Replacement velocity: freezing the water

Sometimes you are lucky and you know the velocity. Maybe you know it because you are dealing with synthetic data. Maybe you know it because you have already drilled 300 shallow holes. Or maybe you can make a good estimate because you have a profile of water depth and you are willing to guess at the sediment velocity. Often the velocity problem is really a near-surface problem. Perhaps you have been dragging your seismic streamer over the occasional limestone reefs in the Red Sea.

Assuming that you know the velocity and that the lateral variations are near the surface, then you should think about the idea of a replacement velocity. For example, suppose you could freeze the water in the Red Sea, just until it is hard enough that the ice velocity and the velocity of the limestone reefs are equal. That would remove the unnecessary complexity of the reflections from deep targets. Of course you can't freeze the Red Sea, but you can reprocess the data to try to mimic what would be recorded if you could.

First, downward continue the data to some datum beneath the lateral variations. Then upward continue it back to the surface through the homogeneous replacement medium.

While in principle the DSR could be used for this job, in practice it would be expensive and impractical. The best approach is to study the two operations--going down, then going up--in combination. Since the two operations are largely in opposition to each other, whatever is done to the data should be just a function of the difference. For example, the equation  

$${\partial P \over \partial z} \eq i \omega \left[ {1 \over v... ...\ {1 \over v (g)} \ \ -\ \ {2 \over v_{\rm avg}} \ \right] \ P$$ (16)

combines the downward continuation with the upward continuation and makes little change to the wavefield P when the velocities are nearly the same. Equation (16) is basically a time-shifting equation. There is an industry process known as static corrections. The word static implies time-invariant--the amount of time shift does not depend on time. When the appropriate corrections are merely static shifts, then the earth model has lateral velocity variations in the near surface only. This is often the case. Equation (16) also has the ability to do time-variable time shifts because v(s) and v(g) can be any function of depth z. Because of the wide-offset angle normally used, it is desirable to extend (16) to a wider angle. Such extensions are found in Lynn [1979]. Lynn also shows how partial differential equations can be written to describe the influence of lateral velocity variation on stacking velocity. Berryhill [1979] illustrated the use of the Kirchhoff method for an irregular datum.

In practice, the problem of estimating lateral velocity variations is usually more troublesome than the application of these velocities during migration. Static time shifts are estimated from a variety of measurements including the elevation survey, travel times from the bottoms of shot holes to the surface, and crosscorrelation of reflection seismograms. Wiggins et al. [1976] provide an analysis to determine the static shifts from correlation measurements.

 

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Figure 19 Data (left) from Philippines with dynamic corrections (right). (by permission from Geophysics, Dent [1983])

Where the lateral variation runs deeper the time shifts become time-dependent. This is called the dynamic time-shift problem. To compute dynamic time shifts, dip is assumed to be zero. Rays are traced through a presumed model with laterally variable velocity. Rays are also traced through a reference model with laterally constant velocity. The difference of travel times of the two models defines the dynamic time shifts. See Figure 19. Where the lateral variation runs deeper still, the problem looks more like a migration problem. Figure 20 illustrates a process called REVEAL by Digicon, Inc., who have not revealed whether a time-shift method or a wave-equation method was used.

 

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Figure 20 Example of processing with a replacement velocity. Observe that deeper bedding is now flatter and more continuous. (distributed by Digicon, Inc.)