Airborne gravity field modelling

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(1)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.

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(3) 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.

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(5) Abstract Regional gravity eld modelling by means of remove-restore procedure is nowadays widely applied in dierent contexts, by geodesists and geophysicists : for instance, 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 practical implementation are explained. GTE showed high performances in computing accurate terrain corrections in a very short time with respect to GRAVSOFT and Tesseroids. 5.

(6) 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 ltering the 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 collocation. 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 performing 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−over analysis within an iterative procedure of ltering and gridding would result in yielding 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.

(7) Acknowledgments Alhamdulillah, all praises to Allah for His blessings in completing this dissertation 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 familyin-law, Abd-Allah, Osama, Ahmed, Fatema, and Iman. 7.

(8) Airborne Gravity Field Modelling Graduate studies has been a wonderful experience for me. It has allowed me to learn, to travel to faraway places, and perhaps most importantly, to make some lasting friendships. I would like to express appreciation for the support I have received over the duration of my PhD journey to:. • Past and present members of the Geomatics team of Politecnico di Milano for encouraging and supporting me; • Past and present members of the Geomatics team of University of Calgary for making my stay at University of Calgary a pleasant and joyful experience; • Victoria Sendureva, Serap Çevirgen, Slobodan Miljatovic, Davide, and Matteo Bianchi; • Carla, Gerardo, and Graziella; • Caroline Minguez-Cunningham for your friendship; • My best friends: Philip Ghaly, Saber El-Sayed, Ahmed Sayed, Asmaa Sayed, Osama Saleh, Khalid Hassan, Ahmed Shanawany, Ahmed Said, Mohammed Said, Mohammed Ali, Okil Mohammed, Mahmoud Serag, Mohammed Reda, Hani El Kadi, Wael El Sawy, Sayed Salah, Mahmoud El-Sayed, and Ramy Basta; • Dina Said, Iliana Tsali, Elaheh Mokhtari, Dimitrios Piretzidis, Babak Amjadiparvar, 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.

(9) 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.

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(11) Declaration I declare that this is my original work and my genuine research.. 11.

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(13) 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. 1.1.4 1.2. 1.3. 1.4. 1.1.3.1. Classical Airborne Gravimetry. . . . . . . . . . . . . 28. 1.1.3.2. Strapdown Airborne Gravimetry . . . . . . . . . . . 28. Satellite data . . . . . . . . . . . . . . . . . . . . . . . . . . . 29. 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. 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. Downward Continuation . . . . . . . . . . . . . . . . . . . . . . . . . 49 13.

(14) Airborne Gravity Field Modelling. 1.5. 1.4.1. The Molodensky Concept . . . . . . . . . . . . . . . . . . . . 50. 1.4.2. Free−Air Downward Continuation . . . . . . . . . . . . . . . . 51. 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−Noisy Data . . . . . . . . . . . . . . . . . . . . . . 55. 1.5.1.1.2 1.5.1.2 1.5.2. Covariance Estimation . . . . . . . . . . . . . . . . . 56. The Stokes0 s Integral . . . . . . . . . . . . . . . . . . . . . . . 56 1.5.2.1. Planar Approximation of Stokes0 s Integral . . . . . . 58. 1.5.2.2. Spherical Approximation of Stokes0 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. 2.2.2 2.3. 2.4. 2.5. First Order Spherical Correction . . . . . . . . . . . 65. The Spherical Corrections . . . . . . . . . . . . . . . . . . . . 70. 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. 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. Remarks on GTE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93. 3 Along-Track Filtering 3.1. 95. The Filtering Schema . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 3.1.1. Downsampling of Gravity Data . . . . . . . . . . . . . . . . . 96. 3.1.2. Wiener Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 3.1.2.1. 14. Least−Squares Collocation for Noisy Data . 55. The Reference Signal . . . . . . . . . . . . . . . . . . 99.

