Handbook of GeoMathematics

Cover\r......Page 1
Preface\r......Page 6
Contents\r......Page 10
1 Geomathematics: Its Role, Its Aim, and Its Potential......Page 21
2 Geomathematics as Cultural Asset......Page 22
3 Geomathematics as Task and Objective......Page 23
5 Geomathematics as Challenge......Page 25
6 Geomathematics as Solution Potential......Page 27
7 Geomathematics as Solution Method......Page 29
8 Geomathematics: Two Exemplary ldquoCircuitsrdquo......Page 32
8.1 Circuit: Gravity Field from Deflections of the Vertical......Page 33
8.1.1 Mathematical Modeling of the Gravity Field......Page 34
8.1.2 Mathematical Analysis......Page 45
8.1.3 Development of a Mathematical Solution Method......Page 46
8.1.4 ldquoBack-transferrdquo to Application......Page 48
8.2.1 Mathematical Modeling of Ocean Flow......Page 51
8.2.3 Development of a Mathematical Solution Method......Page 54
8.2.4 ldquoBack-transferrdquo to Application......Page 56
9 Final Remarks......Page 58
References......Page 59
2 Navigation on Sea: Topics in the History of Geomathematics......Page 61
3 The History of the Magnet......Page 62
4 Early Modern England......Page 64
5 The Gresham Circle......Page 65
6 William Gilberts Dip Theory......Page 66
7 The Briggsian Tables......Page 72
8 The Computation of the Dip Table......Page 80
References......Page 86
3 Earth Observation Satellite Missions and Data Access......Page 87
1 Introduction......Page 89
2.1 Space Segment......Page 90
2.2.1 Flight Operations Segment......Page 91
2.2.2 Payload Data Segment......Page 92
3.2 ERS-1 and ERS-2 Missions......Page 94
3.3 Envisat Mission......Page 95
3.5.2 SMOS (Soil Moisture and Ocean Salinity)......Page 96
3.5.7 Future Earth Explorers......Page 97
3.6.1 Sentinel Missions......Page 98
3.7.1 Meteosat......Page 99
3.8 ESA Third Party Missions......Page 100
4.1 Land......Page 101
4.2 Ocean......Page 102
4.3 Cryosphere......Page 103
4.4 Atmosphere......Page 104
4.5 ESA Programmes for Data Exploitation......Page 105
5 User Access to ESA Data......Page 106
5.1 How to Access the EO Data at ESA......Page 107
References......Page 108
4 GOCE: Gravitational Gradiometry in a Satellite......Page 109
1 Introduction: GOCE and Earth Sciences......Page 110
2 GOCE Gravitational Sensor System......Page 111
3 Gravitational Gradiometry......Page 113
References......Page 117
5 Sources of the Geomagnetic Field and the Modern Data That Enable Their Investigation......Page 121
2 Sources of the Earthaposs Magnetic Field......Page 122
2.1.1 Core Field......Page 124
2.1.2 Crustal Field......Page 125
2.2.1 Ionospheric Contributions......Page 127
2.2.3 Induction in the Solid Earth and the Oceans......Page 128
3.1 Definition of Magnetic Elements and Coordinates......Page 129
3.2 Ground Data......Page 130
3.3 Satellite Data......Page 131
4 Making the Best of the Data to Investigate the Various Field Contributions:Geomagnetic Field Modelling......Page 136
References......Page 138
6 Classical Physical Geodesy......Page 141
1.1 Preliminary Remarks......Page 144
2.1 Gravitational Potential and Gravity Field......Page 145
2.2 The Normal Field......Page 146
2.3 The Geoid and Height Systems......Page 147
2.4.1 Separability of Gravitation and Inertia......Page 148
2.4.3 Separability in Second-Order Gradients......Page 149
2.4.4 Satellite Orbits......Page 151
2.4.6 Gravitation and Time......Page 152
3.1 Harmonic Functions and Spherical Harmonics......Page 154
3.2 Convergence and Analytical Continuation......Page 155
3.3 More About Convergence......Page 157
3.4 Krarupaposs Density Theorem......Page 158
4.1.1 Nonlinear Inverse Problems of Functional Analysis......Page 159
4.1.3 Linearization......Page 161
4.1.4 Solution by Spherical Harmonics: A Useful Formula for Spherical Harmonics......Page 163
4.1.5 Solution by Stokes-Type Integral Formulas......Page 164
4.2.1 Principles......Page 165
4.2.2 Least-Squares Collocation......Page 167
4.2.3 Concluding Remarksast-2pc......Page 168
6 Conclusion......Page 169
6.1 The Geoid......Page 170
References......Page 171
7 Spacetime Modeling of the Earth\'s Gravity Field by Ellipsoidal Harmonics......Page 173
1 Introduction......Page 175
2 Dirichlet Boundary-Value Problem on the Ellipsoid of Revolution......Page 177
2.1 Formulation of the Dirichlet Boundary-Value Problem on an Ellipsoidof Revolution......Page 178
2.2 Power-Series Representation of the Integral Kernel......Page 180
2.3 The Approximation of O(e02)......Page 182
2.4 The Ellipsoidal Poisson Kernel......Page 184
2.5 Spatial Forms of Kernels Li(t,x)......Page 186
2.6 Residuals Ri(t,x)......Page 187
2.7 The Behavior at the Singularity......Page 189
2.8 Conclusion......Page 190
