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E-raamat: Geoid Determination: Theory and Methods

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This book will be based on the material of the lecture noties in several International Schools for the Determination and Use of the Geoid, organized by the International Geoid Serivice of the International Association of Geodesy. It consolidates, unifies, and streamlines this material in a unique way not covereed by the few other books that exist on this subjext. More specifically, the book presents (for the first time in a single volume) the theory and methodology of the most common technique used for precise determination of the geoid, including the computation of the marine geoid from satellite altimetry data. These are illustrated by specific examples and actual computations of local geoids. In addition, the book provides the fundamentals of estimating orthometric heights without spirit levelling, by properly combining a geoid with heights from GPS. Besides the geodectic and geophysical uses, this last application has made geoid computation methods very popular in recent years because the entire GPS and GIS user communities are interested in estimating geoid undulations in order to convert GPS heights to physically meaningful orthometric heights (elevations above mean sea level). The overall purpose of the book is, therefore, to provide the user community (academics, graduate students, geophysicists, engineers, oceanographers, GIS and GPS users, researchers) with a self-contained textbook, which will supply them with the complete roadmap of estimating geoid undulations, from the theoretical definitions and formulas to the available numerical methods and their implementation and the test in practice.
Part I Theory
Fernando Sanso
1 The Forward Modelling of the Gravity Field
3(70)
1.1 Outline of the
Chapter
3(1)
1.2 Newton's Gravitation Law
4(1)
1.3 The Newtonian Gravitational Attraction of Bodies
5(9)
1.4 The Gravity Field
14(2)
1.5 Gauss, Poisson, Laplace
16(4)
1.6 Dirichlet, Green
20(1)
1.7 Elements of Geometry of the Gravity Field and Related Definitions
21(9)
1.8 The Laplace Operator in Curvilinear Coordinates
30(5)
1.9 Simple Mathematical Models of the Gravity Field
35(8)
1.10 Anomalous Quantities of the Gravity Field and a More Precise Definition of the Geoid
43(10)
1.11 Summary of Height Systems and Their Relation to the Geodetic Datum
53(4)
1.12 Exercises
57(6)
Appendix
63(10)
A.1
63(1)
A.2
64(2)
A.3
66(2)
A.4
68(5)
2 Observables of Physical Geodesy and Their Analytical Representation
73(38)
2.1 Outline of the
Chapter
73(2)
2.2 Observables and Observation Equations: Linearization
75(5)
2.3 The Linearized Observation Equations of Physical Geodesy
80(11)
2.4 On the Relation Between Height Anomalies and Geoid Undulations
91(3)
2.5 The Remove-Restore Concept
94(3)
2.6 The Spherical Approximation Procedure
97(4)
2.7 A Review of Observation Equations with Unknown Reference Potential
101(3)
2.8 Exercises
104(1)
Appendix
105(6)
A.1
105(2)
A.2
107(4)
3 Harmonic Calculus and Global Gravity Models
111(58)
3.1 Outline of the
Chapter
111(2)
3.2 The Newton Integral Representation of the Anomalous Potential
113(4)
3.3 Legendre Functions
117(7)
3.4 Spherical Harmonics
124(11)
3.5 Downward Continuation and Krarup's Theorem
135(3)
3.6 Ellipsoidal Harmonics
138(7)
3.7 Global Models as Approximate Solution of Boundary Value Problems
