INTRODUCTION

The ultimate goal of seismic inversion is rock parameter estimation and mapping. Should a successful inversion be achieved, a byproduct is regridding on a mesh of arbitrary refinement. In an attempt to bridge the gap between theory and practice here I set a goal lower than full inversion: regridding post-stack crossline marine reflection data. The problem has real economic significance and is shared by seismology in all its scales, from earthquake seismology to engineering seismology. Indeed, this problem transcends seismology and permeates all of reconnaissance geophysics. ``Geophysical inversion'' purports to offer a solution to the regridding problem. Since it is so rarely used successfully for that purpose, we see a measure of the departure of theory from practice that tells us that there is much work to do and many remaining opportunities. I chose crossline marine data because inline stacks typically give data of high quality, principally marred by the irregularity of the crossline spacing.

 

gapmig
Figure 1 Top is data. Left is without gaps. Right is with gaps. Middle is linear interpolation over gaps. Bottom is migration.

The top two frames of Figure 1 show the synthetic data I made for this study. The synthetic data in Figure 1 is on a $200\times 200$ mesh. The right frame shows the data with 150 of 200 channels deleted by an algorithm that considers traces 4 at a time and then randomly selects one to be kept while the other 3 are deleted. The middle row of the figure shows that the method of Claerbout 1992c is not powerful enough to do a fully satisfactory job of filling in the missing traces because of the many smiles in migration shown in the bottom row. My hypothesis is that I should be able to improve the data interpolation based on the concept that the migration on the bottom right should have smiles removed by the principle that the migration should leave a local monoplane as described in Claerbout 1992d.

The number of missing data points is $150 \times 200= 30,000$.The migration-diffraction program for the synthetic data runs both ways in a few seconds. Thus, theoretically, a linear least squares solution for the unknowns should run in a day or two. On the discouraging side, the problem is not one of linear least squares, and the code has not yet been written. Nonlinear least squares arises because we must simultaneously estimate the model covariance matrix. On the encouraging side, SEP has a parallel computer and the times quoted above are for my desktop workstation. Also, satisfactory results may be found long before the theoretically required 30,000 iterations.

To be a success, I believe I should achieve useful results in 100 or fewer iterations, preferably in a handful of iterations. Fundamentally, many issues arise. Is a preconditioning strategy essential? Damping is an important part of any inversion formulation. Does damping speed the arrival of a useful solution? Figure   suggests that nearest-neighbor interpolations could frustrate iterative migrations. Will it? Is a conversion to the antialias method of Claerbout (1992a) mandatory or advisable? (That would require vectorizing that code over the midpoint axis, a non-trivial undertaking.) How can/should the covariance matrices be bootstrapped? Because of this bewildering array of imponderables, and because of my many frustrating experiences with iterating least-squares problems of high dimensionality Claerbout (1990) and others, unpublished, I chose not to attack the problem frontally, but begin by examining the ingredients to any inversion, the gradient, the gradient after filtering with the inverse-covariance filters, etc.