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1 The Forward Modelling of the Gravity Field |
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3 | (70) |
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1.1 Outline of the Chapter |
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3 | (1) |
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1.2 Newton's Gravitation Law |
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4 | (1) |
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1.3 The Newtonian Gravitational Attraction of Bodies |
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5 | (9) |
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14 | (2) |
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1.5 Gauss, Poisson, Laplace |
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16 | (4) |
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20 | (1) |
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1.7 Elements of Geometry of the Gravity Field and Related Definitions |
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21 | (9) |
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1.8 The Laplace Operator in Curvilinear Coordinates |
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30 | (5) |
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1.9 Simple Mathematical Models of the Gravity Field |
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35 | (8) |
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1.10 Anomalous Quantities of the Gravity Field and a More Precise Definition of the Geoid |
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43 | (10) |
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1.11 Summary of Height Systems and Their Relation to the Geodetic Datum |
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53 | (4) |
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57 | (6) |
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63 | (10) |
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63 | (1) |
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64 | (2) |
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66 | (2) |
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68 | (5) |
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2 Observables of Physical Geodesy and Their Analytical Representation |
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73 | (38) |
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2.1 Outline of the Chapter |
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73 | (2) |
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2.2 Observables and Observation Equations: Linearization |
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75 | (5) |
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2.3 The Linearized Observation Equations of Physical Geodesy |
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80 | (11) |
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2.4 On the Relation Between Height Anomalies and Geoid Undulations |
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91 | (3) |
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2.5 The Remove-Restore Concept |
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94 | (3) |
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2.6 The Spherical Approximation Procedure |
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97 | (4) |
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2.7 A Review of Observation Equations with Unknown Reference Potential |
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101 | (3) |
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104 | (1) |
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105 | (6) |
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105 | (2) |
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107 | (4) |
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3 Harmonic Calculus and Global Gravity Models |
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111 | (58) |
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3.1 Outline of the Chapter |
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111 | (2) |
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3.2 The Newton Integral Representation of the Anomalous Potential |
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113 | (4) |
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117 | (7) |
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124 | (11) |
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3.5 Downward Continuation and Krarup's Theorem |
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135 | (3) |
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3.6 Ellipsoidal Harmonics |
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138 | (7) |
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3.7 Global Models as Approximate Solution of Boundary Value Problems |
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145 | (6) |
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3.8 Commission and Omission Errors. Kaula's Rule |
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151 | (10) |
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161 | (1) |
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162 | (7) |
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162 | (2) |
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164 | (1) |
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165 | (2) |
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167 | (2) |
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4 The Local Modelling of the Gravity Field: The Terrain Effects |
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169 | (34) |
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4.1 Outline of the Chapter |
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169 | (1) |
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4.2 High Accuracy and High Resolution Local Gravity Model |
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170 | (4) |
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4.3 The Smoothing Role of Terrain Correction (TC) |
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174 | (5) |
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4.4 From Terrain Correction (TC) to Residual Terrain Correction (RTC) |
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179 | (6) |
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4.5 Strategies for the Implementation of Terrain Effects |
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185 | (6) |
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4.6 Comparisons and Interpretations |
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191 | (4) |
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195 | (2) |
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197 | (2) |
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199 | (4) |
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199 | (4) |
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5 The Local Modelling of the Gravity Field by Collocation |
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203 | (58) |
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5.1 Outline of the Chapter |
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203 | (1) |
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5.2 An Introduction to the Problem |
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204 | (2) |
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5.3 The Principle of Minimum Square Invariant Prediction Error by a Simple Example |
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206 | (6) |
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5.4 On Collocation Theory, or the Wiener-Kolmogorov Principle Applied in Physical Geodesy |
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212 | (4) |
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5.5 The General Collocation Problem |
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216 | (6) |
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5.6 Covariance and Spectral Harmonic Calculus |
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222 | (6) |
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5.7 The Estimate of Global Covariance Functions |
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228 | (3) |
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5.8 The Estimate of Local Covariance Functions |
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231 | (6) |
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5.9 Covariance Parametric Models |
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237 | (3) |
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5.10 The Least Squares Collocation (l.s.c.) Solution |
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240 | (4) |