145(6)
3.8 Commission and Omission Errors. Kaula's Rule
151(10)
3.9 Exercises
161(1)
Appendix
162(7)
A.1
162(2)
A.2
164(1)
A.3
165(2)
A.4
167(2)
4 The Local Modelling of the Gravity Field: The Terrain Effects
169(34)
4.1 Outline of the
Chapter
169(1)
4.2 High Accuracy and High Resolution Local Gravity Model
170(4)
4.3 The Smoothing Role of Terrain Correction (TC)
174(5)
4.4 From Terrain Correction (TC) to Residual Terrain Correction (RTC)
179(6)
4.5 Strategies for the Implementation of Terrain Effects
185(6)
4.6 Comparisons and Interpretations
191(4)
4.7 An Open Issue
195(2)
4.8 Exercises
197(2)
Appendix
199(4)
A.1
199(4)
5 The Local Modelling of the Gravity Field by Collocation
203(58)
5.1 Outline of the
Chapter
203(1)
5.2 An Introduction to the Problem
204(2)
5.3 The Principle of Minimum Square Invariant Prediction Error by a Simple Example
206(6)
5.4 On Collocation Theory, or the Wiener-Kolmogorov Principle Applied in Physical Geodesy
212(4)
5.5 The General Collocation Problem
216(6)
5.6 Covariance and Spectral Harmonic Calculus
222(6)
5.7 The Estimate of Global Covariance Functions
228(3)
5.8 The Estimate of Local Covariance Functions
231(6)
5.9 Covariance Parametric Models
237(3)
5.10 The Least Squares Collocation (l.s.c.) Solution
240(4)
5.11 On the Optimal Combination of Global Coefficients and Local Observations
244(7)
5.12 Exercises
251(4)
Appendix
255(6)
A.1
255(1)
A.2
256(5)
Part II Methods and Applications
6 Global Gravitational Models
261(50)
6.1 Outline of the
Chapter
261(1)
6.2 Introduction
262(3)
6.2.1 Local and Regional Gravimetric Models
264(1)
6.2.2 Global Versus Local Gravimetric Models: Similarities and Differences
264(1)
6.3 Signal Representation and Data Characteristics
265(4)
6.4 The New Satellite Missions
269(5)
6.5 Beyond the Sensitivity of Satellite Data
274(3)
6.6 State-of-the-Art Global Gravitational Modeling
277(27)
6.6.1 EGM96
279(14)
6.6.2 EGM2008
293(11)
6.7 Data Requirements and Data Availability
304(3)
6.7.1 Elevation Data
304(1)
6.7.2 Terrestrial Gravity Anomaly Data
305(1)
6.7.3 Altimetry-Derived Gravity Anomalies
306(1)
6.7.4 The Merged 5' × 5' Area-Mean Gravity Anomaly File
306(1)
6.8 Use of Global Gravitational Models and of Their By-Products
307(2)
6.9 Temporal Variations
309(1)
6.10 Outlook
309(2)
7 Geoid Determination by 3D Least-Squares Collocation
311(26)
7.1 Outline of the
Chapter
311(1)
7.2 Introduction
311(1)
7.3 Theory
312(4)
7.4 The Remove-Restore Method
316(3)
7.5 Covariance Function Estimation and Representation
319(5)
7.6 Conversion from Geoid Heights to Height Anomalies
324(1)
7.7 LSC Geoid Determination from Residual Data
325(4)
7.8 Conclusion
329(8)
8 Topographic Reductions in Gravity and Geoid Modeling
337(64)
8.1 Outline of the
Chapter
337(1)
8.2 Introduction
338(2)
8.3 Topographic Reductions and Gravity Field Modeling
340(23)
8.3.1 The Potential and the Attraction of the Earth's Topography
340(3)
8.3.2 Terrain Reductions for Gravity Densification and Gridding
343(10)
8.3.3 Topographic/Isostatic Effects on Gravity and Airborne Gravity and Gradiometry
353(3)
8.3.4 Terrain Reductions and Physical Heights
356(1)
8.3.5 The Treatment of the Topography in Geoid and Quasi-geoid Determination
357(6)
8.4 Terrain Effects in Geoid and Quasi-geoid Determination
363(11)
8.4.1 Helmert's Second Method of Condensation
363(2)
8.4.2 Rudzki's Inversion Scheme
365(1)
8.4.3 Residual Terrain Model (RTM)
