There are several approaches that utilise first motion polarity for source inversion including FPFIT (Reasenberg and Oppenheimer, 1985), HASH (Hardebeck and Shearer, 2002, 2003) and FOCMEC (Snoke, 2003).
FPFIT uses a two-step grid search, finding the double-couple source model which min- imises a normalised weighted sum of first motion polarity misfits. The weighting consists of the estimated variance of the data combined with the theoretical P-wave amplitude including a source to station distance term where available. This reduces the weight of observations close to the nodal plane as well as at distant receivers.
Rather than solely finding the minimum misfit solution, FPFIT obtains other significant minima in the misfit, which can correspond to different fault geometries. The uncertainties
estimated by FPFIT are calculated as the range of parameters corresponding to the confidence interval for the misfit. However, as Hardebeck and Shearer (2002) mention, the source mech- anism changes non-linearly with uncertainties, therefore the range of possible mechanisms will rarely form an easily parameterised distribution.
FPFIT does account for possible errors in the observed polarities, but it does not include any uncertainties in the location, or the associated ray path uncertainty.
HASH can include velocity model uncertainties using random sampling from the velocity models to estimate the ray paths. HASH also uses a grid search approach over strike, dip and rake, and rather than estimating errors on the parameters, it produces a range of possible solutions. The misfit minimisation is split between the impulsive and emergent polarities, first reducing the misfit on the impulsive polarities, and then on the emergent polarities. To generate the range of possible solutions, those with a misfit less than some threshold are included.
Different ray paths are tried from various possible one-dimensional velocity models and source depths, and the acceptable solutions are combined. The RMS difference from the average solution is used as a crude indicator of the range of possible solutions. Hardebeck
and Shearer(2003) extended this to include S/P amplitude ratios in the misfit calculation.
FOCMEC (Snoke, 2003) uses more data types, such as P, SH and SV polarities and P/SH, P/SV and SH/SV amplitude ratios, in a grid search approach similar to the others. Again, this generates samples of possible solutions within allowed observation errors. As with HASH, the number of polarity errors allowed can be tuned, and the errors can be weighted in several different ways. However, in using multiple data types it is important to preserve data independence, as described in section 4.3.
Julian(1986) introduced a linear programming approach to seismic source inversion. This
uses polarities by treating them as inequalities. Amplitudes can be included by constructing pairs of inequalities equivalent to an equality. Linear programming approaches solve for the solution by maximising a cost function, subject to the observational constraints (the inequalities). Julian and Foulger (1996) extended this approach to amplitude ratios, using an L1 norm of the residuals as the cost function.
Many other approaches use inversions based on Green functions (e.g. Kim et al.,
2000; Bernardi, 2004; Bernardi et al., 2004; Hjörleifsdóttir and Ekström, 2010; Kim, 2011), although these are not always full waveform based inversions, but instead use other data such as amplitude spectra (Cesca et al., 2006). Godano et al. (2009) used amplitudes rather than the full-waveform to invert for a double-couple source. Source inversions that are a joint inversion between the source and location are usually Green functions based inversions (e.g.
Wéber, 2006; Rodriguez et al., 2012; Kaeufl et al., 2013). O’Toole (2013) have adapted the
CMT approach (Dziewonski et al., 1981) to micro-earthquakes using near-field and geodetic observations, resulting in estimates of the centroid and the moment tensor of the event.
3.7
Summary and Discussion
The Gaussian noise model is a valid noise model for a seismogram provided the low frequency noise, such as the ocean microseisms, are filtered out. There is some directionality to real noise, although this does not appear to be inconsistent with the Gaussian noise model.
Both the Q correction and the near-surface effects alter the seismogram, although the effects are not always strong. Furthermore, site effects and instrument coupling also affect the seismogram, especially when rotating into different components.
The polarity of the first arrival is the easiest observation to make, although it is non-linear, requiring a forward model based approach.
The arrival amplitude can be used in a simple matrix formulation to invert for the source. However, this chapter has shown that since the measurement of the amplitude is related to the extrema of the arrival, the value is more sensitive to background noise than expected from a simple noise model. Both the mean value and the uncertainty increase as the noise level increases, leading to very uncertain measurements, with an arrival specific systematic uncertainty. This uncertaintyaffects the amplitude ratio in complex ways.