3.1 Formulation of the Stokes Problem on an Ellipsoid of Revolution......Page 191
3.2 The Zero-Degree Harmonic of T......Page 193
3.3 Solution on the Reference Ellipsoid of Revolution......Page 194
3.4 The Derivative of the Legendre Function of the Second Kind......Page 195
3.6 The Approximation up to O(e02)......Page 196
3.7 The Ellipsoidal Stokes Function......Page 199
3.8 Spatial Forms of Functions Ki(cos)......Page 200
3.9 Conclusion......Page 204
4.1 Representation of the Actual Gravity Vector as Well as the Reference GravityVector Both in a Global and a Local Frame of Reference......Page 205
4.2 The Incremental Gravity Vector......Page 207
5 Vertical Deflections and Gravity Disturbance in Geometry Space......Page 210
5.1 Ellipsoidal Coordinates of Type Gauss Surface Normal......Page 211
5.2 Jacobi Ellipsoidal Coordinates......Page 213
6.1 Ellipsoidal Reference Potential of Type Somigliana–Pizzetti......Page 219
6.2 Ellipsoidal Reference Gravity Intensity of Type Somigliana–Pizzetti......Page 222
6.3 Expansion of the Gravitational Potential in Ellipsoidal Harmonics......Page 226
6.4 External Representation of the Incremental Potential Relative to theSomigliana–Pizzetti Potential Field of Reference......Page 228
7 Ellipsoidal Reference Potential of Type Somigliana–Pizzetti......Page 229
7.1 Vertical Deflections and Gravity Disturbance in Vector-Valued EllipsoidalSurface Harmonics......Page 230
7.2 Vertical Deflections and Gravity Disturbance Explicitly in Terms of EllipsoidalSurface Harmonics......Page 235
8 Case Studies......Page 237
9 Curvilinear Datum Transformations......Page 239
10 Datum Transformations in Terms of Spherical Harmonic Coefficients......Page 246
11 Datum Transformations of Ellipsoidal Harmonic Coefficients......Page 249
12 Examples......Page 258
References......Page 262
8 Time-Variable Gravity Field and Global Deformation of the Earth......Page 267
1 Introduction......Page 268
2 Mass and Mass Redistribution......Page 269
3 Earth Model......Page 274
4 Analysis of TVG and Deformation Pattern......Page 277
5 Future Directions......Page 280
References......Page 281
9 Satellite Gravity Gradiometry (SGG): From Scalar to Tensorial Solution......Page 283
1 Introduction......Page 284
2 SGG in Potential Theoretic Perspective......Page 285
3 Decomposition of Tensor Fields by Means of Tensor Spherical Harmonics......Page 291
4 Solution as Pseudodifferential Equation......Page 295
4.1 SGG as Pseudodifferential Equation......Page 297
4.2 Upward/Downward Continuation......Page 298
4.4 Pseudodifferential Operator for SST......Page 299
4.7 Survey on Pseudodifferential Operators Relevant in Satellite Technology......Page 301
4.9 A Short Introduction to the Regularization of Ill-Posed Problems......Page 304
4.10 Regularization of the Exponentially Ill-Posed SGG-Problem......Page 309
5 Future Directions......Page 310
6 Conclusion......Page 314
References......Page 315
10 Gravitational Viscoelastodynamics......Page 317
1 Introduction......Page 318
2.1 Kinematic Representations......Page 320
2.2 Total, Initial, and Incremental Fields......Page 322
2.3 Interface Conditions......Page 323
3.1 Equations for the Total Fields......Page 324
3.2 Equations for the Initial Fields......Page 326
3.3.1 Material Form......Page 327
3.3.2 Material–Local Form......Page 328
3.3.3 Local Form......Page 329
3.3.4 Constitutive Equation......Page 330
3.4 Continuity and State Equations......Page 331
4 Asymptotic Incremental Field Theories......Page 332
4.1 Relaxation Functions......Page 333
4.2.2 Small-s Asymptotes......Page 334
4.3.1 Small-t Asymptotes: Field Theory of GED......Page 335
4.3.2 Large-t Asymptotes: Field Theory of GVD......Page 336
5 Approximate Incremental Field Theories......Page 337
5.1.2 Equations for the Incremental Fields: Local Form......Page 338
5.2.2 Equations for the Incremental Fields: Local Form......Page 340
6 Summary......Page 341
References......Page 344
11 Multiresolution Analysis of Hydrology and Satellite Gravitational Data......Page 347
1 Introduction......Page 348
2.1 Preliminaries......Page 349
2.2 Multiresolution in Hilbert Spaces......Page 350
2.3.1 Legendre Wavelets......Page 352
3.1 Tensorial Time–Space Multiresolution......Page 353
3.2 Correlation Analysis Between GRACE and WGHM......Page 356
4 Fundamental Results......Page 357
5 Future Directions......Page 359
References......Page 364
12 Time Varying Mean Sea Level......Page 367
1 Introduction......Page 368
2 Theoretical Considerations......Page 369
4 Data and Methodology of Numerical Treatment......Page 372
5 Fundamental Results......Page 374
6.1 Accuracy of Sea Level Observation......Page 377
6.5 Prediction of Sea Level Evolution......Page 380
