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Book Chapter

Solving Non-Linear Inverse Problems

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Published:
January 01, 1995

Abstract

In most of the interesting inverse problems in geophysics the data are related to the model parameters in a nonlinear way (i.e., not in the explicit form d=Gm). Recall that even the simple refraction delay-time problem examined under Chapters 4 and 5 is actually non-linear since the travel paths of seismic waves through a layered medium are proportional to the path length in the layer but inversely proportional to the velocity, i.e.,

where ti is the total travel time for the ith ray, Lij is the ith ray path in the jth layer and υi is the velocity in the jth layer. The model parameters are inversely, rather than linearly, related to the travel time data; and we had to use the slowness (c = 1/u) instead of v as the parameter of the linear problems of the previous sections. It may be remarked here that the way that a given problem is posed may sometimes determine whether it is in effect li-nrar or not. For instance, if we are interested in a gravity or magnetic model where the model parameters are the anomalous density or susceptibility in cells of fixed position, then such a problem may be considered as being effectively linear. Another set of problems that may be easily manipulated are those in which the forward theory involves simple exponential functions. For instance, in carbon dating (using radioactive decay data for Carbon-14), the forward theory states that the fraction f of an original amount of Caxbon-14 remaining

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Society of Exploration Geophysicists Course Notes

Geophysical Data Analysis: Understanding Inverse Problem Theory and Practice

Max A. Meju
Max A. Meju
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Society of Exploration Geophysicists
Volume
6
ISBN electronic:
9781560802570
Publication date:
January 01, 1995

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