3, etc.). This requirement is usually met by extending the DTM grid with zero
height nodes, the so−called zero padding. More rened solutions are given by Sansò
and Sideris (2013) in order to avoid jumps on the edges. Furthermore, the fast
version of the Discrete Fourier Transform implies the result to be computed on the
same grid on which data are given. For this reason the main algorithm of GTE is
computing terrain corrections on grids. We will discuss in details the main cases in
the upcoming sections.
Ahmed Hamdi Mansi 75
Airborne Gravity Field Modelling
2.3.1.1 GTE for a Grid on the DTM Itself
As mentioned earlier, the main advantage of the Fourier approach is great while
computing convolutions, accordingly let us workout the planar approximation of the
gravity disturbance, Eq. 2.16, in order to reformulate it in the shape of convolution
integrals; in particular we shall concentrate on the top part, because the lower part
can be computed exactly as the lower part of a prism. To simplify the writing let us
agree that ζ is the position vector of P0 in the (x, y) plane (see Fig. 2.5) while η is
the position vector of the running point Q0. Note also that hP = HP ≡ Hζ , when P
is on the DTM, so that δgt can be considered as function of the 2D vector ζ. Then
using Eq. 2.17, the value of 1LPQ
can be written as in Eq. 2.27.
1LPQ
= 1[|ζ−η|2+(Hζ−Hη)2]1/2
∼= 1|ζ−η| −
12
(Hζ−Hη)2
|ζ−η|3 + 38
(Hζ−Hη)4
|ζ−η|5 + · · ·(2.27)
Eq. 2.27: The value of 1LPQ
where h = HP .
In fact, the expanded series stopped at the fourth order term is convergent if the
condition reported in Eq. 2.28 is met everywhere on the DTM. Hence, the maximum
admissible inclination of the DTM, in order to meet the convergence condition,
should be less than 45.|Hζ−Hη ||ζ−η| < 1 (2.28)
Eq. 2.28: The The convergence condition of 1LPQ
where h=HPofwhereh=HP .
The integration of Eq. 2.27 is explicitly written in Eq. 2.29, which shows that each
individual integral is in fact converging thanks to Eq. 2.28 of the the convergence
condition.
δg+t = µ
∫d2ηLPQ
= µ∫d2η( 1
|ζ−η| −12
(Hζ−Hη)2
|ζ−η|3 + 38
(Hζ−Hη)4
|ζ−η|5 )
= µ∫
d2η|ζ−η| −
µ2
∫ (Hζ−Hη)2
|ζ−η|3 d2η + 3µ8
∫ (Hζ−Hη)4
|ζ−η|5 d2η + · · ·
(2.29)
Eq. 2.29: The integration of Eq. 2.27.
However none of these individual integrals is represented as a convolution while,
taking as an example the second order term, Eq. 2.30, shows that each of the integrals
76
CHAPTER 2. GRAVITY TERRAIN EFFECTS
of the right hand side has the form of a convolution integral. Unfortunately, none
of the terms of Eq. 2.30 is any more convergent.
µ2
∫D
(Hζ−Hη)2
|ζ−η|3 d2η = µ2H2ζ
∫D
d2η|ζ−η|3 − µHζ
∫D
Hηd2η
|ζ−η|3 + µ2
∫D
H2ηd2η
|ζ−η|3 (2.30)
Eq. 2.30: The second order term represented as a set of convolution integral.
The problem can be disregarded by explaining that in the left hand side of
Eq. 2.30 we can isolate a small area around ζ because we would consider Hζ ≡ Hµ
in case of ζ and µ belong to the same cell.
In our GTE software, on the contrary, we prefer to adopt a dierent strategy
similar to that proposed in TcLight (Biagi and Sansò, 2000) by dening a "small"
area in the plane around the center (O) as Dε(O); in our case, with the subsequent
discretization in mind, we use a Dε(O) as graphically illustrated in Fig. 2.6. This is
mathematically dened by the characteristic function in Eq. 2.31, with the condition
reported in Eq. 2.32.
Iε(η) = χε(|η1|)χε(|η2|) (2.31)
Eq. 2.31: The characteristic function.
χε(|t|) =
1 → for |t| ≤ ε
0 → elsewhere(2.32)
Eq. 2.32: The conditions of characteristic function.
A note must be taken that Dε(O) can be translated around any point ζ in the
plane generating the set explained in Eq. 2.33. The characteristic function for this
case, reported in Eq. 2.34, is similar to the generic case.
Dε(ζ) = ζ +Dε(0) (2.33)
Eq. 2.33: The translation of Dε(O) around point ζ in the plane.
Iε(ζ − η) =
1 → for (ζ − η) ∈ Dε(0)
0 → elsewhere(2.34)
Eq. 2.34: The characteristic function for the translated Dε(O) around point ζ in
the plane.
Ahmed Hamdi Mansi 77
Airborne Gravity Field Modelling
Figure 2.6. The set used to isolate the singularity.
Therefore, we can rewrite δg+t , Eq. 2.29, as in Eq. 2.35:
δg+t = µ
∫D\Dε(ζ)
d2ηLPQ
+ µ
∫Dε(ζ)
d2ηLPQ
≡ δg+out(P ) + δg+
in(P )
(2.35)
Eq. 2.35: The value of δg+t .
On the one hand, the integral part δg+in(P ) is easily computed by the discretiza-
tion on the prisms (in fact, it is the contribution of only the upper faces of the
prisms). In this case, for each grid node and the corresponding prism, we do not
have to compute a full grid of values, but only (2ε + 1)2 values, thus considerably
reducing the number of computations and consequently, the computational time.
On the other hand, the integral part δg+out(P ) can be written as in Eq. 2.36 and
for a further manipulation of this integral we can use the approach of the series
development seen in Eq. 2.27.
δg+out(P ) = µ
∫D
[1−Iε(ζ−η)]LPQ
d2η (2.36)
Eq. 2.36: The value of δg+out.
78
CHAPTER 2. GRAVITY TERRAIN EFFECTS
However, the one has to point out that now the convergence condition reported
in Eq. 2.28 has to be computed for points ζ and µ distant apart with at least (ε+ 1)
from one another. For the following few passages, we will x ∆x = ∆y = 1 for
the sake of simplicity. As a result, the condition of Eq. 2.28 is much more easily
satised, specially choosing judiciously ε as function of the terrain roughness. Also,
the number of terms required by Eq. 2.27 in order to obtain a good approximation
could be decreased. Furthermore, referring for instance to the second order terms of
Eq. 2.29 elaborated in Eq. 2.30, can be furtherer processed in order to have all the
terms as convolution integrals as Eq. 2.37 that none of its terms is anymore singular.
δg+out,2(P ) = µ
2
∫D\Dε(ζ)
(Hζ−Hη)2
|ζ−η|3 d2η
= µ2H2ζ
∫D
[1−Iε(ζ−η)]|ζ−η|3 d2η − µHζ
∫D
Hη [1−Iε(ζ−η)]
|ζ−η|3 d2η
+µ2
∫D
H2η [1−Iε(ζ−η)]
|ζ−η|3 d2η
(2.37)
Eq. 2.37: The second order terms of δg+out.
A little thought shows that by using the kernel, Kk(ζ − µ, expressed in Eq. 2.38
and by recalling Eq. 2.29, one can elaborate Eq. 2.36 collecting all the terms up to
the maximum power 2N , the computations has been reduced to one expression to
be evaluated like Eq. 2.39.
KK(ζ − η) = 1−Iε(ζ−η)|ζ−η|2K+1 (2.38)
Eq. 2.38: The kernel.
δg+out(P ) =
N∑k=0
2k∑j=0
CKjH2K−1ζ Kk ∗Hj
η (2.39)
Eq. 2.39: δg+out represented as a spherical harmonics expansion, as Ckj represent
the known constants.
Now, the evaluation of δg+out can be accomplished by performing a discretization
step and then apply a FFT algorithm. The last remark is that both values of ε and
N (as integers) can be chosen by the user of the GTE software.
Kk ∗Hjη = F−1FKk ∗ FHj
η (2.40)
Eq. 2.40: Fourier formula for the evaluation of the convolution of δg+out.
Ahmed Hamdi Mansi 79
Airborne Gravity Field Modelling
2.3.1.2 GTE for a Grid at a Constant Height
The situation of using GTE to compute the terrain correction eects at a constant
height h = H where H is totally above the topography, uses a modied and tuned
version of the equations discussed earlier in section 2.3.1.1. By recalling Fig. 2.5,
replacing hP ≡ h = H, and by replacing the notation LPQ0 of Eq. 2.18 that is used
in the case of GTE on the DTM, with LP0Q0 , the new version of Eq.
Also, Eq. 2.27 will be adjusted to consider the new situation, as reported in
Eq. 2.41. Consequently, its power series expansion in terms of Hµ is characterized
by always having convergent terms, as seen in Eq. 2.42.
1LPQ
= 1[s2+H2−2HHη+H2
η ]1/2
= 1[s2+H2]1/2
· 1
[1−2H
s2+H2Hη+1
s2+H2H2η ]1/2
(2.41)
Eq. 2.41: The value of 1LPQ
where h = H.
1LPQ
= 1LP0Q0
+ HL3P0Q0
Hη +3H2−L2
P0Q0
2L5P0Q0
H2η +
H(5H2−3L2P0Q0
)
2LP0Q0H3η + · · · (2.42)
Eq. 2.42: The power series expansion of 1LPQ
where h = H.
By performing the integration of the term 1
L2k+1PQ
over D, several terms are gen-
erated each of which can be written in the form of Eq. 2.43, which could be easily
computed by Fourier, using the convolution theorem.∫D
( HLP0Q0
)2k+1(HηH
)jd2η = Fkj(ζ) → for (j ≥ 2k) (2.43)
Eq. 2.43: The general form of each term of the output of∫D
( 1LP0Q0
)2k+1.
A note has to be taken that both terms under the integral in Eq. 2.43 are smaller
than or equal to 1 and in particular (HLP0Q0
)2k+1, as a bounded function that can be
put to zero when s = |ζ − µ| is larger than the diameter of D, has always a regular
Fourier transform.
Hence, within Eq. 2.41 and Eq. 2.42 the singularity problem is overcome; yet
when H is only a small distance above the top of the topography, HTOP , (see Fig. 2.6
and Fig. 2.7). There are values of HLP0Q0
and HµH
that in some areas can be very close
to 1 strongly degrading the eectiveness of the approximation by series development
in Eq. 2.42. To counteract such an eect, we have implemented in GTE a particular
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CHAPTER 2. GRAVITY TERRAIN EFFECTS
Figure 2.7. The Slicing the topographic body to compute the grid at height H.
algorithm, which is called slicing, schematically exemplied in Fig. 2.7. Simply, the
idea is that GTE computes the terrain correction of the topographic body at the
constant height, H, in a slice−by−slice manner and then sum up these contributions
together. In order to do so, the reference plane is brought up at the base of the slice
in such a way that the ratio of the height of the slice to the height of the computation
grid is never close to 1. To further illustrate the slicing technique, in Fig. 2.7 the
middle slice has a ratio H2−H1
H−H1, which is smaller than 1/2. Then when we move to
the upper slice we still have HTOP−H2
H−H2smaller than 1/2.
A remark about GTE and the slicing step is that the number of slices can be be
either xed by the user, or automatically estimated by the software. For the second
case, the choice of the dierent heights is done following the schema reported in
Eq. 2.44, as follows: H1 = 0.4H,
H2 = 0.4(H −H1),
H3 = 0.4(H −H2),
· · · , etc
(2.44)
Eq. 2.44: The automatic slicing heights schema done by GTE.
2.3.1.3 GTE for Sparse Points
There are cases in which the terrain eects have to be computed at sparse points,
but, given the dimension of the problem, we still want to take advantage of some
Fourier algorithms. We still distinguish the two cases, when the computation points
Ahmed Hamdi Mansi 81
Airborne Gravity Field Modelling
Figure 2.8. The Spatial interpolation at P .
are on DTM or when they are in space. For the rst case GTE computes a simple
bi−linear interpolation to each computation point, from the values at the four cor-
ners of the cell to which P belongs. If on the contrary the sparse computation points
Pk are above the topography, but indeed not all at the same height, the software
computes two grids in correspondence of the minimum (Hmin) and the maximum
(Hmax) heights of the sparse computation points Pk. Then let the computation point,
P , be as in Fig. 2.8. We take the circumscribing cell and interpolate the terrain ef-
fect from G+i and G−i to P+ and P−, respectively. Such horizontal interpolations
are performed by bi−linear functions.
Finally we interpolate linearly from P+ and P− to P . Several experiments have
revealed that instead of computing more grids and to use a higher order polynomial
in the vertical direction, it is always preferable to split the sparse points into several
subsets according to their altitude, then compute a couple of grids for each subset,
and then interpolate linearly.
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CHAPTER 2. GRAVITY TERRAIN EFFECTS
Figure 2.9. The geometry of the body composed by Bt (topographic body), Br
(basement with rock density), Bw (basin lled with water); Bw maximum depthof Bw , H is the height of the grid above the reference surface where we want tocompute δg.
2.3.2 GTE for The Bathymetry
For the bathemetric case, where h = −H ≤ 0, we isolate the rock and the water
bodies of Fig. 2.1, as in Fig. 2.9 in order to develop the corresponding theory. First
of all, let us recall that the main goal for this section is to compute the gravimetric
eect of the body Bt∪Bw. Indeed we already have a software capable of computing
the gravity eect of Bt; we want to show how to use the same software in order to
compute the gravity eect of Bw. Let us use the notation δg(ρ |B)= gravity eect
of a generic density ρ distributed in the generic body B. Statistically speaking we
can write the following equation:
δg(ρw|Bw) = δg(ρw|B0)− δg(ρw|Br) (2.45)
Eq. 2.45: The mathematical formulation of δg(ρw |Bw) .
