WEMVA cost

In the most general case, the data storage required by wave-equation migration velocity analysis is roughly proportional to the size of the wavefield space, i.e. the size of the data times the number of depth steps used in downward continuation. The computational cost is roughly proportional to two migrations by wavefield extrapolation for every linear iteration. Thus, for most exploration 3D datasets, the storage and computational cost involved are tremendous such that the method may reach the limits of even the largest computers available today. However, as computational power increases, WEMVA becomes more and more feasible. The real question is not whether we can use this method today, but rather if it provides additional information which is not available from conventional traveltime-based methods.

WEMVA can exploit the efficiency of distributed cluster computers, with independent processing and storage units. Parallelization is done over frequencies distributed to separate cluster nodes, in a distribution pattern similar to that of wave-equation migration. For example, one frequency of a full 3-D prestack dataset with midpoint samples n_mx=1000 and n_my=500, offset samples n_hx=64 and n_hy=64 and depth samples n_z=1000, requires storage of approximately 1.9 Tb on every cluster node, assuming processing of one frequency/node. This volume is large, even for today's largest computers, although it is not completely infeasible. Wide application of WEMVA, however, requires techniques that reduce substantially the computational and storage cost. Several options, which can be used alone or in combinations, are the following:

Restrict data to a common-azimuth.

This is a common procedure used to reduce data volumes for wave-equation migration. In many instances, reducing data to a common-azimuth is an appropriate approximation, for example for marine data acquired with a narrow azimuth. This approximation was demonstrated to be accurate in imaging complex structures. It is, furthermore, appropriate for velocity analysis, since the background velocity is likely to be inaccurate, degrading the accuracy wave-equation migration more than the common-azimuth approximation does. Datum the recorded data above the region with inaccurate velocity

This procedure effectively sinks the survey from the surface to an arbitrary depth in the subsurface, thus reducing the number of depth steps needed for velocity analysis. The assumption is that the velocity in the upper portion of the model can be effectively constrained using traveltime tomography methods. Reduce the frequency band.

Each frequency of the downward continued wavefield interacts with the velocity to create a different scattered wavefield. In principle, each frequency can be used independently to estimate velocity. Naturally, the low frequencies produce smoother velocities than the high frequencies. For a given size and smoothness of the estimated anomalies, we can limit the maximum frequency to less than what is needed to define the velocity anomalies. A narrow frequency band also acts as a regularizer or smoother applied to the inverted velocity. Furthermore, if we estimate velocities by regularized inversion, the sharper anomalies constrained by the high frequencies are smoothed-out, which is another reason to reduce the frequency band to an effective range. Limit the wavefield to a volume of interest.

We could synthesize a dataset and wavefield by windowing the image in a region of interest and then de-migrating by upward continuation. Then, we can use the data generated this way to create the wavefield used in WEMVA. This method is related conceptually to data datuming from the surface which also limits the volume of investigation. However, if we generate data by de-migration we are also able to limit other characteristics of it, for example the frequency content or the maximum reflection angle, further reducing cost.

The theoretical justification for this process is that the background wavefield is the band-limited equivalent of a rayfield. In traveltime tomography, we use rays obtained by ray tracing which represents an approximate data modeling process. Here, we can model the equivalent of a rayfield, but by using wavefield extrapolation methods. Limit the wavefield to normal incidence.

A drastic reduction of the WEMVA cost can be achieved by limiting the reflection angles represented in the wavefield to normal incidence. This method is similar to the normal incidence velocity analysis involving traveltimes. However, normal-incidence WEMVA is superior to normal-incidence traveltime tomography since the rays used by the wave-equation method are wider than asymptotic rays, and thus more stable and better samplers of the velocity space.

All methods described above reduce the cost of WEMVA. However, each one of these approximations reduces the accuracy of the method. A judgment needs to be made a priori as of which method to use and how. Various velocity analysis scenarios require more or less accuracy and resolution. In the examples presented in example, I use one or more of the techniques described above, either alone or in combination.