Methodology and Results

Two synthetic data sets were generated with 20 sources and 40 receivers from a basin structure (BASIN type) and from a basin structure with two low velocity box regions (BBOX type). Figure 4 shows the two synthetic velocity models. We use genetic algorithms to reconstruct these models.

  

Figure 4: The two synthetic models: BASIN (top) and BBOX (bottom)

Table 1 provides a summary of the cases discussed in this section. Here the first two columns present the case number and the investigated seismic synthetic model type. The next two columns identify the model sizes N_x and N_z. The fifth column presents the type of crossover used, and the sixth column gives the population size used by the genetic algorithm.

 

Table 1: Summary of Cases

Case	Model Type	Nx	Nz	Crossover Type	Popsize
1	BASIN	40	9	2D	150
2	BASIN	40	9	2D	400
3	BASIN	40	9	2D	800
4	BASIN	40	9	2Ds	800
5	BASIN	40	9	1D	800
6	BASIN	40	9	1Ds	800
7	BBOX	40	8	2D	800
8	BBOX	40	8	2Ds	800
9	BBOX	40	8	1D	800
10	BBOX	40	8	1Ds	800

Cases 1 to 3 use a BASIN type synthetic model and a 2D crossover operator to study the influence of population size on the performance of genetic algorithm. Cases 3 to 6 and cases 7 to 10 investigate the performance of different crossover operators on both synthetic models. We will compare results from cases 3 to 10 with earlier results from simulated annealing.

We tested different probabilities of crossover and block mutation. We found that choosing 0.9 as the probability of crossover (P_c) and 0.001 as the probability of mutation (P_m) provided good results. Note that P_m is the probability that a member of the population will undergo mutation. For all cases, we run the GA ten times with different random seeds for 100generations. Every ten generations we smooth the newly created models by averaging values from neighboring cells.

We use cases 1 to 3 to investigate the relation between the population size and the performance of GAs. In each case, the initial minimum error is about 0.1- 0.5 sec², and the average error is in the range 3 - 6 sec². The GA starts to converge at about generation 50. The minimum error stops decreasing at generation 50, but the average error continues to decrease until generation 60. Figure 5 shows the average minimum error over ten random seeds as a function of time for three different population sizes. We can see that the calculated error decreases when the population size increases. We believe that large population sizes increase the performance of GAs by providing more diversity, especially considering our initialization procedure.

  

Figure 5: Error versus time for population size N=150, N=400, and N=800.

We investigated GA performance on the BASIN model with the four different crossover operators. Our results indicate that there is no significant difference between between real crossover and simple real crossover. There are, however, some differences between 1D crossover and 2D crossover. The final error for 2D crossover is about 0.003 sec², which is lower than the error of 0.005 sec² for 1D crossover. This is a statistically significant difference. We have similar results with the BBOX model.

 

Figure 6: Comparing crossover operators on the basin model.

To compare the performance of the four crossover operators and simulated annealing we defined several performance metrics. Let us define model error as the cell-by-cell velocity difference between generated models and the synthetic (true) model, and the em normal model error as the average of the square of cell-by-cell velocity difference. We then compare models, model errors, calculated error E in sec², average model error E_v in km/sec, and the normal model error E_v2 in km²/sec².