(15) CONTENTS. 3.2. 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. 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 Direction) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106. 3.3. 3.2.3. Case-Study 3: Filtering Full Airborne Gravimetric Survey . . 107. 3.2.4. Case-Study 4: Comparison with DTU10 Model Data . . . . . 112. Remarks on Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . 114. 4 Gridding 4.1. 116. 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 4.1.2.3.1. 4.1.2.4 4.2. . . . . . . . . . . . . . . . 123. The Henkel-Fourier Transformation . . . . . 125. The Covariance Matrix . . . . . . . . . . . . . . . . . 127. 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 5.1. 5.2. Flight Tracks Modelling . . . . . . . . . . . . . . . . . . . . . . . . . 143 5.1.1. Intersection Point Computations. 5.1.2. Estimation of the Noise Covariance . . . . . . . . . . . . . . . 145. . . . . . . . . . . . . . . . . 143. Case-Study: The CarbonNet Project . . . . . . . . . . . . . . . . . . 146 5.2.1. 5.3. 142. The Realization of the Cross-Over Noise . . . . . . . . . . . . 146. Remarks on the Cross-Over Analysis . . . . . . . . . . . . . . . . . . 148 Ahmed Hamdi Mansi. 15.

(16) Airborne Gravity Field Modelling. 6 Geoid Determination 6.1. 150. 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.

(17) 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 ` λ) ` . . . . . . . . . . . . . . . . . . . . . . . . . . 43 integral point P` (` r, ϑ,. 1.5. Sketch map of the denition of the prism (after. 1.6. The tesseroid representation in the spherical coordinates system. . . . 46. 1.7. The geometric conventions used in the expression of the gravitational. Nagy et al. (2000)). . 44. 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¯ is the height of the grid above the reference imum depth of Bw , H surface where we want to compute δg .. . . . . . . . . . . . . . . . . . 83. 2.10 The Digital Terrain Model used for the rst test.. . . . . . . . . . . . 87 17.

(18) Airborne Gravity Field Modelling 2.11 TC computed with the SLOW prole and its dierences with respect to the gravitational eects computed by means of dierent proles/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. 3.21 The dV_ELL_RET2012. EIGEN −6C4 Lmax. . . . . . . . . . . . . . 111. (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 18. The gridding scheme. . . . . . . . . . . . . . . . . . . . . . . . . . . . 117.

(19) 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. . . . 143. 5.2. 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).. 5.7. The realization of the noise on the CarbonNet grid (mGal). . . . . . . 148. 5.8. The iterative procedure. . . . . . . . . . . . . . . . . . . . . . . . . . 149 Ahmed Hamdi Mansi. . . . . 147. 19.

(20) Airborne Gravity Field Modelling 6.1. The computed CarbonNet−based geoid heights. . . . . . . . . . . . . 151. 6.2. The error associated to the estimation of the CarbonNet−based geoid heights. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151. 6.3. The geoid dierences between the CarbonNet-based geoid and the ocial AUSGEOID09. . . . . . . . . . . . . . . . . . . . . . . . . . . 152. 20.

(21) List of Tables 1.1. Summery of acceleration terms. . . . . . . . . . . . . . . . . . . . . . 33. 1.2. Examples of dierent reference ellipsoids and their geometrical parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 different proles and software tested. VERY 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 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.

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(23) Chapter 1 Classical Processing of Gravitational Data

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(25) Ï [(47) àñ P@ YË@] [ HAK ª ñÜ. AK@ ð. ð] YK AK . AëA J J K . ZAÒ Ë@. [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 computations 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. Generally speaking, the Earth0 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.

(26) Airborne Gravity Field Modelling Up to the late twenties, pendulum was essential instrument for acquiring absolute 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 gravimeter 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 absolute gravimeter. Because of the very weak nature of the gravity forces, the gravitational 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 gravitational 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.

(27) 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 traditionally 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 performing 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 submarines 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 submaAhmed Hamdi Mansi. 25.

