Gradient direction of the energy flow functional

Now we will show how the electromagnetic field migration introduced above is related to minimization of the energy flow functional. The important step in the solution of the functional minimization problem (11.13) is calculating the steepest ascent direction (or the gradient) of the functional. To solve this problem, let us perturb the conductivity distribution crj, (x, z) = at, x, z)+Sa x, z). Actually, we have to perturb the conductivity only within the inhomogeneous domain F of the lower half-plane  

The first variation of the misfit functional with respect to the perturbation of the background conductivity can be calculated as 

Here 8Ey, SH are the first variations of the residual electric and magnetic fields  

According to the integral representations (9.25) and (9.26), the first variations of the anomalous electric and magnetic fields can be calculated as 

At the same time, the residual magnetic field can be expressed as the vertical derivative of the residual electric field Ey using the equation 

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