366(3)
8.4.4 Terrain Effects and High-Resolution Global Geopotential Models
369(2)
8.4.5 The Remove-Restore Methodology and the Different Reduction Schemes
371(3)
8.5 Methods for the Numerical Estimation of Direct and Indirect Topographic Effects
374(11)
8.5.1 The Mass Prism Topographic Model and the Numerical Integration Method (NIM)
376(4)
8.5.2 The Fast Fourier Transform (FFT) Method
380(5)
8.6 Numerical Examples
385(13)
8.6.1 Effects of Terrain Reductions on Gravity Anomalies and Geoid Heights
386(5)
8.6.2 Determination and Evaluation of Gravimetric Geoid Models
391(7)
8.7 Summary and Concluding Remarks
398(3)
9 Marine Gravity and Geoid from Satellite Altimetry
401(52)
9.1 Outline of the
Chapter
402(1)
9.2 Altimetry Data
403(2)
9.3 Retracking
405(2)
9.4 Sea Surface Height Observations
407(6)
9.4.1 Mean Sea Surface and Mean Dynamic Topography
410(2)
9.4.2 Remove-Restore for Satellite Altimetry
412(1)
9.4.3 Dynamic Sea Surface Topography
412(1)
9.5 Crossover Adjustment
413(5)
9.6 Data Editing, Data Quality and Error-Budget
418(3)
9.7 Gravity Recovery from Altimetry
421(1)
9.8 Least Squares Collocation for Altimetry
422(4)
9.8.1 Interpolation Using Least Squares Collocation
425(1)
9.9 Deterministic Methods
426(2)
9.10 Fast Spectral Methods for Altimetric Gravity Prediction
428(4)
9.10.1 Fast Fourier Techniques for Altimetric Gravity
429(2)
9.10.2 Filtering
431(1)
9.11 Practical Computation of Global High Resolution Marine Gravity
432(7)
9.11.1 North Sea Example
436(3)
9.12 Accuracy of Present-Day Altimetric marine Gravity Fields
439(2)
9.13 Integrating Marine, Airborne and Satellite Derived Gravity
441(2)
9.13.1 East Greenland Airborne and Altimetric Gravity Example
442(1)
9.14 Altimetric Gravity Research Frontiers
443(7)
9.14.1 ICESat and Cryosat-2
444(1)
9.14.2 Altimeter Range Corrections
445(1)
9.14.3 Ocean Tides
446(1)
9.14.4 Retracking in Coastal and Polar Regions
447(3)
Appendix A Data Resources
450(3)
A.1 Altimetry Data
450(1)
A.2 Altimetric Gravity Field Resources
450(3)
10 Geoid Determination by FFT Techniques
453(64)
10.1 Outline of the
Chapter
453(1)
10.2 Review of Stokes's Integral and Its Evaluation
454(6)
10.2.1 Stokes's Boundary Value Problem
454(1)
10.2.2 Geoid Undulations and Terrain Reductions
455(2)
10.2.3 Practical Evaluation of Stokes's Integral
457(2)
10.2.4 The Need for Spectral Techniques
459(1)
10.3 Geoid Undulations by FFT
460(8)
10.3.1 Planar Approximation of Stokes's Integral
460(4)
10.3.2 Spherical Form of Stokes's Integral
464(3)
10.3.3 Elimination of Edge Effects and Circular Convolution
467(1)
10.4 FFT-Evaluation of Terrain Effects
468(8)
10.4.1 2D Formulas for Terrain Effects
468(5)
10.4.2 Terrain Corrections by 3D FFT
473(3)
10.5 Optimal Spectral Geoid Determination
476(2)
10.5.1 Error Propagation
476(2)
10.6 Other Examples of FFT Evaluation of Geodetic Operators
478(3)
10.6.1 The Vening Meinesz Integral
478(1)
10.6.2 The Analytical Continuation Integrals
479(1)
10.6.3 The Inverse Stokes and Inverse Mening Meinesz Formulas
480(1)
10.7 Concluding Remarks
481(2)
Appendix
483(34)
A.1 Basic Definitions
483(1)
A.1.1 Sinusoids
483(1)
A.1.2 Fourier Series
483(2)
A.2 The Continuous Fourier Transform and Its Properties
485(1)
A.2.1 Definition of the Continuous Fourier Transform
485(1)
A.2.2 The Impulse Function
486(2)
A.2.3 The Rectangle and the Sinc Functions
488(1)
A.2.4 Interpretation of the Fourier Transform and the Fourier Series
