Cooling schedule

As Rothman (1985) points out, the selection of the cooling function is very important and can greatly speed convergence. In my method, I treat the sample interval function as a cooling function. To find a good cooling function, I first created a linear cooling function and plotted how the energy decreased as a function of sample interval. This is shown in Figure 12. In this particular example, there appear to be three sample intervals associated with large drops in energy: 185, 130, and 70. I decided to create a cooling schedule that treats the sized 70 sample interval as the critical temperature. This is shown in Figure 13. The critical temperature is where convergence is most significant.

 

trialschedule
Figure 12 Trial cooling schdule. Top: input cooling schedule drops linearly with iterations. Middle: energy($\bf E$) drops in steps with iterations. Bottom: energy($\bf E$) drops in step with sample interval. This is used to determine the more efficient cooling schedule in Figure 13.

 

coolingschedule
Figure 13 Cooling schedule. Top: input cooling schedule based on trial cooling schedule designed to spend more iterations near the critical tempurature. Middle: energy($\bf E$) drops in steps with iterations. Bottom: energy($\bf E$) drops in step with sample interval.

Notice that in Figure 13, the energy drops off much quicker than in Figure 12 and therefore requires fewer iterations.

Figure 14 shows the result of applying the cooling schedule in Figure 13. It has converged to the desired event. Figure 15 shows the application of the calculated displacements to the left side of the fault. Its results are about the same quality as Figure 16, which shows the results of applying the actual known displacement to the left side of the fault.

 

anneal_final
Figure 14 Cross-correlagram with overlay of final solution and known displacement.

 

makefinalmod Figure 15 Applied result, the ``model'' in the center shows the result of applying the calculated displacement to the left side of the fault.

 

makefinalideal Figure 16 Ideal applied result, the ``ideal'' in the center shows the result of applying the known displacement to the left side of the fault.