Abstract
An important part of the processing of vertical seismic profiling (VSP) data is the separation of upgoing and downgoing waves. I introduce a new method for separation based on a time-domain recursive linear filter. The separation method uses an approximation to an optimal, frequency-domain, nonlinear filter solution as the starting point. The time-domain recursive linear (approximate) filter converges to the optimal (exact) solution. Since the computation is in the time domain and since this filter is linear, some of the temporal aliasing and other problems resulting from the forward and inverse Fourier transforms are avoided. Specifically, instability for some frequencies (spectral singularities) is not experienced here.This method uses a priori information of the opposite stepouts of the upgoing and downgoing waves. Equal spacing between borehole measurement points is not required. Further, the computational time may be controlled according to the desired accuracy. An important feature of this method is that it locates the reflecting boundaries of the subsurface. Having located the homogeneous layers, it allows variable-length windows of traces for separation, which eliminates the undesirable effects of smearing and extending wave fields beyond their origins. Also, knowledge of acoustic impedances for accurate implementation of the optimum filter is no longer required.
