Inverse non-stationary convolution and combination

As with the stationary convolution described above, formulae for non-stationary recursive inverse convolution and combination follow simply by rearranging the equations in (5) and (6). Similarly, their adjoints can be obtained by rearranging the equations in (7) and (8). The recursive formulae describing these inverse processes are given in Table 1.

$$x_k = y_k - \sum_{i=1}^{\min(N_a-1, k-1)} a_{i,(k-i)} \; x_{k-i}$$ Inverse NS convolution: (9)

$$x_k = y_k - \sum_{i=1}^{\min(N_a-1, k-1)} a_{i,k} \; x_{k-i}$$ Inverse NS combination: (10)

$$y_k = x_k - \sum_{i=1}^{\min(N_a-1, N-1-k)} a'_{i,k} \; y_{k+i}$$ Adjoint inverse NS convolution: (11)

$$y_k = x_k - \sum_{i=1}^{\min(N_a-1, N-1-k)} a'_{i,(k+i)} \; y_{k+i}$$ Adjoint inverse NS combination: (12)

Table 1: Recursive formulae for non-stationary (NS) inverse operators.

As with the stationary inverse convolution described above, it is apparent that subject to numerical errors, non-stationary inverse filtering with these equations in Table 1 is the exact, analytic inverse of non-stationary filtering with the corresponding forward operator described in equations (5) through (8): they are true inverse processes. If operator ${\bf A}$ represents filtering with a non-stationary causal-filter, and ${\bf B}$ represents recursive inverse filtering with the same filter then

$${\bf A} {\bf B} = {\bf B} {\bf A} = {\bf I} \hspace{0.5in} ... ...\hspace{0.5in} {\bf A}' {\bf B}' = {\bf B}' {\bf A}' = {\bf I}.$$

The nhelicon module Claerbout (1998a) implements the non-stationary combination operator/adjoint pair, described by equations (6) and (8), while npolydiv implements the corresponding inverse operators, described by equations (10) and (12).