The Bayesian framework in Chapter 4 can be extended to relative amplitudes and relative waveforms. As shown in sections 7.2 and 7.2, the constraint produced by the relative inversion is a ratio of a sum of the moment tensor components. However, there is the full moment tensor space for two events to explore, extending the time required to sufficiently sample the moment tensor space.
The joint probability density function (PDF) for an observed relative amplitude or wave- form ratio for two events j and k at station i is
pRjki = rjki | Γjk, Aj i, Aki, σ j i, σik = RN rijk, ΓjkAji, Aik, σij, σki+RN −rijk, ΓjkAji, Aki, σij, σki, (7.47)
similar to Eq. 4.40,where Aj
i is the modelled amplitude for event j at receiver i.
Including relative information in the Bayesian approach from Chapter 4, requires extend- ing the data likelihood (Eqs 4.53 and 4.54) to give the joint likelihood. The joint likelihood for R events for the non relative data is given by
p d0| M , t, , k = ¨ M X k=1 R Y j=1 N Y i=1 h pYij| Ajik, σji, πi pRji | Ajik, σji, πi i ! p (σ) p (π) dσdπ, (7.48) p d0| M , t, , k = ¨ Q X l=1 M X k=1 R Y j=1 N Y i=1 h pYij| Ajiklσij, πi pRji | Ajikl, σij, πi i ! p (σ) p (π) dσdπ, (7.49)
although this assumes that each event is co-located with the others, such that the location uncertainty is not independent between the events.
The joint likelihood can be extended to include the corresponding data for the relative amplitude ratio or the relative waveform ratio (Eq. 7.47), with appropriate limits so that only independent combinations of events are taken.
p d0 | {M } , x, σ, π, k = ¨ M X k=1 R Y j=1 R Y l=j+1 h pRjli = rijl| Γkjl, Ajik, Alik, σji, σlii N Y i=1 h pYij| Ajik, σji, πi pRji | Ajik, σji, πi i ! p (σ) p (π) dσdπ (7.50)
shows the likelihood for a known velocity model, and it is easy to extend this to an unknown velocity model.
The joint posterior PDF is again obtained by multiplying by the prior PDF distributions for each event’s moment tensors:
p ({M } | {d0} , xk, σ, π, k) ∝ p ({d0} | {M } , x, σ, π, k) R
Y
i=1
To calculate the posterior probability for a given event, the joint posterior needs to be marginalised (Sivia, 2000, Section 1.3) with respect to the other events:
p (Mi| {d0} , xk, σ, π, k) =
ˆ
p ({M } | {d0} , xk, σ, π, k) dMj6=i, (7.52)
where the integral is carried out over all possible values of the moment tensors for each event except event 1, so for the case of three events in a relative moment tensor inversion, Eq. 7.52 would become
p (M1 | {d0} , xk, σ, π, k) =
ˆ ˆ
p (M1, M2, M3| {d0} , xk, σ, π, k) dM2dM3.
(7.53) The relative amplitude data provides additional constraint in these marginalised PDFs (Eqs 7.52 and 7.53) because it is including additional data (as long as data independence is preserved) in the inversion, which has not been used in the absolute inversion.
The increased moment tensor space can become computationally prohibitive to explore, however appropriate algorithmic approaches are described in Chapter 8.
The relative moment (Γ ) can be estimated by calculating µΓ and σΓ (Eqs 7.23 and 7.24)
for each receiver and combining to obtain the mean and standard deviation of the combined truncated Gaussian (section 7.3.1). The joint PDF can be evaluated for this combined mean, and the corresponding probabilities for other values of the scale factor can be determined using the approximation from Eq. 7.26.
It is also possible to obtain the marginalised relative moment distribution from the joint PDF samples by reconstructing the truncated Gaussians, scaling them by the sample probab- ility, and then marginalising.
7.5
Summary and Discussion
Relative information can help constrain the sources for co-located events (Dahm, 1993, 1996). Estimating whether an event is co-located depends on the P-S travel time differences, which can be used to estimate the probability of an event being co-located (Eq. 7.8).
The source inversion uses separately measured arrival amplitudes, although the constant multiplier between two waveforms at the same station could be used given a suitable estimate of the resultant distribution. As in previous sections, the combinations of data used for the inversion must maintain independence, for two events there is a limited number of combinations of observations that maintain data independence.
An important factor in the relative likelihood is the relative moment magnitude Γ . The likelihood has an approximately Gaussian distribution in Γ for each receiver. These can be combined to give a resultant estimate of the mean and standard deviation of Γ , removing
the need for a grid search over Γ . There are two prior models for the relative moment mag- nitude dependent on either a Gutenberg-Richter (Gutenberg and Richter, 1949) magnitude distribution or a log-normal magnitude distribution (Eaton et al., 2014).
The joint posterior PDF includes the relative information in the Bayesian approach from Chapter 4.