(28) Airborne Gravity Field Modelling rine gravimetry due to its reduced cost, reliable measurements, and in recent years, the high accuracy achieved. With the steady advancement of the technology, the capabilities to mute the acceleration of the ship, and the dierent contributions, modications, and adaptations made on the original LaCoste & Romberg marine gravimeter, the current state of marine gravimetry has been achieved (Hildebrand. et al., 1990; Zumberge et al., 1997; Sasagawa et al., 2003).. On the one hand, the great capabilities to sail in a non−rough sea states, the steady slow−motion of the ship and the installed gravity platform, the technological tools to average the data over very large intervals, a better gain in the accuracy and higher resolutions were achieved. In additional the majority of marine gravity data are collected in conjunction with other expensive survey methods such as seismic surveys, EM surveys, Remotely Operated Vehicles (ROV) projects, multi−beam bathymetric 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−discipline enhanced 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 limited. The only technique available to bridge the gap in spatial resolution between satellites and ground−based gravimeters is airborne gravimetry, i.e., the measurement of the gravitational eld signal using gravimeters installed on board of aircrafts. 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. Lundberg0 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 real26.

(29) CHAPTER 1. CLASSICAL PROCESSING OF GRAVITATIONAL DATA. Figure 1.2.. The airborne gravity measurement schema.. izing the present−day airborne gravity system (Schwarz , 1980; Brozena. Fonberg , 1993).. et al., 1988;. The early 1960s witnessed a successful attempt to collect gravity data from a xed−wing aircraft (Gumert , 1998), while the rst successful trial acquiring gravity measurements 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 better follow the terrain, the abilities to y at a low altitude that increases the spatial resolution, and the fact that a helicopter is less aected by turbulent conditions than most other types of aircraft (Lee. et al., 2006). A schematic layout for the. airborne gravity measurements is shown in Fig. 1.2. The classical airborne gravity system could easily collect the data with 0.5 to 1 mGal accuracy through integrating the observations made with many dierent sensors and systems installed within one aerogravimetric platform, such as: Ahmed Hamdi Mansi. 27.

(30) Airborne Gravity Field Modelling 1. Gravity sensor system that comprises the airborne gravimeter and the containing 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 28. et al., 1991). The.

(31) CHAPTER 1. CLASSICAL PROCESSING OF GRAVITATIONAL DATA. Figure 1.3.. The aircraft orientation layout.. strapdown inertial navigation systems do not have a gravimeter on board but it does have an IMU that is xed on the plane. With certain software, the oscillations and the acceleration of the aircraft would be computed and then used in order to lter the collected data. The main advantages of this system over the traditional one are its smaller size and relatively low cost (Wei. and Schwarz , 1998) in additional to it has shown that it can reach the same level of accuracy (Bruton et al., 2002) and that the full gravity vector can be obtained (Jekeli , 1994) and that it has the potential to increase the spatial resolution in the future (Alberts et al., 2008).. 1.1.4 Satellite data CHAMP (CHAllenging Minisatellite Payload) is a small German low-Earthorbiting satellite mission for geoscientic and atmospheric research and applications. CHAMP that operated from July 2000 to September 2010 had generated simultaneously 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 mission, a new long−wavelength global gravity eld model, called EIGEN−1S, has been derived solely from analysis of satellite orbit perturbations (Reigber Ahmed Hamdi Mansi. et al., 2002). 29.

(32) Airborne Gravity Field Modelling The Gravity Recovery and Climate Experiment (GRACE) mission by NASA was launched in March of 2002. The GRACE mission is accurately mapping variations in the Earth0 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 Earth0 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 Earth0 s gravity eld with unprecedented accuracy. The GRACE follow−on mission scheduled for 2017 will continue the work of monitoring the Earth (for more details, see (Adam , 2002;. Aguirre-Martinez and Sneeuw ,. 2003)). Integrating the data between CHAMP and GRACE has been done to produce 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 stationary 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 accuracy (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,. 1.2. Mayer-Gürr et al. (2015)).. 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 computed 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 vibration 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 acceler30.

