Computational Cost

Each iteration of iterative LSM costs about two migrations, so the cost of iterative LSM is about $2I$ times that of standard migration. Assuming an ideal land acquisition geometry where the geophones are fixed and $S$ shot gathers are recorded, the total computational cost in computing the migration image is $Cost^{conv}\approx S\alpha$ for conventional prestack migration, where $\alpha$ is the cost of one wave-equation migration. In comparison, if $N$ supergathers are migrated, then the cost per iteration of LSM is only $2N\alpha$ . If $I$ iterations are needed then the total cost of LSM is $Cost^{multi}\approx 2N\alpha I$ . Therefore I conclude that the cost of MLSM can be less than standard migration if

$$2NI < S.$$ (13)

In the empirical results, a high quality image is obtained after 60 iterations for a 320-shot supergather, which translates to about 2.7 times speedup if the numerical tests are performed with wave-equation migration or reverse time migration. Meanwhile, the image is free of migration artifacts and with balanced amplitudes (Figure c).

Another important saving is the reduction of I/O cost. For Kirchhoff migration, the I/O cost can be the dominant factor for the run time. By statically encoding $S$ shots into a supergather, the I/O cost is reduced to $1/S$ of the original cost, which allows significant saving in run time of MLSM. For dynamic encoding, if $I$ iterations are needed, $I$ supergathers with different encoding functions are required at input, so that the I/O cost is reduced to $I/S$ of the original cost. Therefore, MLSM with dynamic encoding does not enjoy a large I/O cost reduction if the number of iterations is large. An optional strategy is to periodically stop the iterations in static iterative LSM and restart them at the stopping model but with a new encoding function in the supergathers. In the above calculation, the cost of computation and I/O of preprocessing step is not considered.