Assume v is medium velocity, is trace interval and is dip angle, the maximum threshold frequency that is not aliased is as follows Yilmaz (1987):
Figure 1 shows the RHS of equation (3) versus frequency. The solid curve represents the frequency-independent grids. The dash line represents the frequency-dependent grids. From this figure, we can see that all the frequency lower than have oversampled the data. Therefore, the next question is what is the disadvantage of oversampling the data?
Figure 1 versus f. The solid curve represents the frequency-independent grids. The dash line represents the frequency-dependent grids.
Unless carefully constrained, an over-parameterized model may contain illegal components that will fit noisy data better than a better model which does not include high wavenumbers for low frequencies.
Spatial prediction filters become the same for all frequencies Nichols (1996). Since the prediction filter is independent of frequency, it will become more stable and more insensitive to the noise. Also, we do not need to estimate independent prediction filters for each frequency slice separately.
In any case, oversampling will not bring any profit to noise suppression. So we will use , not , when . Figure 2 shows that when , is actually the base of another pyramid. So we can regard the input data as frequency- dependent grids directly.
In other words, we can get a series of pyramids corresponding to the frequency which is higher than . From each pyramid, we can estimate a prediction filter. The difference between the first prediction filter ( ) and the other () , is that we can apply the first prediction filter to all the frequencies which are lower than . Each of the remaining filters can only be applied to one frequency slice, which is also the base of that pyramid.
The basic and most important requirement for pyramid transform is that this transform is invertible. In other words, ``forward'' transform can turn a rectangular f-x domain to a pyramid domain and ``inverse'' transform can turn a pyramid domain back to a rectangular f-x domain, as shown in Figure 3.
How do we guarantee this transform invertible? The key is to find a good complex-valued interpolation scheme. Here, we use Dave Hale's 8-point sinc interpolation program in SU. Some more accurate and complicated interpolation scheme can be found in Wade's paper 1988. Figure 4 is one example using 8-point sinc interpolation. The result shows that our scheme is very accurate. There is little difference between input and output.