Testing

A cube of seismic data with an interesting set of dips was used as a test case. Half the traces were replaced by zeroes and input to some different interpolation schemes based on the equations above.

The important thing is the difference between the true data and the interpolated data we invent to take its place. Naturally, in a real problem the true data are unknown, but it is interesting to see if we can infer anything useful about that difference from the residuals.

Convergence of the filter estimation step for the different implementations is shown in Figure curves.nrp. The curve labels are:

• Smoothed means that the filters were calculated using the preconditioned optimization of equation goodleak2, but that $\epsilon$ was zero, and so the damping (equation dumbdamp2) was effectively turned off.
• Both is similar but with $\epsilon \neq 0$.
• Damped used equations basicpefest and dumbdamp2, except the damping equation really reads $\bold 0 \approx \bold I \bold a$, because there is no change of model variables. No preconditioning, but still damped with an identity. It seems sensible to replace this with a curve damped with a reasonable roughener, but that is just a slower way to get the results of the Both curve above.
• Neither was estimated using equation basicpefest alone, which is fine so long as the problem is overdetermined.

This test uses filters estimated in radial patches, with patch boundaries along various lines of constant radius or angle. The density of filters was chosen to give a good interpolation result, and to give patches that create an overdetermined problem at longer offsets and later times where the patches are largest, and an underdetermined problem where the patches are smaller.

Not surprisingly, the residual of the filter estimation step goes down fastest when there is no restriction on the filters (the Neither curve). While a large filter estimation residual does imply something bad (that the filters do not make a good estimate of the data spectrum), a small filter estimation residual is not necessarily good. In this case, it just means the filters have too many degrees of freedom.

The curves all start off about the same, with the two damped algorithms flattening out earlier and higher, because the damping simply prohibits them from putting enough energy in the filter coefficients to fit much of the data. The Damped curve looks bad, and continues to look bad in the later figures as well. Simply limiting the energy in the filter coefficients is not as sensible as limiting their roughness, which is what the damping effectively does for the curve labeled Both. In the Both curve, damping is applied to the preconditioned (roughened) variable $\bold p$, while in the Damped curve, the damping is on the actual filter coefficients $\bold a$.Changing the $\bold I$ to a roughener just reprises the previous curve, but without the change of variables.

The residual for the missing data calculation step is shown in Figure curves.nrd. A pleasing thing about all of these curves is that they converge after a handfull of iterations.

 

curves.nrp Figure 1 Filter calculation residual as a function of iteration. Curves represent different filter calculation schemes.

 

curves.nrd Figure 2 Missing data residual as a function of iteration. Curves represent different filter calculation schemes.

Figure curves.nm shows the real quantity of interest. Each curve is the norm of the difference between the interpolated data and the true data as a function of iteration in the second step of the interpolation. In every case (though just barely in a couple), the difference increases at later iterations. In principle the true data are unknown, so there is no reason for the difference not to increase. The PEFs are certain not to be perfect, and will eventually begin to add in some undesirable components. The misfit starts to go up after the residual from Figure curves.nrd has bottomed out.

Figure curves.nm.filtniter shows the same curve, along with several others that all use the same algorithm (preconditioning, no damping), but different numbers of iterations, to calculate PEFs. With many iterations, the minimum value on a given curve increases. Too many iterations, either in calculating the filter coefficients or in calculating the missing data, degrades the result. In the case of the missing data iterations, the residual has bottomed out, as shown in Figure curves.nrd.30, which plots the data residual and norm of the misfit between interpolated and true data, as a function of iteration. In the case of the filter coefficient iterations, it is not obvious when to stop.

Damping with the appropriate value of $\epsilon$ helps. Figure curves.nm.slightdamp.fn shows the same sorts of curves as Figure curves.nm.filtniter, with a reasonable $\epsilon$.Here the curves are fairly close together, so that running too many filter calculation iterations does not degrade the final result too much. The wrong value of $\epsilon$ causes the same problems as the wrong number of iterations without damping, however. Figure curves.nm.muchdamp shows the misfit between the true and interpolated data for different values of $\epsilon$.With the damping, you need to find the correct value of $\epsilon$,and without it, you need to find the correct number of iterations for calculating the PEF coefficients. Results tend to be somewhat less sensitive to $\epsilon$,and choosing $\epsilon$ to make the two components of the filter calculation residual roughly balance is usually a safe choice Lomask (1998).

 

curves.nm Figure 3 Norm of the misfit between the true data and the interpolated data. Horizontal axis displays number of iterations in the missing data calculation step. Curves represent different filter calculation schemes.

 

curves.nm.filtniter Figure 4 Norm of the misfit between the true data and the interpolated data. Horizontal axis displays number of iterations in the missing data calculation step. Curves represent different numbers of iterations in the filter calculation step.

 

curves.nrd.30 Figure 5 Norm of the missing data residual and the misfit between true and interpolated data. Misfit rises after the missing data residual flattens out.

 

curves.nm.slightdamp.fn Figure 6 Norm of the misfit between true data and interpolated data. Curves represent different numbers of iterations in the filter calculation step.

 

curves.nm.muchdamp Figure 7 Norm of the misfit between true data and interpolated data. Curves represent different amounts of damping.

 

102.smooth.out
Figure 8 Sample interpolation output. This is one result using the test data from all the previous graphs.