6.6 Mathematical Representation of Time-Varying Sea Level......Page 381
7 Conclusions......Page 382
References......Page 383
13 Unstructured Meshes in Large-Scale Ocean Modeling......Page 385
1 Introduction......Page 386
2 Dynamic Equations and Typical Approximations......Page 388
3 Finite-Element and Finite-Volume Methods......Page 391
3.1 FE Method......Page 392
3.2 Finite Volumes......Page 393
3.3 Discontinuous FE......Page 394
4.1 Consistency Between Elevation and Vertical Velocity......Page 395
4.2 Consistency of Tracer Spaces and Tracer Conservation......Page 396
4.3 Energetic and Pressure Consistency......Page 397
5.1 Preliminary Remarks......Page 399
5.2 Solving the Dynamical Part with NC Elements......Page 400
5.3 Solving the Dynamical Part with the CL Approach......Page 403
5.4 Vertical Velocity, Pressure, and Tracers......Page 404
6.1 C-Grid......Page 405
6.2 P0-P1-Like Discretization......Page 407
7 Conclusions......Page 409
References......Page 410
14 Numerical Methods in Support of Advanced Tsunami Early Warning......Page 413
1 Introduction......Page 414
2 Tsunami Propagation and Inundation Modeling......Page 415
2.2 Unstructured Grid Tsunami Simulation......Page 416
2.3 Adaptive Mesh Refinement for Tsunami Simulation......Page 417
3.1 Existing Tsunami Early Warning Systems......Page 421
3.2 Multisensor Selection Approach......Page 422
3.3 Implementing the Matching Procedure......Page 424
3.4 Experimental Confirmation......Page 426
4 Future......Page 428
References......Page 429
15 Efficient Modeling of Flow and Transport in Porous Media......Page 431
1 Introduction......Page 433
2.1 Definition of Scales......Page 436
Pseudo Functions......Page 438
Variational Multiscale Method......Page 439
2.3 Multiphysics Methods......Page 440
Components......Page 441
Density......Page 442
Saturation......Page 443
Capillarity......Page 444
Relative Permeability......Page 445
Dalton\'s Law......Page 446
3.1.7 The Reynolds Transport Theorem......Page 447
3.2.1 The Immiscible Case......Page 448
3.2.2 The Miscible Case......Page 449
Global Pressure Formulation for Two-Phase Flow......Page 450
Saturation Equation......Page 451
Phase Pressure Formulation for Two-Phase Flow......Page 452
3.3.2 The Miscible Case......Page 453
3.4 Non-Isothermal Flow......Page 454
4.2.1 The Immiscible Case......Page 455
Flash Calculations......Page 457
5.1.1 Single-Phase Transport......Page 458
5.1.2 Model Coupling......Page 459
5.1.3 Practical Implementation......Page 460
5.2 A Multiscale Multiphysics Example......Page 461
5.2.1 Multiscale Multiphysics Algorithm......Page 463
5.2.2 Numerical Results......Page 464
6 Conclusion......Page 465
References......Page 466
16 Numerical Dynamo Simulations: From Basic Concepts to Realistic Models......Page 473
1 Introduction......Page 474
2.1 Basic Equations......Page 477
2.2 Boundary Conditions......Page 479
2.3 Numerical Methods......Page 480
2.4 Force Balances......Page 483
3.1 Dynamo Regimes......Page 487
3.2 Dynamo Mechanism......Page 491
3.3 Comparison with the Geomagnetic Field......Page 493
3.3.1 Dipole Properties and Symmetries......Page 495
3.3.2 Persistent Features and Mantle Influence......Page 498
3.3.4 Time Variability......Page 501
3.4.1 The Strong Field Branch......Page 504
3.4.2 Simulations Results at Low Ekman Numbers......Page 507
4 Conclusion......Page 510
References......Page 512
17 Mathematical Properties Relevant to Geomagnetic Field Modeling......Page 517
1 Introduction......Page 518
2 Helmholtz\'s Theorem and Maxwell\'s Equations......Page 519
3.1 Magnetic Fields in a Source-Free Shell......Page 520
3.2 Surface Spherical Harmonics......Page 522
3.3 Magnetic Fields from a Spherical Sheet Current......Page 525
4.1 Helmholtz Representations and Vector Spherical Harmonics......Page 526
4.2 Mie Representation......Page 530
4.3 Relationship of B and J Mie Representations......Page 532
4.4 Magnetic Fields in a Current-Carrying Shell......Page 534
5 Spatial Power Spectra......Page 536
6.1 Uniqueness of Magnetic Fields in a Source-Free Shell......Page 537
6.2 Uniqueness Issues Raised by Directional-Only Observations......Page 540
6.3 Uniqueness Issues Raised by Intensity-Only Observations......Page 541
6.4 Uniqueness of Magnetic Fields in a Shell Enclosing a Spherical Sheet Current......Page 544
6.5 Uniqueness of Magnetic Fields in a Current-Carrying Shell......Page 545
7 Concluding Comments: From Theory to Practice......Page 549
References......Page 550
18 Multiscale Modeling of the Geomagnetic Field and Ionospheric Currents......Page 553
1 Introduction......Page 554
2 Scientifically Relevant Function Systems......Page 556
