Inversion based on the quasi-analytical method

In practice we usually solve forward and inverse problems in the space of discrete data and model parameters. For a numerical formulation of QA inversion we can use the matrix formula for QA approximation (9.241), reproduced here for convenience  

Matrix C depends on the matrix of the background electric field introduced in Chapter 9 (formula (9.219))  

Let us consider the derivation of the Frechet derivative matrix of the discrete forward operator (10.103). Noting that the model parameters are the anomalous conductivity values in the cells of the anomalous body, that matrix A is independent of the model parameters, and that B is a diagonal matrix, one can express the perturbation of the forward operator (10.103) with respect to the model parameters in the form 

The inverse problem (10.103) is usually ill-posed, i.e., the solution can be non-unique and unstable. The conventional way of solving ill-posed inverse problems, according to regularization theory (Chapter 2), is to minimize the Tikhonov parametric functional  

Note again that numerical computations based on formula (10.107) are very-fast and efficient, because the full matrices A and C are precomputed for the background model and are fixed we update only the diagonal matrix B(m) on each iteration of the inverse process. 

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