Chapter 7 proposed an extension to the Bayesian framework to include relative data between co-located events. The algorithms to explore this joint PDF for multiple events can be
developed from those described above. The resultant joint PDF can be marginalised for individual events by merely ignoring the dependence on the others (cf. Fig. 4.7).
Inverting for multiple events increases the dimensions of the source space for each event. This leads to a much reduced probability of obtaining a non-zero likelihood sample, because sampling from the n-event distribution leads to multiplying the probabilities of drawing a non-zero samples, resulting in sparser sampling of the joint source PDF.
Random sampling of the joint PDF (Fig. 8.21) shows that many samples are required to produce good sampling of the PDF, with 1011samples only just having a significant number
of non-zero samples to represent the source PDF. This is expected since 10 samples in each coordinate for two events would correspond to 1010samples of a two-event joint PDF. In fact,
the results in section 8.3 suggest that 1015samples are required to generate a good sampling
of the joint PDF, which can lead to very long durations (Fig. 8.22).
Figure 8.21: Lune plot of the variation in marginalised source PDF for a Monte-Carlo random
sampling inversion including P-relative amplitudes for two events. The top row corresponds to the first event and the bottom row shows the second event. There were 10 overlapping stations between the events used for the relative amplitude inversion.
The elapsed time for the random sampling (Fig. 8.22) is longer per sample than the individual sampling shown in Fig. 8.11, and longer than the sampling for both events due to evaluating the relative amplitude PDF. Moreover, increasing the number of samples 10-fold raises the required time by a factor of 10, requiring some method of reducing the running time for the inversion, since, given current processor speeds, 1015samples would take many years
to calculate on a single core. As a result, more intelligent search algorithms are required for the full moment tensor case.
0 5 10 15 x 108 0 2 4 6
Number Random Samples
Time (hours) 0 5 10 15 x 105 0 2 4 6
Markov Chain Length
Figure 8.22: Elapsed Time for different sample sizes of the random sampling and McMC algorithms
for the relative amplitude joint inversion. See also Fig. 8.11.
Markov chain approaches are less dependent on the model dimensionality. To account for the fact that the uncertainties in each parameter can differ between the events, the Markov chain shape parameters can be scaled based on the relative non-zero percentages of the events when they are initialised. The initialisation approaches also need to be adjusted to account for the reduced non-zero sample probability, such as by initialising the Markov chain independently for each event. The trans-dimensional McMC algorithm needs to allow model jumping independently for each event.
Figure 8.23: Lune plot of the variation in source PDF for a Markov chain Monte-Carlo sampling
inversion including P relative amplitudes for two events. The top row corresponds to the first event and the bottom row shows the second event. There were 10 overlapping stations between the events used for the relative amplitude inversion.
Tuning the Markov chain acceptance rate is difficult, as it is extremely sensitive to small changes in the proposal distribution widths, and with the higher dimensionality it may be necessary to lower the targeted acceptance rate to improve sampling. Consequently, care
needs to be taken when tuning the parameters to effectively implement the approaches for relative amplitude data.
Fig. 8.23 shows that the Markov chain provides a much improved sampling of the joint PDF compared to the random sampling approaches, especially at lower chain lengths than the equivalent random sampling. Moreover the time required for 106 samples is much improved
compared to the random sampling approach (Fig. 8.22). The acceptance rates for these solutions varied more, between 0.02 and 0.17, although all within the targeted range.
Figure 8.24: Lune plot of the variation in source PDF for a trans-dimensional Markov chain
Monte-Carlo sampling inversion including P relative amplitudes for two events. The top row
corresponds to the first event and the bottom row shows the second event. There were 10 overlapping stations between the events used for the relative amplitude inversion.
Chain Length Event 1 pDC Event 2 pDC
1 × 103 0.0 1.0 2 × 103 1.0 0.42 5 × 103 1.0 1.0 1 × 104 1.0 0.80 5 × 104 1.0 1.0 1 × 105 0.78 0.99 5 × 105 0.98 0.97 1 × 106 0.95 0.97
Table 8.1: Trans-dimensional pDCvalues for the
solutions shown in Fig. 8.24
The trans-dimensional McMC solutions are largely double-couple, with strong es- timates of double-couple probability (Table 8.1) for both of these events, consistent with the synthetic double-couple source. The resultant source PDFs are nearly all strongly spiked at the double-couple point (Fig. 8.24). The acceptance rate for these examples was mainly less than 0.1, which introduces a requirement for longer chain lengths compared with the single event solu- tions.
The Bayesian posterior model probabilities estimate from the random sampling solutions corresponds to the probability that both events are double-couple. Estimating the individual model probabilities using the Bayesian posterior model probabilities cannot easily be done as it would require evaluation of all combinations of the two models for the events. For two events, this means evaluating the mixed models, such as event one constrained to the double-couple, while event 2 is a full moment tensor source, adding further to the required time. From these values it is possible to evaluate the individual probabilities, although this requires many samples and is very time consuming, unlike the trans-dimensional approach.