(33) CHAPTER 1. CLASSICAL PROCESSING OF GRAVITATIONAL DATA ation contributions (Dehlinger. et al., 1966). On the contrary, the Eötvös correction. is well known as the change of the centrifugal force of the earth rotation due to measuring the gravity from a moving platform (Glicken , 1962). Moreover, when the airborne gravity platform is not strictly parallel to the level surface that does not only aect the gravitational acceleration but also impact on the vertical component of the horizontal acceleration known as the inclination to the horizontal acceleration correction (Lu. et al., 2014). The lever arm eect happens physically when the Iner-. tial Measurement Unit (IMU) of an INS, the GPS antennas, and the odometers are not located at the same position. The lever arm is the distance between the sensing points of the sensors (Seo. et al., 2006).. Later on, the manipulation of the airborne gravitational data can be done on two phases, the former is the so-called the preprocessing phase and the latter is the data inversion phase. The preprocessing phase consists in many steps such as low−pass ltering, 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−over analysis 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 disturbances by the gravimeter launched on board of an aerogravity platform must be distinguished from the non−gravitational accelerations. Therefore, the accelerations measured or derived from the GPS data are used in order to produce uncorrelated gravity measurements. Also, knowing that the gravimeter attached to the platform 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−o in order to correct the raw data observed. However nowadays due to the improvements in the gravity eld modelling from satellite data, the use of absolute ground gravimeter can be avoided. In the presence work, for instance, as reference eld a global model containing CHAMP, GRACE and GOCE satellite data will be used. This will not only improve the low frequency of the retrieved gravitational eld, but also assure the global consistency of the obtained results. Finally, any instantaneous deviation from the ideal measuring layout such as instrumentation drift, o−level Ahmed Hamdi Mansi. 31.

(34) Airborne Gravity Field Modelling . . . and/or tilt of the platform must be corrected. In order to proceed, one must point out the basic observation equation to recover the gravity disturbances at the ight altitude using a stabilized platform system (Eq. 1.1): (1.1). δg = gm − z¨ + E o¨tvös + tilt − gm0 + ga − γh 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, E o¨tvös is the Eötvös correction, tilt 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 Newton0 s law of motion as in Eq. 1.2:. ~a =. d2 ~ r dt2. r + 2~ω d~ + dt. d~ ω dt. (1.2). × ~r + ω ~ ×ω ~ × ~r. 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 considered. 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 components (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 Earth0 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.

(35) CHAPTER 1. CLASSICAL PROCESSING OF GRAVITATIONAL DATA Acceleration component. X axis (East). Y axis (North). Z axis (Vertical). Coriolis component. 2νN ωEarth sin λ νE νN tan λ R. 2νE ωEarth sin λ νE2 tan λ R. 2νE ωEarth cos λ νE2 + νN2 tan λ R 2 R g0 (R + H)2. Centrifugal component Gravitational component. 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): 2. (1.3). R g = g0 (R+H) 2. 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) and to demonstrate the large impact of the navigation parameters0 errors on it.. Harlan. (1968) expressed mathematically the Eötvös correction as reported in Eq. 1.4 as follows:. E o¨tvös =. ν2 a. 1−. h a. − ε(1 − cos2 ϕ(3 − 2 sin2 α)) + 2ν ωEarth cos ϕ sin α. (1.4). Eq. 1.4: The Eötvös correction, where ν = νE + νN with ν is the aircraft speed, νE and νN the East and the North components of the aircraft speed, a is the semi-major axis of the reference ellipsoid, h is the altitude of the aircraft, ωEarth is the angular velocity of the Earth, ϕ and α are the latitude and the azimuth angles of the aircraft, with ε =. ν2 a. · sin2 ϕ + νωEarth .. 1.2.3 Vertical Acceleration Correction When the vertical axis of the platform of the aircraft is deviated and misaligned from the instantaneous vertical vector, errors due to the horizontal accelerations contaminate the collected airborne gravitational data (Lu. et al., 2014). Aircraft. vertical accelerations for airborne gravimetry have been determined using radar and Ahmed Hamdi Mansi. 33.

