Quasi-linear inversion

In full analogy with the electromagnetic case, we can apply the quasi-linear (QL) approximations introduced in Chapter 14 for acoustic and vector wavefield inversion. We begin our discussion of the basic principles of QL inversion with the simpler scalar case of acoustic waves. 

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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... 

Thc method of quasi-linear inversion has been developed in collaboration with S. Fang (Zhdanov and Fang, 1996b, 1999 Zhdanov et at, 2000). 

The quasi-linear inversion, introduced above, cannot be used for interpretation of multi-transmitter data, because both the reflectivity tensor A and the material property tensor in depend on the illuminating background electromagnetic field. However, in many geophysical applications, for example, in airborne EM and in well-logging, the data are collected with 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. 

Note that the quasi-linear inversion method, outlined above, can be easily extended to the case of the vector wavefield. We leave the detailed derivation of the corresponding formulae (which look very similar to the analogous formulae for electromagnetic field QL inversion) as an exercise for the interested reader. 

Localized quasi-linear inversion increases the accuracy and efficiency of wave-field data interpretation because it is based on a much more accurate forward modeling solution than the Born approximation, used in the original Bleistein method. An example of successful application of the localized QL approximation in radar-diffraction tomography can be found in (Zhou and Liu, 2000). 

Zhdanov, M. S. and S. Fang, 1996b, 3-D quasi-linear electromagnetic inversion Radio Science, 31 No. 4, 741-754. 

Zhdanov, M. S., and E. Tartaras, 2002, Inversion of multi-transmitter 3-D electromagnetic data based on the localized quasi-linear approximation Geophys. J. Int., 148, No 3. 

Therefore, the linear inverse problem (10.4) can have an infinite number of equivalent solutions. All these nonradiating currents form a so-called null-space of the linear inverse problem (10.4). In principle, we can overcome this difficulty using a regularization method, which restricts the class of the inverse excess currents j in equation (10.4) to physically meaningful solutions only. We will discuss an approach to the solution of this problem below, considering a quasi-linear method. 

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. 

One approach, which can be used in 3-D inversion, is based on principles of quasi-linear (QL) approximation. According to QL approximation, the anomalous electric field E inside the inhomogeneous domain D is linearly proportional to the background field E " through some electrical reflectivity tensor A ... 

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. ... 

Zhdanov, M, S., Fang, S., and G. Hursan, 2000, Electromagnetic inversion using quasi-linear approximation Geophysics, 65, No. 5, 1501-1513. 

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. 

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