Quasi-analytical inversion

Quasi analytical approximations (9.90) and (9.91) provide another tool for fast and accurate electromagnetic inversion. This approach leads to a construction of the quasi-analytical (QA) expressions for the Prechet derivative operator of a forward problem, which simplifies dramatically the forward EM modeling and inversion for inhomogeneous geoelectrical structures.  

By introducing a perturbation of the anomalous conductivity SAa (r) we can calculate the corresponding perturbation of the electric field E(rj) on the basis of equation (9.90)  

Substituting equation (10.98) into the second integral in (10.97) and changing the notation for the integration variables, r r and r — r, wo obtain 

In particular, considering the infinitely small domain of the conductivity perturbation, we arrive at the following formula for the Frechet derivative of the electric field  

The last formula provides an analytical expression for computing the Frechet derivative for the forward modeling operator. Note that, in this case, the amount of calculation for the forward modeling solution and for the Frechet derivative is equivalent to computing the Born approximation. 

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The modified conductivity matrix and vector play an important role of quasi-linear inversion, which we will discuss in the next chapter. 9.4-8 Matrix form of quasi-analytical approximation in the method... 

The last formulae serve as a basis for the method of quasi-analytical electromagnetic inversion, which I will present in the next Chapter. 

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. 

The sections on quasi-analytical (Q. l) inversion and magnetotelluric QA inversion were prepared in collaboration with G. Hursan. 

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.

Zhdanov, M. S., and G. Hursan, 2000, 3-D electromagnetic inversion based on quasi-analytical approximation Inverse Problems, 16, 1297-1322. 

By full analogy with the electromagnetic case, one can consider different ways of introducing the reflectivity coefficient A. In particular, two of these solutions play an important role in inversion theory. One is the so-called quasi-analytical (QA) solution, and the other is the localized quasi-linear (LQL) approximation. In this section I will introduce the QA approximation for the acoustic wavefield. 

Newton s method and quasi-Newton techniques make use of second-order derivative information. Newton s method is computationally expensive because it requires analytical first-and second-order derivative information, as well as matrix inversion. Quasi-Newton methods rely on approximate second-order derivative information (Hessian) or an approximate Hessian inverse. There are a number of variants of these techniques from various researchers most quasi-Newton techniques attempt to find a Hessian matrix that is positive definite and well-conditioned at each iteration. Quasi-Newton methods are recognized as the most powerful unconstrained optimization methods currently available. 

The objective of this study is to develop an analytical model for a soil column s response to a sinusoidally varying tracer loading function by applying the familiar Laplace transform method in which the convolution integral is used to obtain the inverse transformation. The solution methodology will use Laplace transforms and their inverses that are available in most introductory texts on Laplace transforms to develop both the quasi steady-state and unsteady-state solutions. Applications of the solutions will be listed and explained. 

Calculation of the inverse Hessian matrix can be a potentially time-consuming operation that represents a significant drawback to the pure second derivative methods such as Newton-Raphson. Moreover, one may not be able to calculate analytical second derivatives, which are preferable. The quasi-Newton methods (also known as variable metric methods) gradually build up the inverse Hessian matrix in successive iterations. That is, a sequence of... 

The most efficient methods that use gradients, either numerical or analytic, are based upon quasi-Newton update procedures, such as those described below. They are used to approximate the Hessian matrix H, or its inverse G. Equation (C.4) is then used to determine the step direction q to the nearest minimum. The inverse Hessian matrix determines how far to move along a given gradient component of f, and how the various coordinates are coupled. The success of methods that use approximate Hessians rests upon the observation that when f = 0, an extreme point is reached regardless of the accuracy of H, or its inverse, provided that they are reasonable. 

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