(36) Airborne Gravity Field Modelling pressure altimeters (Brozena. et al., 1986), laser altimeters (Bower and Halpenny , 1987), and most recently GPS (Brozena et al., 1988). Because the vertical accelera-. tion due to aircraft motion is inseparable from the gravitational acceleration sensed by the installed gravimeter, the navigation data must be utilized in order to derive independent estimate of the vertical acceleration of the platform or to estimate the o−level angle of the platform horizontal acceleration, in order to obtain a correction 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 such as 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).tilt = 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). (1.6). tilt = g(1 − cos θx . cos θy ). 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.. tilt =. q. 2 gaccelerometer + A2 − a2 − ggravimeter. (1.7). with A2 = (A2X + A2L ) and a2 = (a2E + a2N ) 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.

(37) CHAPTER 1. CLASSICAL PROCESSING OF GRAVITATIONAL DATA East and North directions derived from the navigation (GPS) data, and ggravimeter is the gravimeter observed data.. tilt =. A2 −a2 2ggravimeter. (1.8). with A2 = (A2X + A2L ) and a2 = (a2E + a2N ) Eq. 1.8: The approximate tilt correction, where AX and AL are the along the cross and long axis output of the accelerometer, aE and aN are the horizontal acceleration in East and North directions derived from the navigation (GPS) data, and ggravimeter is the gravimeter observed data.. 1.2.4 Lever Arm Eect Physically, the Inertial Measurement Unit (IMU) of an INS, the GPS antennas, and the odometers cannot be located at the same position thus generating what is called the lever arm eect. The result of the lever arm eect is seen in the dierence between the vertical accelerations computed from the GPS data and those observed directly by the gravimeter. In order to end−up with accurate navigation data, a compensation for the eect of lever arm that could be dened as the horizontal distance between the sensing points of the dierent sensors must be utilized (Seo. et. al., 2006). While Olesen (2002) recommend neglecting the lever arm eect for scalar gravity if it is below 1 meter, De Saint-Jean et al. (2007) advised to accurately model. it for vector gravity. Similar to the lever arms of the GPS antennas−gravimeter, lever arms of the INS and altimeter instruments must be corrected, if present. All adjustments are made to the location of the gravimeter, which better be located close to the center of gravity of the aircraft (Hong. et al., 2006).. 1.2.5 Low−Pass Filter Airborne gravity measurements are characterized by its low signal-to-noise ratio as the measurements are collected in a very dynamic environment. A typical value for the noise−to−signal ratio of 1000 or higher could be easily found (Schwarz. and Li ,. 1997). This high noise−to−signal ratio contaminating the raw sensor measurements makes the extraction of the gravitational disturbances a hard and a challenging task. Low-pass ltering is an essential processing step that is applied to the acceleration data in order to separate the high frequency receiver measurement noise from the low frequency acceleration data. On the one hand, to design the optimum lter Ahmed Hamdi Mansi. 35.

(38) Airborne Gravity Field Modelling for airborne use, we must determine the gravity signal waveband. Therefore, the optimum lter must imply that:. • It does not aect or distort the low frequency content of the acceleration signal obtained by airborne gravimetric surveys, as narrowing the transition band of the lter produce distortion to the characteristics of the low frequency acceleration signal; • It lters out the high frequency noise contaminating the gravity observations (Peters and Brozena , 1995). The main advantage of using the low−pass lter is the easy design and implementation of such lter. A secondary advantage of using the low−pass lter is that the band−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 parameters 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−o frequency, ωcut−of f asf ollows :. ( H(ω) =. 1 0. if 0 < ω ≤ ωcut−of f if ωcut−of f < 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−of f 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 the data 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.

