The interval velocity estimation problem can be formulated as a least-square optimization scheme. With this formulation, we search for the velocity parameter that best fits equation (1) considering several points of two adjacent reflections. To avoid division by zero, and to increase the contribution from large horizontal slownesses (where the velocity information is better resolved), I define the objective function as follows:
Here, equation (3) is implemented, with the help of an event-picking algorithm (Zhang, 1991) to define the several reflection curves. The picking algorithm defines the relative time-shifts between samples of adjacent traces. This information is used to trace events in the x-t domain starting at all samples of the first trace. Because the events should coincide with the true reflections wherever they are strong enough, the algorithm computes the stacking-power (or semblance) for each event to select the best candidates for reflections. Figure shows synthetic data generated from a velocity model (with moderate velocity contrasts to avoid strong multiples contamination) overlaid with the six selected picked events. The selection was based entirely on the stacking power, but some interactive procedure will be required when strong multiples or other coherent noises are present.
Figure 3 Synthetic data overlaid with automatically-picked reflection events.
This algorithm also uses the picked reflection-curves to compute also the horizontal slownesses at the sampled offsets. For any two pairs of adjacent reflections, the terms inside the sum of equation (3) are defined by using the sampled points of the bottom reflection as reference. For each of these points the algorithm finds the matching point in the upper curve, that is, the point with the same horizontal slowness as that of the reference point in the bottom curve. To find the matching point, the algorithm uses a spline interpolation of the inverse function xi=xi(pi) (defined for each sampled point i in the upper curve). Figure a shows the functions p(x) corresponding to all six selected events shown in Figure . Clearly the horizontal slowness evaluated with this method is very sensitive to even small errors in the ti(xi) picked curve associated with each reflection. This sensitivity requires the p-curves to be filtered before the velocity estimation step. First, the values of p that substantially differ from the local mean are removed. Then the gaps are filled by interpolation, and convolutional smoothing is applied. Figure b shows the resulting filtered version of Figure a.
It is worth mentioning that the horizontal slowness curves p(x) need to have a monotonic behavior, so that the inverse functions x(p) can be uniquely defined. In principle, this condition should always be satisfied when the plane-layered assumption is valid, but the intervention of noise may result in a nonmonotonic p(x).
After the points sharing the same horizontal slownesses are matched, application of equation (3) is straightforward. Both the resulting estimated model and the true model are represented in Figure . Although the relative errors in the estimated velocities are not large, we should expect to achieve much worse results when dealing with real data because of the high sensitivity of the picked-based horizontal slowness evaluation used in this algorithm.