Gravity Interpretation

Fundamentals and Application of Gravity Inversion and Geological Interpretation

By: Peter L. Smilde, Wolfgang Jacoby

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CD-ROM | 13 February 2009

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Gravity interpretation involves inversion of data into models, but it is more. Gravity interpretation is used in a "holistic" sense going beyond "inversion". Inversion is like optimization within certain a priori assumptions, i.e., all anticipated models lie in a limited domain of the a priori errors. No source should exist outside the anticipated model volume, but that is never literally true. Interpretation goes beyond by taking "outside" possibilities into account in the widest sense. Any neglected possibility carries the danger of seriously affecting the interpretation. Gravity interpretation pertains to wider questions such as the shape of the Earth, the nature of the continental and oceanic crust, isostasy, forces and stresses, geol- ical structure, nding useful resources, climate change, etc. Interpretation is often used synonymously with modelling and inversion of observations toward models. Interpretation places the inversion results into the wider geological or economic context and into the framework of science and humanity. Models play a central role in science. They are images of phenomena of the physical world, for example, scale images or metaphors, enabling the human mind to describe observations and re- tionships by abstract mathematical means. Models served orientation and survival in a complex, partly invisible physical and social environment.

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Published: 23rd September 2014

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Introduction	p. 1
The Subject and Scope	p. 1
Gravity	p. 1
Motivation	p. 2
Aims	p. 3
Special Aspects	p. 4
The Book and the Reader	p. 4
Historical Review	p. 5
Astronomy, Geodesy, Geophysics, 18th and 19th Centuries	p. 5
20th Century	p. 6
Geodesy and Geophysics	p. 7
Purposes of Gravity Measurements	p. 8
Gravity and Gravity Anomalies	p. 9
Some Important Aspects of the Terrestrial Gravity Field and Internal Mass Distribution	p. 10
General Considerations	p. 10
The Earth's Figure and Constitution	p. 11
Continents and Oceans	p. 12
Plate Tectonics and Mantle Flow	p. 13
Associated Gravity Anomalies	p. 16
Other Large-Scale Gravity Features	p. 18
Smaller-Scale Gravity Anomalies Relevant to Exploration for Economic Minerals	p. 19
Harmonic Spectrum of the Gravity Field	p. 19
References	p. 21
Fundamentals of Gravity, Elements of Potential Theory	p. 23
Introduction	p. 23
Units	p. 23
Elements of g	p. 24
Coordinate Systems	p. 25
Spherical Coordinates	p. 25
Vertical Cylinder Coordinates	p. 26
Cartesian Coordinates	p. 27
Newton's Laws: Gravitation and Inertia Plus Centrifugal Acceleration = Gravity	p. 29
Gravity Potential and Equipotential Surfaces	p. 31
Laplace Equation, Field Quantities, Equivalent Stratum; Derivation of Some Field Quantities, Surface Integrals, Poisson Equations, Gravitational Flux	p. 34
Source-Free Space: Laplace Equation	p. 35
The Field Quantities	p. 36
The Equivalent Stratum	p. 39
Applications: Estimation of Field Quantities as Wx, Wy, Wzzz	p. 40
Source Space: Poisson Equation and Gravitational Flux	p. 43
Surface Integrals: Total Mass, Centre of Mass	p. 44
The Gravity Tensor (Eöet;tvöet;s Tensor)	p. 45
Gravity Effects and Anomalies - Summation and Integration	p. 46
General Considerations	p. 46
Coordinate Systems and Integration	p. 50
Special Mass Elements: Integration in One and Two Dimensions, Mass Lines and Mass Planes	p. 53
Disks	p. 61
Shells	p. 67
Uniform Massive Volumes	p. 70
Two-Dimensional Bodies	p. 74
Two-and-a-half Dimensional Models (21/2D)	p. 84
Some Theoretical Aspects of Anomaly Analysis	p. 86
Goals of Post-reduction Data Analysis	p. 86
Smoothing of Spatial Series	p. 87
Polynomials	p. 88
The Field Quantities: Differentiation and Integration	p. 89
Harmonic Functions	p. 90
Special Functions	p. 94
Spherical Harmonics	p. 99
Wavelets	p. 103
Stochastic Representation of Anomalies	p. 104
Aspects of Magnetostatics	p. 105
References	p. 110
Observations and Field Activities	p. 113
Introduction	p. 113
Principles of Gravity Measurement and Instrument Types	p. 114
General Considerations	p. 114
Pendulums	p. 115
Spring Gravimeters	p. 115
Vibrating String Gravity Meters	p. 119
Beam Balances	p. 119
Absolute Gravity Meters	p. 120
Superconducting Gravity Meters	p. 120
Artificial Satellites	p. 121
Torsion Balance and Gradiometer	p. 122
Special Task Gravity Meters	p. 122
Scale and Drift of Gravimeters	p. 124
Instrument Scale	p. 124
Instrumental Drift	p. 124
