Seismic source wavelet estimation and the random reflectivity assumption

In the February 1991 issue of GEOPHYSICS, Anton Ziolkowski gives a scathing criticism of statistical wavelet estimation methods. Among other points, Ziolkowski questions the validity of the randomness assumption. This assumption allows statistical methods to estimate the seismic source wavelet autocorrelation from the seismic trace autocorrelation. In this study, we examine this traditional assumption of a random reflectivity sequence. The validity of this assumption is examined by using well-log synthetic seismograms and by using a procedure for evaluating the resulting deconvolutions. With real data, we compare our wavelet estimations with the in-situ recording of the wavelet from a vertical seismic profile (VSP). As a result of our examination of the randomness assumption, we present a fairly simple test that can be used to evaluate the validity of a randomness assumption. INTRODUCTION Ziolkowski’s work criticizes the conventional assumptions of wavelet deconvolution as being generally invalid. These assumptions included wavelet stationarity, minimum phase and zero phase, and the assumption of random reflectivity. Ziolkowski advocates that seismic source signatures be measured rather than estimated when performing deconvolution. Although it is difficult to ever argue against using physical measurements, the expense and difficulty of the measurements may lead one to use statistical wavelet estimation. It is then worthwhile to consider how accurate these statistical measurements might be. In particular, we seek to evaluate the common assumption of reflectivity randomness by use of model synthetics and real data. Since one would expect this assumption to be dependent on geology, we have examined data from various oil fields. Statistical wavelet estimation methods usually assume that the reflectivity is a random uncorrelated signal (that is, the reflectivity has an autocorrelation which approximates a delta function) and this effectively means that the seismic trace autocorrelation is approximately equal to the seismic wavelet autocorrelation, ) ( ) ( ) ( ) ( * * z W z W z X z X ≅ . Consider the following figure from Russell (1994).