Interval velocities from slant stacks

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  

$$P( \omega , p,z_2 ) \eq P( \omega , p,z_1 ) \ e^{ - i\, {\omega \over v }\ \sqrt { 1\ -\ p^2 v^2 }\ (z_2 \ -\ z_1 )}$$ (13)

Downward continuing to the first reflector, we find that the first reflections should arrive at zero time. In migration it is customary to retard time with respect to the zero-dip ray. So downward continuation in retarded time flattens the first reflection without changing the zero-dip ray. Time shifting the data to align on the first-layer reflection is illustrated by the third panel in Figure 8. The first panel shows the velocity model, and the second panel shows the slant stacks at the surface. After the first reflector is time aligned, we have the data that should be observed at the bottom of the first layer. Now the next deeper curve is an exact ellipse. Estimate the next deeper velocity from that next deeper ellipse. Continue the procedure to all depths. This method of velocity estimation was proposed and tested by P. Schultz [1982].

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

 

schultz
Figure 8 Schultz flattening on successive layers.

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.