Iterative Born inversions

Before moving to the iterative Born inversion technique, we introduce a fast imaging technique based on a Born approximation. Let us recall formula (5.91) for an approximate regularized solution of the linear inverse problem. In the case of equation (10.16), this formula takes the form... 

Note that each equation, (10.41) and (10.42), is bi-linear because of the product of two unknowns. Ad and E. However, if we specify one of the unknowns, the equations become linear. The iterative Born inversion is based on subsequently determining Ad from equation (10.42) for specified E, and then updating E from equation (10.41) for predetermined Ad, etc. Within the framework of this method the Green s tensors Ge and Gh, and the background field stay unchanged. 

The iterative Born inversion based on the modified Green s operator involves subsequently determining A from equation (10.42) for specified E, and then updating E from equation (10.47) for predetermined A5, etc. 

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.

The ideas of iterative Born inversion, introduced in Chapter 10 for an electromagnetic field, can be extended to wavefield inversion as well. Following the basic principles of this method, we first write the original integral equations for the acoustic or vector wavefields (14.20) and (14.81) as the domain equations for the wavefield inside the anomalous domain D ... 

Note in conclusion of this section that iterative Born inversion requires the application of the regularizing methods to make the solution stable. 

DQA approximation in magnetotelluric inverse problem The DQA approximation is particularly suitable for constructing massive 3-D magnetotelluric inversion schemes, because of the low cost and simplicity of the expressions for forward modeling. In this section I discuss the implementation of the DQA approximation in MT inversion, following the paper by Hursan and Zhdanov, 2001. The main advantage of the QA method over the iterative Born method is that now... 

Figure 10-5 A model of a dipping dike (left panel) and inversion results obtained by the iterative Born method mth focusing (right panel).

---