In Eq. 2.45, the term B0 is a big prism, the upper and the lower faces of which
lay on the planes h = 0 and h = −H0 (i.e. B0 = Br ∪ Bw). The rst term on the
R.H.S., δg(ρw |B0) , is just the eect of the prism B0 that the software can easily
handle. The second term on the R.H.S. of Eq. 2.45, namely δg(ρw |Br) is just the
Ahmed Hamdi Mansi 83
Airborne Gravity Field Modelling
eect of the body Br, with a suitably reduced density, computed at a grid on the
plane at height H = H0 + H above the base of Br.
This is exactly what GTE software already does if we just use a new multiplicative
constant equals to µw = G ·ρw and change the computation height from height from
H to H0 + H. Note that if we want to make Bt ∪Br ∪Bw uniform, because then its
eect is that of a Bouguer plate (or better of a prism), once we removed δg(ρr |Bt) ,
we simply have to compute δg(ρw |Bw) , using Eq. 2.45 but substituting (ρw) with
(ρr−ρw). In other words, if we want to get the eect of a prism given by Br∪Bw we
have to remove the gravitational eect of the topography, Bt, with density ρw and
to ll the ocean with a density equal to (ρr − ρw) (i.e. δg(ρr − ρw |BW )).
It is worth mentioning that with the same method but with just changing the
density ρ in the multiplicative constant µ, one could also take into account that the
density below the sea oor is that of a sediment layer instead of rocks. Consequently,
the problem of the bathymetry is fully solved.
2.3.3 GTE for Moho and Sediments
To handle the Moho eects we can use exactly the same tuning and reordering
done within the case of bathymetry, by suitably changing the density constant ρ.
Only since the Moho is deeper and generally smoother than the sea oor, depending
on the resolution of the model available, the user can apply the simple prism algo-
rithm implemented in GTE (in case of low resolution), or the FFT routine without
slicing and with a small ε. Similar is the situation with sedimentary layers, where
the algorithm would be applied twice, the rst time for the lower surface and the
second time for the upper surface of the sediments.
2.4 GTE Performances
Firstly, in order to facilitate the use of the GTE software some proles have been a
priori set. In Table 2.1 we report the main characteristics of each prole implemented
for the computation of the gravitational eect of topography and bathymetry.
Secondly, in order to evaluate the performances of the GTE software, some nu-
merical tests have been executed. These tests will be mainly focused on the compu-
tation of the terrain correction for airborne gravimetry (the algorithm presented in
section 2.3.1.3), which represents the most important feature of the GTE software.
In particular these tests aim to compare the accuracy and the computational time of
the GTE algorithms with respect to those implemented in standard scientic soft-
84
CHAPTER 2. GRAVITY TERRAIN EFFECTS
Topography BathymetryProle names Slices ε Slices εVERY FAST 0 3 0 3FAST 1 5− 10 1 5− 10TRADEOFF 2 5− 15− 30 2 5− 10− 20SLOW Prisms / 2 5− 10− 20VERY SLOW Prisms / Prisms /
Table 2.1. Number of slices and number of prisms used for each slice to reduce theFFT singularity for dierent proles. Parameters are reported in case of computationof topographic and bathymetric eects
ware such as the GRAVSOFT package (Tscherning et al., 1992) and the Tesseroids
(Uieda et al., 2011).
GRAVSOFT is a suite of Fortran programs developed to model the gravitational
signal, its main features allow to:
• evaluate spherical harmonic coecients, modelled by means of least−squarescollocation (GEOCOL software);
• estimate isotropic covariance functions (EMPCOV software);
• t with an analytic expression one or more empirical covariance functions
(COVFIT software);
• compute terrain eect (TC software);
• compute terrain eects by means of Fourier algorithm (TCFOUR software);
• evaluate Stokes formula using spline densication (STOKES software).
The GRAVSOFT package is widely used for scientic and production purposes.
For instance it has been used for geoid determination of the Nordic Area ( see
(Tscherning and Forsberg , 1987; Forsberg et al., 1997), parts of UK (Dodson and
Gerrard , 1990), Italy (Benciolini et al., 1984), Catalonia (Andreu and Simo, 1992),
the Mediterranean Area (Arabelos and Tziavos , 1996), Turkey (Ayhan, 1993), and
in other numerous smaller projects for local detailed geoid determination. Among
the dierence functionalities of the GRAVSOFT package, here, we will concentrate
on the TC software only. Just note that the TCFOUR software, as many FFT based
algorithms, permits to compute only the gravitational eect of topographic masses
on the surface dened by the DTM itself (as in the rst case presented in section
2.3.1.1), it is therefore not suitable for airborne gravity surveying but can be used
Ahmed Hamdi Mansi 85
Airborne Gravity Field Modelling
for ground as well as shipborne surveys. The TC software is more exible allowing
the computation of the gravitational eects of a DTM at any arbitrary point in
the space (outside the masses). In order to improve its speed, TC can consider two
DTM grids (one at high and one at low resolution), for cells close to the computation
point, the terrain eect should be computed using the high resolution model, while
for distant cells, the low resolution one will be considered. The threshold can be set
by the user. Moreover, in order to compute the eect of a single cell of the DTM the
user can choose between dierent solutions: one can use the prism equation (Nagy ,
1966) or its approximate solution with MacMaillian's formula (MacMillan, 1930) or
even with the eect the model of the point mass approximation. The approximation
used is automatically chosen by the software as a function of the geometry of each
specic computation.
As for Tesseroids, it is a collection of command−line C programs to model the
gravitational potential, acceleration, and gradient tensor of topographic masses.
Tesseroids supports models and computation grids in Cartesian and spherical co-
ordinate systems. The main geometric element used in the modelling process is
a spherical prism, also called a tesseroid, the gravitational eect of which can be
computed by means of approximated formula (Asgharzadeh et al., 2007).
The Tesseroids software basically computes the gravitational eect of each tesseroid
by summing up the eect of a number of point masses optimally distributed and
weighted inside the tesseroid. Indeed, the accuracy of the solution remains essen-
tially unchanged for dierent numbers of point masses as long as the node spacing
is smaller that the distance to the observation point (Asgharzadeh et al., 2007),
therefore while using the Tesseroids software, a particular attention has been paid
to respect this simple law. Tesseroids is mainly used for geophysical studies at dif-
ferent scales from the very local one, such as the reconstruction and analysis of the
Grotta Gigante cave (a Karstic cave in the Northern part of Italy) signal (Pivetta
and Braitenberg , 2015), to the regional ones such as the study of the crustal struc-
ture in the Andean region (Alvarez et al., 2014) or the study of the European Alps
orogenetic belt (Braitenberg et al., 2013). All the tests have been performed on a
single node of a supercomputer equipped with two 8− cores Intel Haswell 2.40 GHz
processors (for a total of 16 cores) with 128 GB RAM.
The rst dataset used for the testing purpose is located in the south part of New
Mexico, between 31.5 and 35 S and 105 and 108 W. The digital terrain model,
Fig. 2.10, is a grid with 351 rows and 301 columns with a spatial resolution of 36
arc−second. It is a mountainous region characterized by a mean elevation of 1670 m
with a minimum and a maximum heights of 1050 and 3445 m, respectively.
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CHAPTER 2. GRAVITY TERRAIN EFFECTS
Figure 2.10. The Digital Terrain Model used for the rst test.
Ahmed Hamdi Mansi 87
Airborne Gravity Field Modelling
Prole name Time (s) Mean (mGal) STD (mGal)SLOW 144.0 176.53 40.4FAST 11.9 -0.37 0.09
VERY FAST 29.4 -0.14 0.02TRADEOFF 83.3 -0.13 0.01GRAVSOFT 511.1 -6.95 2.41TESSROIDS 31120.0 2.01 0.29
Table 2.2. The statistics and the computational time on a grid at 3500 m for thedierent proles and software tested. SLOW prole shows statistics on the computedsignal. For the other rows the statistics refer to the dierence between each resultand the terrain eect computed with the SLOW prole
2.4.1 Test 1: TC at a Constant Height
The rst test performed consists in comparing, in terms of accuracy and compu-
tational time, the results computed on a regular computational grid at H = 3500
m obtained by the dierent GTE proles as well as by dierent softwares. Note
that since the test area is completely onshore both proles SLOW and VERYSLOW
are computing the gravitational eect by using only the prism equation. It should
also be observed that, since we are in a planar approximation, the solution obtained
by means of the pure prism equation represents the exact solution of the problem
and can be used for comparisons. The altitude of the computational grid has been
chosen assuring that the computations of the gravitational eect of the topography
is always performed outside the masses, therefore as already said it has been xed
at 3500 m, only 55 m above the highest peak.
Results of this rst test are graphically presented in Fig. 2.11 and numerically
summarized in Table 2.2, where the computational time required for each compu-
tation and some statistics on the dierences between each solution and the SLOW
prole (used as a reference) are reported. Starting from the comparison between the
SLOW prole and the GRAVSOFT software (with the standard compilation) it can
be seen that the computational time required by GTE (144 s) is less than one-third of
that required by GRAVSOFT (about 511 s). This is due to the fact that some of the
GTE routines have been parallelized, thus exploiting the maximum computational
power of the machine. The dierence between the two solutions shows a mean value
of −6.95 mGal and a standard deviation of 2.41 mGal; removing a border of about
40 km where border eects of the FFT can have some importance, the standard
deviation drops to 0.7 mGal while the average remains practically unchanged.
To improve the mean value of the GRAVSOFT terrain correction, one has to force
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CHAPTER 2. GRAVITY TERRAIN EFFECTS
Figure 2.11. TC computed with the SLOW prole and its dierences with respectto the gravitational eects computed by means of dierent proles/software.
Ahmed Hamdi Mansi 89
Airborne Gravity Field Modelling
the software to use only the high resolution grid and compute the eect by means
of prism equation. In this case, the dierence on the average dropped to −4 mGal
but the computational time required to reach the GRAVSOFT solution considerably
increased to more than 6 hours. This new mean dierence of −4 mGal between the
averages of the GTE SLOW prole and GRAVSOFT solutions could be explained
by the dierent algorithms used from the two software to map geodetic coordinates
in Cartesian ones: in fact, while GTE uses the mapping presented in section 2.2,
GRAVSOFT denes its Cartesian reference system simply as x = 111195 ·∆λ cos ϕ)
and y = 111195 ·∆ϕ with ∆λ and ∆ϕ representing the DTM resolution in longitude
and latitude direction, respectively (in radians) and ϕ is the average latitude of the
DTM itself.
As for the Tesseroids results, it can be seen that they are closer than GRAVSOFT
to those of GTE with an average dierence of about 2 mGal and a standard deviation
of 0.29 mGal which dropped to only 0.17 mGal if the border region is removed from
the statistics. It should be observed that the Tesseroids computation is performed
directly in spherical approximation, this is the main cause of the 2 mGal dierence
(as can be observed from the graphical representation of the results in Fig. 2.11.
However, even for such a small example, the Tesseroids takes more than 8 hours to
compute the solution which is 2 orders of magnitude more than the slowest GTE
computational time.
Considering the other GTE proles, namely the VERYFAST, FAST and TRADE-
OFF, it can be observed that the use of FFT speeds up the computation of a factor
ranging between 2 and 10 giving practically the same result (the standard deviation
of the dierences is always smaller than 0.1 mGal).
2.4.2 Test 2: TC at the Surface of the DTM
The same dataset has been used also to test the performances of GTE in com-
puting of the terrain eect directly on a grid on the surface of the DTM itself (i.e.
the algorithm explained in section 2.3.1.1). In this case, Tesseroids cannot be used
since its solution became unstable when the observation point is close to the masses.
GTE computes the solution in 9.8 s with dierences smaller than 1−3 mGal with re-
spect to the pure prism solutions while GRAVSOFT takes more than 580 s to reach
a solution giving dierences of few mGal (a mean of −6.17 mGal and a standard
deviation of 2.72 mGal).
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CHAPTER 2. GRAVITY TERRAIN EFFECTS
Prole name Time (s) Mean (mGal) STD (mGal)SLOW 312.7 180.30 41.7FAST 23.8 -0.11 0.96
VERY FAST 58.8 -0.21 0.95TRADEOFF 197.6 0.20 0.95GRAVSOFT 7.1 -7.18 2.53TESSROIDS 309.7 -2.2 0.30
Table 2.3. The statistics and the computational time on a 1000 points for thedierent proles and software tested. SLOW prole shows statistics on the computedsignal. For the other rows the statistics refer to the dierence between each resultand the terrain eect computed with the SLOW prole
2.4.3 Test 3: TC at the Sparse Points
The Third experiment performed has computed the gravitational eects using the
same digital elevation model on a set of 1000 sparse points with a variable altitude,
which was set to 150 m above the DTM. The results of this test were summarized
and reported in Table 2.3.
The comparisons between GRAVSOFT and Tesseroids with the SLOW prole
are practically similar to the results of the rst test. It should be stated that the
computational time required by GTE does not depend on the number of the sparse
points since, once the two grids are computed, it is just a matter of linear inter-
polations that is not computationally demanding. On the contrary in the case of
GRAVSOFT and Tesseroids doubling the number of the sparse points would at least
double the computational time (note that classical airborne gravimetry surveys can
reach more than 106 points, i.e. 1000 times the number of points used in this ex-
periment). As for the other proles, basically they show the same statistics, which
are however degraded (the standard deviation increased from less than 0.1 mGal to
about 0.9 mGal) due to the closeness of the computation points to the DTM.
In any case, we should keep in mind that these results refer to an extreme DTM
with an unrealistic situation, since it is probably unsafe to y so close to the ground
with such a rough topography. It should also be observed that in this test GRAV-
SOFT is the fastest software, however, it gives quite inaccurate results with respect
to the SLOW prole and to Tesseroids with standard deviations larger than 2 mGal
(in both cases). In both comparisons, it should be emphasized that the fact of
having always a positive topography, and of computing the gravitational eect just
over the highest peaks, is not helping. In fact the 3 TC algorithms tested adopt
dierent reference frames which implies dierent masses for the same DTM cell. As
Ahmed Hamdi Mansi 91
Airborne Gravity Field Modelling
a consequence, the gravitational signal due to this inconsistency has the same sign
for all the grid nodes and therefore, this cumulates up giving high dierences in
terms of mean value when comparing the results. In any case, it is important to
note that in geophysical exploration applications the average value of the eld is not
so important and is usually disregarded.
