Inverse problem

The least-squares inverse problem is formulated as follows. A constant-offset l₂ misfit energy functional can be defined as

 

$$E = \int_w \int_{{\bf x}_m} \left[ D({\bf x}_m;{\bf x}_h,w) - U({\bf x}_m;{\bf x}_h,w) \right]^2 \,d{\bf x}_m\,dw \;,$$ (3)

where $D({\bf x}_m;{\bf x}_h,w)$ is a recorded constant offset section. Minimizing (3) with respect to $\grave{P}\!\acute{P}$ and $\Theta$ leads to two coupled normal equations. The equations can be decoupled by the stationary phase (high-frequency) approximation, in which the major contribution to (2) occurs near the specular point when $\phi_s \approx \phi_r \approx 0$.In this case, the $\grave{P}\!\acute{P}$ equation can be solved independently of $\Theta$, and the result can be backsubstituted into the original normal equation for $\Theta$.