Nonlinear inversion

Tarantola, A., 1987, Inverse Problem Theory, Elsevier Tarantola, A., Valette, B., 1982, Generalized nonlinear inverse problems solved using the least squares criterion. Reviews of Geophysics and Space Physics 20, 219... 

Alifanov O, Artyukhin E (1976) Regularized numerical solution of nonlinear inverse heat-conduction problem. J Engin Phys 29 934-948... 

Carasso AS (1992) Space marching difference schemes in the nonlinear inverse heat conduction problem. Inverse Problems 8 25-43... 

Khan A. and Mosegaard K. (2002) An inquiry into the lunar interior a nonlinear inversion of the Apollo lunar seismic data. J. Geophys. Res. 107, E63.1-E63.23. 

Of course, it is usually not enough to use only one iteration for the solution of a nonlinear inverse problem in the framework of the Newton method (because we used the linearized approximation (5.38)). However, we can construct an iterative process based on the relationship (5.43) ... 

Regularized gradient-type methods in the solution of nonlinear inverse problems... 

Approximate regularized solution of the nonlinear inverse problem We can find an approximate solution of the regularized normal equation (5.86) for the optimum step, using the same idea which we applied for the approximate. solution of the linear inverse problem in Chapter 3. Let us assume that the regularization parameter tv is big enough to neglect the term with respect to the term... 

For a regularized solution of a nonlinear inverse problem, let us introduce a parametric functional. 

Wc have shown in this section how regularization tlicory can bo applied to nonlinear inverse solutions. In the following chapters of the book, we will illustrate the general theory for some important geophysical inverse problems. 

McGillivray, P. R., and D. W. Oldenburg, 1990, Methods for calculating Frechet derivatives and sensitivities for the nonlinear inverse problem a comparative study Geophys. Prosp., 38, 499-524. 

Considci a general nonlinear inverse problem for anomalous electric or magnetic fields ... 

One practical way to overcome this difficulty is to abandon the integral equation approach for nonlinear inverse problems and to consider the finite difference or finite element methods of forward modeling. We will present this approach in Chapter 12. Another way is based on using approximate, but accurate enough, quasi-linear and quasi-analytical approximations for forward modeling, introduced in Chapter 8. We will discuss these techniques in the following sections of this chapter. 

After determining m and A it is possible to evaluate the anomalous conductivity distribution Aa from equation (10.63). This inversion scheme reduces the original nonlinear inverse problem to three linear inverse problems the first one (the quasi-Born inversion) for the tensor m, the second one for the tensor A, and the third one (correction of the result of the quasi-Born inversion) for the conductivity Act. This method is called a quasi-linear (QL) inversion. ... 

Figures 10-5 - 10-6 present the results of inversion for an xy polarized field using, respectively, the iterative Born and the diagonalized quasi-analytical (DQA) inversion methods, with focusing. Both methods produce good results. Note that inversion of theoretical MT data takes just few minutes on a personal computer. A remarkable fact is that it is possible for this polarization to separate the upper and lower parts of the dike. The position and shape of the dike are also reconstructed quite well. However, the resistivity of the dike is not recovered correctly, which is a common problem in nonlinear inversion. The first method (iterative Born inversion) underestimates the anomalous conductivity, while the second (DQA inversion) produces a more correct result. The reason for the underestimation of the conductivity by the iterative Born inversion is very simple. The linearized response, which is used in this technique, usually overestimates the anomalous field, which results in an underestimation of the conductivity in inversion. The quasi-analytical approximation produces a more accurate electromagnetic response, so the inversion works more accurately as well.

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