2.4.4 Test 4: TC at the Sparse Points of the CarbonNet
Project
The last test has been performed considering a more realistic case (with less
extreme dataset): a real airborne acquisition performed in the framework of the
CarbonNet project (CarbonNet Project Airborne Gravity Survey , 2012; Department
of Primary Industries , 2012) has been used. The dataset is made of 404384 airborne
observations acquired in 2011 by Sander Geophysics Ltd. to provide a better under-
standing of the onshore, near-shore and immediate oshore geology of the Gippsland
Basin, a sedimentary basin situated in South−Eastern Australia, about 200 km east
of the city of Melbourne.
A DTM with spatial resolution of 250 m, based on AusGeo model (Whiteway ,
2009) that covers the region between 37.3 and 39.3 S and 146.2 and 148.9 E
represented as a total number of 819 by 1093 grid cells has been used. The height of
the DTM ranges between 1700 m of the Mount Howitt and −2754 m in correspon-
dence of the beginning of the Bass Canyon with a mean altitude of only 20 m and
a standard deviation of 503 m. The aircraft ew oshore at 165 m above the ocean
and it ew onshore following the topography with a maximum altitude of 369 m.
The DTM used as well as the survey tracks are shown in Fig. 2.12.
The dierent results of this test were reported in Table 2.4, where it can be seen
how the use of the FFT allows to compute the terrain correction for the considered
dataset in less than 1 hour. Classical software, like GRAVSOFT or Tesseroids,
requires at least few hours up to more than 1 day to compute the TC eects. Actually
the VERY FAST prole gives a quick−overview for the terrain eects with standard
deviation of the errors of the order of 0.1 mGal in less than 10 minutes, while the
FAST prole will need about 20 minutes to give the results, which are one order
of magnitude more accurate (standard deviation of 0.016 mGal) than any prole
specially with respect to the prism solution, thus conrming the goodness of GTE
software for this kind of application. On the other hand, the VERY SLOW prole
completely exploits the potentiality of the processing machine used for the test needs
only 4 hours to compute the solution.
92
CHAPTER 2. GRAVITY TERRAIN EFFECTS
Prole name Time (s) Mean (mGal) STD (mGal)VERY SLOW 1.5 · 104 -0.67 4.44
SLOW 7457 -0.034 0.14FAST 459 -0.66 0.11
VERY FAST 1112 -0.043 0.016TRADEOFF 2632 -0.035 0.016GRAVSOFT 2 · 104 1.2 0.31TESSROIDS 5 · 105 0.062 0.021
Table 2.4. The statistics and the computational time on 404384 points for thedierent proles and software tested. VERY SLOW prole shows statistics on thecomputed signal. For the other rows the statistics refer to the dierence betweeneach result and the terrain eect computed with the VERY SLOW prole
On the one hand, GRAVSOFT gives the highest standard deviation results in a
time comparable to that of the VERY SLOW prole solution. On the other hand,
the Tesseroids software is the slowest with more than 5 days of computational time
with results very close to those of the GTE software. Again, it should be stated that
the large part of these dierences are probably due the fact that Tesseroids works
in a spherical approximation environment.
2.5 Remarks on GTE
Concluding this chapter, let us recall here that we have presented all the theory
that was adopted and implemented in a new software, called GTE, for fast and
accurate computation of the gravitational terrain eect. In details, GTE has been
developed addressing two major issues required by modern geodetic and geophysical
applications, namely high accuracy and high computational performances in order
to nd a solution, which is basically an innovative combination of FFT techniques
and the classical prism modelling aiming to keep errors lower than 0.1 mGal.
As proven in section 2.2, the planar approximation can in general be used, thus
simplifying the problem, when dealing with regions smaller than 200 × 200 km,
which is the typical situation of airborne gravimetric surveys for local geophysical
applications. Then, some correction terms to account also for the main eects of
spherical approximation have been also illustrated. In order to compute the terrain
correction by means of Fourier algorithms, Newton′s integral has been expanded
in a Taylor series. Some solutions to address the problems of the convergence of
the series (slicing) and of its singularity (prism−FFT mixed algorithm) have been
explained in great depth.
Ahmed Hamdi Mansi 93
Airborne Gravity Field Modelling
Figure 2.12. The Digital terrain model used for the fourth test and the black linesrepresent the dierent ight tracks followed to acquire the data.
Finally, all the choices done in designing the software have been driven by nu-
merical tests, with the purpose of guaranteeing an accuracy in the computation of δg
at the level of 102 mGal, meanwhile always assuring a fast computation. Many dif-
ferent comparisons have been performed showing that the results obtained by GTE
are very close to those obtained by prism equation. The dierences of the results
slightly increase to 0.06 mGal and 0.021 mGal (as mean and standard deviation
values, respectively) when GTE algorithm is compared compared with Tesseroids
that works in a spherical approximation environment.
In any case, since the nal accuracy of the terrain correction largely depends on
the specic geometry of the problem (i.e. on the specic digital terrain model and
on the distance between topography and the observation points), all the parameters
of the GTE software, like the number of slicing sections as well as the ε parameter
or even the number of convolutions can be set by the user.
94
Chapter 3
Along-Track Filtering
[ AJ.
JË @
èPñ] [(7) @
XA
Kð
@
ÈA
J. m.
Ì'@
ð]
[And the mountains as pegs? (7)] [Quran, An−naba]
This chapter is dedicated to give a detailed discussion about the ltering tech-
nique applied on the along−track airborne gravimetric data. Because the spectral
techniques provide excellent means of extracting gravity eld information contained
in each of the gravity eld data with the view of determining the contribution to the
gravity spectrum of each data type ( (Sideris , 1987a; Forsberg , 1984, 1986; Kotsakis
and Sideris , 1999).
The ltering methodology discussed within this chapter consists in 2 computa-
tion milestones. The rst milestone focuses on downsampling the collected gravity
data and the result of this computation milestone would reduce the spatial resolu-
tion from 50 m to 250 m. In order to complete the proposed ltering methodology,
the availability of 2 dierent GGMs that are simultaneously exploited is essential to
perform the second computation milestone which could be thought as a remove−likestep performed in a completely dierent manner if compared to the classical remove
step of the Remove−Compute−Restore procedure (for more information, see sec-
tion 1.3). On the one hand, the rst GGM is used to remove the low frequencies
while the other GGM will be used to suppress the medium−to−high frequencies of
the observed signal.
The second computation milestone involves applying a Wiener lter in the fre-
quency domain. The Wiener lter revokes the observation noise contaminating the
collected gravimetric data from a dynamic platform (e.g., shipborne and/or airborne
gravity data). At this step, we would obtain the ltered signal that is essential to
perform data gridding and Least Squares Collocation.
This strategy is especially advantageous for the local geoid determination. Since
95
Airborne Gravity Field Modelling
Figure 3.1. Schematic representation of the ltering procedure.
satellite models are low−frequency, the satellite spectrum does not necessarily over-
lap with the local data spectrum. The integration of satellite data with a combined
geopotential model increases the bandwidth of the satellite model. The information
content of the satellite data is somewhat stretched into higher frequencies giving a
new model of better quality in the lower frequencies. Hence, a rened local geoid
can be expected exploiting such a combination.
3.1 The Filtering Schema
This section will discuss in details the procedure implemented to lter the gravity
data (see Fig. 3.1), which could be thought as a preparation step for the data gridding
and/or for the data combination using the LSC.
3.1.1 Downsampling of Gravity Data
Generally speaking, the resolution of the output of all methods concerned with
aerogravimetry data processing, is limited by several factors such as the sampling
rate, the altitude of the platform the collects the data, and the spatial extent of
the gravity survey. As mentioned in section 1.1.3, the classical speed of the aircraft
ranges between 180 km/h (50 m/s) and 720 km/h (200 m/s) while the commonly
used sampling frequency is 1 Hz (ωs = 1 Hz) for gravity sensors, therefore, the
spatial resolution of the collected data varies between 0.05 km (50 m) and 0.20 km
96
CHAPTER 3. ALONG-TRACK FILTERING
Figure 3.2. The procedure to compute the Reference Signal and the nal lteredsignal.
(200 m) (Hehl , 1995).
On the one hand, the classical systems of airborne gravimetry, as discussed in
section 1.1.3.1, showed that relative gravity surveys can be accomplished yield-
ing accuracy of a 2 − 3 mGal at a half−wavelength resolution of 5 km (Wei and
Schwarz , 1998). While, the strapdown systems, as in 1.1.3.2, can yield slightly
higher accuracies of about 1.5 mGal at a half−wavelength of 2 km and 2.5 mGal
at a half−wavelength of 1.4 km, demonstrating the potential of this approach for
high−resolution applications (Alberts et al., 2005).
Consequently, based on the aforementioned paragraphs, in case the user did not
specify any particular downsampling frequency of interest, the ltering algorithm
will automatically apply a downsampling rate , ωds = 1/5 Hz, on the collected data
keeping out an observation every 5 observations, reducing the spatial resolution from
50 m to 250 m.
3.1.2 Wiener Filter
At this step, we will apply the Wiener lter (Wiener , 1949), mathematically
represented in Eq. 3.1, on the downsampled observation vector, on a track−by−trackbasis. The Wiener lter, which is known to be the optimal lter in order to increase
the signal to noise ratio, will also decrease the bias and variance simultaneously as
compared to other ltering techniques (Ghael et al., 1997). Additionally, the Wiener
lter removes the systematic errors potentially present within the data, hence, it
Ahmed Hamdi Mansi 97
Airborne Gravity Field Modelling
Figure 3.3. The SH coecients of the EIGEN − 6C4 GGM.
improves the low frequencies of the downsampled observation vector (Zaroubi et al.,
1999). Finally, both the space domain and frequency domain methods were carefully
examined and the results show that the frequency domain method is superior in
estimating the covariance functions for a local area.
Wf =Ss
Ss + Sν(3.1)
Eq. 3.1: The mathematical representation of the Wiener lter, where Ss is the
power spectral density function of the downsampled signal and Sν is the power
spectral density function of the observation noise.
In order to apply the Wiener lter, the signal,SS, will be considered a known
component. This known component, SS, will be computed by exploiting a combined
satellite gravity model GGM to compute the 1D PSD function, named as the refer-
ence signal. The advantage of using the reference signal as SS is to be able to get a
proper collocation length in order to perform the ltering of the noise observations.
On the other hand, any errors exist in computing the reference signal will propagate
to generate error within the ltered signal.
On the other hand, the noisy observation vector will be used as the denominator
of Eq. 3.1, SS + Sν , as explained afterwards. Usually, the gravimetric noise, Sν ,
is computed on a static environment not on a ight mode and in such case, Sν is
considered fully known and there would be no need to perform the Wiener lter but
a much simpler lter techniques could be implemented.
Both, the signal and the observation noise power spectral density function will
be used to evaluate the covariance matrix, as explained in Chapter 4.
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CHAPTER 3. ALONG-TRACK FILTERING
The ltering software requires 2 global gravity eld models; a combined satellite
gravity model (e.g., EIGEN−6C4 (Förste et al., 2014)) and a Global Gravity Field
Models related to Topography (e.g., dV_ELL_RET2012 (Claessens and Hirt , 2013)),
while the interested user can use a dierent set of models.
Figure 3.4. The degree variances of EIGEN − 6C4 model.
3.1.2.1 The Reference Signal
The reference signal, SS, could be easily obtained through performing 2 pieces
of computations following the schema presented in Fig. 3.2. The rst piece of com-
putation is that the ltering software would implement the spherical harmonics
coecients from d/o 0 to the maximum d/o (LEIGEN−6C4max ) of the combined satellite
(EIGEN − 6C4) model that equals to 2190, as seen in Fig. 3.3 in order to synthet-
ically obtain the gravitational signal at the ight track. For better illustration, the
degree variances of the EIGEN − 6C4 model is reported in Fig. 3.4 in which the
one can realize that EIGEN − 6C4 model has a continuous contributions all over
the spectrum with a sudden drop at the high degrees. This drop is to be enhanced
using the RTC signal.
The second main computation is to evaluate twice the TC for a high resolution
DTM and a smoothed DTM computed via using a smoothing window that coincides
with the maximum spatial resolution (full wavelength) of the Earth surface obtain-
able when using the SH synthesis up to LEIGEN−6C4max that can be evaluated from
Eq. 3.2 (Lambeck , 1990).
Maximum Spatial Resolution = 40000 km/(Lmax + 0.5) (3.2)
Ahmed Hamdi Mansi 99
Airborne Gravity Field Modelling
Eq. 3.2: The maximum spatial resolution at the Earth surface.
Now, the one can compute the RTC evaluating the dierences between both the
TC of the high resolution DTM and the the smoothed DTM. Fig. 3.10 shows that
the RTC contributes mainly to the medium−to−high frequency zone with relatively
low power values if compared to the power values of the GGM. Then and there, the
RTC is added up to the synthetically evaluated gravitational signal to produce the
reference signal as seen in Fig. 3.11.
Figure 3.5. The development of the SH coecients of the model to be removed.
3.1.2.2 The Noisy Observation Signal
The airborne gravimetric observations vector that is characterized with a high
noise−to−signal ratio (NSR) will be used as (SS +Sν) (e.g., the denominator of the
Wiener lter). The spectral range of the noisy observation vector is truncated to
the same d/o range as the reference signal. From Fig. 3.9, the one can see that the
signal is not easily distinguishable due to the contaminating noise, which has a high
NSR ratio specially within the medium−to−high frequency region.
Generally speaking for airborne gravimetric data, the observations suer from
a linear drift of the instrument used to register the gravity signal as discussed in
details in Chapter 1. Also, the observations are nothing but the summation of
all the contributions of all wavelengths that might be of larger wavelengths than
the extent of the ight track that would be reected on the maximum retrievable
wavelength of the airborne gravimetric campaign. These larger wavelengths can not
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CHAPTER 3. ALONG-TRACK FILTERING
be isolated to be corrected for, therefore replacing the low frequency components
of the observed signal with the very solid and well−estimated low frequency signal
obtained from the implementation of the GGMs would enhance the observed signal.
