Reflectivity tensor

The last formula gives us the tensor quasi-linear (TQL) equation with respect to the electrical reflectivity tensor A,... 

We shall analyze in the next sections the different techniques of the TQL equation solution hich result in different analytical expressions for the electrical reflectivity tensor A (r ). In particular, one of these solutions gives rise to the extended Born, or localized nonlinear (LN) approximation, introduced above. Note that these approximations may be less accurate than the original QL approximation with a fine grid for the discretization of A(rj). 

In the framework of the quasi-linear approach, the electrical reflectivity tensor can be selected to be a scalar (Zhdanov and Fang, 1996a) A=A. In this case, integral equation (9.80) can be cast in the form... 

Taking into account that we are looking for a scalar reflectivity tensor, it is useful to introduce a scalar equation based on the vector equation (9.84). We can obtain a scalar equation by taking the scalar product of both sides of equation (9.84) with the complex conjugate background electric field ... 

The QA solutions developed in the previous section were based on the assumption that the electrical reflectivity tensor was a scalar. This assumption reduces the areas of practical applications of the QA approximations because in this case the anomalous (scattered) field is polarized in a direction parallel to the background field within the inhomogeneity. However, in general cases, the anomalous field may be polarized in a direction different from that of the background field, which could generate additional errors in the scalar QA approximation. To overcome this difficulty, we introduce in this section a tensor quasi-analytical (TQA) approximation. The TQA approximation uses a tensor A, which permits different polarizations for the background and anomalous (scattered) fields. 

The solution of equation (9.115) gives us a localized electrical reflectivity tensor Al (r), which is obviously source independent. Expression (9.111) with (r) determined according to (9.115), is called a localized quasi-linear (LQL) approximation (Zhdanov and Tartaras, 2001) ... 

However, the scattering tensor is defined explicitly through an integral of the anomalous conductivity. On contrast, the reflectivity tensor in the LQL approximation is determined as the solution of the minimization problem (9.115). Note also that the... 

Another important difference between the LQL approximation and LN approximation is that for the former one can choose different types of reflectivity tensors. For example, one can introduce a scalar or diagonal reflectivity tensor. 

In the special case of a diagonal reflectivity tensor X/, = diagA, ... 

Formula (9.154) shows that the minimum of

electrical reflectivity tensor ... 

Note that minimization problem (9.155) is equivalent to the minimization problem which we have used in determining the electrical reflectivity tensor for the original QL approximation. This means that we can use exactly the same reflectivity tensor in both cases. 

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 ... 

Note that equation (10.62) is linear with respect to in. The reflectivity tensor A can be determined from the following linear equation inside the inhomogeneous domain D, as long as we know m ... 

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. 

In the case of a scalar reflectivity tensor, we introduce a scalar parameter mi (r) ... 

Following the basic ideas of the original QL approximation for an electromagnetic field, introduced by Zhdanov and Fang (1996), we determine the reflectivity tensor by solving the minimization problem ... 

We can determine the reflectivity tensor A by solving the minimization problem (14.95) on a coarse grid. The accuracy of the QL approximation depends only on the accuracy of this discretization of A, and, in principle, can be made arbitrarily good. 

---