By J. Claerbout, J. Black
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Extra resources for Basic Earth Imaging
The inverse conversion, like an adjoint, retraces the steps of the forward transform but it does the inverse at every stage. There is however, a messy problem with nearly all field data that must be handled along the inverse route. The problem is that the observed RMS velocity function is generally a rough function, and it is generally unreliable over a significant portion of its range. To make matters worse, deriving an interval velocity begins as does a derivative, roughening the function further.
Thus, matrix inversion is easy here. ” Least squares is a general method for solving sets of equations that have more equations than unknowns. 5) presumes the existence of the inverse of B′ B. As you might expect, this matrix is nonsingular when B stretches the data, because then a few data values are distributed among a greater number of locations. Where the transformation squeezes the data, B′ B must become singular, since returning uniquely to the uncompressed condition is impossible. We can now understand why an adjoint operator is often an approximate inverse.
Vela-velvel [ER] There is one final complication relating to weighting. The most symmetrical approach is to put w into both A and A′ . This is what subroutine velsimp() on the current page does. 8 is nonphysical. An alternate view is to define A (by the pseudo code above, or by some modeling theory) and then for reverse transformation use w 2 A′ . 54 CHAPTER 4. 9. 9: Transformation of data as a function of offset (left) to data as a function of slowness (velocity scans) on the right using subroutine velsimp().
Basic Earth Imaging by J. Claerbout, J. Black