Figure 3.6. The observation versus the reduced observation of track #204800.
3.1.2.3 The Removal-Like Step
The signal to be removed could be described as a synthetic gravitational sig-
nal computed through the implementation of the SH coecients of the 2 GGM
required by the software. This signal covers the same d/o range as the reference
signal. The removal-like step is so important as it would reduce the strength of the
constrains of the Wiener lter that forces the 1D PSD of the noisy observation to
follow the this of the reference signal and removing a big portion of the reference
signal is reected on decreasing the strength of this constrain. The superposition of
the long wavelength signal from LEIGEN−6C4min = 0 to a LEIGEN−6C4
max = 720 obtained
from the EIGEN − 6C4 model and the medium−to−short wavelength signal from
LdV_ELL_RET2012
min = 720 to a LdV_ELL_RET2012max = 2190 obtained from the dV_ELL_RET2012,
fully describe the gravitational signal to be removed (3.5) in terms of degree vari-
ances. The decision of the threshold value of the d/o that separates the low fre-
quencies from the medium−to−high frequencies (e.g., d/o 720 in our case) is a very
subjective matter, therefore the values used within this research are not compul-
sive for other researches and other values could be adequately tuned within other
applications.
Finally, the signal to be removed would be restored later in terms of any grav-
itational functionals after obtaining the ltered signal in a store−like step, thanksto the well known SH coecients of the model to be removed.
Ahmed Hamdi Mansi 101
Airborne Gravity Field Modelling
Figure 3.7. Schema of the ltering software: it computes the ltered signal fordierent tracks then it computes the nal ltered signal at all the track points byinterpolating the values computed for the dierent tracks.
3.1.2.3.1 The Reduced Reference Signal
The long wavelength contributions of both the reference signal and the signal to
be removed (Fig. 3.4) are constructed from the same EIGEN − 6C4 model from
LEIGEN−6C4min = 0 to a LEIGEN−6C4
max = 720, consequently the reduced reference sig-
nal is characterized with zero contribution within the frequency that ranges from
0 to 720. On the other hand, the reference signal has a medium−to−short wave-length contributions constructed from the EIGEN−6C4 model from LEIGEN−6C4
min =
720 to LEIGEN−6C4max = 2190 but the signal to be removed is constructed from the
dV_ELL_RET2012 model from LdV_ELL_RET2012
min = 720 to LdV_ELL_RET2012max = 2190 (the 1D
representation of the degree variances of the dV_ELL_RET2012 LEIGEN−6C4max model
is shown in Fig. 3.5), therefore the contributions of the reduced reference signal
within the medium−to−short wavelength spectrum is nothing but the pure dier-
ences of the two models in addition to the contributions of the RTC within the
medium−to−short wavelength spectrum.
3.1.2.3.2 The Reduced Noisy Observation Signal
The reduced noisy observations signal can not be described to follow a particular
behavior (as seen in Fig. 3.6) but in general the reduced noisy observation signal still
suers from a high NSR ratio and that the noise level dominates both the original
and the reduced observation signals.
One last comment to be made here, is that the the observed signal is at least a
couple of orders of magnitude higher than that of the signal to be removed, as seen
in the statistics of the following case−studies.
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CHAPTER 3. ALONG-TRACK FILTERING
3.2 The Filtered Signal
In this section we will report the ltered signal of 2 ight tracks #1040 and
#204800 and for the whole airborne gravimetric survey done within the framework
of the CarbonNet project, which was used earlier in section section 2.4.4 to perform
Test 4: TC at the Sparse Points. The ltering software allows the interested user to
compute the along−track ltered signal with a higher accuracy through computing
the ltered signal for dierent downsampled tracks then compute the nal estimate of
the ltered signal by interpolating the values computed at the dierent downsampled
tracks. Hence, at the end of the computation of the ltered signal will be computed
at each point of the ight track, see Fig. 3.7.
Figure 3.8. The gravity observations (Signal+Noise) of track #1040.
3.2.1 Case-Study 1: Filtering Short Track #1040 (Perpendic-
ular Direction)
The rst case study is done over a short ight track #1040 that has collected its
data in the perpendicular direction. It has a length of 56967 meters and its noisy
observations reported in Fig. 3.8, where it is so evident to suer very high noise
levels as seen in the 1D PSD representation of the observations (see Fig. 3.9).
On the one hand, summing up both gravitational signals of the EIGEN − 6C4
model (3.11) and the RTC signal (3.10) results in the reference signal, SS, as reported
in Fig. 3.11. The reference signal is further reduced using the signal to be removed
over track #1040 before applying the Wiener lter and obtaining the ltered signal
Ahmed Hamdi Mansi 103
Airborne Gravity Field Modelling
Figure 3.9. The 1D PSD representations of track #1040.
Min (mGal) Max (mGal) Mean (mGal) STD (mGal)Obs. Signal -3544.2846 3516.3191 -622.6788 713.9003
Filtered Signal -51.0038 -21.6939 -41.6072 8.4355EIGEN−6C4 -52.1484 -20.4117 -40.3187 10.1120RTC Signal -1.0406 1.1453 -0.0672 0.3420REF. Signal -51.9001 -20.9638 -41.6072 9.5794
dV_ELL_RET2012 -2.1160 3.2099 0.5450 1.6248Reduced Ref. -4.9760 1.5006 -1.8334 1.9861
Table 3.1. The statistics of all the signals aecting track #1040
(3.8) whose PSD function is reported in Fig. 3.11.
The statistics of all the signals involved within the ltering step are summarized
and reported in Table 3.1.
Table 3.1 shows that the constrains of the Wiener lter is evident from the statis-
tics of the reference (input) signal and the ltered (output) signal. Also, Fig. 3.12
conrms the statistics and the mathematical theory by showing that both signals
have the same shape.
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CHAPTER 3. ALONG-TRACK FILTERING
Figure 3.10. The PSD of the RTC signal of track #1040.
Figure 3.11. The 1D PSD function of all the signals over track #1040.
Ahmed Hamdi Mansi 105
Airborne Gravity Field Modelling
Figure 3.12. The Reference and Filtered signals of track #1040.
3.2.2 Case-Study 2: Filtering Long Track #204800 (Reference
Direction)
The second case study is made for the data collected over the track #204800
own in the reference direction that has a full length of 127219.5 meters. The
collected data (3.13) is characterized with a minimum, maximum, mean, and stan-
dard deviation of −2962.3896, 2860.2111,−666.6880, and 665.3906 mGal, respec-
tively and the PSD of the observation is shown in Fig. 3.14. The nal ltered
signal is characterized with a minimum, maximum, mean, and standard deviation
of −49.9667,−1.7820,−35.7970, and 14.2131 mGal, respectively. The observation
and ltered signals are graphically plotted in Fig. 3.13.
The dierent pieces of computations such as the reference signal and the reduced
reference signal (3.15) were computed in order to be able to apply the Wiener lter
on the data−vector of track #204800.
The resulted ltered signal has the same shape as the reference signal as expected
as shown in Fig. 3.16 and as happened within Case−Study 1: Filtering Short Track
#1040 (Perpendicular Direction). The conclusion made from studying the dierent
long and short lines is that the longer the track the higher the medium−to−highfrequency content of the ltered signal (compare Fig. 3.12 and Fig. 3.16) and this is
reected in the values of the standard deviation (i.e., the standard deviation values
is 8.4355 mGal for the short track#1040 versus a 13.4231 mGal for the long track
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CHAPTER 3. ALONG-TRACK FILTERING
Figure 3.13. The gravity observations (Signal+Noise) of track #204800.
Min (mGal) Max (mGal) Mean (mGal) STD (mGal)Obs. Signal -2962.3896 2860.2111 -666.6880 665.3906
Filtered Signal -49.9443 -5.1613 -35.3564 13.4231EIGEN − 6C4 -3.9107 1.3830 -0.1596 0.8686RTC Signal -53.3673 -2.4276 -35.7970 14.7815REF. Signal -49.9667 -1.7820 -35.7970 14.2131
dV_ELL_RET2012 -4.6588 4.8104 -0.2510 2.6004Reduced Ref. -4.2973 3.2300 -0.1895 2.1717
Table 3.2. The statistics of all the signals aecting track #204800
#204800 ).
The full statistics of all the signals used to obtain the ltered signal are summa-
rized in Table 3.2.
3.2.3 Case-Study 3: Filtering Full Airborne Gravimetric Sur-
vey
Within this subsection, we will present the ltered signal of the full CarbonNet
gravimetric campaign and we will demonstrate that it would become much more
clearer that using the ltering signal would immediately increase our knowledge
about the studied area.
Ahmed Hamdi Mansi 107
Airborne Gravity Field Modelling
Figure 3.14. The 1D PSD representations of track #204800.
Figure 3.15. The 1D PSD function of all the signals over track #204800.
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CHAPTER 3. ALONG-TRACK FILTERING
Figure 3.16. The Reference and Filtered signals of track #204800.
The altitude levels of the CarbonNet project are characterized with a mini-
mum, maximum, mean, and standard deviation of 148.3300, 369.5500, 185.9583, and
38.5608 meters, respectively are plotted in Fig. 3.17.
The gravimetric noisy observations of the full acquisition presented in Fig. 3.18
are characterized with a minimum, maximum, mean, and standard deviation of
−6178.5997, 5967.5731, 1043.4251, and 726.0095 mGal, respectively.
Pointing out that the acquisition has been done on−shore and o−shore, the ex-pert eye can not really distinguish/appreciate the gravity signal presented in Fig. 3.18
because of the very high levels of noise contaminating it. It is also clear that the
gravimetric signal changes from positive to negative values and vice−versa within a
singl track with no clear reason but the high noise levels aecting it.
The reference signal (3.19) has been reduced by the low frequency components
obtained from the EIGEN − 6C4 model (3.20) and the high frequency components
obtained from the dV_ELL_RET2012 model (3.21) in order to compute the reduced
reference signal (3.22).
The same reduction has been applied on the noisy observation to obtain the
reduced observations, which is not so much dierent from the original noisy data of
Fig. 3.18.
Afterwards, the Wiener lter is applied in order lter out the noise, we have ob-
tained the ltered signal (3.23) characterized with a minimum, maximum, mean, and
Ahmed Hamdi Mansi 109
Airborne Gravity Field Modelling
Figure 3.17. The altitude of the ight performed the gravity acquisition of theCarbonNet project.
Figure 3.18. The gravity observations (Signal + Noise) of the CarbonNet project.
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CHAPTER 3. ALONG-TRACK FILTERING
Figure 3.19. The reference signal of the CarbonNet project.
Figure 3.20. The EIGEN − 6C4 (low frequencies) Signal.
Ahmed Hamdi Mansi 111
Airborne Gravity Field Modelling
Figure 3.21. The dV_ELL_RET2012 LEIGEN−6C4max (high frequencies) Signal.
standard deviation of −54.5277, 4.0475,−31.6938, and 11.7934 mGal, respectively.
The full statistics of all the signals used to obtain the CarbonNet ltered signal
are reported in Table 3.3.
3.2.4 Case-Study 4: Comparison with DTU10 Model Data
The gravity signal of the DTU10 global model (Andersen, 2010) data has been
computed for the same studied region (3.24) and then it was reduced by the same
signal (signal to be removed) as the ltered data. On the one hand, the DTU
gravity signal shows a minimum, maximum, mean, and standard deviation values
of −42.6718, 113.3248, 20.9929, and 14.3512 mGal, respectively. On the other hand,
the reduced DTU gravity signal shows a minimum, maximum, mean, and standard
deviation values of −46.9551109.6468,−0.0890, and 14.6599 mGal, respectively.
Because both the ltered observations′ signal and the reduced DTU10 signal do
not have a zero mean, therefore a further reduction step to remove this drift−likesignal from both signals is essential to establish the comparison by performing a least
squares adjustment in order to estimate the parameters dening this 2D drift−likesignal. The nal reduced DTU10 signal shows a very similar behavior with a 13.5718
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CHAPTER 3. ALONG-TRACK FILTERING
Min (mGal) Max (mGal) Mean (mGal) STD (mGal)Obs. Signal -6178.5997 5967.5731 -1043.4251 726.0095
Filtered Signal -54.5277 4.0475 -31.6938 11.7934EIGEN − 6C4 -52.4951 -0.8514 -31.8081 10.8783RTC Signal -3.9239 4.2369 -0.0951 0.5629REF. Signal -53.6160 5.8113 -31.6938 12.0004
dV_ELL_RET2012 -7.1979 7.0322 -0.0597 2.2062Reduced Ref. -8.3060 10.8231 0.1739 3.1525
Table 3.3. The statistics of the CarbonNet airborne gravimetric campaign
Figure 3.22. The reduced reference signal of the CarbonNet project.
Ahmed Hamdi Mansi 113
Airborne Gravity Field Modelling
mGal standard deviation value.
Figure 3.23. The ltered signal of the CarbonNet project.
3.3 Remarks on Filtering
The ltering analyses implement gravity eld spectrum through the computation
of the spectral density function of the observed aerogravimetric signal and also from
several GGMs, local gravity data and heights would provide the necessary gravity
eld signal and the error covariance or PSD functions required for better geoid
prediction techniques.
On the one hand, the Wiener lter is a powerful tool to lter highly contaminated
signals with very high NSR ratio (e.g., the NSR is 2712.8 and 3152.9 for track
#1040 and track #204800, respectively, while the NSR is 3559.8 for the whole
CarbonNet project). On the other hand, the strength of the constrains of the Wiener
lter is reected on the resulted ltered signal, which could be further improved via
the implementation of an iterative ltering procedure introducing some information
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CHAPTER 3. ALONG-TRACK FILTERING
Figure 3.24. The DTU10 gravity signal computed for the region of the CarbonNetproject.
about the covariance matrix of the noise contaminating the observation through
performing a cross−over analysis, as explained Chapter 5.
