Non-stationary recursive filtering

If the velocity varies smoothly in space, then we can extend the stationary theory to cover spatially-variable filtering. Rather than filtering with the stationary forward and inverse convolution equations,

$$\begin{eqnarray} y_k &=& x_k + \sum_{i} a_{i} \; x_{k-i}, \\ {\rm and} \hspace{0.15in} x_k &=& y_k - \sum_{i} a_{i} \; x_{k-i},\end{eqnarray}$$ (46) (47)

we can extend the concept of a filter to that of a filter-bank with one filter for every location in the input/output space Claerbout (1998a); Margrave (1998). Now equations () and () become

$$\begin{eqnarray} y_k &=& x_k + \sum_{i} a_{i,k-i} \; x_{k-i}, \\ {\rm and} \hspace{0.15in} x_k &=& y_k - \sum_{i} a_{i,k-i} \; x_{k-i}.\end{eqnarray}$$ (48) (49)

Non-stationary convolution and recursive inverse convolution are indeed true inverse processes, but like stationary polynomial condition non-stationary inverse filtering has potential stability problems. Appendix  discusses this in more detail; it also demonstrates that, in general, the stability of recursive non-stationary inverse filtering is not guaranteed even if the individual filters that make up the filter-bank are all minimum phase.