According to the Dix approximation, travel time is a unique function of vertical travel time because

(2) |

Figure 1

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.

11/11/1997