Quasi-analytical solutions for a 3-D electromagnetic field

In this section we analyze different approximate solutions of the tensor quasi-linear 

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 

Following ideas of the extended Born approximation outlined above, we use the fact that the Green s tensor Ge (j 1 r) exhibits either singularity or a peak at the point where r = r. Therefore, one can expect that the dominant contribution to the integral Ge [ActAE ] in equation (9.83) is from some vicinity of the point r = r. Assuming also that A (r) is slowly varying within domain D, one can write 

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  

Dividing equation (9.85) by the square of the background field and assuming that 

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