High-resolution velocity gathers and offset space reconstruction

We present a high-resolution procedure to reconstruct common-midpoint (CMP) gathers. First, we describe the forward and inverse transformations between offset and velocity space. Then, we formulate an underdetermined linear inverse problem in which the target is the artifacts-free, aperture-compensated velocity gather. We show that a sparse inversion leads to a solution that resembles the infinite-aperture velocity gather. The latter is the velocity gather that should have been estimated with a simple conjugate operator designed from an infinite-aperture seismic array. This high-resolution velocity gather is then used to reconstruct the offset space. The algorithm is formally derived using two basic principles. First, we use the principle of maximum entropy to translate prior information about the unknown parameters into a probabilistic framework, in other words, to assign a probability density function to our model. Second, we apply Bayes’s rule to relate the a priori probability density function (pdf) with the pdf corresponding to the experimental uncertainties (likelihood function) to construct the a posteriori distribution of the unknown parameters. Finally the model is evaluated by maximizing the a posteriori distribution. When the problem is correctly regularized, the algorithm converges to a solution characterized by different degrees of sparseness depending on the required resolution. The solutions exhibit minimum entropy when the entropy is measured in terms of Burg’s definition. We emphasize two crucial differences in our approach with the familiar Burg method of maximum entropy spectral analysis. First, Burg’s entropy is minimized rather than maximized, which is equivalent to inferring as much as possible about the model from the data. Second, our approach uses the data as constraints in contrast with the classic maximum entropy spectral analysis approach where the autocorrelation function is the constraint. This implies that we recover not only amplitude information but also phase information, which serves to extrapolate the data outside the original aperture of the array. The tradeoff is controlled by a single parameter that under asymptotic conditions reduces the method to a damped least-squares solution. Finally, the high-resolution or aperture-compensated velocity gather is used to extrapolate nearand far-offset traces. INTRODUCTION dow sometimes cause a poor velocity resolution when using Conventional velocity analysis is performed by measuring energy along hyperbolic paths for a set of tentative velocities. The analysis of the results in the (two-way zero offset time and velocity) plane serves to estimate the stacking velocity that is later used to construct the zero-offset section. The semblance (Neidell and Taner, 1971) is one of the most popular measures of coherent energy along hyperbolic trajectories in common-midpoint (CMP) gathers. The semblance measures the ratio of the signal energy within a window to the total energy in the window. Noise with nonzero mean and closely-spaced events in the same winthis measure. The poor resolution of the semblance has lead to more sophisticated techniques based on the eigenstructure of the data covariance matrix (Biondi and Kostov, 1989; Key and Smithson, 1990). In these techniques, the data covariance matrix is decomposed into signal and noise space contributions. Different metrics based on the eigenvector of the signal space are then used to measure coherent energy along hyperbolic paths. The semblance, or any other velocity measure, can be displayed as a contour map where each maximum corresponds to the pairs Mapping the original data back Manuscript received by the Editor February 17, 1994; revised manuscri t received November 17, 1994. Dept. of Geophysics and Astronomy, The University of British Columbia, 129-2219 Main Mall, Vancouver, B.C., V6T 1Z4, Canada. © 1995, Society of Exploration Geophysicists. All rights reserved.