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According to the Dix approximation, travel time $t(\tau )$ is a unique function of vertical travel time $\tau$ because

t^2 \quad =\quad \tau^2 + x^2 / v(\tau)^2
\end{displaymath} (2)

The reverse is not true, however, $\tau(t)$ can be a multivalued function of $t$, and is especially likely to be so where $v(\tau)$ is a rough function of $\tau$. When $\tau(t)$ 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.

Figure 1.
Right shows $t(\tau )$ for many offsets.
[pdf] [png] [scons]

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 $t(\tau )$ at about $t=2.5\tau$ for all $x$. In a constant velocity earth, the ratio $t/\tau=2.5$ corresponds to a propagation angle $\cos\theta = \tau/t$ or about $\theta = 66^\circ$. Thus, a wave with average angle greater than $\theta = 66^\circ$ generally arrives at the same time and offset as another wave with an average angle less than $\theta = 66^\circ$.

The second way of being multivalued is less familiar and hence more interesting, the roughness in the $t(\tau )$ transformation. We see this roughness does give rise to multivaluedness. Disappointingly, the multivaluedness is not found everywhere but mainly along the $\theta = 66^\circ$ 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'' $\theta = 66^\circ$ 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 $\tau$ axis? It happens where a long region on the $t$ axis maps to a short region on the $\tau$ axis, in other words, where the slope $dt/d\tau$ is greatest. This is the opposite of usual NMO in the neighborhood of the diagonal asymptote in Figure 1 where $dt/d\tau<1$. 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 $dt/d\tau$ 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.

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Next: FURTHER STEPS Up: Claerbout: Super resolution Previous: ROUGH VELOCITY(z)