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In previous chapters, when I analyzed methods for imaging zero-offset and prestack data, I assumed that the velocity function was known either for rms velocity or for interval velocity. Of course, in reality that is not true, and velocity must be estimated from the seismic data.

Estimation of velocity from seismic data is an ill-posed inverse problem, in the sense that the data do not contain all the necessary information to define a velocity function that varies arbitrarily with depth and along the horizontal directions. Fortunately, in many practical situations, seismic imagers have a priori knowledge of the behavior of the velocity function. That knowledge complements the information contained in the seismic data, and the combination of these two sets of information — the information contained in the data and the a priori knowledge — often is sufficient to constrain the problem adequately.

A typical example is velocity estimation in a horizontally layered medium using the Dix inversion formula. If one assumes that the subsurface is layered horizontally and is isotropic, one can measure the rms velocity function from the data, and one can apply the Dix formula to estimate the interval velocity function. However, in the presence of structural dips and lateral variations in velocity, one must use more sophisticated analytic methods or risk estimating grossly inaccurate velocity functions. The drawback of using methods that are more general than the Dix formula is that velocity estimation becomes poorly constrained as one adds degrees of freedom to the set of

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