(39) CHAPTER 1. CLASSICAL PROCESSING OF GRAVITATIONAL DATA lter in the frequency domain. In particular the lter is obtaining by studying end exploiting the spatial correlation of the gravitational eld along each airplane track. Further details on this approach will be discussed in Chapter 3.. 1.3. Remove−Compute−Restore. The classical Remove−Compute−Restore (RCR) technique is the most adopted and applied technique for regional gravimetric geoid determination (Schwarz. et al.,. 1990). The RCR is composed by three essential steps; the rst step namely, Remove, targets the removal of the long−wavelength contributions utilizing the maximum degree or a truncated spherical harmonics expansion of a certain global geopotential model (GGM) exploiting either a satellite−only GGM or a combined GGM (Abbak. et al., 2012). In addition to, the removal of the short−to−medium wavelength contributions of the topography existing above the geoid, this computation is called terrain correction (TC). The last piece of the removal step is to compute the residual terrain correction (RTC) and remove it from the observed signal in order to suppress the short−wavelength contributions. The data model can be explained as reported in Eq. 1.10:. ∆gred = ∆g − ∆gGGM − ∆gT C − ∆gRT C. (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,. ∆gT C is the terrain correction gravimetric signal representing the short−to−medium wavelength contributions, and ∆gRT C 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 Stokes0 integral (Stokes , 1849), while evaluating the Stokes0 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 Stokes0 integral (i.e., the most important formula in physical geodesy), the solution of the boundary value problem in the potential theory permitting the determination of the geoid undulation from gravimetric data (Heiskanen and Moritz , Ahmed Hamdi Mansi. 37.

(40) Airborne Gravity Field Modelling 1967) as follows:. N∆gred =. R 4πγ. ZZ (∆gred + gM olodensky ) S(ψ) dσ. (1.11). Eq. 1.11: The Stokes0 integral, where R is the mean Earth radius, γ is the normal gravity on the reference ellipsoid, ∆gred is the reduced gravity anomaly signal,. gM olodensky 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 expansion gM olodensky would be insignicant (Forsberg could be ignored (Schwarz. and Sideris , 1989) and therefore. et al., 1990). The Stokes0 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) =. ∞ X. 2n+1 P (cos ψ) n−1 n. (1.12). n=2. Eq. 1.12: The Stokes0 function, where n is the spherical harmonic degree, and. Pn (cos ψ) 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 (∆gT C ) and the RTC signal in terms of terms of geoid undulation as seen in Eq. 1.13.. ˆ = NGGM + N∆g + Nindirect N red. (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.

(41) CHAPTER 1. CLASSICAL PROCESSING OF GRAVITATIONAL DATA. 1.3.1 Global Geopotential Model (GGM) Simply, a global geopotential model could be dened as the mathematical approximation 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 statistics 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 functional of the potential eld such as geoid undulation, gravity anomalies, deection 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 various 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 = ∆gT C + γEllipsoid. (1.14). Eq. 1.14: The GGM signal, as γEllipsoid = γEllipsoid (P ) + δghP is the gravitational signal of the reference ellipsoid that contains long−wavelength contributions at point P and δghP is the corresponding height correction. Due to the advancement of geodesy and the consequent improvements to the dening parameters of the reference ellipsoid (for instance see Table 1.2, and for the Ahmed Hamdi Mansi. 39.

(42) Airborne Gravity Field Modelling Ellipsoid name Airy 1830 Helmert 1906 International 1924 Australian National GRS 1967 GRS 1980 WGS 1984. Table 1.2. eters. Semi−major axis (a) 6377563.396 6378200.000 6378388.000 6378160.000 6378160.000 6378137.000 6378137.000. Reciprocal of attening (1/f ) 299.324964600 298.300000000 297.000000000 298.250000000 298.247167427 298.257222101 298.257223563. Examples of dierent reference ellipsoids and their geometrical param-. 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.. γEllipsoid P = 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 computations 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 represented 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 ) =. GM R. LX max. l X. ( Rr )l+1 Vn,m Yn,m (ϑ, λ). (1.16). l=2 m=0. 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 polynomials as reported in Eq. 1.17 (as in Torge (1989); Dragomir 40. et al. (1982)) and the.

