Shots versus receivers, 2-D versus 3-D

There are a couple of choices to make concerning how to arrange the input data. The first is really a matter of sorting. Since the data are not really being addressed by physical coordinates but rather by shot number and channel and so forth, it makes sense that the way the data are sorted might affect the output, because sorting changes the distribution of recorded and nonrecorded traces. This is shown in the previous chapter, in Figures traces-crg and traces-csg. One cube can be made from the other by sorting. It is intuitively appealing to have each missing trace surrounded on all sides by known data, as in Figure traces-crg.

Another important choice is how much data to work on at one time, and whether to split the data in such a way that its dimensionality is reduced. Some of the data sets used in the examples are 2-D data sets, and some are 3-D. In the case of marine data, there really is very little difference between 2-D data and 3-D data, and the same routines are used. In the case of land data, 3-D can be vastly more complicated than 2-D.

A 2-D prestack data set is a 3-D cube, with axes for time, midpoint, and offset; or time, shot, and receiver. A 3-D prestack data set is a 5-D cube of data, where the surface axes each have an inline and crossline component. In marine acquisition, the boat typically tows only two sources, and so it is easy enough to consider a sail line to be two 4-D cubes of seismic data. Because the crossline receiver aperture may be small, with only a few streamers, it makes sense to decompose each of those two 4-D cubes into a series of $n_{\rm streamer}$ 3-D cubes. It is helpful to pad a couple of zeroes onto each axis during the filter calculation step. If the data are only a few points wide along one axis, then padding that axis can easily double the size of the data. It also significantly increases the number of filter coefficients, if the filter has some width along the extra axis. Thus for a boat with few streamers, including the crossline receiver axis can easily double the size of the input data, making convolutions that much more expensive, but probably not improving the final results noticeably.

There are benefits to using more dimensions. Having an extra direction for things to be predictable in often means a better interpolation result. Calculating a PEF from a 2-D plane of white noise will produce a filter that works like an identity. That filter will then not insert anything into any missing samples. If the plane of white noise is a slice from a cube or hypercube that contains coherent energy along some other axis, then a PEF that is calculated with some coefficients spread along that coherent axis will be a good interpolator.

My experience has been that a three dimensional input cube (many 2-D gathers) works noticeably better than a single two dimensional gather. Four dimensions work somewhat better than three, but the returns diminish as the misfit gets small. At any rate, on most examples I have done, three dimensions turns out to be plenty to get a good interpolation result. Nearly all of the examples in this thesis are done with three dimensions of input. If there are only a few streamers, working in 4-D brings many extra computations because of padding, and because adding one to the dimensionality of the PEFs significantly increases the number of coefficients in each filter. It is probably not worth the extra computations if a good result comes from 3-D. If there are many streamers, as is the case with some modern boats which may have 10 or 20, or if results with three dimensions are not good enough, then it would make sense to use a 4-D input.

As a final note, the notion of deciding whether or not to filter along the crossline receiver axis assumes that that is not the axis you want to resample. However, one can certainly imagine wanting to interpolate new streamers, to refine the typically sparse crossline receiver sampling. In that case, you would naturally want to predict in that crossline receiver direction. This is done in an example later in this chapter.