Review of Inverse Interpolation

In least-squares fitting goals, the regularized inverse interpolation problem can be stated as follows:

$$\begin{eqnarray} \bf Lm &\approx& \bf d \\ \epsilon_x \bold A_x \bold m &\approx& \bf 0 \\ \epsilon_y \bold A_y \bold m &\approx& \bf 0\end{eqnarray}$$ (7) (8) (9)

Operator $\bf L$ maps traces in a gridded model $\bf m$ to the earth's continuous surface. Operators $\bold A_x$ and $\bold A_y$ are ``steering filters'' Clapp et al. (1997) in the x and y direction, respectively. The steering filters are initialized with a space-variable dip function and decorrelate events which have that dip, and tend to steer the estimated model along the dip direction. In this fashion we impose our prior model covariance estimate on the missing traces. Scalars $\epsilon_x$ and $\epsilon_y$ balance the two model residuals [equations (8) and (9)] with the data residual [equation (7)].