(43) CHAPTER 1. CLASSICAL PROCESSING OF GRAVITATIONAL DATA evaluation of this equation shows that a smoothening eect hits the signal and therefore it loses the high frequency components and gradually damps with the height and would vanish at innity.. V (r, ϑP , λP ) =. GM R. LX max. l X. ( Rr )l+1 [C¯lm cos mλP + S¯lm sin mλP ]P¯lm cos(ϑP ) (1.17). l=2 m=0. 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 Stokes0 formula and consequently enhance the geoid determination (Sun , 2002). The methods which are widely implemented for the computation of the TC are dependent of the typology of the data, the direct integrations of the TC is preferred for point−wise computations(Vannes , 2011) while the FFT is in general an ideal method for grid−wise computation (Schwarz. et al., 1990) and it also requires less. time comparing to the direct integral method. In order to evaluate such correction there is an urgent need for densely sampled DTM (Tsoulis , 2001) (Tsoulis, 2001). To be able to implement such procedure, in case no detailed DTM model are available in the area, both land topography values by SRTM (Farr. et al., 2007) (1 arc−second of about 30 meters of spatial resolution) and oceans bathymetry values by ETOPO1 (Amante and Eakins , 2009) (1 arc−minute grid cell resolution of about 1.8 km) could merged by a procedure of Kriging to build an adequate DTM model assuring at least a 50 km extension in every direction around the studies area. Ahmed Hamdi Mansi. 41.

(44) Airborne Gravity Field Modelling. 1.3.2.1 Point−Mass Model The direct integration is based on the Newtonian volume integral. In the point mass model in order to compute the gravitational potential V at of attracting mass, the attracting body is condensed and represented as a set of point−masses each located at specic pointP` (` x, y`, z`). The eect of each point mass at any arbitrary computation point P (xP , yP , zP ) is reported in Eq. 1.18 in its nal form in Cartesian coordinate system.. ZZZ 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: Newton0 s volume integral for gravitational potential evaluation in the Cartesian coordinate system, where G = 6.67 · 10−11 is Newton0 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 ` λ) ` at the computation point P (r, ϑ, λ) in terms of potential value as in at P` (` r, ϑ, Eq. 1.19.. Z. 2π. Z. 2π. Z. r`max. V (r, ϑ, λ) = G λ=0. ϑ. r=0. ` λ) ` ρ(` r, ϑ, ` rdϑd ` λ ` √ r`2 sin ϑd` r2 + r`2 − 2r` r[cos ψ]. (1.19). Eq. 1.19: Newton0 s volume integral for gravitational potential evaluation in ` . Spherical coordinate system, where cos ψ = cos ϑ cos ϑ` + sin ϑ 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.,. Han and Shen , 2010)), then a new expression of the gravitational potential and its derivatives were elaborated by D0 Urso (2012). The rectangular−prism rep2002;. 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.

(45) CHAPTER 1. CLASSICAL PROCESSING OF GRAVITATIONAL DATA. Figure 1.4.. The Spherical coordinates of the computation point P (r, ϑ, λ) and the ` λ) ` . integral point P` (` r, ϑ,. Ahmed Hamdi Mansi. 43.

(46) Airborne Gravity Field Modelling. Figure 1.5.. Sketch map of the denition of the prism (after. Nagy et al. (2000)).. V (r, ϑP , λP ) = Gρ |||xyln(z + r) + yzln(x + r) + zxln(y + r) − 2 2 − z2 tan−1 xy |x2 |y2 |z2 − y2 tan−1 xz yr zr x1 y1 z1. yz x2 tan−1 xr 2. (1.20). Eq. 1.20: Newton0 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 possible, 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.