2.1 Vector Spherical Harmonics......Page 557
2.2 Green\'s Function for the Beltrami Operator......Page 558
3 Key Issues for Multiscale Techniques......Page 559
3.1 Wavelets as Frequency Packages......Page 560
3.2 Locally Supported Wavelets......Page 563
3.2.1 Regularized Green\'s Function and Single Layer Kernel......Page 564
4 Application to Geomagnetic Problems......Page 566
4.1.1 Wavelets as Frequency Packages......Page 567
4.1.2 Locally Supported Wavelets......Page 570
4.2.1 Wavelets as Frequency Packages......Page 572
4.2.2 Locally Supported Wavelets......Page 573
6 Conclusion......Page 575
References......Page 576
19 The Forward and Adjoint Methods of Global EM Induction for CHAMP Magnetic Data......Page 579
1 Introduction......Page 582
2 Basic Assumptions on EM Induction Modeling for CHAMP Magnetic Data......Page 584
3.1 Formulation of EM Induction for a 3-D Inhomogeneous Earth......Page 585
3.2 Special Case: EM Induction in an Axisymmetric Case......Page 588
3.3 Gauss Representation of Magnetic Induction in the Atmosphere......Page 590
4.1 Classical Formulation......Page 592
4.2.1 Ground Magnetic Data......Page 593
4.2.2 Satellite Magnetic Data......Page 594
4.3 Frequency-Domain and Time-Domain Solutions......Page 595
4.4 Vector Spherical Harmonics Parameterization Over Colatitude......Page 596
4.5 Finite-Element Approximation over the Radial Coordinate......Page 597
4.6 Solid Vector Spherical Harmonics Parameterization of A0......Page 598
5 Forward Method of EM Induction for the ExternalGauss Coefficients......Page 600
5.1 Classical Formulation......Page 601
6 Time-Domain, Spectral Finite-Element Solution......Page 602
7.2 Two-Step, Track-by-Track Spherical Harmonic Analysis......Page 604
7.2.2 Extrapolation of Magnetic Data from Mid-Latitudes to Polar Regions......Page 606
7.2.3 Selection Criteria for Extrapolation......Page 607
7.2.4 Examples of Spherical Harmonic Analysis of the CHAMP Magnetic Data......Page 608
7.3 Power-Spectrum Analysis......Page 609
8 Adjoint Sensitivity Method of EM Induction for the Z Componentof CHAMP Magnetic Data......Page 613
8.1 Forward Method......Page 614
8.2 Misfit Function and Its Gradient in the Parameter Space......Page 615
8.3 The Forward Sensitivity Equations......Page 616
8.4 The Adjoint Sensitivity Equations......Page 617
8.5 Boundary Condition for the Adjoint Potential......Page 619
8.6 Adjoint Method......Page 620
8.7 Reverse Time......Page 621
8.8 Weak Formulation......Page 622
9.2 Misfit Function and Its Gradient in the Parameter Space......Page 623
9.3 Adjoint Method......Page 624
9.5 Summary......Page 626
10.2 Model Parameterization......Page 627
10.3.2 Conjugate Gradient Inversion......Page 628
10.4.2 Conjugate Gradient Inversion......Page 630
11 Conclusions......Page 631
References......Page 635
20 Asymptotic Models for Atmospheric Flows......Page 639
1 Introduction......Page 640
2.1 QG-Theory......Page 641
2.2 Sound-Proof Models......Page 642
3.1 Characteristic Scales and Dimensionless Parameters......Page 644
3.2 Distinguished Limits......Page 646
4 Classical Single-Scale Models......Page 647
4.1.2 Some Revealing Transformations......Page 648
4.2 Midlatitude Internal Gravity Wave Models......Page 650
4.3 Balanced Models for Advection Time Scales......Page 651
4.3.2 Synoptic Scales and the Quasi-Geostrophic Approximation......Page 652
4.4 Scalings for Near-Equatorial Motions......Page 654
4.4.2 Equatorial Synoptic and Planetary Models......Page 655
4.5 The Hydrostatic Primitive Equations......Page 656
5.1 Shaw and Shepherd\'s Parameterization Framework......Page 658
6 Conclusions......Page 659
References......Page 660
21 Modern Techniques for Numerical Weather Prediction: A Picture Drawn from Kyrill......Page 663
1.1 What Is a Weather Forecast?......Page 664
2 Data Assimilation Methods: The Journey from 1d-Var to 4d-Var......Page 665
2.2 Variational Analysis......Page 666
3.1.1 -Coordinates......Page 667
3.1.2 -Coordinates......Page 668
3.2 The Eulerian Formulation of the Continuous Equations......Page 669
3.3.1 Clouds and Precipitation......Page 670
3.3.3 Precipitation......Page 671
3.4 The Discretization......Page 672
4 Ensemble Forecasts......Page 673
5 Statistical Weather Forecast (MOS)......Page 675
6 Applying the Techniques to Kyrill......Page 676
6.1 Analysis of the Air Pressure and Temperature Fields......Page 678
6.2 Analysis of Kyrills Surface Winds......Page 679
6.3 Analysis of Kyrill\'s 850hPa Winds......Page 680
6.4 Ensemble Forecasts......Page 683
6.5 MOS Forecasts......Page 685
6.6 Weather Radar......Page 688
7 Conclusion......Page 690
References......Page 691
22 Modeling Deep Geothermal Reservoirs: Recent Advances and Future Problems......Page 693
