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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.



 
next up previous print clean
Next: Modifications of basic equations Up: Jedlicka: Velocity analysis by Previous: COMPUTATION OF ADJOINT OPERATORS
Stanford Exploration Project
1/13/1998