In addition, the estimates of the data sampling density derived from the degree
variances of the gravity signal would give a better picture of the data required for
geoid estimation with sub−decimeter accuracy.
Ahmed Hamdi Mansi 115
Chapter 4
Gridding
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[When the sky is rent asunder, and it becomes red like ointment :(37) Then which
of the favours of your Lord will ye deny? (38)] [Quran, Ar−rahman]
At this point, the analyses of the gravimetric data have to go further by gridding
the ltered signal. The main aim of the gridding step is to obtain the gravitational
eld directly on a regular grid at a constant altitude. This step is generally applied
for geophysical applications due to the fact that when regular grids are available some
computations can be eciently performed in the frequency domain by exploit the
convolution theorem. Moreover, it is strictly required in geodetic applications, where
gridded geoid undulations can be used for instance to convert ellipsoidal heights into
orthometric heights. In order to have this step fullled, the one has to interpolate
all the ltered signals on a regular grid by a proper algorithm, e.g. by using a
stochastic interpolator. In this research, the LSC will be used. Section 4.1 intro-
duces the mathematics for a new methodology to compute the covariance function,
by implementing Bessel functions of the the rst order and zero degree to t the
covariance function. Consequently, the estimate of the covariance matrix could be
obtained to be used for the LSC. Section 4.2 will present the results obtained in the
framework of the CarbonNet project (CarbonNet Project Airborne Gravity Survey ,
2012; Department of Primary Industries , 2012).
The owchart shown in Fig. 4.1 represents the schema followed to perform the
gridding using the LSC. Each element within this owchart will explained in detailed
within the sequel.
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CHAPTER 4. GRIDDING
Figure 4.1. The gridding scheme.
4.1 The Mathematical Arguments
For the mathematical development of the equations of this chapter (recall the
terms used within section 1.4), let the generalized form of the the observation equa-
tion of Eq. 1.35 written as in Eq. 4.1:
yi = (li,∆g0) + νi (4.1)
Eq. 4.1: The observation equation.
By pointing out that the footprint of ∆g(xP , yP , zi) or ∆g0(xP , yP , 0) is harmonic
for zi ≥ 0, so that the problem that seems 3D is in reality nothing but a 2D problem.
The evaluation of ∆g is possible employing Eq. 1.27. Therefore, with the upper
continuation kernel stated as `P (x, y, zi) = zi2π[x2+y2+z2
P ]3/2, the observation equation
could be extended to Eq. 4.2.
(lk,∆g0) = ∆g(xP , yP , zk) = ∆g0(xP , yP , 0) ∗ lk(xP , yP , zk) (4.2)
Eq. 4.2: The extension of the observation equation.
Ahmed Hamdi Mansi 117
Airborne Gravity Field Modelling
As for the downward continuation, it should be observed that in a typical survey
such as the one from the CarbonNet project, the problem is much more simplied.
In fact only the reduced signal (which has a standard deviation of 11.7934 mGal
see Fig. 3.23 should be "moved". Moreover the observations should be downward
continued only of few hundred meters. As a result, having both the signal and the
dierence in heights as small quantities, the downward continuation could be per-
formed empirically just by computing the radial derivative of the reduced reference
signal. From the one side, the statistics of the noise vector that is assumed to be a
white noise are presented in Eq. 4.3.
Eν = 0
EννT = Cν
(4.3)
Eq. 4.3: The statistics of the noise vector.
On the other side, the statistics of the ∆g0 signal computed at zP = 0 are
presented in Eq. 4.4.
E∆g0(xP , yP , 0) = 0
E∆g0(µ, 0) ·∆g0(ξ, 0) = E∆g0(µ) ·∆g0(ξ)
= C0(|µ− ξ|)
(4.4)
Eq. 4.4: The statistics of ∆g0 vector.
Last but not least, the signal and the noise have no correlation as in Eq. 4.5.
CSν = E∆g0(xP , yP , 0) · ν = 0 (4.5)
Eq. 4.5: The independence condition between the observation and noise vectors.
Now, we want to predict the signal using Eq. 4.6.
y = (L,∆g0)
L = l(x, y, z)
(4.6)
Eq. 4.6: The equation of the predicted signal, where (x, y) is a point on the grid
with altitude z.
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CHAPTER 4. GRIDDING
4.1.1 The Formulation of the Least Squares Collocation So-
lution
Among the various techniques available to nd the solution, we would use the
Wiener˘Kolmogorov (W−K) Best Linear Unbiased Predictor (BLUP) principle (Bhansali ,2004). The advantages of using the BLUP is that it is able to estimate the random
eects of the targeted signal (Henderson C. , 1986) in addition to BLUP is shrinkage
towards the mean, which is often a desirable statistical property of an estimator, as
it increases accuracy (Hill and Rosenberger , 1985). The BLUP is explicitly explained
in Eq. 4.7.
y = λTy = (λTL,∆g0) + λTν Linear
Ey = λTEy = 0 Unbiased
λ = Argmin||ε2(λ)|| Best
(4.7)
Eq. 4.7: The BLUP principle.
The problem of minimizing the errors using the BLUP principle could be implic-
itly explained as in Eq. 4.8.
ε2(λ) = E(y − y)2
= E(L,∆g0)2 − 2E(L,∆g0)(λTL,∆g0)+ E(λTL,∆g0)2+ E(λTv)2(4.8)
Eq. 4.8: The implicit form of the BLUP principle.
With the implementation of Krarup's solution (Krarup, 1968) and using L(∆g0) =
(L,∆g0) and M(∆g0) = (M,∆g0) , then the rst term of Eq. 4.8 could be written
Ahmed Hamdi Mansi 119
Airborne Gravity Field Modelling
as follows:
E(L,∆g0)(M,∆g0) = E(L(µ),∆g0(µ))(M(µ),∆g0(µ))
= E(L(µ))(M(µ),∆g0(µ)∆g0(µ))
= (L(µ))(M(µ), E∆g0(µ)∆g0(µ))
= (L(µ))(M(µ), C0(|µ− µ|))
= (L(µ), C0(µ,M))
= C0(L,M)
(4.9)
Eq. 4.9: The development of the rst term of Eq. 4.8.
In the same manner, the second term could be written as Eq. 4.10.
E(L,∆g0)(λTL,∆g0) =∑λkE(L,∆g0)(Lk,∆g0)
=∑λkC0(L,Lk)
=∑λkC0(L)
= λTC0(L)
(4.10)
Eq. 4.10: The development of the second term of Eq. 4.8.
Furthermore, the third term could be simplied as Eq. 4.11.
E(λTL,∆g0)(λTL,∆g0) =∑
k,j λkλjE(Lk,∆g0)(Lj,∆g0)
=∑
k,j λkλjC0(Lk, Lj)
= λTC0λ
(4.11)
Eq. 4.11: The development of the third term of Eq. 4.8.
By substituting Eq. 4.9, Eq. 4.10, and Eq. 4.11 in Eq. 4.8, the BLUP solution
could be written in Eq. 4.12, which has an argmin as the solution of Eq. 4.13.
ε2(λ) = C0(L,L)− 2λTC0(L) + λTC0λ+ λTCνλ (4.12)
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CHAPTER 4. GRIDDING
Eq. 4.12: The explicit form of the BLUP principle.
Argmin||ε2(λ)|| = (C0 + Cν)λ = C0(L) (4.13)
Eq. 4.13: The argmin of Eq. 4.12.
The prediction errors can be easily evaluated via Eq. 4.14.
ε2(λ) = C0(L,L)− C0(L)T [C0 + Cν ]−1C0(L) (4.14)
Eq. 4.14: The prediction error of the BLUP.
Figure 4.2. The spectral estimate of the reduced-ltered signal.
4.1.2 The Estimation of the Covariance Matrix
The added−values of the available dierent gravimetric data to the LSC solution
is not the main focus of the current work but we will concentrate on highlighting the
Ahmed Hamdi Mansi 121
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contribution of the newly added−value from the airborne gravity data, consequently,
the covariance estimation will be estimated based on δgA. The airborne gravity can
be described by a general observation equation (Eq. 1.1) that could be written in a
similar form to Eq. 4.1 as follows:
δgA = bA + ν (4.15)
Eq. 4.15: Gravity disturbances at the ight level.
Two essential remarks must be done now before proceeding with the mathematics
of the covariance estimation. The rst remark is that the noise by hypothesis is a
homogeneous random eld with zero mean and a standard deviation equivalent to
σν . The other remark is the existence of the bias in all the gravimetric data, therefore
the airborne gravimetric data used in this section are generally biased for dierent
reasons but mainly due to the data reduction schemes used in earlier stages of the
data processing and due to cutting the data in a particular area.
Figure 4.3. The 1D PSD representation of the data.
4.1.2.1 Data Reduction
The ltered signal must be reduced for a 2D planar signal that counteracts the
existing bias and transforms the signal to a zero mean signal. Then, the challenge is
to nd the biggest inscribed rectangle inside the geometry of the airborne acquisition
to be treated as a grid. The reduced ltered signal and the selected rectangle/grid
will be used only for the estimation of the covariance.
The reduced ltered signal of Eq. 4.16 will be interpolated to the inscribed rect-
angle gridded with the same grid size of the DTM or courser. If the inscribed
rectangle is too small dierent sources, as for instance the reference model described
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CHAPTER 4. GRIDDING
in Chapter 3, itself (e.g., DTU10 model, EIGEN − 6C4, or dV_ELL_RET2012) can
be used instead.
δgredA = δgA − δgA − δg(xP , yP ) (4.16)
Eq. 4.16: The data reduction for the mean value and for the 2D gravitational
signal, where δg(xP , yP ) = ax+ by + c, knowing that c = 0 due to the removal of
the average airborne gravitational ltered signal δgA.
4.1.2.2 The Spectral vs. PSD Analysis
Then, we would perform the spectral estimation for the data because this process
can be automatized. This estimate is achievable by applying a 2D FFT for the data,
as explained in Eq. 4.17.
δgredA = FFT (δgredA ) (4.17)
Eq. 4.17: The spectral estimation for the reduced ltered gravitational signal.
The next step is to move from the 2D spectral representation (Fig. 4.2) to 1D
PSD (Fig. 4.3) representation of the data by performing an averaging of the data as
a function of P and dP , satisfying the condition of Eq. 4.18, so that we obtain an
estimate for the empirical 1D spectrum. The advantage of this averaging step is to
clean any residual unltered noise.
Sest(P ) = |δgredA (P )|2 for P ≤ |P | ≤ P + dP (4.18)
Eq. 4.18: The averaging scheme of the spectral data.
4.1.2.3 The Covariance Function
At this point, the one targets a better modelling for the empirical 1D spectrum by
nding a way to best t it. This operation can be fullled by using the well−knowncovariance functions/models (such as the Bilinear, Circular, Spherical, Gaussian,
Whittle, Exponential models, etc.) or by performing a classical spline interpolation.
For this research, we will use a set of Bessel functions of the rst kind , Jn(x), (see
Fig. 4.4) in order to best t the empirical spectrum so that the computations of the
Henkel−Fourier transformation, discussed in the sequel, becomes much easier.
Knowing that the 1D PSD values can be mathematically computed exploiting
Eq. 4.19, which can be rearranged in order to have a formulation for the S0(P )
term. the one must check that Eq. 4.20 does not explode before proceeding with the
computation.
Sest,H(P ) = e−4πPH · S0P +σ2
0
P+ ∆2
(4.19)
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Figure 4.4. The graphical representation of Bessel functions of the rst kind.
Eq. 4.19: The mathematical representation to compute the 1D PSD value, as H is
the mean height of the grid, ∆ is the spacing of X-axis for the 1D PSD representation,
and σ20 = σ2
ν
2π∆.
S0(P ) = e4πPH(Sest,H(P )− σ20
P+∆2
) for P ≤ Pcut−off (4.20)
Eq. 4.20: The mathematical condition to compute the S0(P ) value, as Pcut−off is
the maximum frequency where there is no signicant signal beyond it.
Now, we have to compute the inverse FFT of the S0(P ) in order to be able to
compute the covariance function in terms of distance as in Eq. 4.21, which is true if
and only if Eq. 4.22 is a real function.
C0(r) = FFT−1(S0(P )) (4.21)
Eq. 4.21: The analytical form of the covariance function.
S0(P ) = FFT (C0(r)) for P = |P | (4.22)
Eq. 4.22: The relationship between S0(P ), as a FFT of the covariance function
C0(r).
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CHAPTER 4. GRIDDING
In order to perform these FFT and inverse FFT, we must implement the Henkel−Fouriertransformation.
4.1.2.3.1 The Henkel-Fourier Transformation
The use of Bessel function of the zero order, J0(x), allows us to write Eq. 4.20
as:
S0(P ) =
∫ ∞0
2πJ0(2πρr)C(r)rdr
=
∫ ∞0
J0(ρr)C(r)rdr
(4.23)
Eq. 4.23: The analytical form of S0(P ) in terms of J0(x).
Therefore, the inverse of the Henkel−Fourier transform of Eq. 4.21 can be gotten
from Eq. 4.24 through the implementation of the special formulas of Watson (1966)
that we recall in Eq. 4.25 and Eq. 4.26.
C0(r) =
∫ ∞0
2πJ0(2πPr)S0(P )PdP
=
∫ ∞0
J0(Pr)S0(P )PdP
(4.24)
Eq. 4.24: The inverse of the Henkel−Fourier transform of Eq. 4.21.
J0(Z) = 12π
∫ 2π
0
e−iZ sin θdθ
= 12π
∫ 2π
0
eiZcosθdθ
(4.25)
Eq. 4.25: Bessel function as an integral over a sphere.
∫ ∞0
J0(2πPr)J1(2πP r)
rrdr =
0 when P > P
14πP
when P = P1
2πPwhen P < P
(4.26)
Eq. 4.26: Special formula for Bessel function.
