By Michael S. Zhdanov

This e-book offers cutting-edge geophysical inverse conception built in smooth mathematical terminology. The e-book brings jointly basic effects constructed by way of the Russian mathematical tuition in regularization concept and combines them with the comparable examine in geophysical inversion performed within the West. It provides a close exposition of the tools of regularized answer of inverse difficulties according to the tips of Tikhonov regularization, and exhibits the several sorts of their functions in either linear and nonlinear tools of geophysical inversion. this article is the 1st to regard many varieties of inversion and imaging options in a unified mathematical manner.The booklet is split in 5 elements overlaying the principles of the inversion idea and its functions to the answer of other geophysical inverse difficulties, together with power box, electromagnetic, and seismic equipment. the 1st half is an creation to inversion idea. the second one half encompasses a description of the elemental equipment of resolution of the linear and nonlinear inverse difficulties utilizing regularization. the subsequent elements deal with the applying of regularization equipment in gravity and magnetic, electromagnetic, and seismic inverse difficulties. the main connecting inspiration of those utilized elements of the e-book is the analogy among the suggestions of the ahead and inverse difficulties in numerous geophysical tools. The ebook additionally comprises chapters relating to the trendy expertise of geophysical imaging, in keeping with seismic and electromagnetic migration.This quantity is exclusive in its specialise in offering a hyperlink among the equipment utilized in gravity, electromagnetic, and seismic imaging and inversion, andrepresents an exhaustive treatise on inversion conception.

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**Extra info for Geophysical Inverse Theory and Regularization Problems**

**Sample text**

Theorems of uniqueness A. Gravity field The inverse gravity problem is actually an inverse source problem, because the masses are the sources of the gravity field. Thus, it may have many equivalent solutions. For example, all spherical material balls with the same center and the same total mass (but with different radii) produce the same gravity field (outside these balls). 77- M = P,—R\ 4-7r = P2-jRl. 39) Existence and uniqueness of the inverse problem solutions Figure 1-8 19 Non-uniqueness in a gravity inverse problem.

In order to develop these methods, one should study carefully the physics of seismic waves. The simplest model of these waves is one based on acoustic principles. Assume that the earth can be treated as an acoustic medium and the influence of variations in density can be ignored. 22) where P ( r , t) is a pressure field, F{r,t) is the strength of a source of energy, and c(r) is the velocity of wave propagation. 23) Formulation of forward and inverse problems for different geophysical fields 15 where A^ is the acoustic forward modeUng operator.

Interpretation of electromagnetic data collected by the receiver is another typical inverse geophysical problem. 19) where e and /i are the dielectric constant (permittivity) and the magnetic permeability, respectively. Note that magnetic permeability is related to magnetic susceptibility Xm by a simple formula: M = Mo(l + Xm), (1-20) where /iQ = 47r X 10"'' Henry/meter is the free space magnetic permeability. 21) where a is the electric conductivity of a medium which is the reciprocal of its resistivity p , (7 - 1/p.