AIRBORNE GRAVITY FIELD MODELLING by Ahmed … · AIRBORNE GRAVITY FIELD MODELLING by Ahmed Hamdi...

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


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


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


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


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



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


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-



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


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:



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


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).



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


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



. . . 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


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



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


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



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


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 ,



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


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



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


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.



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


Figure 1.4. The Spherical coordinates of the computation point P (r, ϑ, λ) and theintegral point P (r, ϑ, λ).



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


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)



Figure 1.6. The tesseroid representation in the spherical coordinates system.

46


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



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


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).



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


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



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


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



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


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



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


∆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.



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

Ñº

Êª

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


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.



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


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



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


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



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


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

.



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


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)



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


(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.



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.

72


I2 = µ2

∫d2x

s2QR2

∫ HQ

0

∂∂hQ

1LPQ

dhQ

∼= µ

∫d2x

s2Q2R2 ( 1

LPQ− 1

LPQ0)

(2.21)


I3 = µ2R

∫d2x

∫ HQ

0

s2PQL3PQdhQ

∼= µ2

∫d2x

HQR

s2PQL3PQ0

(2.22)


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)


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)



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


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.



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


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.



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


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.



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

80


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



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.

82


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



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


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



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.

86


Figure 2.10. The Digital Terrain Model used for the rst test.



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

88


Figure 2.11. TC computed with the SLOW prole and its dierences with respectto the gravitational eects computed by means of dierent proles/software.



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).

90


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



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


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.



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


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



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.

98


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)



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

100


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.



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.

102


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



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.

104


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.



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

106


Figure 3.13. The gravity observations (Signal+Noise) of track #204800.


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.



Figure 3.14. The 1D PSD representations of track #204800.

Figure 3.15. The 1D PSD function of all the signals over track #204800.

108


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



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.

110


Figure 3.19. The reference signal of the CarbonNet project.

Figure 3.20. The EIGEN − 6C4 (low frequencies) Signal.



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

112



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.



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

114


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.


Chapter 4

Gridding

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à

AK.

Y

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K A

Ò

º

K.

P ZB@

ø

AJ.

¯ (37)

à

Aë

YËA

¿

èX P

ð

I

KA

¾

¯

ZA

Ò

Ë@

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@ @

XA

¯]

[ áÔgQË@

[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.

116

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.



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.

118

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



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)

120

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



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

122

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)



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).

124

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) =



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 ).

126

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)



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

128

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



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)

130

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



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

132

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



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

134

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



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

136

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.



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.

138

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



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.


Chapter 5

The Cross-Over Analysis

[

HA« PAJË @

èPñ] [ (31) A

ëA

« Q

Ó

ð A

ë

ZA

Ó A

î

DÓ

h.

Q

k

@ (30) A

ëA

g

X

½Ë

X

Y

ª

K.

P

B@ð]

[And the Earth, moreover, hath He extended (to a wide expanse); (30) He draweth

out therefrom its water and its pasturage (31)] [Quran, An−nazia′t]

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



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


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.



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


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.



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


Figure 5.8. The iterative procedure.


Chapter 6

Geoid Determination

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[And you see the mountains, thinking them rigid (rmly xed), while they will

pass as the passing of clouds. [It is] the work of Allah, who perfected all things.

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.



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.


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



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.


Chapter 8

Recommendations and Future Work

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[It has been observed that if a body is left to its (body-related) characteristics

(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.


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


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.


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