Synthetic Example

To illustrate the homothetic scaling approach we built a synthetic model (Figures 1 and 2), and downward continued to the reflector. If we start with an initial constant velocity model equal to the background velocity we see good coherency at zero time at CMP locations far away from the anomaly (Figure 3). For CMP's closer to the anomaly, focusing deteriorates and deviates from zero time.

 

layer.gauss Figure 1 Input velocity model. Gaussian anomaly at 4 km/s in an otherwise homogeneous half-space (3km/s).

 

layer.shot Figure 2 A sample shot gather illustrating the non-hyperbolic move-out (hyperbolic move-out indicated by `*') caused by the velocity anomaly.

 

layer.gama.1
Figure 3 Result of downward continuing at three different locations. The left panel is far away from anomaly(CMP 8); the center, at the edge of the anomaly(CMP 7); and the right directly above anomaly(CMP 6).

We then found which scaled version of the travel-time field best focused the data at various CMP locations. Figure 4 shows the optimal focusing result for the same three CMP's as Figure 3. As expected, away from the anomaly the best focusing occurs at zero time when $\gamma=1.$ At the edge of the anomaly, best focusing is achieved at positive time, with $\gamma<1.$ (outer offset ray-paths pass through the anomaly, decreasing curvature, indicating a faster velocity). While directly above the anomaly we see maximum focusing at negative time, at $\gamma\gt 1.$ (inner offsets faster, more curvature, indicating lower velocity and increased travel-times).

 

layer.gama.best
Figure 4 The best focusing $\gamma$.The left, CMP 8, $\gamma=1.$;center, CMP 7, $\gamma=.88$; and right, directly above, CMP 6 $\gamma=1.10$.