1 Introduction......Page 694
2 Scientific Relevance......Page 695
2.1 Reservoir Detection......Page 696
2.2 Stress and Flow Problems......Page 697
3.1 Mathematical Models of Seismic Migration and Inversion......Page 699
3.2.1 Hydrothermal Systems......Page 702
3.2.2 Petrothermal Systems......Page 703
Discrete Models......Page 705
4.1.1 Reverse-Time Migration Using Compact Differences Schemes......Page 708
4.1.2 Wavelet Approach for Seismic Wave Propagation......Page 711
4.2 Heat Transport......Page 716
5 Future Directions......Page 718
6 Conclusion......Page 719
References......Page 720
23 Phosphorus Cycles in Lakes and Rivers: Modeling, Analysis, and Simulation......Page 727
1 Introduction......Page 728
2 Mathematical Modeling......Page 729
3 Numerical Method......Page 731
3.1 Finite Volume Method......Page 732
3.2 Positivity Preserving and Conservative Schemes......Page 735
3.2.1 Numerical Results for Positive Ordinary Differential Equations......Page 741
3.3 Practical Applications......Page 743
4 Conclusion......Page 748
References......Page 750
24 Noise Models for Ill-Posed Problems......Page 753
2 Noise in Well-Posed and Ill-Posed Problems......Page 754
2.1 Well-Posed and Ill-Posed Problems......Page 755
2.2 Fredholm Integral Equations with Strong Noise......Page 756
2.3 Super-Strong and Weak Noise Models......Page 758
3 Weakly Bounded Noise......Page 759
4.1 Strongly Bounded Noise......Page 761
4.2 Weakly Bounded Noise......Page 763
5.1 Discrepancy Principles......Page 765
5.2 Lepskiı\'s Principle......Page 767
6 An Example: The Second Derivative of a Univariate Function......Page 768
7 Conclusions and Future Directions......Page 770
8 Synopsis......Page 771
References......Page 772
25 Sparsity in Inverse Geophysical Problems......Page 775
1.1 Case Example: Ground Penetrating Radar......Page 776
2 Variational Regularization Methods......Page 777
3 Sparse Regularization......Page 779
3.1 Convex Regularization......Page 780
3.2 Nonconvex Regularization......Page 783
4.1 Iterative Thresholding Algorithms......Page 784
4.2 Second Order Cone Programs......Page 785
5.1 Mathematical Model......Page 786
5.1.1 Born Approximation......Page 787
5.1.2 The Radiating Reflectors Model......Page 788
5.2 Migration Versus Nonlinear Focusing......Page 789
5.2.1 The Limited Data Problem......Page 790
5.2.2 Application of Sparsity Regularization......Page 791
5.4 Application to Real Data......Page 794
References......Page 795
26 Quantitative Remote Sensing Inversion in Earth Science: Theory and Numerical Treatment......Page 797
1 Introduction......Page 799
2.1 Land Surface Parameter Retrieval Problem......Page 801
2.2 Backscatter Cross-Section Inversion with Lidar......Page 802
2.3 Aerosol Inverse Problems......Page 803
3.1 What Causes Ill-Posedness......Page 804
3.3 Tikhonov/Phillips–Twomey\'s Regularization......Page 806
3.3.1 Choices of the Scale Operator D......Page 807
3.3.2 Regularization Parameter Selection Methods......Page 808
3.4 Direct Regularization......Page 809
3.5 Statistical Regularization......Page 810
4.1 Sparse/Nonsmooth Inversion in l1 Space......Page 811
4.2.1 Newton-Type Methods......Page 813
4.2.2 Gradient-Type Methods......Page 814
5.1.3 Land Surface Parameter Retrieval Results......Page 815
5.2 Inversion of Airborne Lidar Remote Sensing......Page 817
5.3 Particle Size Distribution Function Retrieval......Page 818
6 Conclusion......Page 821
References......Page 822
27 Multiparameter Regularization in Downward Continuation of Satellite Data......Page 825
1 Introduction......Page 826
2 A Functional Analysis Point of View on Satellite Geodetic Problems......Page 829
3 An Appearance of a Multiparameter Regularization in the Geodetic Context:Theoretical Aspects......Page 834
4 Computational Aspects of Some Multiparameter Regularization Schemes......Page 837
5 Numerical Illustrations......Page 840
References......Page 843
28 Correlation Modeling of the Gravity Field in Classical Geodesy......Page 845
1 Introduction......Page 846
2 Correlation Functions......Page 847
2.1 Functions on the Sphere......Page 848
2.2 Functions on the Plane......Page 849
2.3 From the Sphere to the Plane......Page 851
2.4 Properties of Correlation Functions and PSDs......Page 852
3 Stochastic Processes and Covariance Functions......Page 855
3.1 Earth\'s Anomalous Gravitational Field......Page 857
3.2 The Disturbing Potential as a Stochastic Process......Page 859
4 Covariance Models......Page 862
4.2 The Reciprocal Distance Model......Page 863
4.3 Parameter Determination......Page 865
5 Summary and Future Directions......Page 866
References......Page 874
29 Modeling Uncertainty of Complex Earth Systems in Metric Space......Page 877