It may not be obvious that the relative information can provide an additional constraint on the source, but it is important to remember that these observations are new data, not included in the main inversion, which uses only amplitude ratios and polarities, although care must be taken if additional data types are included in an inversion. The relative P-amplitude ratios are an additional independent measure, because five independent amplitude ratios can be constructed from six amplitude observations for two events, three for each event, so a possible combination of amplitude ratios is P/SH and P/SV for both events and the relative
P-amplitude ratio.
This approach, while improving the constraint on the source by the addition of new data to the inversion, increases the dimensionality of the inversion problem, and although these dimensions are independent, this can drastically affect the approaches used to solve the forward model. Consequently, the method can not be used indiscriminately due to the increasing computational burdens, requires judgement before carrying out the joint inversion.
8 Moment Tensor Sampling Algorithms
The Bayesian approach described in chapter 4 uses forward-modelling, which requires a method of sampling and exploring the source space to evaluate the posterior PDF. The choice of sampling algorithm can have a large effect on the computational resources required (CPU time and memory) to gain a sufficient sampling of the source PDF.
The six independent moment tensor components (Section 2.2) describe a basis for the six-dimensional moment tensor space. Each point in this space uniquely describes a moment tensor. If the scalar moment magnitude is considered as a separate parameter the moment tensor can be normalised to unity (Eq. 3.12). In a coordinate system based on the six-vector form of the moment tensor (Eq. 2.8), this is a vector normalisation, which reduces the normalised moment tensor solution space to the five dimensional surface of a unit six-sphere. Since the double-couple source model (Section 2.3.1) is a prime source model candidate, the search space can be limited to its sub-space. It has three free parameters, rather than the five of the normalised moment tensor space. This constraint prevents spurious non-double- couple solutions, but the double-couple space is a very small subspace of the full moment tensor space. Consequently, sampling from the full moment tensor space is very unlikely to sample the double-couple subspace at all.
The probability of a double-couple source-type is large when sampling using algorithm B.1 (Fig. B.4). However, the double-couple point is a single value, consequently since the source type parameters are continuous values, the probability of obtaining a double-couple source is zero, as the probability that a variable x is in the range a 6 x 6 b is
P (a 6 x 6 b) = ˆ b
a
p (x) dx, (8.1)
so if a = b, the probability goes to zero. However, computers use floating point arithmetic on a discrete number line, so there is some very small probability that a double-couple source can be sampled. Sampling from a PDF distribution can be performed by taking a sample from the uniform distribution between 0 and 1 and using the inverse CDF for the target distribution. Generating an exact double-couple solution therefore requires sampling the correct CDF value for the two distributions which is exactly 0.5 in both cases. The probability of obtaining this sample from the uniform distribution will be very low1. As a result, it is extremely unlikely that any of the random moment tensor samples will be a double-couple, despite this 1The exact value depends on both the approach for generating random samples and the number of bits used for the calculation.
being at the peak of the PDF in Fig. B.4. This is an inherent problem of sampling the moment tensor space.
Although random sampling of the full moment tensor space is very unlikely to produce a double-couple sample, there will be a lot of samples very close to this (Figs 2.28 and B.4). Random sampling approaches do not easily allow for a qualitative evaluation of whether the source type is double-couple or not, since it is unlikely to be directly sampled and certainly not well-sampled without running such an algorithm for a very long time.
This chapter introduces three different sampling algorithms, and examines the advantages and disadvantages of each, evaluating the computation resources required for satisfactory sampling of the source PDF.
There are alternate approaches for forward model sampling, such as neural-network approaches (e.g Kaeufl et al., 2013), along with sampling approaches which increase in density in the area of interest, such as the neighbourhood algorithm (e.g. Sambridge, 1999a,b), and oct-tree (e.g. Lomax and Curtis, 2001). These approaches have advantages and disadvantages: neural-network approaches produce a fast inversion at the expense of pre-calculating a training set of possible events and ray paths, and both the neighbourhood algorithm and the oct-tree approach can struggle with large numbers of dimensions. However, this chapter is presenting three out of many possible algorithms which could be used for sampling the forward model, and is not intended as a deep overview of every different possible algorithm. The performance of these three algorithms is evaluated using three synthetic events, generated using a finite difference approach (Bernth and Chapman, 2011).
8.1
Monte Carlo Random Sampling
Stochastic Monte-Carlo sampling introduces no biases and provides an estimate for the true PDF, but requires a sufficient density of sampling to reduce the uncertainties in the estimate. The sampled PDF approaches the true distribution in the limit of infinite samples. However, this approach is limited both by time and memory. Some benefits can be gained by only keeping the samples with non-zero probability.
The dimensions of the moment tensor space require large numbers of samples. 100,000 samples from the five-dimensional surface of a six-sphere amounts to 10 in each dimension, which can only be thought of as coarse sampling. Doubling the sampling density will increase the number of samples 32-fold.
The approach can be parallelised and the samples can be evaluated iteratively, discarding those with zero probability (Algorithm 8.1).
Algorithm 8.1Monte Carlo Random Sampling Search Algorithm
1. Draw new samples x0.
2. Evaluate likelihood for samples x0.
3. Store moment tensors from x0with non-zero likelihoods.
4. Return to 1 until sufficient samples are drawn.