Eq. 4.26 can be rewritten in the shape of Eq. 4.23 in order to get an idea about
the shape of the output of the Henkel−Fourier transformation, using J0(P · r) =
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J0(2π · P · r) and J1(P · r) = J1(2π · P · r) as follow:
P2π
∫∞0
2πJ0(2πPr)2πJ1(2πP r)r
rdr
= P2π
∫∞0J0(Pr) J1(P r)
rrdr
= FFT P2π· J1(P r)
r =
0 when P > P1
1 when P < P1
(4.27)
Eq. 4.27: The basic solution of the Henkel−Fourier transformation.
Therefore,
FFT P2
2π· J1(P2r)
r− P1
2π· J1(P1r)
r =
0 when P > P2
1 when P1 < P < P2
0 when P < P1
= χP1,P2,∆(P )
(4.28)
Eq. 4.28: The FFT of the Bessel functions, as ∆ = P2 − P1.
Accordingly, we can apply the inverse FFT for both sides of the equation, which
allows us to get the following expression:
FFT−1χP1,P2,∆(P ) =P2
2π· J1(P2r)
r− P1
2π· J1(P1r)
r
(4.29)
Eq. 4.29: The IFFT.
Consequently, we obtain the following approximated expression for Eq. 4.20:
S0(P ) =n∑k=0
Sk · χP1,P2,∆(P ) (4.30)
Eq. 4.30: The mathematical expression for the estimate of S0(P ).
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CHAPTER 4. GRIDDING
As a result, Eq. 4.21 could be elaborated to get its solution as follows:
C0(r) = FFT−1(∑n
k=0 Sk · χk,k+1,∆(P ))
=∑n
k=0 Sk(k + 1)∆
2π· J1((k + 1)∆r)
r− k∆
2π· J1(k∆r)
r
= S0∆2π· J1(∆r)
r+∑n−1
k=1(Sk − Sk−1)(k + 1)∆
2π· J1((k + 1)∆r)
r
+Sn(n+ 1)∆
2π· J1((n+ 1)∆r)
r
(4.31)
Eq. 4.31: The mathematical expression for the calculations of C0(r).
Eq. 4.31 is nothing more than a linear summation of the n Bessel function′s
coecients used to t the covariance function we obtained from averaging the 2D
spectrum along the radius.
4.1.2.4 The Covariance Matrix
Recalling that the Molodensky concept expresses the computations of the grav-
itational anomaly outside the masses as convolution integrals between the gravity
anomaly observed on the surface of the mass and the upper continuation kernel,
as discussed previously. Consequently, we could extend elaborating Eq. 4.2 by im-
plementing the formulas reported in Eq. 1.28 and Eq. 1.29 to obtain the following
formula:(Lk,∆g0) = ∆g0(xP , yP , 0) ∗ lk(xP , yP , zk) = G0(P ) ∗ Lk
=∫∫G0(P ) ∗ Lkd2P
=∫∫G0(P ) ∗ e−2πzkPd2P
(4.32)
Eq. 4.32: The Molodensky concept (convolution integrals).
Consequently,
C0(Lk, Lj) = E(∫∫G0(P ) ∗ e−2πzkPd2P )(
∫∫G0(Q) ∗ e−2πzjQd2Q)
= E∫∫G0(P )G0(Q) ∗ e−2π(zkP−zjQ)d2Pd2Q
=∫∫e−2π(zkP−zjQ)d2Pd2Q · EG0(P )G0(Q)
(4.33)
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Airborne Gravity Field Modelling
Eq. 4.33: The mathematical representation of the covariance matrix.
Where, the rst term of Eq. 4.33 can be simplied to the following formula:
EG0(P )G0(Q) = E∫∫ei2π(Pµ−Qξ)∆g0(µ) ·∆g0(ξ)d2µd2ξ
=∫∫ei2π(Pµ−Qξ)d2µd2ξ · E∆g0(µ) ·∆g0(ξ)
=∫∫ei2π(Pµ−Qξ)d2µd2ξ · C0(|µ− ξ|) let τ = ξ − µ
=∫∫ei2π(P (ξ−τ)−Qξ)d2(τ)d2ξ · C0(|τ |)
=∫∫ei2πPτei2π(P−Q)ξd2(τ)d2ξ · C0(|τ |)
=∫d2ξ∫ei2πPτei2π(P−Q)ξd2(τ)d2ξ · C0(|τ |)
= S0(P )∫ei2π(P−Q)ξd2ξ
= S0(P )δ(P −Q)
(4.34)
Eq. 4.34: The rst term of Eq. 4.33.
By substituting Eq. 4.34 into Eq. 4.33, the one can formulate a nal expression
to be exploited to build the covariance matrix, as in Eq. 4.35.
C0(Lk, Lj) =∫∫e−2π(zkP−zjQ)d2Pd2Q · EG0(P )G0(Q)
=∫∫e−2π(zkP−zjQ)d2Pd2Q · S0(P )δ(P −Q)
=∫e−2πP (zk−zj)d2P · S0(P )
(4.35)
Eq. 4.35: The nal expression of the covariance matrix.
4.2 The CarbonNet Case-Study
The biggest inscribed rectangle inside the geometry of the CarbonNet airborne
gravimetric data has been selected, then the values of the grid point has been com-
puted via a linear interpolation of the reduced data. Because of the small size of
this biggest inscribed rectangle, the resulted 2D spectral estimation that was used to
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CHAPTER 4. GRIDDING
Figure 4.5. The 2D spectral estimation of the reduced observations.
build the 1D covariance function did not show a good performance as the correlation
length obtained was not so reliable. Therefore, the DTU10 data has been used.
At this step, we applied the 2D FFT on the values of grid nodes of the DTU10
model in order to perform the 2D spectral estimation, shown in Fig. 4.5. Simply,
the 2D spectral estimation is computed by evaluating the multiplication of the 2d
FFT of the signal and its complex conjugate.
By averaging the 2D spectral values, we can move from the 2D into the 1D rep-
resentation of the data as seen in Fig. 4.6. The one can realize that the empirical
covariance function is characterized with an oscillating tail that has a correlation
length of the order of 10 kilometers, therefore, a mutation step has been introduced
in order to have a tail that goes to zero at a correlation length higher than 150 kilo-
meters that is the maximum length of a single ight track. The empirical covariance
function was interpolated using a set of n Bessel functions of the rst order and zero
degree, as shown in Fig. 4.6.
The tting curve that uses the n coecients of the Bessel function matches the
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Airborne Gravity Field Modelling
inclining part of the theoretical covariance in a good manner, and both covariances
show to have a correlation length of around 10 kilometers at half − C0.
A few remarks must be pointed out about the advantages of using the Bessel
function as a base function to re-represent any other arbitrary function, in our
case the empirical covariance function. First of all, it allows us to t almost every
known−functions with just few parameters if compared to the splines interpolator.
The main other advantage is the simplicity of extrapolate the the empirical covari-
ance, as this is done in terms of a linear superposition of the few coecients used
to dene and map the Bessel function. Last and most important, the estimated
covariance matrix will be a positive denite matrix, consequently, its inversion does
exist, which is reected on reaching a LSC solution.
Figure 4.6. The 1D empirical Covariance ([red]) and the theoretical Covariance ([blue]) by tting the empirical Covariance with set of Bessel functions.
Now, we are ready to compute the covariance matrix exploiting the linear su-
perposition of the coecients of the Bessel function obtained in the previous step
exploiting the discrete form of Eq. 4.35 as reported in Eq. 4.36.
C0(Li, Lj) =n∑k=1
akJ0(Pkr)
r(4.36)
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CHAPTER 4. GRIDDING
Eq. 4.36: The discrete expression of the covariance matrix, where ak is the nth
coecient of the Bessel function, Pk is the nth parameter that equals 2πPk, and r
is the Euclidean distance between the observation points i and j.
The o−diagonal entities of the observation−observation covariance matrix (Cyy)
is estimated by evaluating Eq. 4.36 depleting the distance matrix of the observations
that is a symmetric matrix with zeros along the main diagonal. Because Eq. 4.36
explodes for the main diagonal therefore we can not use it. Instead, we would use
the variance of the signal exploiting the information of the empirical covariance
function. Now, the square observation−observation covariance matrix (Cyy) is fully
estimated, as reported in Eq. 4.37.
CSig,Sig Cobs1,obs2 Cobs1,obs3 . . . Cobs1,obsn−2 Cobs1,obsn−1 Cobs1,obsn
Cobs2,obs1 CSig,Sig Cobs2,obs3 . . . Cobs2,obsn−2 Cobs2,obsn−1 Cobs2,obsn
Cobs3,obs1 Cobs3,obs2 CSig,Sig . . . Cobs3,obsn−2 Cobs3,obsn−1 Cobs3,obsn...
......
. . ....
......
Cobsn−2,obs1 Cobsn−2,obs2 Cobsn−2,obs3 . . . CSig,Sig Cobsn−2,obsn−1 Cobsn−2,obsn
Cobsn−1,obs1 Cobsn−1,obs2 Cobsn−1,obs3 . . . Cobsn−1,obsn−2 CSig,Sig Cobsn−1,obsn
Cobsn,obs1 Cobsn,obs2 Cobsn,obs3 . . . Cobsn,obsn−2 Cobsn,obsn−1 CSig,Sig
(4.37)
Eq. 4.37: The general shape of the observation−observation covariance matrix
(Cyy).
While the rectangular grid−observation covariance matrix Cxy is obtained using
the distance matrix between the observations and the prediction grid nodes, as seen
in Eq. 4.38.
Cobs1,grd1 Cobs1,grd2 Cobs1,grd3 . . . Cobs1,grdm−2 Cobs1,grdm−1 Cobs1,grdm
Cobs2,grd1 Cobs2,grd2 Cobs2,grd3 . . . Cobs2,grdm−2 Cobs2,grdm−1 Cobs2,grdm
Cobs3,grd1 Cobs3,grd2 Cobs3,grd3 . . . Cobs3,grdm−2 Cobs3,grdm−1 Cobs3,grdm...
......
. . ....
......
Cobsn−2,grd1 Cobsn−2,grd2 Cobsn−2,grd3 . . . Cobsn−2,grdm−2 Cobsn−2,grdm−1 Cobsn−2,grdm
Cobsn−1,grd1 Cobsn−1,grd2 Cobsn−1,grd3 . . . Cobsn−1,grdm−2 Cobsn−1,grdm−1 Cobsn−1,grdm
Cobsn,grd1 Cobsn,grd2 Cobsn,grd3 . . . Cobsn,grdm−2 Cobsn,grdm−1 Cobsn,grdm
(4.38)
Eq. 4.38: The general shape of the grid−observation covariance matrix Cxy .
The noise covariance matrix (Cνν) is estimated by assigning the value of the
observational noise variance (σ2νobs
) to the diagonal location coinciding with the ob-
servations and assigning the value of the observational noise variance (σ2νDTU10
) to
Ahmed Hamdi Mansi 131
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Figure 4.7. The nal gridded data.
the diagonal location coinciding with the points where DTU10 GGM data have been
used. The noise matrix is characterized by being a diagonal matrix with zero value
everywhere but the main diagonal entities, as shown in Eq. 4.39.
σ2νobs
0 0 0 0 0
0 σ2νobs
0 0 0 0
0 0 σ2νobs
0 0 0
0 0 0 σ2νobs
0 0
0 0 0 0 σ2νDTU10
0
0 0 0 0 0 σ2νDTU10
(4.39)
Eq. 4.39: The general shape of the noise covariance matrix Cνν .
The Gridding software allows the user to dene dierent values for the observa-
tional noise variance (σ2νobs
) and the GGM noise variance (σ2νDTU10
) than the values
adopted for this research, which equal to 1 and 9 mGal2, respectively.
To obtain the nal grid, the observation vector is in general downsampled.In fact
airborne surveys usually acquire up to 1: 2 million raw observations that cannot be
contemporary used for the LSC solution. In order to exploit the full dataset, we
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CHAPTER 4. GRIDDING
Figure 4.8. The prediction error associated with the nal gridded signal.
apply the following strategy: rst of all, we downsample the observations from the
original size to about 40000 data, which is more or less our actual limit considering
the computational power available, after that we compute several grids changing
the input downsampled data. The nal grid is obtained just by averaging all the
computed solutions.
The nal grid, shown in Fig. 4.7, was achieved by averaging 3 dierent LSC
solutions, which we obtained by adopting a downsampling frequency ωds = 1/100
Hz, each. Then, we have restored the signal that we removed during the remove−likestep of the ltering algorithm. Also, the prediction error associated with the nal
gridded signal is reported in Fig. 4.8. On the one hand, it is clear that the error
values are minimal where that data are collected while on the other hand, the error
values explode and show high values where there is no data collected.
4.2.1 Comparison between the Dierent Grids
Here, we will compared the dierent nal−grids (without applying the restore−likestep), which we obtained by elaborating dierent numbers of LSC solutions (i.e. 1
and 3 grids), which is calculated through adopting dierent downsampling frequency
Ahmed Hamdi Mansi 133
Airborne Gravity Field Modelling
Figure 4.9. The reduced gridded signal obtained using 6743 observations and 3LSC solutions.
for the observation vector (i.e. ωds = 1/100, 1/50, and1/10 Hz).
4.2.1.1 Comparison 1: 1 Grid Vs. 3 Grids
The rst grid that would be considered as the reference grid for the dierent
comparisons was obtained with a downsampling frequency, ωds = 1/100 Hz, there-
fore, using only a vector of 6743 observations out of the 440000 observations of the
CarbonNet dataset for the LSC prediction in order to compute 3 dierent reduced
grids, which were averaged in order to obtain the nal reduced grid (e.g., the ref-
erence grid). The averaging step can also demolish and reduce the impact of any
residual errors generated from the dierent approximations done earlier such as the
simple mapping, the simplied downward continuation, . . . etc.
Here the airborne data have been integrated with the DTU10 data. The ref-
erence grid plotted in Fig. 4.9 is characterized with a minimum, maximum, mean,
and standard deviation of −11.6351, 9.8681,−0.0254, and 3.3448 mGal, respectively.
While the prediction error adopting a single LSC solution is shown in Fig. 4.10.