1 Introduction......Page 878
2 Nomenclature......Page 879
3.1 Characteristics of Modeling in the Earth Sciences......Page 880
3.2 Distances, Metric Space, and Multidimensional Scaling......Page 881
3.3 Kernels and Feature Space......Page 884
3.4 Model Expansion in Metric Space......Page 888
4.1 The Pre-Image Problem......Page 891
4.2 The Post-Image Problem......Page 894
5.1 Illustration of the Post-Image Problem......Page 897
6 Conclusions and Future Directions......Page 898
References......Page 901
30 Slepian Functions and Their Use in Signal Estimation and Spectral Analysis......Page 903
1 Introduction......Page 905
2.1.1 General Theory in One Dimension......Page 906
2.2.1 General Theory in Two Dimensions......Page 908
2.2.2 Sturm–Liouville Character and Tridiagonal Matrix Formulation......Page 909
2.3.1 General Theory in ``Three\'\' Dimensions......Page 913
2.3.2 Sturm–Liouville Character and Tridiagonal Matrix Formulation......Page 916
2.4 Mid-Term Summary......Page 917
3 Problems in the Geosciences and Beyond......Page 918
3.1.1 Spherical Harmonic Solution......Page 919
3.1.3 Bias and Variance......Page 920
3.2 Problem (ii): Power Spectrum Estimation from Noisyand Incomplete Spherical Data......Page 921
3.2.3 Variance of the Periodogram......Page 922
3.2.5 Bias of the Multitaper Estimate......Page 923
4 Practical Considerations......Page 924
4.1.2 Slepian Basis Solution......Page 927
4.1.3 Bias and Variance......Page 929
4.2.1 The Spherical Periodogram......Page 930
5 Conclusions......Page 931
References......Page 932
31 Special Functions in Mathematical Geosciences: An Attempt at a Categorization......Page 937
1 Introduction......Page 938
2.1 Spherical Harmonics......Page 939
2.2 Transition to Zonal Kernel Functions......Page 941
2.3 Transition to the Vector and Tensor Context......Page 944
3 The Uncertainty Principle as Key Issue for Classification......Page 946
3.2 Localization in Frequency......Page 947
3.3 Uncertainty......Page 949
4 Fundamental Results......Page 950
4.1 Localization of Spherical Harmonics......Page 951
4.2 Localization of Nonbandlimited Kernels......Page 952
4.3 Quantitative Illustration of Localization......Page 953
5.2 Multiscale Approach......Page 955
6 Conclusion......Page 958
References......Page 960
32 Tomography: Problems and Multiscale Solutions......Page 961
2.1 In the Univariate Case......Page 962
2.2 On the 2-Sphere......Page 964
2.3 On the 3d-Ball......Page 965
3.1 Product Series......Page 966
3.2.1 General Aspects of Reproducing Kernels......Page 968
3.2.2 Reproducing Kernel Hilbert Spaces on the 3d-Ball......Page 969
4 Splines......Page 972
5.1 For the Approximation of Functions......Page 974
5.2 For Operator Equations......Page 977
6.1 The Inverse Gravimetric Problem......Page 978
6.3 Traveltime Tomography......Page 980
6.4 Inverse EEG and MEG......Page 982
References......Page 983
33 Material Behavior: Texture and Anisotropy......Page 985
2 Scientific Relevance......Page 986
3 Rotations and Crystallographic Orientations......Page 987
3.1 Parametrizations and Embeddings......Page 989
3.2 Harmonics......Page 990
3.4 Crystallographic Symmetries......Page 992
3.5 Geodesics, Hopf Fibres, and Clifford Tori......Page 993
4 Totally Geodesic Radon Transforms......Page 994
4.1 Properties of the Spherical Radon Transform......Page 995
Darboux Differential Equation......Page 996
Range......Page 997
Localization......Page 998
5 Texture Analysis with Integral Orientation Measurements: TextureGoniometry Pole Intensity Data......Page 999
6 Texture Analysis with Individual Orientation Measurements: ElectronBack Scatter Diffraction Data......Page 1002
7.1 Effective Physical Properties of Crystalline Aggregates......Page 1005
7.2 Properties of Polycrystalline Aggregates with Texture......Page 1008
7.3 Properties of Polycrystalline Aggregates—An Example......Page 1010
8 Future Directions......Page 1011
9 Conclusions......Page 1012
References......Page 1013
34 Dimensionality Reduction of Hyperspectral Imagery Data for Feature Classification......Page 1017
1 Introduction......Page 1019
2 Euclidean Distance and Gram Matrix on HSI......Page 1020
2.1 Euclidean and Related Distances on HSI......Page 1021
2.2 Gram Matrix on HSI......Page 1024
3 Linear Methods for Dimensionality Reduction......Page 1027
3.1 Principal Component Analysis......Page 1028
3.2 Linear Multidimensional Scaling......Page 1031
4 Nonlinear Methods for Dimensionality Reduction......Page 1032
4.2 Isomap......Page 1034
4.2.2 Isomap Algorithm......Page 1035
4.2.3 Conclusion......Page 1036
4.3.2 SDP Algorithm......Page 1037
4.4.1 Description of the LLE Method......Page 1038
