The phase arrival times are used to estimate the hypocentre of the seismic event. The location is combined with the observed characteristics of the arrivals (polarity, amplitude etc.) to estimate the source parameters. The accuracy of the event location also depends on the network coverage, requiring good constraint to estimate the hypocentre. There is often a trade-off between estimating the earthquake time and the location. Uncertainties in the phase arrival times (Section 6.1) can have a large effect on the hypocentre uncertainties.
Most location approaches use a travel-time look-up table to locate the events, although the type of search algorithm used to estimate the location varies. The HYPO* approaches, including Hypo71 (Lee and Lahr, 1975), HYPOELLIPSE (Lahr, 1989), HYPOINVERSE (Klein, 2002) and HYPOCENTER (Lienert et al., 1986), use iterative linearised approaches based on that of Geiger (1912). These minimise the linearised travel-time residuals, producing a single solution and linear estimates of the uncertainties. The results can be poor if there are multiple possible solutions due to weak constraints or where the complete solution is irregular (Husen and Hardebeck, 2010).
More modern approaches employ the increase in computer power to search the complete space. NonLinLoc (Lomax et al., 2000, 2009) uses a grid search approach. CMM (Drew
et al., 2013) follows a similar approach, undertaking a spatio-temporal search for the peak
in the coalescence function. These approaches produce complete location PDFs, but can be computationally demanding compared to the simpler linearised approaches. Nevertheless, they produce better estimates of the location uncertainties, as well as allowing the full location PDF to be sampled, so that the location uncertainty can be included as in section 4.2.5.
Hypocentre determination can be improved using relative relocation, either master event (Deichmann and Garcia-fernandez, 1992) or double-difference (Waldhauser and Ellsworth, 2000; Waldhauser, 2001). These methods are both linearised, and can dramatically reduce the relative uncertainty between hypocentre locations, yet the absolute uncertainties are not necessarily similarly reduced. Menke and Schaff (2004) show that the use of waveform cross- correlation and differential travel-times in relative relocation can produce accurate absolute locations. However, the uncertainty in the relative relocation is often estimated from the pick time differences or the cross-correlation and not the corresponding absolute uncertainties in the initial hypocentres. Alternately, additional information in the waveform such as the P-wave coda can be used to estimate the separation between events (Robinson et al., 2011), using coda wave interferometry (Snieder and Vrijlandt, 2005; Snieder, 2006).
If the relative information could be correctly combined with the location PDF, the loc- ation uncertainties could be reduced further, but care must be taken to maintain the data independence, otherwise the results could be artificially sharpened1.
Figure 6.1: Marginalised xy and xz plots of the location PDF from NonLinLoc for different
background noise levels. Red corresponds to high probability and blue to low probability. The station geometery for this event is shown in Fig. 6.2, and the velocity models used to generate the synthetic and locate it are given by the red and blue lines in Fig. 6.3.
1A cursory examination suggests that the data-types used in relative relocation are not independent of those used in the inital location.
Noise can affect the detection and picking of an event by masking an arrival or making the onset of an arrival emergent rather than impulsive and thus difficult to precisely pick the arrival. Consequently, the noise level can affect the location PDF, as shown in Fig. 6.1. Increasing the noise level increases the lateral uncertainty, and the vertical uncertainty appears largest in the SNR = 3 case, which may be due to an improved depth constraint for the SNR = 2 case, due to the different distributions of receivers that it was possible to manually pick arrivals for. The station geometry for this synthetic is shown in Fig. 6.2.
−5 −2.5 0 2.5 5 −5 −2.5 0 2.5 5 Eastings (km) Northings (km)
Figure 6.2: Location geometry for the synthetic events shown in Fig. 6.1, the source depth was
3.5 km.
Figure 6.3: Plot of the marginalised location uncertainty of a synthetic event, arising from velocity
model uncertainty. The first pair of plots correspond to a percentage variation of ∼ 3%, while the second correspond to a percentage variation of ∼ 10% from the true models. For each pair, the first plot shows the range of velocity models used, with the red and blue corresponding to the VPand VS models used to generate the synthetic source. The second plot shows the xy and xz marginalised location PDFs generated by NonLinLoc, including the model uncertainty. Red indicates high location probability and blue is low location probability.