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5.11 On the Optimal Combination of Global Coefficients and Local Observations |
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244 | (7) |
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251 | (4) |
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255 | (6) |
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255 | (1) |
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256 | (5) |
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Part II Methods and Applications |
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6 Global Gravitational Models |
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261 | (50) |
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6.1 Outline of the Chapter |
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261 | (1) |
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262 | (3) |
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6.2.1 Local and Regional Gravimetric Models |
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264 | (1) |
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6.2.2 Global Versus Local Gravimetric Models: Similarities and Differences |
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264 | (1) |
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6.3 Signal Representation and Data Characteristics |
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265 | (4) |
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6.4 The New Satellite Missions |
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269 | (5) |
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6.5 Beyond the Sensitivity of Satellite Data |
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274 | (3) |
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6.6 State-of-the-Art Global Gravitational Modeling |
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277 | (27) |
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279 | (14) |
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293 | (11) |
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6.7 Data Requirements and Data Availability |
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304 | (3) |
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304 | (1) |
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6.7.2 Terrestrial Gravity Anomaly Data |
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305 | (1) |
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6.7.3 Altimetry-Derived Gravity Anomalies |
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306 | (1) |
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6.7.4 The Merged 5' × 5' Area-Mean Gravity Anomaly File |
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306 | (1) |
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6.8 Use of Global Gravitational Models and of Their By-Products |
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307 | (2) |
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309 | (1) |
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309 | (2) |
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7 Geoid Determination by 3D Least-Squares Collocation |
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311 | (26) |
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7.1 Outline of the Chapter |
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311 | (1) |
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311 | (1) |
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312 | (4) |
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7.4 The Remove-Restore Method |
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316 | (3) |
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7.5 Covariance Function Estimation and Representation |
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319 | (5) |
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7.6 Conversion from Geoid Heights to Height Anomalies |
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324 | (1) |
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7.7 LSC Geoid Determination from Residual Data |
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325 | (4) |
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329 | (8) |
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8 Topographic Reductions in Gravity and Geoid Modeling |
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337 | (64) |
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8.1 Outline of the Chapter |
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337 | (1) |
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338 | (2) |
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8.3 Topographic Reductions and Gravity Field Modeling |
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340 | (23) |
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8.3.1 The Potential and the Attraction of the Earth's Topography |
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340 | (3) |
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8.3.2 Terrain Reductions for Gravity Densification and Gridding |
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343 | (10) |
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8.3.3 Topographic/Isostatic Effects on Gravity and Airborne Gravity and Gradiometry |
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353 | (3) |
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8.3.4 Terrain Reductions and Physical Heights |
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356 | (1) |
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8.3.5 The Treatment of the Topography in Geoid and Quasi-geoid Determination |
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357 | (6) |
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8.4 Terrain Effects in Geoid and Quasi-geoid Determination |
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363 | (11) |
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8.4.1 Helmert's Second Method of Condensation |
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363 | (2) |
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8.4.2 Rudzki's Inversion Scheme |
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365 | (1) |
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8.4.3 Residual Terrain Model (RTM) |
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366 | (3) |
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8.4.4 Terrain Effects and High-Resolution Global Geopotential Models |
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369 | (2) |
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8.4.5 The Remove-Restore Methodology and the Different Reduction Schemes |
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371 | (3) |
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8.5 Methods for the Numerical Estimation of Direct and Indirect Topographic Effects |
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374 | (11) |
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8.5.1 The Mass Prism Topographic Model and the Numerical Integration Method (NIM) |
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376 | (4) |
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8.5.2 The Fast Fourier Transform (FFT) Method |
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380 | (5) |
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385 | (13) |
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8.6.1 Effects of Terrain Reductions on Gravity Anomalies and Geoid Heights |
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386 | (5) |
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8.6.2 Determination and Evaluation of Gravimetric Geoid Models |
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391 | (7) |
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8.7 Summary and Concluding Remarks |
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398 | (3) |
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9 Marine Gravity and Geoid from Satellite Altimetry |
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401 | (52) |
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9.1 Outline of the Chapter |
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402 | (1) |
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403 | (2) |
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405 | (2) |
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9.4 Sea Surface Height Observations |
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407 | (6) |
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9.4.1 Mean Sea Surface and Mean Dynamic Topography |
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410 | (2) |
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9.4.2 Remove-Restore for Satellite Altimetry |
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412 | (1) |