There are several different approaches to estimating the amplitude, and both the maximum absolute amplitude and peak-to-peak amplitude estimates produce very similar results. The RMS amplitude often has much lower uncertainties, but has a larger percentage deviation from the true value. Amplitude ratios can also be used in a linear formulation of source inversion to within some constant scaling factor on the source.
There are several existing approaches to source inversion, many using just first motion polarities and limiting the source type to the double-couple model. However, some in- clude amplitudes and amplitude ratios, and can even include some of the velocity model uncertainties.
4 Bayesian Source Inversion
Source inversion approaches use the observed data to estimate the source properties. Amp- litudes and waveforms are linear with respect to the moment tensor, and so the source can be calculated by using an appropriate left inverse of the station propagation coefficient matrix (Chapter 3). However, the station propagation coefficients are often uncertain. Furthermore, the easily observed polarity information is non-linearly related to the source and therefore cannot be inverted using a matrix form.
A Bayesian approach to solving for the moment tensor and source parameters can sur- mount some of these problems. Probabilities allow different types of observations to be easily combined to produce a probability density function (PDF) for the source.
There are several Bayesian approaches, including that of Brillinger et al. (1980) and
Walsh et al.(2009) for polarity inversion, as well as full waveform approaches, such as those
proposed Wéber (2006) and Kaeufl et al. (2013).
In this chapter, a new Bayesian approach to source inversion is described using polarity data, which has been derived separately but is consistent with the methods of Brillinger et al. (1980) and Walsh et al. (2009), and amplitude ratio data, as well as the full moment tensor source description. Additionally, two different approaches to representing the resultant source distribution on source plots are compared. Throughout the chapter, the Bayesian definition of probability is used, so the probability represents the state of belief in a result, rather than the frequentist approach where the probability reflects the relative frequency of a result, when the measurement or experiment is repeated.
4.1
Bayes’ Theorem
Optimisation processes aim to find the best fitting model parameters for the given data. For source inversion, the likelihood of the observed data being correct is evaluated for different possible sources. The resulting estimates of the probability density function (PDF) can be combined for all the data to approximate the true PDF for the source. This PDF describes the likelihood of observing the data for a given source, p (data | model). However, the value of interest in such an inversion is the probability of the model given the observed data, the posterior PDF. This can be evaluated from the likelihood using Bayes’ Theorem (Bayes and
Price, 1763; Laplace, 1812; Sivia, 2000)
p (model | data) = p (data | model) p (model)
Bayes’ theorem links the two conditional PDFs using prior probabilities. The prior rep- resents the information known before the experiment, and can be used to include constraints such as fault geometries or source type.
The choice of prior is not trivial, because it does have an effect on the resultant posterior PDF (Sivia, 2000, Chapter 2), although the uniform prior is often used.
The normalised full moment tensor has five independent components, compared to three in the normalised double-couple tensor. These extra free parameters provide an improved fit, for example to both data and noise. A possible choice of prior could be one that reduces the tendency for full moment tensor inversions not to fit an exact double-couple solution, reflecting the commonly held view that most earthquakes are double-couple, without taking the step of forcing the solution to be double-couple.
4.1.1
Bayesian Marginalisation
Source inversion is particularly sensitive to uncertainties and, because it is usually carried out after several other required steps, the effect of the uncertainties may be difficult to understand. The effects of uncertainties on the inversion results are discussed in Chapter 6.
Although interdependencies between the uncertainties can be explored, a quantitative relationship is not known. For such a treatment to be truly rigorous, the variations in these errors must be included throughout the inversion.
The Bayesian formulation allows rigorous inclusion of uncertainties in the problem using marginalisation (Sivia, 2000, Section 1.3). Marginalisation removes the dependence of the joint PDF, p (A, B), on one of the variables by integrating over the PDF for that variable:
p (A) = ˆ
p (A , B) dB = ˆ
p (A | B) p (B) dB. (4.2) Consequently, marginalisation can be used to remove the dependence of the final PDF on the uncertainties in the inversion. With discrete sampling from an unknown PDF, Eq. 4.2 can be evaluated as a Monte Carlo integration (Caflisch, 1998), using samples randomly drawn from the conditional and prior PDFs:
p (A) =
N
X
p (A | B) p (B) ∆B, (4.3)
where ∆B is the space of B divided by the number of random samples, N.