Further thoughts on the stacking response in seismic data processing

In a recent technical article in First Break, Gausland (2004) made the case that the result of stacking is not limited to the often quoted factor of reduction in noise where n is the fold of a CMP-gather. Through his figures and illustrations, Gausland showed that stacking also acts as a frequency and wavenumber filter. Although the intentions of the article were not to, as Gausland put it, ‘give methods or formulae’ and that a ‘simplified analysis can be made using simulation’ instead of mathematics, we could not help but see a connection between his main points and the method of stationary phase (Born & Wolf, 1980). In this comment, we give our interpretation of Gausland’s results within the language of stationary phase. Gausland’s conclusions are supported by the stationary phase analysis, save for some details concerning the time-delay induced by mis-stacking. In addition, we find other factors affecting the stacking response that were not pointed out by Gausland. The issues on fold versus spreadlength brought up by Gausland cannot be addressed explicitly within the stationary phase analysis; we return to them at the end of this comment. Arguments based on stationary phase have been used extensively in the literature on imaging and migration (Bleistein et al., 2001). Hence, a result of this mathematical excursion is a clear connection between stacking and migration, which Gausland alluded to briefly when he stated that ‘a further analysis of the similarities between . . . stacking and migration is necessary for a full understanding of these important aspects of seismic data processing . . .’. Suppose that in a CMP-gather there is a single event, with a zero-offset waveform ƒ(t) at zero-offset two-way-time T0, that has hyperbolic moveout with NMO-velocity v and a wavelet that does not change with offset (see Fig. 1). The event is stacked with a hyperbola whose apex is at T0 using a stacking velocity vst not necessarily equal to v. When vst does not equal v, a time shift ∆tk occurs at the k-th offset trace before stacking. Hence, the normalized stacking response, neglecting NMO-stretch, is