All these approaches rely on accurate knowledge of the velocity model, at a suitable resolution for the data. Both CMM and many of the HYPO* type models use simple one- dimensional models which will not be accurate for the entire region. Those that can use full
three dimensional models, such as NonLinLoc, can have a trade-off between memory usage and model resolution.
Velocity model uncertainty is often a large (and frequently overlooked in commonly used software such as the HYPO* approaches and CMM, although others such as NonLinLoc don’t directly include velocity model uncertainty, but it can easily be included) source of hypocentre uncertainty. Additionally, the uncertainty on three dimensional models can be hard to quantify and, therefore, difficult to include in a location inversion. Fig. 6.3 shows that for well constrained models and low noise levels, the effects are minimal. However, as the range of models increases, the effect becomes larger.
6.3
Source Inversion
Uncertainties can have a large effect on source inversion results. It is important to understand how all the different uncertainties interact in the inversion process and their resultant effects. A set of synthetic seismograms, computed from three source types using a finite difference approach (Bernth and Chapman, 2011), are used to explore the dependency of the PDF solutions on different types of uncertainty, for both the fully constrained double-couple space and the full moment tensor space. The three source types used (Fig. 6.4) are a double-couple source (event A), an opening tensile crack (event B), and a full moment tensor source (event C). Random Gaussian noise was added to the traces at varying noise levels.
The synthetic data were manually picked for both P and S arrival times and P polarities, and then located using NonLinLoc (Lomax et al., 2000, 2009) and a simplified version of the known velocity model. The arrival time picks were used to automatically window and measure P, SH, and SV amplitudes.
A simple Monte Carlo based random search algorithm (Algorithm 8.1) using the posterior PDFs calculated in Section 4.2.5 was used in both double-couple space and the unconstrained full moment tensor space to produce samples of the solution PDFs. The chosen prior for each case was the uniform prior, so no preferred source mechanism, or orientation was specified.
Figure 6.4: Lower hemisphere equal area projections, and lune plots (Section 2.5.6) of the example
sources used for the uncertainty tests. Event A is a double-couple source, event B an opening tensile crack and event C is a full moment tensor source.
Figure 6.5: Lower hemisphere equal area projections, and marginalised lune plots of the source PDF
for three synthetic sources (Fig. 6.4), inverted using polarities for a range of data uncertainties, given by the inverse of the SNR σ = 0, σ = 0.1, σ =1/7, σ = 0.2, σ =1/3and σ = 0.5. For each source (pair of columns) The first column shows the source PDF for the solution constrained to be
double-couple only, and the second column shows the source PDF for the full moment tensor solution. Manually picked station first motions are given by upward red or downward blue triangles. For the focal sphere plots, possible fault planes are given by dark lines. The most likely fault planes are given by the darkest lines. High probability regions are shown in red and low probability in blue on the lune plot. The positions of the different source types on the lune plot are shown in Fig. 2.21. The posterior model probabilities for the double-couple source model estimated from the Bayesian evidence are shown in Table 6.1.
6.3.1
Background Noise
The simplest source of uncertainty is the ambient noise at the receivers. This can affect the detection and picking of an event by masking an arrival or making the onset of an arrival emergent rather than impulsive and thus difficult to precisely pick the arrival polarity. The
background noise level affects the uncertainty in the polarity observation (Eq. 4.9) and the amplitude observations for the amplitude ratio PDF (Eq. 4.35) (Sections 3.2 and 3.3).
Noise has a deep effect on the inversion workflow, as it leads to uncertainty in the arrival time picks, and therefore uncertainty in the hypocentre location. Consequently, it is difficult to isolate the effects of the noise on the inversion. Adding uncertainty to the observations in the source inversion step provides some indication of its effect on the resultant PDF (Figs 6.5 and 6.6). A B C SNR i ii i ii i ii ∞ 0.77 1.0 ? ? ? ? 10 0.78 0.01 0 0 0 0 7 0.80 0 0 0 0 0 5 0.84 0 0 0 0 0 3 0.91 0.02 0 0 0.01 0 2 0.94 0.01 0 0 0.09 0
Table 6.1: Posterior DC model probabilities
for the source PDFs shown in Fig. 6.5 and 6.6. Each event is split into two columns for P polarity only (i), and P polarity and P/SH and P/SV amplitude ratios (ii). ? indicates events with no double-couple solutions found.
