The question of amplitude versus offset (AVO) arises in two ways: first, what is the best AVO on the earth's impulse response? and second, what is the best AVO to put on the migration operator (which should be the inverse operator)? I'm afraid some people assume these two AVO's are the same while in truth the relationship is much more complicated and is more like an inverse relation. These questions may be answered by a careful theoretical study. The same questions arise in dip-moveout analysis, and a variety of different answers have been given by able researchers--it is not an easy problem. The same question again arises in residual depth migration where various techniques such as ray tracing allow a fairly uncomplicated determination of the proper amplitudes on the forward modeled impulse response, but the determination of the inverse operator is again not easy.
The desire for correct amplitudes led me to consider 2-D imaging operators in the same mathematical framework as 1-D inverse filters. Such an approach seems hardly controversial from a theoretical point of view. Perhaps it is being ignored because the computability requirements seem wholly unrealistic. I undertook this study to attempt to show that the computability can be kept manageable and to see what kinds of practical gains can arise from designing imaging filters in much the same way as optimum 1-D filters. If this study is successful, it should obviate the need for an oppressive amount of theoretical work, and it should yield better results because many practical factors such as noise and acquisition arrays are readily incorporated.
First notice that the 2-D imaging filter is sparse--it is mostly zeros, clustering around a hyperbola of nonzero values. In this introductory study I choose the trajectory to be hyperbolic. Other trajectories could be more important in practice because a purely numerical approach is more necessary when theoretical approaches are difficult.
The first step is to write a subroutine to convolve two-dimensional filters in some way that avoids multiplying by the many embedded zeros. I started out by representing the filter in a compressed way--without the zeros. The compression is not required for memory saving, but to lay the groundwork for the next stage which is applying the conjugate-gradient subroutines described in my recent book.
It is too early to draw conclusions, but the broad outline of the costs are already apparent. To crosscorrelate a sparse 2-D filter with itself costs the square of the number of nonzero points in it. In practice one such crosscorrelation would add a cost to ordinary migration equal to padding the end of a seismic section by a space equal to the width of the migration operator. This is not a large additional cost. But solving for the inverse filter is an iterative process of steps each as costly as the crosscorrelation. In principle the number of iterations required equals the number of unknowns in the filter, but in practice we usually find that a much smaller number is generally satisfactory. By doing only two iterations we should be doing better than conventional processing. Thus the proposed method is very promising. Early examples showed little change after about five iterations. Taking a seismic section about five hyperbola widths wide implies a cost of optimum migration filter design about matching the cost of conventional migration.