MODELING NEAR SURFACE LATERAL VELOCITY VARIATION

The velocity model used for this study consists of two reflectors and a near surface velocity perturbation (Figure 1). The velocity perturbation consists of a low velocity zone bounded by two Gaussian shaped surfaces.

 

model Figure 1 Two horizontal reflectors at $-0.4\; km$ and $-0.8\; km$ are represented by solid lines. Synthetic data are initially generated at the surface represented by the dotted line. This data is then upward continued through the two surfaces represented by the dashed lines up to the solid line at $0.4\; km$. The medium velocity is $2\; km/s$ everywhere except between the two dashed lines where it is $1.5\; km/s$.

Synthetic data are generated using two Kirchhoff modeling programs. I first generate constant velocity synthetic shot gathers (Figure 2a) using a constant offset migration and modeling program similar to the program described in a previous report Bevc and Claerbout (1993). These synthetic gathers are generated at a depth of $0\;km$(Figure 1). The synthetic data are then upward continued to an elevation of $0.4\; km$using a data parallel Kirchhoff datuming algorithm Bevc (1993). The upward continuation is performed in three steps, using constant velocities in each step. The first upward continuation transforms the data from the flat surface at $0\;km$ to the Gaussian surface $0.19\;km$. The data are then upward continued with a low velocity to the next Gaussian surface. The complete synthetic data set is generated by upward continuing the data to the level surface at elevation $0.4\; km$.

The effect of the low velocity perturbation can be seen in Figure 2b as a timing and focusing anomaly. Because of the focusing effect, a static shift of the data is not adequate to correct for the velocity perturbation. As the shot number increases, the anomaly moves from right to left across the gathers. There are some artifacts in the synthetics because of data truncation effects and the rapid undulation of the Gaussian boundary.

 

shot
Figure 2 A synthetic shot gather (a) before upward continuation and (b) after upward continuation through the near surface velocity anomaly. The effect of the velocity perturbation is to cause a timing shift and a focusing anomaly in the synthetic data.