The measurement error has little effect on the polarity-only inversion if the solutions are well constrained (Fig. 6.5). As the uncertainty level increases, there is some expansion of the cor- responding range of solutions and an increase in differentiation between possible solutions due to the change in the likelihood function (Fig. 4.1), but the peak remains close to the true source. There is some deviation in the full moment tensor solutions, especially for events B and C, although event B is affected by the random sampling ap- proach (Fig. 2.28).
Whilst some double-couple solutions are pos- sible for event B, these are all very low probabil-
ity, as reflected in the posterior double-couple model probabilities (Table 6.1). These posterior model probabilities agree with the true sources, with values bigger than 0.5 for event A , and non-zero solutions for event C only at the very high noise level solutions.
Including the amplitude ratio observations improves the constraint on the solutions (Fig. 6.6), with the range of possible solutions expanding much less as the measurement uncertain- ties increase. Event A shows some rotation away from the true source for the double-couple constrained model, and a deviation from the double-couple point for the full moment tensor source for the SNR 6 3 examples. The amplitude ratios can suffer from more deviation as the noise level increases (Section 3.4), which may cause some of these deviations. The solutions for event C tend towards the closing tensile crack point, with very low probability double-couple solutions. The double-couple orientations are controlled by the amplitude ratios, especially at high noise levels, as is evident when compared to Fig. 6.5, and the polarity misfits in the solutions for event C.
Event A has very low posterior double-couple model probabilities (Table 6.1), despite the solutions appearing to be good double-couple solutions from a visual inspection. The full moment tensor solutions are dominated by a few high probability samples, which can lead to uncertainties in the Monte Carlo integration approach used to calculate the Bayesian evidence. Events B and C show an increasing range of solutions, and even with some possible double-couple solutions, the posterior probability of the double-couple source is 0 for both events, although these distributions are again dominated by a few high probability samples, however the possible double-couple solutions are much lower probability.
Figure 6.6: Lower hemisphere equal area projections, and marginalised lune plots of the source PDF
for three synthetic sources (Fig. 6.4), inverted using polarities and amplitude ratios at a range of data uncertainties. The amplitude ratios uncertainties were equivalent to SNR = ∞, SNR = 10,
SNR = 7, SNR = 5, SNR = 3 and SNR = 2, with the polarity uncertainty given by the inverse of the SNR. The posterior model probabilities for the double-couple source model estimated from the Bayesian evidence are shown in Table 6.1. See also Fig. 6.5.
The noise has a larger overall effect than that shown in Figs 6.5 and 6.6, as it affects the ability to make the arrival pick and estimate the location. Reprocessing these synthetic events at different noise levels shows that adding noise reduces the number of confident picks, increasing the range of possible solutions, partially due to the network distribution (Section 6.3.2).
Figs 6.7 and 6.8 show the effect of different noise levels on the entire workflow using manual arrival picking of the synthetic events.
A B C SNR i ii i ii i ii ∞ 0.77 0 ? ? ? ? 10 0.71 0.99 ? ? 0.45 0 7 0.71 0.70 ? ? 0.43 0 5 0.77 0.84 0 ? 0.48 0 3 0.64 0.82 ? ? 0.62 0 2 0.62 0.58 0 0 0.59 0.47
Table 6.2: Posterior DC model probabilities for
the source PDFs shown in Figs 6.7 and 6.8. Each event is split into two columns for P polarity only (i), and P polarity and P/SH and P/SV amplitude ratios (ii). ? indicates events with no
double-couple solutions found.
The noise reduces the number of stations with polarity picks, leading to a much lar- ger range of solutions for the inversion us- ing only polarity data (Fig. 6.7). The mo- ment tensor solutions remain peaked around the double-couple point for event A, while the distribution for event C shifts towards double-couple as the noise increases and the number of receivers is reduced. The mo- ment tensor solutions for event B remain non-double-couple but deviate from the true source as the noise level increases. The pos- terior double-couple model probabilities are
negligible for event B, showing that no double-couple solutions can fit the event well. Event A has significant values for all of the solutions, consistent with true source. Event C has sig- nificant double-couple posterior values for all the solutions with SNR 6 10, although this is to be expected because the receiver distribution does not prevent the fit of any double-couple solutions.
While including amplitude ratios improves the source constraint, there can be deviation from the true source point, as seen in the several of the solutions for event C. This is possibly due to the deviation in amplitude ratios as the noise level increases (Section 3.4), which effectively corresponds to a systematic error in the amplitude measurements, and may require increasing the uncertainty estimates of the amplitudes from just the noise to properly incorporate it in the source inversion.
