Localized quasi-linear inversion based on the Bleistein method

2 Localized quasi-linear inversion based on the Bleistein method We have noticed already in electromagnetic sections of the book that the quasi-linear inversion, introduced above, cannot be used for interpretation of multi-source data, because both the reflectivity coefficient A and the material property parameter m depend on the illuminating incident wavefield. However, in many geophysical applications, for example in seismic exploration or in cross-well tomography, the data are collected using moving transmitters. In this case one can build an effective inversion scheme based on the localized quasi-linear approximation introduced in Chapter 9, which is source independent (Zhou and Liu, 2000 Zhdanov and Tartaras, 2002). 

We assume now that the scattered acoustic field p (rj, cu) (generated by a source with one or different positions) is measured at a number of observation points, 

We can solve the linear equation (15.160) with respect to m/, (r), which is source independent. Now, a scalar reflectivity coefficient l (r) can be determined, based on condition (14.54)  

This inversion scheme can be used for a multi-source technique because A and mi are source independent. It reduces the original nonlinear inverse problem to the same Born-type inversion we discussed in the previous sections. That is why some methods of Born-type inversion can be applied in this situation as well. In particular, the Bleistein method can be very useful in combination with localized QL inversion. 

For example, in the case of zero-offset data, we can use a formula similar to (15.106) to describe integral equation (15.160) 

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