Earlier we showed that downward continuation of Snell waves is purely a matter of time shift. The amount of time shift depends only on the angle of the waves. For example, a frequency domain equation for the shifting is

(13) |

Figure 8 illustrates the difficulty caused by a shallow, high-velocity layer.

Figure 8

Reflection from the bottom of any deeper, lower-velocity layer
gives an incomplete ellipse.
It does not connect to the ellipse above
because it seems to want to extend beyond.
The large *p*-values (dotted in the figure)
are missing because they are blocked by the high-velocity
(low *p*) layer above.
The cutoff in *p* happens where waves in the high-velocity layer
go horizontally.
So there are no head waves on deeper, lower-velocity layer bottoms.

Schultz's method of estimating velocity from an ellipse
proceeds by summing on scanning ellipses of various velocities
and selecting the one with the most power.
So his method should not be troubled by shallow high-velocity layers.
It is interesting to note that when the velocity
does increase continuously with depth,
the velocity-depth curve can be read directly from
the rightmost panel of Figure 8.
The velocity-depth curve would be the line connecting the ends (maximum *p*)
of the reflections,
i.e. the head waves.

10/31/1997