DATA INVERSION

A common-midpoint gather from Southern California (Figure a) was used to test the algorithm. A velocity panel after one iteration with the operators ${\bf P^T}$ and ${\bf P}$ is shown in Figure a. This corresponds to the usual velocity stack, but instead of stacking along hyperbolas, stretching along parabolas was used. The velocity analysis panel after ten iterations is shown in Figure b. We can see that the resolution has improved and a series of multiples have become apparent. A contour map of both velocity panels is in Figure .

The resolution is better in Figure b, but a better comparison can be seen from Figure , which shows the increase of amplitudes in successive iterations.

By applying a transpose operator on the velocity panel we should get back the input data. The results produced by one and ten iterations are shown in Figure b and Figure a. We can see that amplitudes decrease with offset after one iteration. The explanation is in the geometrical representation of ${\bf H^TH}$. (The explanation should be for the operator ${\bf PP^T}$ in this case.) The error after ten iterations is shown in Figure b.

We can perform velocity analysis for various ways of sampling in the velocity space. We should choose the sampling for which the convergence is the fastest. The comparison of least-squares errors for even sampling in velocity, slowness, and sloth domains is shown in Figure a and Figure b. We can see that the least errors are in the sloth domain. Convergence is faster with operators $(\bold H, {\bf H^T})$ than with $({\bf P^T},\bold P)$for a small number of iterations.

If the lower limit of slowness is chosen to be greater than zero, convergence becomes faster. We will have a look at the cause of this effect later. It appears that even sampling in the velocity domain should not be chosen.