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9.4.3 Dynamic Sea Surface Topography |
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412 | (1) |
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413 | (5) |
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9.6 Data Editing, Data Quality and Error-Budget |
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418 | (3) |
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9.7 Gravity Recovery from Altimetry |
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421 | (1) |
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9.8 Least Squares Collocation for Altimetry |
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422 | (4) |
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9.8.1 Interpolation Using Least Squares Collocation |
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425 | (1) |
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9.9 Deterministic Methods |
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426 | (2) |
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9.10 Fast Spectral Methods for Altimetric Gravity Prediction |
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428 | (4) |
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9.10.1 Fast Fourier Techniques for Altimetric Gravity |
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429 | (2) |
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431 | (1) |
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9.11 Practical Computation of Global High Resolution Marine Gravity |
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432 | (7) |
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436 | (3) |
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9.12 Accuracy of Present-Day Altimetric marine Gravity Fields |
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439 | (2) |
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9.13 Integrating Marine, Airborne and Satellite Derived Gravity |
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441 | (2) |
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9.13.1 East Greenland Airborne and Altimetric Gravity Example |
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442 | (1) |
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9.14 Altimetric Gravity Research Frontiers |
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443 | (7) |
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9.14.1 ICESat and Cryosat-2 |
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444 | (1) |
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9.14.2 Altimeter Range Corrections |
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445 | (1) |
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446 | (1) |
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9.14.4 Retracking in Coastal and Polar Regions |
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447 | (3) |
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Appendix A Data Resources |
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450 | (3) |
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450 | (1) |
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A.2 Altimetric Gravity Field Resources |
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450 | (3) |
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10 Geoid Determination by FFT Techniques |
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453 | (64) |
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10.1 Outline of the Chapter |
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453 | (1) |
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10.2 Review of Stokes's Integral and Its Evaluation |
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454 | (6) |
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10.2.1 Stokes's Boundary Value Problem |
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454 | (1) |
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10.2.2 Geoid Undulations and Terrain Reductions |
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455 | (2) |
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10.2.3 Practical Evaluation of Stokes's Integral |
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457 | (2) |
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10.2.4 The Need for Spectral Techniques |
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459 | (1) |
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10.3 Geoid Undulations by FFT |
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460 | (8) |
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10.3.1 Planar Approximation of Stokes's Integral |
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460 | (4) |
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10.3.2 Spherical Form of Stokes's Integral |
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464 | (3) |
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10.3.3 Elimination of Edge Effects and Circular Convolution |
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467 | (1) |
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10.4 FFT-Evaluation of Terrain Effects |
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468 | (8) |
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10.4.1 2D Formulas for Terrain Effects |
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468 | (5) |
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10.4.2 Terrain Corrections by 3D FFT |
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473 | (3) |
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10.5 Optimal Spectral Geoid Determination |
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476 | (2) |
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476 | (2) |
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10.6 Other Examples of FFT Evaluation of Geodetic Operators |
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478 | (3) |
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10.6.1 The Vening Meinesz Integral |
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478 | (1) |
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10.6.2 The Analytical Continuation Integrals |
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479 | (1) |
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10.6.3 The Inverse Stokes and Inverse Mening Meinesz Formulas |
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480 | (1) |
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481 | (2) |
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483 | (34) |
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483 | (1) |
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483 | (1) |
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483 | (2) |
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A.2 The Continuous Fourier Transform and Its Properties |
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485 | (1) |
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A.2.1 Definition of the Continuous Fourier Transform |
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485 | (1) |
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A.2.2 The Impulse Function |
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486 | (2) |
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A.2.3 The Rectangle and the Sinc Functions |
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488 | (1) |
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A.2.4 Interpretation of the Fourier Transform and the Fourier Series |
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489 | (1) |
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A.2.5 Properties of the CFT |
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489 | (1) |
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A.2.6 Convolution and Correlation |
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490 | (3) |
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A.3 The Discrete Fourier Transform |
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493 | (1) |
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A.3.1 From the Continuous to the Discrete Fourier Transform: Aliasing and Leakage |
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493 | (3) |
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A.3.2 Discrete Convolution and Correlation: Circular Convolution and Correlation |
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496 | (2) |
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A.3.3 Correlation, Covariance, and Power Spectral Density Functions |
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498 | (2) |
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A.3.4 The DFT in Computers |
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500 | (2) |
