Velocity analysis

A diffractor in the subsurface scatters the energy of an incident wave. For a constant velocity subsurface, such a diffractor causes a perfect hyperbola in the recorded time section. The hyperbola's apex is at the lateral position of the diffractor independent of the the actual shot location. The hyperbolic shape of the event is caused by the differences in traveltime of rays from the diffractor to individual geophones. Varying the shot location at the surface changes the traveltime of the ray from the shot to the diffractor, shifting the hyperbola vertically without affecting its shape or lateral position.

 

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Figure 1 Velocity analysis. The different rows of frames describe the same processing sequence for different velocity cases. In all rows, the left frame depicts a subsurface section, two surface shot positions, and several raypaths. The center frames show the diffractor response in each shot gather. The right frame overlays the shot gather s₁ and s₂ after extrapolation.

The left uppermost frame of Figure 1 depicts schematically the raypaths connected with two shots, s₁, s₂, in a constant velocity medium with a single diffractor. The two center frames of the top row show the two recorded shot gathers. After extrapolation of shot gather s₁ to shot gather s₂ using shot continuation, the hyperbolic events coincide, as it is indicated by overlaying the two gathers in the fourth frame.

In the center row of Figure 1, a velocity perturbation along geophone rays distorts the hyperbolic moveout in both shot gathers. Since the events in both shot gathers are distorted identically, the shot gathers nevertheless coincide after shot continuation has been applied. In the bottom row of Figure 1, however, a velocity perturbation along a shot ray causes an additional relative time shift of the two hyperbolic events. After continuation of shot s₁ to shot location s₂, the two events are parallel but shifted in time.

In the case of a more complex subsurface, these shifts are detected by choosing a vertex position, extracting a local hyperbolic window in both equivalent gathers and correlating them by shifting along the time axis. The time shift $\tau$ which yields the maximum correlation is stored at the vertex position.

Since only velocity perturbations along the shot-diffractor ray segment cause these $\tau$ shifts, a simple first step to velocity inversion is back projecting these $\tau$ values along the line between vertex and shot position. Superposing the resulting time sections for different shots yields a map displaying the difference between the actual subsurface velocity distribution in (x,t) and the constant velocity model.

In a velocity variable medium, the outlined velocity analysis scheme only approximates the true velocity. Tests on complex models and real data will have to prove its usefulness.