Eigenstructure analysis for the seismic analytical signal in stacking velocity estimation

Abstract

Stacking velocity estimation in seismic processing may be benefit by the use of the complex analytic signal obtained by the Hilbert transform combined with eigenstructure-based methods. In this paper we investigate how the Multiple Signal Classifier is affected by the Hilbert transform. We propose for the complex seismic data a model that approximates the arrival delays as a steering vector for velocities close or equal to the stacking velocity. This approximation helps to interpret the higher-resolution of the eigenstructure-based methods combined with the Hilbert transform. We observed that the eigendecomposition of the complex traces presents a larger concentration of energy in the first eigenimage when compared to the decomposition without the Hilbert transform. Introduction Seismic data acquisition is performed in shot-receiver coordinate (Yilmaz, 2001). The recorded data form a common-shot gather with the same shot recorded at different receivers. At each receiver, the recorded data are known as seismic trace. For processing seismic data we can sort the recorded traces in different ways, being one of the most popular the common-midpoint (CMP) method (Mayne, 1962). A CMP gather is formed by putting the traces with the same midpoint between the shot and receiver locations together, with the characteristic that all the traces will be reflections from the same point in depth. In reflection seismic processing, stacking is the operation which uses the diversity property of CMP gathers and generates for each CMP a single zero-offset (ZO) trace. Each sample of the zero-offset trace at time t0 is generated by summing the amplitudes of the traces in the CMP gather along the normal-moveout (NMO) equation (Dix, 1955), which models the arrival time of reflections in the traces and which depends on the stacking velocity. The process of estimating the stacking velocities in the framework of the CMPmethod is called velocity analysis (Taner and Koehler, 1969). In this procedure, for each t0 and each candidate stacking velocity vk a coherence function is computed. The trial velocity which corresponds to the highest coherence value is chosen as the stacking velocity for that t0. The standard coherence function used for velocity analysis is a second-order energy measure called semblance (Neidell and Taner, 1971). In (Biondi and Kostov, 1989) and (Kirlin, 1992), it was shown that coherence measures based on eigenstructure methods, such as the MUltiple SIgnal Classifier (MUSIC) introduced by (Schmidt, 1986), can lead to velocity spectra with higher resolution than semblance. Recently, a seismic MUSIC based on the temporal correlation matrix has been proposed (Barros et al., 2015). On the other hand, the advantages of processing complex analytic seismic traces, obtained by the Hilbert transform, has been discussed since (Taner et al., 1978). In (Sguazzero and Vesnaver, 1986) different coherence measures (some of them using the complex seismic analytical signal) were discussed and it was mentioned in (Biondi and Kostov, 1989) that eigenstructure methods could be benefit by using the complex seismic analytical signal. Although the use of the Hilbert transform improves the resolution of the velocity stacking estimation, this observation is not explained in the literature. In this paper we discuss and try to explain the impact of the Hilbert transform on the resolution of eigenstructure methods such as MUSIC, as illustrated in (Biondi and Kostov, 1989). In order to do that, we model the arrival delays for the complex seismic data as a steering vector, near the region of the correct stacking velocity. We then analyze the eigendecomposition of the complex seismic traces and present some numerical experiments to support our analysis. Our main observation is that the eigendecomposition of the complex traces presents a larger concentration of energy in the first eigenimage when compared to the decomposition without the Hilbert transform. Eigenstructure-based coherence In this section, we first show how the seismic windowed data, which is used by all coherence measures, is formed. Then, we present the spatial and temporal correlation matrices of the windowed data, and show how these matrices yield the MUSIC coherence measures. For further detail, we encourage the reader to see (Barros et al., 2015).