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# ROUGH V(z) MAKES TAU(t) MULTIVALUED

According to the Dix approximation, travel time is a unique function of vertical travel time because
 (2)
The reverse is not true, however, can be a multivalued function of t, and is especially likely to be so where is a rough function of .When is a multivalued function of t the process of offset continuation breaks down. Then extra offsets are providing extra information. We don't yet know if the extra information is a small amount or a large amount or whether that extra information is uniformly or locally distributed. Figure 1 shows an example.

superres
Figure 1
Right shows for many offsets.

Figure 1 shows two kinds of multivaluedness in the transformation. First is the familiar kind that arises whenever dv/dz > 0 where travel times of shallow waves cross those of deep waves. Let us place a line through the broad maxima in at about for all x. In a constant velocity earth, the ratio corresponds to a propagation angle or about .Thus, a wave with average angle greater than generally arrives at the same time and offset as another wave with an average angle less than .

The second way of being multivalued is less familiar and hence more interesting, the roughness in the transformation. We see this roughness does give rise to multivaluedness. Disappointingly, the multivaluedness is not found everywhere but mainly along the trend. We have not yet answered how much extra information we can obtain from this. Clearly though, if multivaluedness is what makes different offsets give us different information, it is along this mute-line'' trend where we must look.

Let us find the high frequency. Where does an observable (low) frequency on the t axis map to a high frequency on the axis? It happens where a long region on the t axis maps to a short region on the axis, in other words, where the slope is greatest. This is the opposite of usual NMO in the neighborhood of the diagonal asymptote in Figure 1 where .From the figure, we see the possibility for frequency boosting does not arise from the roughness in velocity but just beneath the water bottom at any offset, i.e., at the greatest angles. Since is negative there, it gives a kind of upside-down image. To understand this image, think of head waves where the deepest layer is fastest and hence has the earliest arrival with shallower layer arrivals coming later.

It is possible the Dix approximation is breaking down here, a concern that requires further study. Accurate reflection seismograms in this region are easy to make with the phase shift method. Getting correct head waves is more complicated.

Next: FURTHER STEPS Up: Claerbout: Super resolution Previous: ROUGH VELOCITY(z)
Stanford Exploration Project
11/11/1997