Velocity-independent time-domain seismic imaging using local event slopes

Discussion

The main advantage of the oriented approach is speed. The cost of velocity scanning in conventional processing (excluding the manual picking labor involved) can be estimated roughly as the number of scanned velocities $N_v$ times the input data size. The cost increases dramatically in the case of non-hyperbolic approximations when more than one parameter needs to be picked. The cost of local slope estimation with the plane-wave destruction method is roughly the data size times the number of iterations $N_i$ times the filter size $N_f$. Typically, $N_i \approx 10$ and $N_f = 6$, which is approximately equivalent in cost to scanning $N_v=60$ velocities. The next step, however, is dramatically different. Since each data point is mapped directly to the image instead of being spread into a wide impulse response, we save the factor in cost proportional to the size (in samples) of the migration impulse response. The prestack migration result shown in Figure 11 was accomplished in under 10 seconds on a single-node PC.

The cost savings will be reduced somewhat if we take into account more than one local slope (crossing reflection events, diffractions, multiple reflections, etc.) The plane-wave destruction algorithm (Fomel, 2002) can be applied for estimating several interfering data slopes simultaneously. In order to take full advantage of it, data decomposition into local slope components may be required. The curvelet transform (Douma and de Hoop, 2006; Herrmann, 2003) suggests a possible data decomposition approach. To extend the method of curvelet imaging developed by Douma (2006), each local slope component would need to be imaged separately by an oriented approach with its contribution stacked into the final image.

Seismic imaging and velocity estimation is inherently an uncertain process because of limitations in the data acquisition geometry and signal bandwidth. In the oriented approach, the uncertainty in the velocity estimation and in the positioning of seismic reflectors comes directly from the uncertainty in estimating local event slopes. Such uncertainty is much easier to estimate and analyze in the oriented rather than in the traditional approach thanks to the explicit time-domain imaging equations 18 and 19 that transform uncertainties in the local event slopes $p_h$ and $p_y$ directly into uncertainties of the image point positioning.

Velocity-independent time-domain seismic imaging using local event slopes