Planning a Survey	p. 127
General	p. 127
Base Stations	p. 127
Base Station Networks	p. 128
Field Stations	p. 128
Field Procedures	p. 129
Setting Up Stations	p. 129
Surveying Requirements	p. 129
Additional Field Operations	p. 130
Instrument Calibration	p. 130
Surveying or Levelling and Recording of Earth Tides	p. 131
Rock Densities	p. 132
Preparing the Data for Reductions and Analysis	p. 138
Error Assessment and Accuracy	p. 138
Outliers	p. 139
Systematic Errors	p. 139
Random Errors	p. 140
Conclusion	p. 141
References	p. 142
Gravity Anomalies and Disturbances: Reductions and Analyses	p. 151
Introduction	p. 151
Earth Tide Reduction	p. 152
The Time-Invariant Gravity Anomalies and Their Fundamental Properties	p. 153
Components of Observed Gravity	p. 156
Normal Gravity	p. 157
Deviations From the Normal Earth	p. 159
The Reductions	p. 159
The Normal Reduction	p. 160
The Height Reduction	p. 160
Topographic Mass Reduction	p. 160
The Result of the Reductions: Gravity Anomalies and Gravity Disturbances	p. 164
FA: Free Air Anomaly, Faye Anomaly	p. 164
BA: Bouguer Anomaly	p. 165
IA: Isostatic Anomalies	p. 165
Preliminary Data Analysis	p. 166
General Aspects	p. 166
Data Snooping or Identifying Outliers	p. 166
Smoothing, Averaging, Filtering	p. 167
Functional Fitting	p. 168
Statistical Approach, Correlations, Regression	p. 171
Derived Field Quantities	p. 174
Regional-Residual Separation	p. 174
Directional Analysis	p. 176
Evaluation of Reduction Errors	p. 177
Conclusion	p. 178
References	p. 178
Qualitative Interpretation	p. 181
Fundamental Ideas, Principles	p. 181
Qualitative and Quantitative Interpretation	p. 181
The Ambiguity Problem and a priori Information	p. 182
Information Content of Gravity Anomalies	p. 184
Data Representation and Interpretation Constraints	p. 185
Anomaly and Model Effect	p. 185
Digital and Visual Modes of Representation	p. 188
Geometrical Constraints: Patterns	p. 189
Three-Dimensional - Two-Dimensional	p. 189
Spatial Frequency	p. 189
Physical Constraints: Realistic Limits, Integral Relations	p. 191
Geological Constraints: Visual and Statistical Analyses, Structures, Densities	p. 192
General Remarks	p. 192
Scale	p. 192
Gravity in Relation with Other Geological Quantities	p. 192
Some Simple Estimates of Gravity Effects	p. 193
Bouguer Plate	p. 193
Scale Rule	p. 194
Half Width Rules	p. 195
Use of the Solid Angle $$: Vertical Templates	p. 198
Undulated Boundaries	p. 199
Diagrams	p. 199
Maximum Depth Rules	p. 200
Edge Effects	p. 200
Vertical Dipoles	p. 202
Examples	p. 210
Messel Maar Crater and Fault Zone (MFZ) and Meerfeld Maar	p. 210
Salt Diapir: Helgoland	p. 214
Granite Batholiths: Bancroft Area, Canada	p. 215
Rhine Graben	p. 217
SE Iceland Shelf Edge	p. 218
Spreading Ridges, Reykjanes Ridge	p. 220
Plumes, the Iceland Plume	p. 222
Tonga-Kermadec Trench, Subduction and Back arc Basin	p. 224
Mantle Convection	p. 226
Error Discussion and Conclusions	p. 229
References	p. 229
Quantitative Interpretation	p. 233
Introduction: From Qualitative to Quantitative Interpretation	p. 233
Principal Considerations: Qualitative and Quantitative Interpretation	p. 234
General Methodological Aspects	p. 235
Philosophy of Modelling: Detailed Description Versus Catching the Fundamental Features and Their Uncertainties	p. 236
Model Types: Two and Three-Dimensional; Large Model Bodies Versus Small Mass Elements	p. 237
Density	p. 238
Two-Dimensional (2D) Approximations and Modelling	p. 241
Few Large 2D Model Units	p. 241
Many Small 2D Model Units	p. 246
Two-and-a-Half Dimensional (21/2D) Models	p. 246
Three-Dimensional (3D) Approximation and Modelling	p. 247
Few Large 3D Model Units	p. 247
Many Small 3D Model Units	p. 251
Summary: Strategies of Model Building; from Trial and Error to Inversion	p. 253
Examples	p. 253
Messel Maar Crater and Fault Zone (MFZ)	p. 254
Salt Structure of Helgoland	p. 256
Anstruther Batholith: Bancroft Area, Canada	p. 258
Rhine Graben	p. 260
The SE Iceland Shelf Edge	p. 261
Spreading Ridges	p. 263
Mantle Plumes	p. 264
Tonga-Kermadec Trench, Subduction and Back Arc Basin	p. 265
Mantle Convection	p. 268
Summary of Chapter 6	p. 268
References	p. 268
Optimization and Inversion	p. 271
Introduction	p. 271
Optimization	p. 273
Theory	p. 274
Practice	p. 285
Inversion	p. 304
Overview	p. 305
Direct Algorithms	p. 308
Other Algorithms	p. 328
Case Studies	p. 332
Meerfeld Maar	p. 333
SE Iceland Shelf: Edge Effect	p. 362
Outlook	p. 378
References	p. 381
Appendix: Analytical Geometry	p. 383
Index	p. 387
Table of Contents provided by Ingram. All Rights Reserved.

Gravity Interpretation

Fundamentals and Application of Gravity Inversion and Geological Interpretation

At a Glance

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