Numerical Results

The least-squares phase-encoded GDM method is tested on synthetic data associated with the 2D SEG/EAGE salt model (see Figure ). These traces were computed by FD solutions to the 2D acoustic wave equation. The data include 160 shot gathers with a peak-frequency of 15 Hz, with 176 traces in each shot gather, the shot and receiver intervals are 97.6 m and 24.4 m, respectively. To save computation time, I down-sampled the traces from 4001 time samples with a time interval of 0.001 to 1001 time samples with a time interval of 0.004 s in each trace.

Figure: The 2D SEG/EAGE velocity model and the corresponding reflectivity model.

For comparison, I implement the single-source GDM first and the results are shown in Figure  and . Figure  shows the first-arrival GDM results, in which the migration kernel is calculated by convolution of the first-arrivals in the source-side Green's function and receiver-side Green's function. In Figure , the migration kernel is obtained by a full length convolution of two Green's functions. Compared to the LSM image using the full Green's function (Figure c), the one only using the first-arrival Green's function (Figure c) looks more noisy and less resolved below the salt body; however, it is much cheaper due to fewer samples in the Green's function for convolution.

Figure: Comparison of single-source GDM results using the first-arrival Green's function.

Figure: Comparison of single-source GDM results using the full Green's function.

Figure: Formation of phase-encoded supergathers with different phase-encoding functions.

Before applying the least-squares method, I did some tests on different phase-encoding functions. There were three types of phase-encoding functions, source statics, receiver statics and random polarity. Figure  shows such a supergather with different phase-encoding functions.

Figure  shows the 10-fold multisource GDM results using the first-arrival Green's function and the full Green's function. Only source statics are applied while generating phase-encoded gathers. It is obvious that the shorter duration of the source-side Green's function, the less cross-talk contamination in the migration image.

Figure: Comparison of multisource GDM results without iterations using the first-arrival Green's function and the full Green's function. Ten shot gathers are phase-encoded in a supergather using source statics phase-encoding.

Figure: Comparison of multisource GDM results without and with applying receiver statics.

Figure: Comparison of multisource GDM results without and with applying random polarity.

Figures  and demonstrate the effectiveness of the other two phase-encoding functions, receiver statics and random polarity. From the comparisons of without and with these phase-encoding functions, I can conclude that both of them are valid ways to reduce the cross-talk noise in the migration image. If I combine the source statics together with these two phase-encoding functions, the cross-talk noise in the migration image will be further reduced.

A preconditioned conjugate gradient LSM method is then implemented with Figure a as the starting model. All of the multisource migration kernels are available, so no new simulations are needed to get the updated migration image at each iteration. This is the key point for an efficient implementation of iterative LSM, and the result after 20 iterations is shown in Figure b. I note that the low-frequency migration artifacts shown above the salt are successfully eliminated by the least-squares iterations, while faults and small structures below the salt dome are better resolved in Figure b. The multisource least-squares GDM image (Figure b) is comparable with the single-source version (Figure c) but with less random noise; however, the storage cost for the migration kernel and the corresponding I/O cost of it is about 1/10 of the single-source version.

Figure: The least-squares phase-encoded GDM image after 20 iterations(b) with the starting model of (a). (c) is the least-squares GDM image after 20 iterations without phase-encoding which is 10 times slower than (b). Source and receiver statics and random polarity are all employed in this test.