What is the Transpose Operation?
, by Jon Claerbout
Geophysical problems generally involve linear operators. Optimization theory demands the transpose to these operators. Examples drawn from NMO, migration, and related areas show that computation of the transpose is usually a straightforward adjunct to the computation itself. An operator may represent only an approximation to reality. Never-the-less, for use in optimization, the computed transpose should be, and generally can be, the exact transpose (within machine precision) of the approximate operator. The transpose is also useful because it is often a practical approximation to the inverse. For example, Kirchhoff migration is really the transpose to Kirchhoff modeling, not the inverse. Also, the transpose to convolving with a filter response is crosscorrelating the with response. So the transpose isn't the inverse filter, but it does subtract all the phase. The transpose to NMO is much like inverse NMO. The transpose removes all time shift. NMO itself is trivially invertible when nearest neighbor interpolation is used. One-dimensional stretching deformations, such as NMO, that are dons by linear interpolation are easily invertible by means of the tridiagonal solver.
STANFORD