Introduction

Thorson and Claerbout (1985) extensively studied the use of velocity stacks, of their pseudo-inverse and stochastic inverse, to solve the problem of missing data restoration, and to design a more robust stacking operator for data with missing traces. They stressed the importance of estimating the inverse covariance matrix of the model, and provided a theoretical basis for justifying the use of an iterative process to update the estimation. Harlan (1988) modeled primaries and multiples using two beams of velocity functions that were manually designed using the semblance spectrum as a guide. The modeled multiples were then convolved with a wavelet and subtracted from the original data. Some limitations of this approach include the absence of diffractions in the modeled data, the assumption that multiples and primaries are restricted to non-overlapping thin regions, and the need for human interaction in the design of the beams.

Although the velocity space is a good place to distinguish multiple from primary reflections, attempts to suppress energy associated with multiples in this domain are usually spoiled not only by the cost of the method, but also by the presence of artifacts and significant changes in the amplitude and phase of primary events. Filtering a region in the velocity domain and transforming back to the space domain would be an expensive procedure if an exact (pseudo)inverse were to be used, because the velocity transform lacks any of the special structures that would allow for a less expensive exact inverse. Fortunately, only a few iterations are required to obtain a very good approximation for the least-squares inverse. Also, the filtering can be formulated as an optimization problem (power minimization) that can be solved by a fast-converging conjugate gradients method. In general, the method converges after a few iterations, which makes it affordable for routine processing.

The other undesirable aspects, artifacts and changes in the primaries, are caused by the fact that although primaries and multiples are concentrated, respectively, in the high- and low-velocity regions, the contribution from the near offset traces spreads through the whole velocity domain. Attempts to separate the velocity domain into two or three regions (filter design) involve two adverse factors: complete dependence on the designer's judgement, and the unavoidable truncation of both primary and multiple events. To overcome these limitations we use the predictability of the multiples in the velocity domain to automatically design a window function that separates primaries from multiples. The method uses a multichannel prediction error filter, which is also designed using conjugate gradients.