Examples for the trial-and-error method

1. Examples for the trial-and-error method • gravity modeling

2. Waveform modeling Sidao Ni et al. Science 296, 1850 (2002)

3. Chen et al. 2007

4. SA exmaple F(x,y) has its global maximum value of 1.0 at x = 0, y = 0. However, it also has several secondary maxima Error function global maximum at (0, 0)

5. Effect of T on pdf distribution T = 1 T = 10 T = 0.1 T = 0.01 The effect of this temperature T is to exaggerate or accentuate the differences between different values of the error function.

6. Figure 4.5. For a model with 8 model parameters (each having 8 possible values) heat bathalgorithm starts with a randomly chosen model shown by shaded boxes in (a). Each modelparameter is then scanned in turn keeping all others fixed. (b) shows scanning through themodel parameter m 2. Thus m26 is replaced with m23 and we have a new model described byshaded boxes in (c). This process is repeated for each model parameter.

7. Figure 4.6. Model generation probabilities (left) and their corresponding cumulativeprobabilities (right) at different temperatures. At high temperature, the distribution is nearly flatand every model parameter value is equally likely. At intermediate temperature, peaks start toappear and at low temperature, the probability of occurrence of the best solution becomes veryhigh. A random number is drawn from a uniform distribution and is mapped to the cumulativedistribution (fight) to pick a model.

8. Application of heat bath SAseismic waveform inversion Tk = T0(0.99)k (k: number of iteration) 100 iterations

12. Real data inversion: SA

13. 11 models that gave a correlation value of 0.74 or greater

14. Neighborhood Algorithm(Sambridge, 1999, GJI)