The cases with SNR 6 3 in Fig. 6.8 show poor constraint on the double-couple con- strained solutions for event A, with rotation of the likely orientation from the true source. The full moment tensor solutions for event A show that adding noise increases the range of pos-
sible solutions, and can move the maximum probability source away from the double-couple point. This may explain some reported non-double-couple type sources.
Figure 6.7: Lower hemisphere equal area projections, and marginalised lune plots of the source PDF
for three synthetic sources (Fig. 6.4), inverted at a range of noise levels using P polarities only. The noise levels were SNR = ∞, SNR = 10, SNR = 7, SNR = 5, SNR = 3 and SNR = 2. The posterior model probabilities for the double-couple source model estimated from the Bayesian evidence are shown in Table 6.2. See also Fig. 6.5.
The posterior double-couple model probabilities (Table 6.2) for the SNR = ∞ example for event A is a poor estimate due to the domination of the full moment tensor PDF by a few high probability samples. The other double-couple examples have posterior probabilities that are more consistent with the expected values. The probabilities for event B are negligible. Event C has non-negligible posterior values for the double-couple source when SNR = 2, which is
understandable given the close similarity of the radiation pattern to a double-couple, and the reduced number of receivers.
Figure 6.8: Lower hemisphere equal area projections, and marginalised lune plots of the source PDF
for three synthetic sources (Fig. 6.4), inverted at a range of noise levels using P polarities and P/SH and P/SV amplitude ratios. The noise levels are SNR = ∞, SNR = 10, SNR = 7, SNR = 5, SNR = 3and SNR = 2. The posterior model probabilities for the double-couple source model estimated from the Bayesian evidence are shown in Table 6.2. See also Fig. 6.5.
There is some variation in the orientation between the double-couple and moment tensor solutions (Fig. 6.9), although the general characteristics are consistent between the two. However, in most cases there are additional peaks in the moment tensor orientations, which are not present in the double-couple solutions. These results suggest that the full moment tensor solutions often have a well-constrained orientation, with deviations in the source type
corresponding to the improved fit of the data due to the extra free parameters, rather than any large trade-off between orientation and source type.
0 Event A 0 Event B 0 π 2π 0 κ Event C 0 0.5 1 h −π/2 0 π/2 σ 0 π 2πκ 0 0.5 1h −π/2 0 π/2 σ DC MT
Figure 6.9: Marginalised distribution of the Tape orientation parameters (strike, dip cosine, and
rake) for the SNR = 2 examples from Fig. 6.8. The vertical axis scales are independent, as the distributions are normalised. The red lines show the values for the synthetic sources (Fig. 6.4).
The deviation and sparseness in some of these solutions arises because the P/SH and P/SV amplitude ratio solutions often occupy different regions of the source plot due to the measurement errors (Fig. 6.10). Consequently, the combined distribution can be multimodal, which is an explanation for why some of the distributions using both the amplitude ratio data- types can be multimodal (e.g. Figs 6.6 and 6.8). Nevertheless some of the solutions are multimodal without this effect, such as the SNR = 2 example for P polarities and P/SH amplitude ratios in Fig. 6.10.
Fig. 6.10 suggests that care must be taken when incorporating amplitude ratios in an inversion, and they should only be used if consistent and well measured. Without this check, it is often hard to identify whether the distribution is correct or an artefact of disparate amplitude ratio distributions.
6.3.2
Network Distribution
In many cases the network of receivers is not ideally distributed for locating the source, leading to increased location uncertainties. Furthermore, the sampling of the focal sphere by the receivers often does not constrain the source well, leading to an increased range of possible solutions (Vavryčuk, 2007; Godano et al., 2009; Kim, 2011). Increasing the number of receivers does not always improve the solution, since the receivers need to improve the
Figure 6.10: Lower hemisphere equal area projections, and marginalised lune plots of the source
PDF for synthetic source A (Fig. 6.4), inverted at a range of noise levels using P polarities and either P/SH or P/SV amplitude ratios. The noise levels are SNR = ∞, SNR = 10, SNR = 7, SNR = 5, SNR = 3and SNR = 2.
sampling of the focal sphere to improve the constraint (Fig. 6.12). It is possible to invert for