Phase encoding with Gold codes for wave-equation migration

Introduction

Biondi (2007,2006) introduced the concept of the prestack exploding-reflector modeling. This method synthesizes source and receiver wavefields along the entire survey at the surface, in the form of areal data, starting from a prestack migrated image cube computed with wave-equation migration. For migration velocity analysis, the aim is to generate a considerably smaller dataset than the one used in the initial migration, while maintaining the necessary kinematic information to update the velocity.

Conceptually, the synthesized areal data are computed by upward propagating source and receiver wavefields using subsurface-offset-domain common-image gathers (SODCIGs) as initial conditions. To decrease the number of experiments to migrate, we take advantage of the linearity of the wave propagation to combine several experiments into a set of composite records. Combining several experiments gives rise to crosstalk during imaging (Guerra and Biondi, 2008; Biondi, 2006). Guerra and Biondi (2008) use pseudo-random-phase encoding (Romero et al., 2000) during the modeling step to attenuate crosstalk.

It is common, in the exploration geophysics community, to employ pseudo-random codes using intrinsic functions specific to the programing language. These pseudo-random codes present, generally, a uniform distribution. Their autocorrelation and cross-correlation functions have no special properties. The autocorrelation function presents nearly periodic side lobes with additive low-amplitude random variations. The peak-to-side lobe ratio is around 30. The cross-correlation function is pseudo-random, and its amplitudes are of the same order of magnitude as those of the non-zero lags of the autocorrelation function. Herein, theses codes are called conventional random codes.

In wireless communication, especially for systems using Code Division Multiple Access (CDMA), a class of different pseudo-random codes have been widely used (Shi and Schelgel, 2003). These codes are binary sequences and have unique autocorrelation and cross-correlations properties which make them more suited to achieve the above-mentioned objectives with minimal crosstalk. The autocorrelation function is represented by a large peak, whose amplitude equals the number of samples in the code, and the cross-correlation peaks, at non-zero lags, with the same amplitudes as that of the autocorrelation. Examples of binary pseudo-random codes used by these communities are Golay (Golay, 1961; Tseng, 1972), Kasami (Kasami, 1966) and Gold codes (Gold, 1967). Medical imaging (Gran, 2005) and radar communities (Levanon and Mozeson, 2004) also exploit the statistical properties of these pseudo-random codes to increase bandwidth, signal-to-noise ratio and pulse compression.

Quan and Harris (1991) analyze orthogonal codes to encode simultaneous source signatures for cross-well surveys, and conclude that m-sequences and Gold codes provide the best results on the separation of the seismograms. Here, we exploit the properties of the Gold codes to encode the prestack exploding reflector modeling experiments.

In the next section we give a brief description of the prestack exploding-reflector modeling. Then we discuss how to compute the Gold codes. To illustrate the effectiveness of phase encoding with Gold codes, we compare the migration of prestack-exploding-reflector modeled data encoded with conventional random codes and Gold codes.

Phase encoding with Gold codes for wave-equation migration