Elsewhere in this report, ClaerboutClaerbout (1993) introduces a technique for extrapolating three-dimensional data based on his one-dimensional extrapolation given in Claerbout (1992). This extrapolation is done by calculating three-dimensional annihilation filters from the available data and then minimizing the residuals after filtering a larger data volume that includes the data to be extrapolated.
The most immediate application of this method appears to be the extrapolation of data past the boundaries of seismic lines to reduce migration end-effects. Another application is to predict areas of missing data within a line to complete a partially filled grid so techniques that operate on regular grids may be used. This application is not limited to the finite-difference and f-k techniques that require regular grids, since even integral methods that can operate on irregular grids involve assumptions about the regularity of the input and may benefit from filling in the missing parts of a grid.
In this paper, I show that a purely three-dimensional operator makes a significant improvement over a prediction operator that is not purely lateral. A purely lateral filter is one in which the samples in a given output trace are not affected by the input samples at that same trace position.
I also show that the prediction is improved by using a combination of several prediction filters. Several prediction filters are used to reduce bad predictions at the edges of the volume and to produce good predictions throughout the volume for dips of all directions. There are several methods of incorporating the results of the multiple prediction filters. An unexpected difference was found in predicting the residual by simultaneously minimizing the four separate results of applying four filters and predicting the residual by minimizing the sum of the four results. Minimizing the sum of the four results produced a reasonable result; simultaneously minimizing the four separate results produced a very poor extrapolation.
Comparisons between a purely lateral operator result and the result of using an operator containing vertical components are shown to demonstrate how lateral operators improve predictions. Then the application of a single operator is compared to that of multiple operators to demonstrate how multiple operators can remove poor edge predictions and how the summed filter results improve predictions over the simultaneous minimization with all the prediction operators.
The results presented in this paper are preliminary. Most of this work has been done only recently and requires further analysis.