A recursive method for the separation of upgoing and downgoing waves of vertical seismic profiling data

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.

A STUDY OF DIFFERENT METHODS OF WAVEFIELD SEPARATION FOR APPLICATION TO VSP DATA1

“…However, their ability to preserve discontinuities and avoid smearing is unique. Aminzadeh's (1986) recursive filter, the optimum array filters, the f-k filters and the z-p filters are all useful for VSP wavefield separation, but the amount of filtering and smearing must be weighed against the distortion of the waveforms.…”

“…The probable explanation for this is that the P-wave, vertical raypath, horizontally-layered model on which the method is based does not describe the events of our survey well enough. When an interpretation is made and the layer boundaries found, either by using Aminzadeh's (1986) automatic picking method or by other means, we should still have a sufficient number of levels inside each layer. As a rule of thumb, seven levels is the minimum number of input traces necessary to reject the direct downgoing waves.…”

“…The section in Fig. 17 shows Aminzadeh's (1986) recursive filter applied within these layer boundaries. A five-point sliding window is used inside the layers with a change of reference trace when approaching boundaries.…”

VSP wave field separation: An optimization method based on block relaxation and singular value thresholding

“…Many VSP wave field separation approaches (Aminzadeh, 1986;Seeman and Horowicz, 1983;Simaan, 1983;Simaan and Love, 1984) can successfully separate wave fields that propagate uniformly within a chosen analysis window. Besides these approaches, signal processing methods such as median filter (Stewart, 1985), frequency-wave number domain filter (Treitel et al, 1967), and Radon transform filter (Moon et al, 1986) are also applied to VSP wave field separation.…”

“…This is different from the traditional SVD method. In the past research, it was generally assumed that the wave forms of a same wave type recorded at different traces were the same in an analysis window (Aminzadeh, 1986;Seeman and Horowicz, 1983;Simaan, 1983;Simaan and Love, 1984). In practice, the wave forms are not the same but similar because of reflection, absorption and scattering.…”