Quasi-analytical solutions for the vector wavefield

As for the electromagnetic case, the reflectivity tensor, can be selected to be a scalar, A=AI, in the quasi-linear approach. In this case, formula (14.93) can be cast in the form  

Substituting this expression for the scattered field on the left-hand side of the integral representation (14.94), we obtain the following integral equation for the reflectivity coelficient A  

Following the ideas of the extended Born approximation outlined above, we can take into account that the Green s tensor G (r r, uj) exhibits either a singularity or a peak at the point where Vj = r. Therefore, one can expect that the dominant contribution 

Calculating the scalar product of both sides of equation (14.98) with the incident wavefield and dividing the resulting equation by u u, we finally obtain an analytical expression for the refiectivity coefficient  

Formula (14.102) gives the quasi-analytical (QA) approximation for a 3-D vector wavefield, which can be treated as an analog of the corresponding QA approximation for the electromagnetic case (Zhdanov, et ah, 2000) 

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