The prediction error has a minimum, maximum, mean, and standard deviation of
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CHAPTER 4. GRIDDING
Figure 4.10. The prediction error of the reduced gridded signal obtained using6743 observations and 3 LSC solutions.
1.8461, 4.9598, 2.1901, and 0.3101 mGal, respectively.
The one can see that the prediction error has the minimum value located at the
center of the gravity campaign and it increases slowly within the area covered with
data, then it changes rapidly beyond the border of the CarbonNet project where the
values increase dramatically.
The second grid was obtained with the same downsampling frequency, therefore
using 6743 observations to perform a single LSC prediction. The reduced gridded
signal is characterized with a minimum, maximum, mean, and standard deviation
of −11.6363, 9.8661,−0.0246, and 3.3438 mGal, respectively. While the prediction
error adopting a single LSC solution has a minimum, maximum, mean, and standard
deviation of 1.8461, 4.9598, 2.1901, and 0.3101 mGal, respectively.
In order to make it easy to appreciate the very small dierences of the compared
grids, we will present directly the plots of the dierences. Fig. 4.11 shows the dier-
ences of the gridded signals that have a minimum, maximum, mean, and standard
deviation of −0.1578, 0.0359,−0.0008, and 0.0159 mGal, respectively, which reects
that the footprints of the gridded signals are very similar. In the research that we
Ahmed Hamdi Mansi 135
Airborne Gravity Field Modelling
Figure 4.11. The dierence of the reduced gridded (3 LSC solutions single LSCsolution).
performed to identify the source of the big dierences, we found that this area co-
incides with a big anomaly of the Gippsland Basin (it is named the Central Deep
region) and an o−shore site for Oil/Gas production (Brien et al., 2008).
Fig. 4.12 reports the very small dierences of the prediction error (with 10−3
mGal order of magnitude) that have a minimum, maximum, mean, and standard
deviation of −0.0011, 0.0017, 0, and 0.0001 mGal, respectively, which means that
the prediction error values are almost identical because they are below the precision
of the gravimetric acquisition. The one can see 2 adjacent spots with relatively
anomalous values with respect to their neighborhood, they coincide with the region
with the big dierences of the gridded values of Fig. 4.11.
4.2.1.2 Comparison 2: Downsampling Frequency 1/100 Vs. 1/50
In this section, we will compare the reference grid with the average signal of
the 3 dierent reduced LSC solutions executed for the observation vector, down-
sampled with ωds = 1/50 Hz, therefore using 10786 observations instead of 6743
for the reference grid. On the one hand, Fig. 4.13 reports the dierences of the
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CHAPTER 4. GRIDDING
Figure 4.12. The dierence of the prediction error (3 LSC solutions single LSCsolution).
gridded signals that have a minimum, maximum, mean, and standard deviation
of −1.9469, 1.3900, 0.0224, and 0.2638 mGal, respectively. The small values of the
mean and the standard deviation of the dierences reect the similar footprints of
both gridded signals. While, the deviation of the minimum and the maximum values
of the dierences from zero could be explained by the new data introduced for the
new LSC solutions (e.g., additional 4000 observations corresponding to 60% of the
data used for the reference grids) that were not available due to the high sampling
frequency, ωds = 1/100 Hz.
On the other hand, Fig. 4.14 shows the dierences of the prediction error, which
have a minimum, maximum, mean, and standard deviation of−0.0007, 0.1516, 0.0165,
and 0.0148 mGal, respectively. The one can realize that the prediction error within
the central region, where the data were collected, does not show evident improve-
ments. While the extra data, introduced along the border of the region, specially
along the Northern face/edge and the Southern−West corner, impact the prediction
error and improves the results, which is reected in having the maximum values
correspond to the Southern−West corner.
Ahmed Hamdi Mansi 137
Airborne Gravity Field Modelling
Figure 4.13. The dierence of the reduced gridded (ωds = 1/100 ωds = 1/50) Hz.
Figure 4.14. The dierence of the prediction error (ωds = 1/100 ωds = 1/50) Hz.
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CHAPTER 4. GRIDDING
Figure 4.15. The dierence of the reduced gridded (ωds = 1/100 ωds = 1/10) Hz.
4.2.1.3 Comparison 3: Downsampling Frequency 1/100 Vs. 1/10
The dierences of the reference signal and that obtained adopting a sampling fre-
quency, ωds = 1/10 Hz, shown in Fig. 4.15, are characterized with a minimum, max-
imum, mean, and standard deviation of −8.7415, 6.1873, 0.1182, and 1.2963 mGal,
respectively. While the dierences of the prediction error, reported in Fig. 4.16,
are characterized with a minimum, maximum, mean, and standard deviation of
0.0024, 0.5474, 0.0476, and 0.0462 mGal, respectively.
In the same manner, similar to section 4.2.1.2, the new data are introduced
mainly along the borders of the region, therefore, the improvements in terms of
gravitational disturbance, as shown in Fig. 4.17, are located along the borders and
beyond. These improvements are clear if the dierences of the gridded signals and
the prediction error are reported simultaneously.
4.3 Remarks on Gridding
From the rst comparison, it is clear that using few data and computing many
LSC solutions does not impact the nal computed grid. Also, computing many
LSC solutions and averaging it, is reected in having a homogenized nal grid. The
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Airborne Gravity Field Modelling
Figure 4.16. The dierence of the prediction error (ωds = 1/100 ωds = 1/10) Hz.
Figure 4.17. The improvements in terms of gravity disturbances are located wherethe new data are introduced (i.e., on the border of the gravimetric campaign andbeyond).
140
CHAPTER 4. GRIDDING
second and the third comparisons showed that the more the data utilized to nd the
LSC solution, the better the obtained results would be.
Because of the limitation of the software used to invert the covariance matrix
within the LSC step, several solutions, which exploit the full observation vector must
be computed. Also, a homogeneously−distributed downsampled observations vector
might be selected for the LSC step for better results.
Ahmed Hamdi Mansi 141
Chapter 5
The Cross-Over Analysis
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This chapter will discuss the analyses of the cross−over, i.e. the intersection
between two perpendicular tracks (see Fig. 5.1), pointing out that in general the al-
titude dierences between the two−intersecting tracks have been kept as minimum
as possible while performing the aerogravimetric survey. Then, the point-wise eval-
uations of the observation error have been estimated by computing the dierence of
the observations at the intersection point of the two dierent tracks.
Note that these errors are not only due to the gravimeter observation error, but
also the results of the imperfection of the whole procedure applied to estimate the
nal gravity eld, such as mis−modelling in the Eötvös eect or in the computation
of the aircraft position.
The cross−over study is done mainly to empirically estimate the stochastic char-
acteristics of the error of the ltered signal with the aim of obtaining a realistic
estimate of its covariance function. The results of the cross−over study give us the
capability to distinguish the nature of the contaminating noise (e.g., White or Col-
ored noise) over the classically adopted assumption of having a pure White noise.
Therefore, geophysicists can take advantage of the cross−over output while interpo-lating the anomalous maps to understand the nature of the hot−spots if they are
signal or just the eect of colored noise.
142
CHAPTER 5. THE CROSS-OVER ANALYSIS
Figure 5.1. The graphical explanation of the cross−over of the ight−tracks.
5.1 Flight Tracks Modelling
At the beginning of the cross−over study, each track is modelled as a 3D line
in the space−domain using a 3D least square procedure to minimize the root mean
square dierences between the coordinates of the actual points occupied by the
airborne during the survey and the corresponding modelled points (Fig. 5.2). In
order to reduce the number and the time of the computations, the direction that
has less numbers of tracks is analyzed on a track−by−track basis, hereafter named
as the reference direction.
5.1.1 Intersection Point Computations
In order to better determine the real intersection points, i.e. the point where the
tracks intersect, a two−step procedure has been followed. The rst step is met by
performing a projection of the 3D−modeled track−lines on the 2D plane, then the
intersection points of each track of the reference direction with all the perpendicular
tracks are computed, as shown in Fig. 5.3.
The second step is fullled by moving back to the real tracks domain by per-
forming a perpendicular projection of each computed intersection point on the real
track, then a last renement is done to improve the determination of the intersection
Ahmed Hamdi Mansi 143
Airborne Gravity Field Modelling
Figure 5.2. The 3D original and modeled ight−tracks projected in the 2D space.
Figure 5.3. The intersections of all the ight−tracks projected in the 2D space.
144
CHAPTER 5. THE CROSS-OVER ANALYSIS
Figure 5.4. The results of the 12−cycles renement procedure, the [green lines]represent the actual ight tracks, the [blue lines] represent the 3D LS estimated linesprojected into the 2D space, the [black stars] are the initial intersection points, the[red stars] are the intermediately calculated intersection points, the [black circle] isthe nal intersection point.
points. The renement is done by performing a cyclic projection of the computed
intersection point from one track to the other perpendicular one. By testing the
renement script, a loop of at least 12 cycles showed a 100% convergence behavior
for obtaining the real intersection point, see Fig. 5.4.
5.1.2 Estimation of the Noise Covariance
Within this section, we will elaborate the data to nd the correlation between the
2 values of the gravimetric disturbance measured at each intersection point aiming
to form an estimate for the noise/error covariance function then implement it to be
able to compute the covariance matrix of the noise. The targeted covariance matrix
is expected to enrich our knowledge about the noise error, therefore it could be
used in the LSC discussed in Chapter 4. The empirical covariance function will be
estimated and then it would be modelled as a series of n Bessel functions of the rst
order and zero degree (Watson, 1995). The major dierence between the technique
applied here and the one applied within section 4.1.2.3 is that the coecients will
be estimated by means of a non−negative Least Squares adjustment (Lawson and
Hanson, 1974) in order to obtain, in an easy and in an automatic way, a positive
denite theoretical covariance function with oscillating tail.
Ahmed Hamdi Mansi 145
Airborne Gravity Field Modelling
Figure 5.5. The Empirical covariance function Cνν(d) for the CarbonNet data.
The covariance matrix of the noise will be then evaluated and consequently, the
data could be interpreted on the basis of the results of using the niose covariance
matrix, Cνν .
5.2 Case-Study: The CarbonNet Project
The cross−over study has been carried out using the data collected for the Car-
bonNet project where the airborne gravimetric data have been elaborated as dis-
cussed in Chapter 4. Then, the intersection points were estimated then the dier-
ences of the observations have been computed at the intersection points. Finally,
by adopting the assumption to have a homogeneous and isotropic observation error,
the empirical covariance function, Cνν(d), as seen in Fig. 5.5, has been estimated,
where (d) as the planar distance between any arbitrary couple of points.
5.2.1 The Realization of the Cross-Over Noise
The one can see that the covariance of the error is 2.4 mGal2 with a correlation
length of 2.5 km, approximately. The observed noise on the ight tracks is shown
in Fig. 5.6, where it is so evident that the cross−over exists and it contaminates the
track signal with order of magnitude that might be closer to that of the reference grid
of Fig. 4.9. In order to better visualize the eect of the noise, once the covariance
146
CHAPTER 5. THE CROSS-OVER ANALYSIS
Figure 5.6. The realization of the noise on the CarbonNet tracks (mGal).
function is computed from the crossover analysis, it is possible to simulate, on a given
area, a possible realization of the noise. This can be simply done by computing the
covariance matrix Cνν in correspondence of the simulated points. The noise, ν, can
be simulated as ν = L · u (Franklin, 1965; Demeure and Scharf , 1987): where L is
the lower triangular matrix from Cholesky decomposition of Cνν and u is a vector
containing one random extraction from a normal distribution for each observation
points.
The realization made for the cross−over noise over the same grid where the LSC
solution was computed, is shown in Fig. 5.7. The gridded noise is characterized with
a minimum, maximum, mean, and standard deviation of −4.9237, 6.2014, 0.4744,
and 1.4101 mGal, respectively.
From the values the observational noise variance σ2νobs
of 2.4mGal2, the one can
expect that using a standard deviation for the noise σνobs = 2.04 mGal is more
adequate than the value σνobs = 1 mGal used earlier for the LSC.
Ahmed Hamdi Mansi 147
Airborne Gravity Field Modelling
Figure 5.7. The realization of the noise on the CarbonNet grid (mGal).
5.3 Remarks on the Cross-Over Analysis
In order to obtain better results from the gridding analysis (Chapter 4), the
cross−over analysis could be done on a track−by−track basis to estimate the em-
pirical covariance function of the cross−over noise for each track, then we can com-
pute the 1D PSD of the noise, Sν . Subsequently, integrating the estimated empirical
covariance function of the ltered signal (after applying the Restore−like step), theone can build a more realistic PSD of the signal SS and for the reference signal, SS,
instead of counting on the 1D PSD from the GGM. Using the iterative procedure
described in Fig. 5.8, we can exploit the results of the LSC solutions, as described
within this dissertation, we can update the PSD of the noise, Sν , the signal SS, and
the reference signal, SS, in order to perform a new LSC solution. The one should
expect the solution of the LSC to converge after few trials and that the solution to
be more realistic as we got rid of the strong constrains of the Wiener lter that was
amply discussed in (Chapter 3).
148
CHAPTER 5. THE CROSS-OVER ANALYSIS
Figure 5.8. The iterative procedure.
Ahmed Hamdi Mansi 149
Chapter 6
Geoid Determination
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Indeed, He is Acquainted with that which you do. (88)] [Quran, An−naml]
The geoid determination from the CarbonNet project was not planned at the
beginning of this research, the geoid is simply achieved by applying the Stokes′
integral in order to transform the grids of the gravimetric disturbances (mGal) into
geoid heights (m). The eect of the Stokes′ integral is nothing more than applying
a kernel that transforms gravity data into geodetic heights.
The theory and the mathematical equations related to the the Stokes′ integral
were explicitly discussed in section 1.5.2, consequently, we will directly display and
discuss the results.
