The performance of the different algorithms can be compared, both on the resultant PDF and the elapsed time. Comparable sample sizes and chain lengths produce better solutions for the McMC approaches.
Figure 8.7: Lune plot (Section 2.5.6) of the variation in marginalised source PDF for a synthetic
double-couple source with different sample sizes for the random sampling algorithm. Red corresponds to high probability and blue low probability regions.
The random sampling example (Fig. 8.7) needs sample sizes of the order of 5 × 107 to
produce a good sampling of the PDF. Increasing the number of random samples improves the resolution of the source PDF. The solutions have better constraint of the PDF as the number of samples increases, with the peak becoming clear at 5 × 108samples. If the plots use a lower
resolution for the source PDF histogram, the peak is visible at lower numbers of samples. This holds true for the McMC approaches (Figs 8.8 and 8.9). The results from the different algorithms are consistent, as is to be expected.
Figure 8.8: Lune plot of the variation in marginalised source PDF for a synthetic double-couple
source with different chain lengths for the Markov chain Monte Carlo algorithm.
Figure 8.9: Lune plot of the variation in marginalised source PDF for a synthetic double-couple
source with different chain lengths for the trans-dimensional Markov chain Monte Carlo algorithm. The probability of a double-couple source type is 0.34, 0.91, 0.69, 0.67, 0.77, 0.73, 0.68, 0.74, respectively, for each plot.
A chain length of 50, 000 for the McMC approach is comparable to 1 × 108 random
samples (Fig. 8.8). There is some discrepancy between whether there are one or two peaks in the solution, with some examples having two peaks and others only one. This is probably due to the acceptance rate, which was targeted for these examples at 0.3.
The trans-dimensional McMC solutions (Fig. 8.9) are plotted with a low resolution on the PDF histogram to make the strong peak at the double-couple point clear. The double-couple solution is well sampled, unlike in the normal McMC and random sampling approaches, and there is an associated strong peak at the double-couple point, consistent with the double- couple synthetic source. The targeted acceptance rate for the trans-dimensional approach is 0.15, because of the dimension jumping. Consequently, more of the space should be explored. The probability of the source being a double-couple type source from the trans-dimensional approach varies between 0.24 and 0.99. Despite the sharp peak at the double-couple point, the non-double-couple samples resemble those for the other examples (Figs 8.7 and 8.8), as shown in Fig. 8.10.
Figure 8.10: Lune plot of the marginalised source PDF for the trans-dimensional Markov chain
Monte Carlo results shown in Fig. 8.9, showing the non-double-couple distribution only.
Comparing different sample sizes shows that the McMC approaches require far fewer samples than random sampling. However, the random sampling algorithm is quick to calculate the likelihood for a large number of samples, unlike the McMC approach, because of the extra computations in calculating the acceptance and obtaining new samples. Some optimisations have been included in the McMC algorithms, including calculating the probability for multiple new samples at once, with sufficient samples that there is a high probability of containing an accepted sample. This is more efficient than repeatedly updating the algorithm. Despite these optimisations, the McMC approach is still much slower to reach comparable sample sizes (Fig. 8.11), and is slower than would be expected just given the targeted acceptance rate, because of the additional computational overheads.
0 5 10 15 x 108 0 1 2 3 4
Number Random Samples
Time (hours) 0 2 4 6 x 106 0 5 10 15 20
Markov Chain Length
Figure 8.11: Elapsed time for different sample sizes of the random sampling (left plot) and for the
McMC algorithms (right plot) with different chain lengths. The red dots in the McMC case
correspond to the trans-dimensional McMC algorithm and the blue dots correspond to the standard algorithm.
The random sampling approach can easily be parallelised. N processors reduce the required time N-fold. There are techniques for sampling multiple Markov chains in parallel, by sampling at different acceptance rates and jumping between them (Sambridge, 2013), to speed up the chain, but these are not as effective because the evaluation of the forward model
Including location uncertainty and model uncertainty in the forward model causes a rapid reduction of the available samples for a given amount of RAM and increases the number of times the forward model must be evaluated, lengthening the time for sufficient sampling (Fig. 8.12). 0 5 10 15 x 106 0 50 100 150
Number Random Samples
Time (hours) 0 2 4 6 x 104 20 40 60 80
Markov Chain Length
Figure 8.12: Elapsed time for different sample sizes of the random sampling algorithm and for
McMC algorithms with different chain lengths. The velocity model and location uncertainty in the source was included with a one degree binning reducing the number of location samples from 50,000 to 5, 463. See also Fig. 8.11.
