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The adjoint of non-stationary convolution can be written as
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(7) |

and the adjoint of non-stationary combination can be written as
| |
(8) |

For many applications, the adjoint of a linear operator is the same
operator applied in a (conjugate) time-reversed sense. For example,
causal and anti-causal filtering, integration, differentiation, upward
and downward continuation, finite-difference modeling and reverse-time
migration etc.
For non-stationary filtering, it is important to realize this is
*not* the case: the adjoint of non-stationary convolution is
time-reversed non-stationary combination, and vice-versa.
Therefore, the output of adjoint combination is a superposition of
scaled time-reversed filters, . So for anti-causal
non-stationary filtering, it may be advantageous to apply adjoint
combination, as opposed to adjoint filtering.

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** Up:** Theory
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Stanford Exploration Project

10/25/1999