489(1)
A.2.5 Properties of the CFT
489(1)
A.2.6 Convolution and Correlation
490(3)
A.3 The Discrete Fourier Transform
493(1)
A.3.1 From the Continuous to the Discrete Fourier Transform: Aliasing and Leakage
493(3)
A.3.2 Discrete Convolution and Correlation: Circular Convolution and Correlation
496(2)
A.3.3 Correlation, Covariance, and Power Spectral Density Functions
498(2)
A.3.4 The DFT in Computers
500(2)
A.3.5 The Fast Fourier Transform
502(1)
A.4 The Two-Dimensional Discrete Fourier Transform
503(2)
A.5 Efficient DFT for Real Functions
505(1)
A.5.1 DFT of Two Real Functions Via a Single FFT
505(1)
A.5.2 Simultaneous Convolution of Two Real Functions with the Same Function
506(1)
A.6 Use of the Fast Hartley Transform
507(1)
A.6.1 The Discrete Hartley Transform
508(1)
A.6.2 Definition of the 1D Discrete Hartley Transform
508(1)
A.6.3 Definition of the 2D Discrete Hartley Transform
509(1)
A.6.4 Properties of the Discrete Hartley Transform
509(5)
A.7 Relationship Between the DHT and the DFT
514(1)
A.7.1 Computation of the 1D DFT Via the 1D DHT
514(1)
A.7.2 Computation of the 2D DFT Via the 2D DHT
515(1)
A.7.3 Advantages Unique to the FHT
516(1)
11 Combination of Heights
517(30)
11.1 Outline of the
Chapter
517(1)
11.2 Introduction
517(3)
11.3 Why Combine Geoid, Orthometric and Ellipsoidal Height Data?
520(5)
11.3.1 Modernizing Regional Vertical Datums
520(3)
11.3.2 Global Vertical Datum
523(1)
11.3.3 GNSS-Levelling
523(1)
11.3.4 Refining and Testing Gravimetric Geoid Models
524(1)
11.4 Least-Squares Adjustment Methodology for Combining Heights
525(3)
11.5 Application of MINQUE to the Combined Height Adjustment Problem
528(3)
11.6 Role of the Parametric Model
531(12)
11.6.1 Modelling Options
534(1)
11.6.2 Semi-automated Assessment Procedure
535(4)
11.6.3 Numerical Example
539(4)
11.7 Remarks
543(4)
Part III Advanced Analysis Methods
Fernando Sanso
12 Hilbert Spaces and Deterministic Collocation
547(44)
12.1 Outline of the
Chapter
547(1)
12.2 An Introduction to Hilbert Spaces
548(7)
12.3 Orthogonality, Duality, Bases
555(13)
12.4 Hilbert Spaces with Reproducing Kernel
568(15)
12.5 Exercises
583(8)
13 On Potential Theory and HS of Harmonic Functions
591(54)
13.1 Outline of the
Chapter
591(1)
13.2 Harmonic Functions and Harmonic Polynomials
592(11)
13.3 Spherical Harmonics
603(9)
13.4 Hilbert Spaces of Harmonic Functions and First Theorems of Potential Theory
612(15)
13.5 Green's Function and Krarup's Theorem
627(13)
13.6 Exercises
640(5)
14 A Quick Look to Classical Boundary Value Problems (BVP) Solutions
645(18)
14.1 Outline of the
Chapter
645(1)
14.2 The Classical Molodensky Approach: A Historical Excursus
645(2)
14.3 The Approximate Solution of Molodensky's Problem by Downward Continuation
647(5)
14.4 On the Local Use of Molodensky's Formula
652(5)
14.5 The Helmert Approach: A Short Review
657(2)
14.6 Exercises
659(4)
15 The Analysis of Geodetic Boundary Value Problems in Linear form
663(44)
15.1 Outline of the
Chapter
663(3)
15.2 A Precise Definition of the Two Main BVP's and of Their Solution Spaces
666(6)
15.3 Linearized Molodensky's Problem
672(9)
15.4 The Analysis of the Linearized Fixed Boundary BPV
681(2)
15.5 From Least Squares to Galerkin's Method
683(10)
15.6 Two Geodetic Solutions of Galerkin's System
693(8)
15.7 New Data Sets from Spatial Gravity Surveying
701(3)
15.8 Exercises
704(3)
References 707(20)
Index 727