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A.3.5 The Fast Fourier Transform |
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502 | (1) |
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A.4 The Two-Dimensional Discrete Fourier Transform |
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503 | (2) |
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A.5 Efficient DFT for Real Functions |
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505 | (1) |
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A.5.1 DFT of Two Real Functions Via a Single FFT |
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505 | (1) |
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A.5.2 Simultaneous Convolution of Two Real Functions with the Same Function |
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506 | (1) |
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A.6 Use of the Fast Hartley Transform |
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507 | (1) |
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A.6.1 The Discrete Hartley Transform |
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508 | (1) |
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A.6.2 Definition of the 1D Discrete Hartley Transform |
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508 | (1) |
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A.6.3 Definition of the 2D Discrete Hartley Transform |
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509 | (1) |
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A.6.4 Properties of the Discrete Hartley Transform |
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509 | (5) |
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A.7 Relationship Between the DHT and the DFT |
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514 | (1) |
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A.7.1 Computation of the 1D DFT Via the 1D DHT |
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514 | (1) |
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A.7.2 Computation of the 2D DFT Via the 2D DHT |
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515 | (1) |
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A.7.3 Advantages Unique to the FHT |
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516 | (1) |
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11 Combination of Heights |
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517 | (30) |
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11.1 Outline of the Chapter |
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517 | (1) |
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517 | (3) |
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11.3 Why Combine Geoid, Orthometric and Ellipsoidal Height Data? |
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520 | (5) |
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11.3.1 Modernizing Regional Vertical Datums |
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520 | (3) |
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11.3.2 Global Vertical Datum |
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523 | (1) |
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523 | (1) |
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11.3.4 Refining and Testing Gravimetric Geoid Models |
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524 | (1) |
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11.4 Least-Squares Adjustment Methodology for Combining Heights |
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525 | (3) |
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11.5 Application of MINQUE to the Combined Height Adjustment Problem |
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528 | (3) |
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11.6 Role of the Parametric Model |
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531 | (12) |
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534 | (1) |
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11.6.2 Semi-automated Assessment Procedure |
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535 | (4) |
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539 | (4) |
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543 | (4) |
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Part III Advanced Analysis Methods |
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12 Hilbert Spaces and Deterministic Collocation |
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547 | (44) |
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12.1 Outline of the Chapter |
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547 | (1) |
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12.2 An Introduction to Hilbert Spaces |
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548 | (7) |
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12.3 Orthogonality, Duality, Bases |
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555 | (13) |
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12.4 Hilbert Spaces with Reproducing Kernel |
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568 | (15) |
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583 | (8) |
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13 On Potential Theory and HS of Harmonic Functions |
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591 | (54) |
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13.1 Outline of the Chapter |
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591 | (1) |
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13.2 Harmonic Functions and Harmonic Polynomials |
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592 | (11) |
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603 | (9) |
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13.4 Hilbert Spaces of Harmonic Functions and First Theorems of Potential Theory |
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612 | (15) |
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13.5 Green's Function and Krarup's Theorem |
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627 | (13) |
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640 | (5) |
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14 A Quick Look to Classical Boundary Value Problems (BVP) Solutions |
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645 | (18) |
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14.1 Outline of the Chapter |
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645 | (1) |
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14.2 The Classical Molodensky Approach: A Historical Excursus |
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645 | (2) |
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14.3 The Approximate Solution of Molodensky's Problem by Downward Continuation |
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647 | (5) |
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14.4 On the Local Use of Molodensky's Formula |
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652 | (5) |
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14.5 The Helmert Approach: A Short Review |
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657 | (2) |
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659 | (4) |
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15 The Analysis of Geodetic Boundary Value Problems in Linear form |
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663 | (44) |
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15.1 Outline of the Chapter |
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663 | (3) |
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15.2 A Precise Definition of the Two Main BVP's and of Their Solution Spaces |
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666 | (6) |
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15.3 Linearized Molodensky's Problem |
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672 | (9) |
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15.4 The Analysis of the Linearized Fixed Boundary BPV |
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681 | (2) |
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15.5 From Least Squares to Galerkin's Method |
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683 | (10) |
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15.6 Two Geodetic Solutions of Galerkin's System |
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693 | (8) |
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15.7 New Data Sets from Spatial Gravity Surveying |
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701 | (3) |
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704 | (3) |
References |
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707 | (20) |
Index |
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727 | |