Signal/noise separation and velocity estimation

Abstract

A signal/noise separation must recognize the lateral coherence of geologic events and their statistical predictability before extracting those components most useful for a particular process, such as velocity analysis. Events with recognizable coherence we call signal; the rest we term noise. Let us define “focusing” as increasing the statistical independence of samples with some invertible, linear transform L. By the central limit theorem, focused signal must become more non‐Gaussian; the same L must defocus noise and make it more Gaussian. A measure F defined from cross entropy measures non‐Gaussianity from local histograms of an array, and thereby measures focusing. Local histograms of the transformed data and of transformed, artificially incoherent data provide enough information to estimate the amplitude distributions of transformed signal and noise; errors only increase the estimate of noise. These distributions allow the recognition and extraction of samples containing the highest percentage of signal. Estimating signal and noise iteratively improves the extractions of each. After the removal of bed reflections and noise, F will determine the best migration velocity for the remaining diffractions. Slant stacks map lines to points, greatly concentrating continuous reflections. We extract samples containing the highest concentration of this signal, invert, and subtract from the data, leaving diffractions and noise. Next, we migrate with many velocities, extract focused events, and invert. Then we find the least‐squares sum of these events best resembling the diffractions in the original data. Migration of these diffractions maximizes F at the best velocity. We successfully extract diffractions and estimate velocities for a window of data containing a growth fault. A spatially variable least‐squares superposition allows spatially variable velocity estimates. Local slant stacks allow a laterally adaptable extraction of locally linear events. For a stacked section we successfully extract weak signal with highly variable coherency from behind strong Gaussian noise. Unlike normal moveout (NMO), wave‐equation migration of a few common midpoint (CMP) gathers can image the skewed hyperbolas of dipping reflectors correctly. Short local slant stacks along midpoint will extract reflections with different dips. A simple Stolt (1978) (f-k) type algorithm migrates these dipping events with appropriate dispersion relations. This migration may then be used to extract events containing velocity information over offset. Offset truncations become another removable form of noise. One may remove non‐Gaussian noise from shot gathers by first removing the most identifiable signal, then estimating the samples containing the highest percentage of noise. Those samples containing a significant percentage of signal may be zeroed; what remains represents the most identifiable noise and may be subtracted from the original data. With this procedure we successfully remove ground roll and other noise from a shot (field) gather.