Magnetotellurics in the Context of Theory of Ill-Posed Problems

by Berdichevskii, Mark N.; Dmitriev, Vladimir I.; Keller, George V.

• ISBN13: 9781560801061
• ISBN10: 1560801069
• Format: Hardcover
• Copyright: 2003-02-01
• Publisher: Soc of Exploration Geophysicists

Table of Contents

Preface	ix
Basic Notations	xiii
I General Statements	1	(30)
Ill-posed Problems in Geophysics	3	(17)
Direct and Inverse Geophysical Problems	3	(1)
Elements of Functional Analysis	4	(2)
Conditionally Correct Statement of Inverse Problems in Geophysics	6	(2)
Solution of an Inverse Problem by Optimization	8	(1)
Solution of an Inverse Problem Using the Method of Regularization	9	(3)
The Method of Regularized Optimization	12	(1)
Adaptive Regularizing Operators	12	(1)
Construction of the Inverse Operator	13	(2)
The Backus-Gilbert Method	15	(3)
Probabilistic Statement of the Inverse Problem	18	(2)
The Magnetotelluric Field	20	(7)
Basic Types of Magnetotelluric Variations	20	(3)
Can Galvanic Coupling Between the Ionosphere and the Earth Be Ignored?	23	(1)
The Magnetotelluric Field in a Quasi-stationary Approximation	24	(2)
Quasi-stationary Magnetotelluric Field in the Frequency Domain	26	(1)
Statement of the Magnetotelluric Problem	27	(4)
The Tikhonov-Cagniard Transformations	27	(1)
Two Approaches to Magnetotelluric Problems	28	(1)
Exploration and Deep Magnetotelluric Surveys	29	(2)
II One-dimensional Magnetotellurics	31	(92)
A Spherical Model of the Earth	33	(6)
Spectral Decomposition of the Field in a Spherical Model	33	(1)
Spectral Impedance of the Earth in a Spherical Model	34	(1)
Spectral Impedance in a Homogeneous Spherical Model	35	(1)
Spectral Impedance in a Layered Spherical Model	35	(1)
The Tikhonov-Cagniard Operators for a Spherical Model	36	(1)
The Tikhonov-Cagniard Transformations as Spatial Filtration	37	(1)
Is It Possible to Ignore the Earth's Sphericity?	37	(2)
A Plane Model of the Earth	39	(15)
Spectral Decomposition of the Field in a Plane Model	39	(1)
Spectral Impedance of the Magnetic Type in a Plane Model	40	(1)
Spectral Impedance of the Electric Type in a Plane Model	41	(1)
General Case: The Field Contains Both Modes	42	(1)
The Tikhonov-Cagniard Impedance	42	(1)
Asymptotics of Spectral Impedances	43	(1)
The Wait-Price Criterion	43	(1)
The Tikhonov-Cagniard Magnetotelluric Operator in a Plane Model	44	(2)
The Tikhonov-Cagniard Magnetovariational Operator in a Plane Model	46	(1)
Effective Responses of the Tikhonov-Cagniard Filters	46	(2)
The Tikhonov-Cagniard Transformations for a Uniform Magnetic Field	48	(1)
The Tikhonov-Cagniard Transformations for a Field with Linear Spatial Variations of Horizontal Magnetic Components	48	(1)
The Tikhonov-Cagniard Transformations for a Field with Quadratic Spatial Variations of Horizontal Magnetic Components	49	(1)
Accuracy of the Tikhonov-Cagniard Model	49	(2)
Practical Use of the Tikhonov-Cagniard Model	51	(3)
Tikhonov-Cagniard Impedance	54	(25)
Basic Model of Magnetotelluric Sounding	54	(1)
The Magnetotelluric Field in the Tikhonov-Cagniard Model	54	(1)
The Law of Total Current	55	(1)
Physical Interpretation of the Impedance	55	(2)
Determination of the Impedance from the Boundary Value Problem for the Electromagnetic Field	57	(1)
The Layered Model	57	(1)