4.4.2 LLE Algorithm......Page 1039
4.5.1 Description of the LTSA Method......Page 1041
4.5.2 LTSA algorithm......Page 1042
4.6.1 Description of the Laplacian Eigenmap Method......Page 1043
4.6.2 Theoretic Discussion of the Laplacian Eigenmap Method......Page 1044
4.7.1 Theoretic Background on HLLE Method......Page 1045
4.7.3 HLLE Algorithm......Page 1047
4.8.2 Theoretic Background of the Diffusion Maps Method......Page 1049
4.8.3 Diffusion Maps Algorithm......Page 1051
4.9.1 Anisotropic Transform Method Description......Page 1053
4.9.2 Theoretic Background of the Anisotropic Transform Method......Page 1054
4.9.3 Anisotropic Transform Algorithm......Page 1055
5 Appendix. Glossary......Page 1056
References......Page 1058
35 Oblique Stochastic Boundary-Value Problem......Page 1061
1 Introduction......Page 1062
2 Scientifically Relevant Domains and Function Spaces......Page 1063
3 Poincaregrave Inequality as Key Issue for the Inner Problem......Page 1066
3.1 The Weak Formulation......Page 1067
3.2 Existence and Uniqueness Results for the Weak Solution......Page 1069
3.3 A Regularization Result......Page 1070
3.4 Ritz–Galerkin Approximation......Page 1071
3.5 Stochastic Extensions......Page 1072
4 Fundamental Results for the Outer Problem......Page 1073
4.1 Transformations to an Inner Setting......Page 1074
4.2 Solution Operator for the Outer Problem......Page 1079
4.3 Ritz–Galerkin Method......Page 1081
4.4 Stochastic Extensions and Examples......Page 1082
4.4.1 Gaussian Inhomogeneities......Page 1083
4.4.3 Noise Model for Satellite Data......Page 1084
5 Future Directions......Page 1085
References......Page 1086
36 Geodetic Deformation Analysis with Respect to an Extended Uncertainty Budget......Page 1087
1 Introduction......Page 1088
2.1 Background......Page 1089
2.2 Modeling of a Deformation Process......Page 1090
2.3 Uncertainty Assessment......Page 1092
3.1 Modeling Chain in Geodetic Deformation Analysis......Page 1093
3.2 Observation Uncertainty and Its Impact......Page 1095
3.3 Uncertainty Treatment Using Deterministic Sets......Page 1098
3.4 Uncertainty Assessment in Geodetic Deformation Analysis......Page 1102
4 Fundamental Results......Page 1103
5 Future Directions......Page 1107
6 Conclusions......Page 1108
References......Page 1109
37 Mixed Integer Estimation and Validation for Next Generation GNSS......Page 1111
1.1.2 Benefits of GNSS......Page 1113
1.2 Mixed Integer Model......Page 1114
1.3 Chapter Overview......Page 1116
2.1.1 Pull-In Regions......Page 1117
2.1.3 Optimal Integer Estimation......Page 1118
2.2.1 Aperture Pull-In Regions......Page 1119
2.2.2 Probability Distribution and Successful Fix-Rate......Page 1120
2.3.1 A Larger Class of Estimators......Page 1121
2.3.2 Best Integer Equivariant Estimation......Page 1122
3.1.1 Scalar and Vectorial Rounding......Page 1123
3.1.2 Rounding Success-Rate......Page 1124
3.2.1 The Bootstrapping Principle......Page 1125
3.2.2 The Bootstrapped PMF and Success-Rate......Page 1127
3.2.3 Z-Transformations......Page 1128
3.3.1 Mixed Integer Least-Squares......Page 1129
3.3.2 The ILS Search......Page 1130
3.3.3 The Least-Squares PMF and Success-Rate......Page 1131
4.1 Fixed and Float Baseline......Page 1132
4.2 Cross-Validation of Mixed Integer Model......Page 1133
References......Page 1135
38 Mixed Integer Linear Models......Page 1139
1 Introduction......Page 1140
2.1 Classical Linear and Nonlinear Models......Page 1142
2.2 Mixed Integer Linear and Nonlinear Models......Page 1143
2.3 The Integer Least Squares Problem......Page 1144
3 Numerical Solution to the Integer LS Problem......Page 1146
3.2 Decorrelation by Gaussian Elimination......Page 1148
3.3 Decorrelation by Integer Gram-Schmidt Orthogonalization......Page 1150
3.4 Decorrelation by Inverse Integer Cholesky Decomposition......Page 1151
3.5 Do Decorrelation Methods Really Work All the Time?......Page 1152
4.1 Representation of the Integer LS Estimator......Page 1153
4.2 Voronoi Cells......Page 1154
4.3 Bounding Voronoi Cell V0 by Figures of Simple Shape......Page 1156
5.1 The Probability That the Integers Are Correctly Estimated......Page 1157
5.2 Shannon\'s Upper Probabilistic Bound......Page 1158
5.3 Upper and Lower Probabilistic Bounds......Page 1159
6 Hypothesis Testing on Integer Parameters......Page 1162
References......Page 1164
39 Statistical Analysis of Climate Series......Page 1169
2.1 Weather Stations......Page 1170
2.3 Precipitation Series......Page 1171
3.1 Comparison of the Last Two Centuries......Page 1173
4.1 Auto-Correlation Coefficient......Page 1175
4.2.1 Application to Climate Data......Page 1179
4.2.2 Folk Sayings......Page 1180
5.1 Differences, Prediction, Summation......Page 1181
