Basin Model

We first look at the basin model. For every case (from cases 3 to 6), the velocity models we get from different random seeds are very similar and their evolution follows a similar path from start to finish. After the first ten generations, the generated velocity models seem to be unorganized but the errors decrease compared to the initial constant models. Selection, crossover, and mutation operators work through every generation. After 20 generations, the GA can separate low velocity regions near the surface from high velocity regions near the bottom and we start seeing sub-surface layers. After 50 generations, we can see the basin within which velocities are lower than rest of the region. The average error of the population continues to decrease until generation 60. Figure 7 shows the best BASIN models generated by the GA and SA while Figure 8 displays their model errors.

  

Figure 7: Best GA (bottom) and SA (top) models

Figures 7 and  8 (top) show the final velocity model and model error from simulated annealing [Pullammanappallil, 1994]. In general, the basin structure was reconstructed but velocities were lower in the right bottom corner compared to the synthetic model. Smoothing and the number and distribution of sources and receivers restrict model resolution.

  

Figure 8: Model error of best GA(bottom) and SA (top) BASIN models.

Both GA and SA models are geologically plausible models acceptable to an expert seismologist. One significant difference is that the GA quickly converges around an acceptable model while the SA takes much longer. In addition, a GA's easy parallelizability makes it feasible to handle much larger models.

Figures 7 and  8 (bottom) show the best velocity model and its model error. As with the the result from SA, we can see the basin structure. In general the middle part of the model, with largest ray coverage, is better reconstructed than the extremities. This is expected since the middle of the model lies in the path of more seismic waves (rays) from sources to receivers distributed on the surface.

Table 2 compares the error between GAs and SA. Each value in the table is the best available for each case with avg indicating the average of the best for ten random seeds. The calculated error for all the cases are smaller than 0.0068 sec². The synthetic model itself has an error of 0.00029 sec²which is from measurement. The average model error for all cases are smaller than 0.63 km/sec. The normal errors for all cases are also smaller than the SA. GAs thus seem to generate better velocity models.

 

Table 2: Error comparisons of different experiments for BASIN model

Model	1D	1D avg	1Ds	1Ds avg	2D	2D avg	2Ds	2Ds avg	SA	Synthetic
E	0.0019	0.0035	0.0019	0.0035	0.00097	0.0025	0.0019	0.0025	0.0068	0.00029
Ev	0.46	0.62	0.46	0.62	0.54	0.50	0.45	0.50	0.63	0
Ev2	0.29	0.53	0.29	0.53	0.37	0.41	0.27	0.44	0.96	0

Figure 9 shows the best velocity models generated by the SA for the BBOX model. The basin structure was also reconstructed in general, but the two low velocity boxes were not.

  

Figure 9: Best SA model for BBOX.

Figure 10 shows the best model generated by the GA using 2D crossover. As with simulated annealing, we can see the basin structure for all cases. The 1D crossover operator can not detect the two low velocity boxes present in the BBOX synthetic model, but, some runs of the GA with 2D or 2Ds crossover do partially detect the two boxes. Note that at the bottom right and left of Figure 10 (near N_z = 6) there are two low velocity regions, which correspond to the two low velocity boxes in the synthetic model.

  

Figure 10: Best GA model for BBOX.

From Figure 11 we can see that the model errors in these regions are significantly lower (darker) than those of the SA¹. This is in general true for the rest of our crossover operators when compared to simulated annealing.

  

Figure 11: Model errors for the best GA (bottom) and SA (top) models for BBOX.

 

Table 3: Error comparisons of different experiments for BBOX model

Model	1D	1D avg	1Ds	1Ds avg	2D	2D avg	2Ds	2Ds avg	SA	Synthetic
E	0.0039	0.0045	0.0029	0.0045	0.0019	0.003	0.0019	0.0025	0.0063	0.0005
Ev	0.49	0.79	0.58	0.79	0.57	0.63	0.45	0.68	0.52	0
Ev2	0.47	1.01	0.59	1.00	0.49	0.75	0.36	0.84	0.56	0

Table 3 compares our crossover operators with the SA. The calculated travel-time error for all the cases are smaller than the value of 0.0063 sec² from simulated annealing and the synthetic model has an error of 0.0005 sec². All average model errors are larger than SA while the best model errors from 1D and 2Ds are smaller than those from SA.