The Gradient Model	58	(2)
Determination of the Impedance from the Riccati Equation	60	(1)
Reciprocal Models	61	(2)
High-frequency Asymptotics of the Impedance	63	(1)
Low-frequency Asymptotics of the Impedance	64	(1)
Impedance on the Plane of Complex Frequency	65	(3)
Frequency Dependence of the Impedance	68	(1)
Relations Between Real and Imaginary Parts of the Impedance	69	(2)
Relations Between Module and Argument of the Impedance	71	(1)
Principle of Similitude	72	(1)
Admittance	73	(2)
Magnetotelluric Transformation in the Time Domain	75	(2)
Determination of the Impedance in the Sea	77	(2)
Transformations	79	(29)
Conversion of Impedance to a Form Useful for Interpretation	79	(1)
Skin Effect in Layered Media	79	(2)
Summing or Scanning?	81	(1)
Apparent Resistivity of the Earth	81	(3)
Apparent-resistivity Curves	84	(3)
Asymptotes of Apparent-resistivity Curves	87	(1)
Properties of Apparent-resistivity Curves for a K-type Model	88	(3)
Properties of Apparent-resistivity Curves for an H-type Model	91	(3)
Evaluating Geoelectric Parameters from Apparent-resistivity Curves	94	(1)
Apparent Conductivity of the Earth	95	(1)
The Niblett-Wittgenstein Transformation	96	(2)
The Molochnov-Viet Transformation	98	(2)
Controlled Transformation	100	(2)
The Depth of Magnetotelluric Investigation	102	(1)
Global Magnetovariational Sounding	103	(4)
Magnetotelluric Profiling	107	(1)
The Inverse Magnetotelluric Problem	108	(15)
One-dimensional Definition of the Inverse Magnetotelluric Problem	108	(1)
Existence and Uniqueness of a Solution to the Magnetotelluric Problem	109	(1)
Equivalent Geoelectric Media	110	(2)
Instability of the Magnetotelluric Inverse Problem	112	(1)
Comparison Criteria	112	(2)
Spline Approximation of MT Data	114	(2)
The Optimization Method	116	(3)
The Method of Regularized Optimization	119	(1)
The S-method	120	(3)
III Multidimensional Magnetotellurics	123	(80)
Magnetotelluric Transfer Functions	125	(14)
Impedance of a Horizontally Inhomogeneous Medium	125	(4)
Polar Diagrams of the Impedance Tensor	129	(1)
Principal Values and Directions of the Impedance Tensor	130	(3)
Separation of Local and Regional Effects	133	(3)
The Wiese-Parkinson Matrix (the Tipper)	136	(3)
Analysis of MT Data	139	(7)
Questions and Stages in Magnetotelluric Interpretation	139	(1)
Analysis of the Impedance Tensor	140	(1)
Analysis of the Wiese-Parkinson Matrix	141	(1)
Analysis of Apparent Resistivity	142	(4)
Multidimensional Magnetotelluric Inversion	146	(21)
Multidimensional Definition of the Inverse Magnetotelluric Problem	146	(1)
Quasi-one-dimensional Inversion	147	(1)
Two-dimensional Inversion	148	(19)
Magnetotellurics in Geological Service	167	(36)
A Bit of History	167	(1)
Magnetotellurics in Exploration for Oil and Gas	168	(5)
Magnetotellurics in Studies of the Earth's Crust and Upper Mantle	173	(1)
Geoelectric Models of the Juan de Fuca Subduction System	174	(4)
Geoelectric Model of the Kirghiz Tien Shan	178	(9)
Geoelectric Model of the Baikal Rift	187	(10)
Interpretation of Global Magnetovariational Sounding	197	(6)
IV Summary	203	(5)
Magnetotellurics in the Twenty-First Century	205	(3)
Restrictions of Magnetotellurics	205	(1)
The Future of Magnetotellurics	206	(2)
References	208	(7)
Index	215