5.2.2 ARIMA-Residuals......Page 1183
5.2.3 GARCH-Modeling......Page 1185
5.3 Yearly Precipitation Amounts......Page 1186
6.1 Trend+ARMA Method......Page 1187
6.2 Comparisons with Moving Averages and with Lag-12 Differences......Page 1190
7 Conclusions......Page 1191
References......Page 1194
40 Numerical Integration on the Sphere......Page 1195
1 Introduction......Page 1197
2 Sphere Basics......Page 1198
2.1 Spherical Polynomials......Page 1199
2.2 Spherical Geometry......Page 1201
3.1 Cubature Based on Partitioning the Sphere......Page 1203
4 Rules with Polynomial Accuracy......Page 1204
4.1 Longitude–Latitude Rules......Page 1205
4.2 Rules with Other Symmetries......Page 1206
4.3 Interpolatory Integration Rules......Page 1207
4.4 Interpolatory Rules Based on Extremal Points......Page 1208
4.6 Number of Points for Rules with Polynomial Accuracy......Page 1210
4.7 Gaubeta Rules for S2 Do Not Exist for L2......Page 1211
4.8 Integration Error for Rules with Polynomial Accuracy......Page 1212
5.1 Cubature Based on Voronoi Tessellation or Delaunay Triangulation......Page 1213
5.2 Cubature Rules with Polynomial Accuracy......Page 1214
5.3 Cubature Based on Spherical Radial Basis Functions......Page 1215
6 Rules with Special Properties......Page 1217
6.2 Centroidal Voronoi Tessellations......Page 1218
7.1 Integration over Spherical Caps and Collars......Page 1219
7.2 Integration over Spherical Triangles......Page 1220
8.1 Error Analysis Based on Best Uniform Approximation......Page 1221
8.2 The Sobolev Space Setting and a Lower Bound......Page 1222
8.3 Sobolev Space Error Bounds for Rules with Polynomial Accuracy......Page 1223
9 Conclusions......Page 1224
References......Page 1225
41 Multiscale Approximation......Page 1229
1 Introduction......Page 1230
2 Wavelet Analysis......Page 1231
2.1 The Discrete Wavelet Transform......Page 1232
2.2 Biorthogonal Bases......Page 1236
2.3 Wavelets and Function Spaces......Page 1237
2.4 Wavelets on Domains......Page 1238
3 Image Analysis by Means of Wavelets......Page 1242
3.1 Wavelet Compression......Page 1243
3.2 Wavelet Denoising......Page 1245
References......Page 1248
42 Sparse Solutions of Underdetermined Linear Systems......Page 1251
1 Introduction......Page 1252
2 Areas of Application......Page 1255
3 Theory Behind Matrix Selection......Page 1259
4 Algorithms......Page 1260
5 Numerical Experiments......Page 1262
6 Ways for Algorithms Improvement......Page 1265
7 Conclusion......Page 1266
References......Page 1267
43 Multidimensional Seismic Compression by Hybrid Transform with Multiscale Based Coding......Page 1269
1 Introduction......Page 1270
2.1 Local Cosine Transform......Page 1272
2.2 Wavelet Transforms......Page 1273
2.3 Multiscale Coding......Page 1274
2.4 Butterworth Wavelet Transforms......Page 1275
2.4.2 Lifting Scheme: Reconstruction......Page 1276
3.1 Outline of the Reordering Algorithm......Page 1277
3.2 The Hybrid Algorithm......Page 1278
3.3.1  Handling the Boundaries......Page 1280
4 Fundamental Results......Page 1282
4.2 Compression of Marine Shot Gather Data Section......Page 1283
4.3 Compression of the Segmented Stacked CMP Data Section......Page 1286
5 Conclusions and Future Directions......Page 1289
A Appendix: LCT Internals......Page 1291
A.2 Discretization of LCT-IV......Page 1292
A.3 Implementation of LCT-IV by Folding......Page 1293
References......Page 1294
44 Cartography......Page 1297
1 From Omnipotent to Omnipresent Maps......Page 1298
2 Ubiquitous Cartography and Invisible Cartographers......Page 1299
3.1 Geospatial Data Modeling......Page 1300
3.3 Map Design......Page 1301
4 Spectrum of Cartographic Products......Page 1303
4.1 Communicative Maps......Page 1304
4.2 Analytical Maps......Page 1305
4.4 Internet Maps......Page 1306
5 The State of the Art of Cartography......Page 1307
5.1 2D Maps Versus 3D Maps......Page 1308
5.2 Map Products Versus Map Services......Page 1309
5.3 Stationary Maps Versus Mobile Maps......Page 1311
5.4 Graphic User Interface Versus Map Symbols......Page 1312
6 Responsibilities of Cartographers......Page 1315
References......Page 1318
45 Geoinformatics......Page 1321
2 Interpretation of Geodata......Page 1322
2.2 Interpretation of 3D Point Clouds......Page 1323
3 Integration and Data Fusion......Page 1324
3.2 Geometric Matching......Page 1326
3.3 Fusion, Homogenization, and Analysis......Page 1327
4 Multiscale Concepts......Page 1328
4.2 Optimization-Based Methods for Generalization......Page 1329
4.3 Multiscale/Varioscale Representation......Page 1330
4.4 Quality of Generalization Process......Page 1332
5 Future Research Directions......Page 1333
References......Page 1334