Introduction

Migration velocity analysis (MVA) is one of the most important problems of seismic imaging Claerbout (1999), and yet it remains one without a conventional solution. Many techniques have been devoted to solving this problem and, generally speaking, they fall into two broad categories: methods which directly use traveltimes computed using the eikonal equation, and methods which use the entire recorded wavefields. The methods in the first category are usually known by the name of traveltime tomography Clapp (2001); Stork (1992), while the methods in the second category are known by the names of wave-equation tomography Tarantola (1984); Woodward (1992) or wave-equation migration velocity analysis Biondi and Sava (1999); Sava and Fomel (2002); Stolk and Symes (2002).

The wave-equation MVA techniques are, in theory, superior to the traveltime-based MVA methods since they make use of the entire recorded data and not only of picked traveltimes at selected events. Some of those methods are also better able to account for multipathing occurring in complicated geological situations, a goal that is difficult to achieve with ray-traced traveltimes. Moreover, wave-equation techniques are more accurately describing wave propagation, since they are not based on high frequency asymptotic assumptions.

However, none of the wave-equation velocity analysis methods has yet been accepted as a practical solution to exploration problems. Part of the reason is cost, which remains high, despite the continually decreasing cost of computing hardware. In addition, many of those velocity analysis techniques become unstable if the data are polluted with coherent noise, if the recorded offsets are too short or if enough low frequencies are not available in the band of the data Pratt (1999).

Wave-equation MVA (WEMVA) is different from wave-equation tomography (WET) with respect to the domain in which each one computes residuals: WET operates in the data space, and estimates velocity by fitting the recorded data, while WEMVA operates in the image space, and estimates velocity by improving the quality of the migrated images. As for the traveltime tomography methods, estimating velocity in the migrated image space is a much more robust approach and more likely to converge to geologically meaningful solutions.

The usual property used for optimization is that of flat events measured along angle-domain common-image gathers. Optimal flatness in the angle-domain is equivalent to optimal focusing at zero-offset Stolk and Symes (2002), therefore explicit conversion to the angle-domain is not necessary. Similarly, we could use focusing along the spatial axes as well as focusing along offset in order to estimate migration velocity Sava and Etgen (2002)

In this paper, we generalize the wave-equation migration velocity analysis technique to include both the target image fitting method of Biondi and Sava (1999) and the differential semblance optimization method of Stolk and Symes (2002) in a unified framework. We show that both methods are just special cases of a more general technique. We discuss these two members of this general class of problems, and we point out that other more or less optimal methods exist.

In the following sections, we present in detail our generalization of the WEMVA method, followed by an example and a brief discussion of the results.