6.1 Case−Study : The CarbonNet Project
Fig. 6.1 reports the computed geoid heights (in meters) that is, as expected, a
smooth eld, which is characterized with a minimum, maximum, mean, and standard
deviation of 2.4610, 11.3515, 6.0654, and 2.3770 m, respectively. The computed geoid
has a gentle slope from the North to the South. On the contrary, the high frequency
component that is easily identiable on the Northern region is most probably due
to the border eects of applying the Stokes′ kernel.
On the other hand, Fig. 6.2 shows the errors of estimating the geoid heights,
which looks similar to the errors of the gridded data used as input for the Stokes′
integral. The statistics of the geoid estimation error have a minimum, maximum,
150
CHAPTER 6. GEOID DETERMINATION
h!
Figure 6.1. The computed CarbonNet−based geoid heights.
h!
Figure 6.2. The error associated to the estimation of the CarbonNet−based geoidheights.
Ahmed Hamdi Mansi 151
Airborne Gravity Field Modelling
mean, and standard deviation of 0.0755, 0.11345, 0.0917, and 0.0091 m, respectively.
It shows low error values where the data were collected and the error values increase
while moving away from the center of the surveyed region.
Figure 6.3. The geoid dierences between the CarbonNet-based geoid and theocial AUSGEOID09.
6.1.1 Geoid Comparison
The ocial Australian geoid, AUSGEOID09, over the Southern Australian region
has been used to compute the geoid over the CarbonNet region through a classical
bi−cubic interpolation. The dierences between the computed geoid based on the
collected airborne gravimetric data and the ocial AUSGEOID09 values are shown
in Fig. 6.3. The geoid dierences have a non zero mean of 0.1798 m that is caused
by the height datum issues of the Australian geoid and the long−wavelengths of thegeoid that are not retrievable by such gravimetric campaign over a limited area (i.e.,
the CarbonNet data have been collected locally not regionally) .While the standard
deviation value of 0.0694 m is so close to the value of standard deviation of the
errors. Also, the border eects that contaminate the geoid results is reected in
the standard deviation value. the maximum value is 0.5548 m coincides with the
152
CHAPTER 6. GEOID DETERMINATION
Northern-West corner that is highly aected by the border eects.
Ahmed Hamdi Mansi 153
Chapter 7
Discussion and Conclusion
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[They ask thee concerning the Spirit (of inspiration). Say: "The Spirit (cometh) by
command of my Lord: of knowledge it is only a little that is communicated to you,
(O men!). (85)] [Quran, Al−Isra]
In Chapter 1, a literature review has been done, where we explained the dierent
types of gravity data (e.g., Land, Marine, Airborne, and Satellite gravity data)
and the main equipments used to acquire gravimetric data and their accuracies,
giving more attention to the airborne garvimetry. Then, the classical data reduction
schema using the classical Remove−Compute−Restore procedure has been discussedenlightening the dierent modelling techniques implemented to compute the terrain
correction with high accuracy (i.e., Point−mass, Prism, Tesseroid, Polyhedral, and
FFT models).
The classical processing of airborne gravity data requires data reduction through
computing the terrain and residual terrain corrections, then data ltering, followed
by the downward continuation, and nally to perform a LSC in order to repre-
sent the data in a grid format that is a must in case the computation of the geoid
computation is required through the evaluation of the Stokes′ integral. In practice,
each single computation involved within the classical processing is very time expen-
sive. Therefore, the work documented within this dissertation has introduced new
techniques that are characterized with being fast and accurate.
In Chapter 2, we introduced the theory of a new hybrid prism and FFT−basedsoftware, named as Gravity Terrain Eects (GTE), from the mathematical point of
view. Here, the expression of the Newtonian volume integral of the gravitational
potential (Eq. 1.18) has been elaborated to be in the form of two terms (Eq. 2.15),
154
CHAPTER 7. DISCUSSION AND CONCLUSION
the former representing the eects of the planar approximation, and the latter is the
spherical correction term. Similarly, the computations of the gravitational eects of
the topographic masses, known as the Terrain Correction, of Eq. 1.26 was mathe-
matically elaborated into a nal expression that consists of two terms, each could
be written as a convolution integral. The planar terrain correction, δgPt , evaluates
the major part of the topography eects while the spherical correction term, δgSCt ,
proved to compute a signal that is at least 3 orders of magnitude less than that of
the planar integral.
Starting from the theory, a new software (called GTE as well) has been imple-
mented. It allows the user to compute the terrain corrections on grids of constant
heights, sparse points, and on the DTM surface. It also permits to compute not
only the eects of the topography but also the eects of the bathymetry, the sedi-
ment layers, and the Moho. The accuracy of the computation (and as a consequence
the computational power required) can be managed by the user who can select be-
tween dierent proles. Using a single node of a supercomputer equipped with two
8 − cores Intel Haswell 2.40 GHz processors (for a total of 16 cores) with 128 GB
RAM, the slow GTE proler needs approximately 2.5 minutes to compute the eects
of a 351×301 DTM on a grid of the same size at a constant height of 3500 m, which
is almost 3.5 times faster than the performance of the GRAVSOFT package (Fors-
berg , 2003) and 150 times faster than Tesseroids (Uieda et al., 2011). Although GTE
and GRAVSOFT require almost the same time (GRAVSOFT is slightly slower) to
compute the eects on sparse points, the standard deviation of the dierence of
the output and the reference prism solution are 0.14 and 0.31 mGal, for GTE and
GRAVSOFT, respectively (in case of a realistic airborne acquisition of almost 440000
sparse points). Tesseroids is 70 times slower, while its results are accurate with a
standard deviation of the dierence below 0.1 mGal for the dierences. While GTE
computes the terrain corrections on the surface of the digital elevation model in
about 10 seconds with dierences smaller than 10−3 mGal with respect to the pure
prism solutions, GRAVSOFT showed to be 60 times slower, while Tesseroids cannot
be used since its solution became unstable as the observation points are very close
to the masses.
In Chapter 3, we tackled the challenge of ltering the gravimetric data collected
from a dynamic platform, which is characterized with a very high noise−to−signalratio through applying a Wiener lter in the frequency domain on a track−by−trackbasis so that the whole dataset could be utilized. In order to use such a lter, we
exploited the power spectral density functions of the reference signal and the noisy
observation vector. The reference signal was computed by means of summing the low
Ahmed Hamdi Mansi 155
Airborne Gravity Field Modelling
to medium frequency contributions obtained from the synthesis of a global gravity
model (i.e., the EIGEN−6C4 (Förste et al., 2014)) and adding a residual correction
term. Before applying the Wiener lter, we performed a remove−like step to elimi-
nate the low frequencies from the EIGEN − 6C4 model and the medium−to−highfrequencies from dV_ELL_RET2012 (Claessens and Hirt , 2013). The remove−likestep was done in a smart way considering the rst and the last observations of each
track, and downsample the inbetween observations. This step helped to minimize
the errors of the ltering. Because the ltered signal and the ight heights are
small quantities, beside that the observations should be downward continued for
few hundred meters, the downward continuation was performed empirically just by
computing the radial derivative of the reduced reference signal. The Wiener lter
enhances the results by substituting the low frequencies of the airborne acquisition
with the well−observed low frequencies of GOCE mission. Filtering the CarbonNet
dataset required almost 7 minutes when the TC and RTC were available and 30
minutes to compute all of the TC and RTC, and to lter the signal.
Chapter 4 introduced a new mathematical tool to be used for the evaluation of
the covariance matrix by using a series n Bessel functions of the rst order and zero
degree that assures to gain a positive denite covariance matrix with existing inverse,
and consequently allowing us to automatize the covariance estimation process. Using
several downsampled ltered signals as input for the LSC, we computed several
grids that were averaged in order to calculate the nal grid. If the size of the
biggest inscribed rectangle inside the geometry of the airborne acquisition is large,
the ltered data will be used to compute the empirical covariance function otherwise,
the data of a GGM reduced to the same signals used within the ltering step would be
utilized (e.g., the DTU10 (Andersen, 2010)). The results illustrated that utilizing
smaller downsampling step and compute few numbers of grids (e.g., using ωds =
1/5 or 1/10 Hz to compute 5: 10 grids) is better than using bigger downsampling
step and compute large numbers of grids (e.g., using ωds = 1/50 or 1/100 Hz to
compute 50]colon100 grids). The area where the data were acquired exhibited small
values for the prediction error, while the areas where the GGM model was used had
high prediction error values. The more the data used, the better the resulted grids
and the lower the estimated prediction errors.
In Chapter 5, we discussed how to perform the cross−over analysis that improves
our knowledge about the noise. e.g., in the case of the CarbonNet project, the
realization of the expected noise illuminated a standard deviation of 2.04 mGal. The
impact of the cross−over analyses could be high if it would be integrated within the
ltering technique explained in Chapter 3 through an iterative procedure to yield
156
CHAPTER 7. DISCUSSION AND CONCLUSION
the best grids of the airborne gravimetric data.
To conclude, the work done within this dissertation suits processing of airborne
gravimetric data, ltering the noisy observations collected from a dynamic platform
in order to perform a LSC to build the grids of the ltered data and its prediction
errors.
Ahmed Hamdi Mansi 157
Chapter 8
Recommendations and Future Work
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(physical properties), and without subjecting it to any external (unbalanced)
eects that force the body to have a certain shape or state, therefore, the body's
physical properties make the body keep and hold its current state, knowing that
resistance to a body is not just the eect of another body but it is whatsoever does
not make the body to keep its state] Ibn Sina (980-1037) [The (signs) signals and
the (patterns) approaches]
A follow−up work for the research documented within the content of this Ph.D.
dissertation is so much recommended. It is recommended to:
• Compare the nal grid computed from the airborne gravity data acquired
within the framework of the CarbonNet project with the nal grid computed
using ground (on−shore) data;
• Compute the nal grid exploiting dierent GGM within the ltering module
and compare the results;
• Evaluate the dierences of the computed nal grids and assess them;
• Implement the cross−over study within the LSC prediction and assess the
impact of such an implementation on the resulted gridded data;
158
CHAPTER 8. RECOMMENDATIONS AND FUTURE WORK
• Assess the performance of the MATlab function that performs the Stokes'
integral via elaborating the milestone data of the GRAVSOFT Tutorials for
the area of New Mexico region;
• Study the height datum problem that exists within the Australian geoid;
• Study the cause of the big dierences between the ocial AUSGEOID09 and
the one computed from the DTU10 data;
• Compare the obtained results with local and regional gravity−based geoids;
• Compare the computed geoid heights the latest available Australian geoid
model.
Ahmed Hamdi Mansi 159
Chapter 9
Appendix A
Here is a complete list of the dierent reference ellipsoids and their geometrical
parameters and a further comparison is done with respect to the most common and
widely used reference ellipsoid WGS84. The rst column represents the name of
the ellipsoid, the second column represents the semi−major axis (equatorial radius
a), the third column reports the reciprocal attening (1/f), the fourth and the fth
columns are dierences of the semi−major axis values and of the reciprocal atten-
ing values of the pointed reference ellipsoid with respect to WGS84, respectively
(Defense Mapping Agency , 1987b).
160
CHAPTER 9. APPENDIX A
Ellipsoidname
Semi−majoraxis (a)
Reciprocalof attening(1/f)
a− aWGS84 ((1/f) −(1/f)WGS84)×104
Airy 1830 6377563.396 299.324964600 573.604 0.119600230
AustralianNational
6378160.000 298.250000000 -23.000 -0.000812040
Bessel1841 6377397.155 299.152812800 739.845 0.100374830Bessel1841(Nambia)
6377483.865 299.152812800 653.135 0.100374830
Clarke 1866 6378206.400 294.978698200 - 69.400 -0.372646390Clarke 1880 6378249.145 293.465000000 -112.145 - 0.547507140Everest 6377276.345 300.801700000 860.655 0.283613680Fischer 1960(Mercury)
6378166.000 298.300000000 -29.000 0.004807950
Fischer 1968 6378150.000 298.300000000 -13.000 0.004807950GRS 1967 6378160.000 298.247167427 -23.000 -0.001130480GRS 1980 6378137.000 298.257222101 0.000 -0.000000160Helmert 1906 6378200.000 298.300000000 -63.000 0.004807950Hough 6378270.000 297.000000000 -133.000 -0.141927020International1924
6378388.000 297.000000000 -251.000 -0.141927020
Krassovsky 6378245.000 298.300000000 -108.000 0.004807950Modied Airy 6377340.189 299.324964600 796.811 0.119600230ModiedEverest
6377304.063 300.801700000 832.937 0.283613680
Modied Fis-cher 1960
6378155.000 298.300000000 -18.000 0.004807950
South Ameri-can 1969
6378160.000 298.250000000 -23.000 -0.000812040
WGS 60 6378165.000 298.300000000 -28.000 0.004807950WGS 66 6378145.000 298.250000000 -8.000 -0.000812040WGS 72 6378135.000 298.260000000 2.000 0.0003121057WGS 1984 6378137.000 298.257223563 0.000 0.000000000
Table 9.1. Full list of the reference ellipsoids and their geometrical parameters
Ahmed Hamdi Mansi 161
Chapter 10
Appendix B
162
CHAPTER 10. APPENDIX B
Name FormulasModel l = Ax+ s+ ν
l Observation vectorA Design matrixx Parameter vectorl Signal vectors Observation vectors Signals to be predictedν Noise vectort = (s, s)T
Covariance functions Css = E(ssT )Cνν = E(ννT )Cνs = CT
sl = E(lsT )Cll = E(llT ) = Css + Cνν
Assumptions E(s) = E(ν) = E(sνT ) = E(tνT )=0E(l) = Ax
Minimum principle tTC−1tt t+ νTC−1
νν ν = minSolutions x = (ATC−1
ll A)−1ATC−1ll )
s = CssC−1ll (l − Ax)
s = CssC−1ll (l − Ax)
ν = CννC−1ll (l − Ax)
Error covariances x = (ATC−1ll A)−1
s = Css − CssC−1ll (I − ATC−1
ll A)−1ATC−1ll )Css
ν = Css − CssC−1ll (I − ATC−1
ll A)−1ATC−1ll )Css
Table 10.1. Details of dierent GGM combinations.
Ahmed Hamdi Mansi 163
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