Event A Event B
Figure 8.13: Lune plots of the
non-double-couple synthetic sources used to test the sampling algorithms.
The location uncertainty has less of an effect on the McMC algorithms, since the number of samples being tested at any given iteration are small. Consequently, as the random sampling approach becomes slower, the fewer samples required to construct the Markov chain starts to produce good samples of the source PDF at comparable times. However, there is an initial offset in the elapsed time for the Markov chain Monte Carlo approaches due to the burn in and initialisation of the algorithm.
Different source types can have an effect on the results, although the results for each approach are consistent. Two different non-double-couple sources were used to test these algorithms (Fig. 8.13). Figs 8.14 - 8.16 correspond to a moment tensor source close to a double-couple (Event A in Fig. 8.13). The solution is multi-modal, with the maximum probability source type close to the synthetic source. Again, as the number of samples increases, the peak becomes clearer.
The McMC solutions are usually consistent with the maximum peak in the random sampling source PDF and the true synthetic source, but they show only one of the regions, and the example with a chain length of 105 samples shows a lower probability region. The
lack of sampling of the multiple modes is again due to the acceptance rate of the algorithm, similar to the example in Fig. 8.8. In this case, the acceptance rate is targeted at 0.3.
Figure 8.14: Lune plot of the variation in source PDF for a synthetic source with different sample
sizes for the random sampling algorithm. The synthetic source (Event A in Fig. 8.13) is close to a double-couple, with a small amount of closing crack.
Figure 8.15: Lune plot of the variation in marginalised source PDF for a synthetic source with
different sample sizes for the Markov chain Monte Carlo algorithm. The synthetic source (Event A in Fig. 8.13) is close to a double-couple, with a small amount of closing crack.
The trans-dimensional McMC approach (Fig. 8.16) has a much better sampling of the multiple peaks in the source PDF and is more consistent with the random sampling source PDF (Fig. 8.14), with good sampling of the three large regions of non-zero probability and the peak close to the double-couple source type.
Figure 8.16: Variation in marginalised source PDF for a synthetic source with different chain
lengths for the trans-dimensional Markov chain Monte Carlo algorithm. The synthetic source (Event A in Fig. 8.13) is close to a double-couple, with a small amount of closing crack.
were accepted. The targeted acceptance rate for the trans-dimensional approach was 0.15, suggesting that a value close to this may produce a better sampling of multi-modal source PDFs.
Figs 8.17 - 8.19 correspond to solutions for a synthetic opening crack source (Event B in Fig. 8.13). The random sampling solutions (Fig 8.17) are consistent with the synthetic source, although they show poor sampling due to the relative sparsity of non-zero solutions, as well as being close to the upper edge of possible random moment tensor samples (Appendix B).
Figure 8.17: Lune plot of the variation in marginalised source PDF for a synthetic source with
different chain lengths for the random sampling algorithm. The synthetic source (Event B in Fig. 8.13) is an opening crack.
The McMC solutions (Fig. 8.18-8.19) show much better sampling and better constraint on the solution, with an improved ability to explore regions with a low prior probability. The source PDF is well constrained for all of the chain lengths as it is a very small region.
Figure 8.18: Lune plot of the variation in marginalised source PDF for a synthetic source with
different chain lengths for the Markov chain Monte Carlo algorithm. The synthetic source (Event B in Fig. 8.13) is an opening crack.
As with the previous example, the only difference between the trans-dimensional McMC approach (Fig. 8.19) and the normal McMC approach (Fig. 8.18) is the acceptance rate, as none of the double-couple jumps were accepted. However, unlike the previous example,
the lower acceptance rate does not seem to improve the sampling of the PDF, since it is so sharply peaked.
Figure 8.19: Lune plot of the variation in marginalised source PDF for a synthetic source with
different chain lengths for the trans-dimensional Markov chain Monte Carlo algorithm. The synthetic source (Event B in Fig. 8.13) is an opening crack.