Computational Methods in Geophysical
Electromagnetics
Eldad Haber
haber@math.ubc.ca
August 9, 2014
2
Contents
Preface
1
2
9
Maxwell’s Equations and their geophysical applications
1.1
Di↵erent representations of Maxwell’s equations . . . . . . . . .
1.1.1
Maxwell’s equations in time and frequency . . . . .
1.1.2
Integral form of Maxwell’s equations and interface
conditions . . . . . . . . . . . . . . . . . . . . . . . .
1.1.3
The equations in weak form . . . . . . . . . . . . . .
1.2
Dimensional Analysis and Geophysical Experiments . . . . . . .
1.2.1
Elliptic case - Statics with magnetic sources: Magnetics . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.2
Statics with electric sources: Direct Current and
Magnetometric Resistivity . . . . . . . . . . . . . . .
1.2.3
Parabolic approximations: Quasi Static regime and
electromagnetic induction . . . . . . . . . . . . . . .
1.2.4
Hyperbolic regime - Ground Penetrating radar and
Radio Imaging . . . . . . . . . . . . . . . . . . . . .
1.2.5
Frequency dependent conductivity and Induced Polarization . . . . . . . . . . . . . . . . . . . . . . . .
1.3
Electromagnetic fields, Sources and the EM coupling problem .
1.4
Some useful analytic solutions . . . . . . . . . . . . . . . . . . .
1.4.1
DC resistivity over a uniform earth . . . . . . . . . .
1.4.2
Magnetotelluric response of a half space . . . . . . .
1.4.3
The frequency response of a half space to a magnetic
dipole . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4.4
The frequency response of full and half space to a
electric dipole . . . . . . . . . . . . . . . . . . . . . .
1.5
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The Finite Volume method in 1D and its application to Maxwell’s
equations
2.1
Deriving the discretization . . . . . . . . . . . . . . . . . . . . .
2.2
Matrix Representation . . . . . . . . . . . . . . . . . . . . . . .
2.3
Implementation of boundary conditions . . . . . . . . . . . . . .
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31
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36
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Contents
2.4
2.5
2.6
2.7
3
4
1D in practice . . . . . . .
Programmer note: Testing
sources . . . . . . . . . . .
The 1D MT problem . . .
Exercises . . . . . . . . . .
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Finite Volume Discretization in 3D
3.1
Discretization of di↵erential operators on a tensor mesh
3.2
Inner Products on tensor mesh . . . . . . . . . . . . . .
3.2.1
Cell center inner product . . . . . . . . . .
3.2.2
Face variables inner product . . . . . . . .
3.2.3
Edge variables inner product . . . . . . . .
3.3
Discretization on logically orthogonal grids . . . . . . .
3.3.1
Di↵erential Operators . . . . . . . . . . . .
3.3.2
Computation of inner products . . . . . . .
3.4
Programming the operators using Kronecker products .
3.4.1
Programming Linear Di↵erential Operators
3.4.2
Programming the inner product matrices .
3.5
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . .
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Discretization of Maxwell’s equations
4.1
Discretization of Maxwell’s equations in frequency . . . . . . . .
4.2
Statics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3
Programming Maxwell’s equations . . . . . . . . . . . . . . . . .
4.4
Setting up sources . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4.1
Mathematical models of the sources and receivers .
4.4.2
Direct discretization of the sources . . . . . . . . . .
4.4.3
Primary secondary discretization . . . . . . . . . . .
4.5
Time stepping . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5.1
A (very) quick review of methods for time dependent
ODE’s . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5.2
Discretization of Maxwell’s equations in time . . . .
4.5.3
Initialization of the Magnetic fields and their divergence . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5.4
Exponential integrators . . . . . . . . . . . . . . . .
4.6
Solving the linear systems . . . . . . . . . . . . . . . . . . . . .
4.6.1
The ill-conditioning of the discrete Maxwell system
4.6.2
Iterative solvers and the discrete Helmholtz decomposition . . . . . . . . . . . . . . . . . . . . . . . . .
4.6.3
Solution using direct solvers . . . . . . . . . . . . . .
4.7
Example - modeling magnetic loop in conductive media . . . . .
4.7.1
Source generation . . . . . . . . . . . . . . . . . . .
4.7.2
Calculation of the data . . . . . . . . . . . . . . . .
4.7.3
The data from a block in a half-space . . . . . . . .
4.8
Exercises for Maxwell’s simulation . . . . . . . . . . . . . . . . .
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94
Contents
5
6
7
8
5
Sensitivity Calculation
5.1
The concept of sensitivities . . . . . . . . . . . . . . . . . . .
5.2
The basic sensitivity equation . . . . . . . . . . . . . . . . .
5.3
Computation of the sensitivities . . . . . . . . . . . . . . . .
5.3.1
Computation stationary linear forward problem .
5.3.2
Computation for time dependent problems . . .
5.4
Sensitivities of complex variables . . . . . . . . . . . . . . . .
5.5
Working with sensitivities in practice . . . . . . . . . . . . .
5.6
Di↵erentiating linear algebra expressions . . . . . . . . . . .
5.7
Programmer note . . . . . . . . . . . . . . . . . . . . . . . .
5.7.1
Derivative test . . . . . . . . . . . . . . . . . . .
5.7.2
The adjoint test . . . . . . . . . . . . . . . . . .
5.8
Example - sensitivities for the time domain model problem .
5.9
Exercises for sensitivity calculation . . . . . . . . . . . . . .
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95
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102
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110
Data fitting, Regularization and Inversion
6.1
The misfit . . . . . . . . . . . . . . . . . . . . . . . .
6.2
Over-fitting and Regularization techniques . . . . . .
6.2.1
Subspace and parametric techniques . . .
6.2.2
L2 based regularization . . . . . . . . . .
6.2.3
Total variation and anisotropic di↵usion .
6.3
L1 based Regularization . . . . . . . . . . . . . . . .
6.3.1
Discretization of regularization operators
6.3.2
Hard constraints . . . . . . . . . . . . . .
6.4
Inversion through optimization . . . . . . . . . . . . .
6.4.1
First order methods . . . . . . . . . . . .
6.4.2
Gauss-Newton and Inexact Gauss-Newton
6.4.3
Bound constraints . . . . . . . . . . . . .
6.4.4
A comment about direct search methods
6.4.5
Optimization over complex variables . . .
6.4.6
Implementation . . . . . . . . . . . . . .
6.4.7
Regularization parameter selection . . . .
6.5
Inversion through inverse scale space methods . . . .
6.6
Exercises . . . . . . . . . . . . . . . . . . . . . . . .
Large
7.1
7.2
7.3
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111
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141
scale problems and problems with many sources
Big data and big models . . . . . . . . . . . . . . . . .
Mesh decoupling . . . . . . . . . . . . . . . . . . . . . .
Stochastic methods for problems with many sources . .
7.3.1
Stochastic Hessians . . . . . . . . . . . . .
7.3.2
The stochastic gradient and Gauss-Newton
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Inversion in practice
153
8.1
Electromagnetic data set . . . . . . . . . . . . . . . . . . . . . . 153
8.2
Understanding the data . . . . . . . . . . . . . . . . . . . . . . . 154
8.3
Solving the inverse problem - first pass . . . . . . . . . . . . . . 156
6
Bibliography
Contents
161