The Bayesian approach discussed in Chapter 4 can easily be extended to include the automated polarity observations. The PDF for the observed polarity at a given time must be considered with respect to the source:
p (Y (t) = pol (t) |Ai, ∆ (t) , (t)) = H pol (t) Ai+
(t) ∆ (t)|Ai|
!!
. (5.9)
This is dependent on the theoretical amplitude, Ai, the measurement uncertainty, (t), and
the observed amplitude change, ∆ (t) > 0.
Given that the standard deviation of the background noise, σ, can be estimated, the simplest noise model for the uncertainty on ∆ (t), (t) is a Gaussian distribution, with standard deviation√2σdue to ∆ (t) corresponding to the amplitude change. Marginalising with respect to the noise model gives
p (Y (t) = pol (t) |Ai, ∆ (t) , σ) = ˆ ∞ −∞ Hpol (t)Ai+∆(t)(t) |Ai| √ 2πσ2 e −(t)2 2σs dδ (t) . (5.10) Following the same approach as in Section 4.2.1, the marginalised PDF is
p (Y (t) = pol (t) |Ai, ∆ (t) , σ) = 1 2 1 + erf pol (t) ∆ (t) sgn (Ai) 2σ !! , (5.11)
although this is not marginalised with respect to time.
Eq. 5.11 can be simplified, because the signum function in the PDF is equivalent to writing the PDF using the Heaviside step function H (x), giving
p (Y (t) = pol (t) |Ai, ∆ (t) , σ) = H (Ai) 1 2 1 + erf pol (t) ∆ (t) √ 2σ !! + H (−Ai) 1 2 1 + erf − pol (t) ∆ (t) √ 2σ !! . (5.12)
The modelled amplitude is independent of the time. Consequently, the time marginalised PDF is given by
p (ψi|Ai, σ, τ ) = H (Ai) ψi+ H (−Ai) (1 − ψi) , (5.13)
where ψiis the time marginalised positive polarity PDF for the arrival (Eq. 5.5).
As with the manually estimated polarity PDF (Eq. 4.9), it is possible that there could be a receiver orientation error with probability πi, leading to a flipped polarity so Eq. 5.13 can
be extended to include this:
This is the polarity probabilities likelihood, equivalent to Eq. 4.10, and can be included in the Bayesian source inversion of Chapter 4, by substituting into the source likelihood:
p (d0 | M , τ , k) = ¨ M X j=1 N Y i=1 h pψi| Aij = aj· ˜M , σi, πi, τi pRi = ri| Aij = aj · ˜M , σi, τi i p (σ) p (π) dσdπ, (5.15) p (d0 | M , τ , k) = ¨ Q X k=1 M X j=1 N Y i=1 h pψi| Aijk = ajk· ˜M , σi, πi, τi pRi = ri| Aijk = ajk · ˜M , σi, τi i p (σ) p (π) dσdπ. (5.16) Care must be taken with the data independence, and the caveats of section 4.3 still apply. Automated polarity observations must not be used in combination with manual polarity observations in the inversion, because this artificially sharpens the resultant PDF.
The quantitative method for including the noise in the polarity estimation is unlike the approaches for manual polarity observations such as that proposed in section 4.2.1 (Brillinger
et al., 1980; Walsh et al., 2009).
The source inversion results for a synthetic double-couple event and a real event from the Upptyppingar dyke swarm in 2007 (White et al., 2011) were evaluated using the Bayesian approach from Chapter 4, adapted for automated results (Eqs 4.53 and 4.54).
The results for the source inversion using automated picking resemble those of the manual picking, although there is usually a wider range of possible solutions, since most of the proposed solutions have at least a small non-zero probability. In some cases, the solution can be improved by additional constraint from receivers with no manual polarity pick but a suitable arrival time pick, as shown by the example from the Upptyppingar dyke swarm, which has a few receivers without manual polarity picks but with a suitable arrival-time pick to estimate the polarity probabilities. The double-couple solutions from the automated polarity inversion tend to have a clearer demarcation between the high-probability solutions and the lower-probability solutions, as can be seen in Figs 5.18 and 5.19.
Figure 5.18: Lower hemisphere equal area fault-plane plots and marginalised full moment tensor
lune plots showing the comparison of automated polarity and manual polarity source inversions for two events, a synthetic event (A) and the event from table 5.1 and Fig. 5.12 (B). The first two columns show the the double-couple constrained and full moment tensor PDFs for the automated polarity probabilities and the second two show the solutions for the manual polarity observations. The fault plane plots show the most likely solutions with the darkest lines. The stations are indicated by circles if there is no manual polarity information used (such as in the polarity probability based inversions), and upwards red triangles or downwards blue triangles depending on the manually observed polarity. The lune plots are normalised and marginalised to show the most likely source type, red regions correspond to high probability and blue low probability.
A common approach to dealing with polarity uncertainties is to allow a blanket probability of a pick being incorrect (pmispick)(Hardebeck and Shearer 2002, 2003). This is equivalent
to setting the value of ψi in Eq. 5.14 to either 1 − pmispickor pmispick depending on whether
the manual pick is positive or negative. Unlike the automated polarity approach, including an arbitrary blanket probability of a mistaken pick does not account for the arrivals that are most difficult to pick being most likely to be incorrect. Consequently the range of solutions is often not well constrained. The double-couple solutions from the automated polarity in Fig. 5.19 show a stronger demarcation between the low and high probability solutions than those using a blanket probability of a mistaken pick, although the ranges are similar. The full moment tensor solutions in Fig. 5.19 have very similar distributions for both of the inversions, with poorer constraint than the ordinary manual polarity solutions from Fig. 5.18.
Figure 5.19: Lower hemisphere equal area fault-plane plots and marginalised full moment tensor
lune plots showing the comparison of results from automated polarity (first two columns) and arbitrary probability of a mistaken pick (pmispick= 0.1)(second two columns) for the synthetic (A) and real (B) events shown in Fig. 5.18. See also Fig. 5.18.
Figure 5.20: Lower hemisphere equal area fault-plane plots and marginalised full moment tensor
lune plots showing the comparison of results from automated polarity using a Gaussian time PDF around the manual time pick (first two columns) and STA/LTA time picking (second two columns) for the synthetic (A) and real (B) events shown in Fig. 5.18. See also Fig. 5.18..
Although the choice of time PDF is independent of the approach, the CMM STA/LTA (Fig. 5.5) was tested as a possible PDF. Fig. 5.20 shows that this arrival-time PDF can work in low-noise environments, but in a higher-noise environment it may not be possible to estimate a solution. This is clear in the solutions for the Upptyppingar event, which have no constraint on the possible source for both the double-couple and full moment tensor inversions because the variations in probability are too low. The low noise synthetic example shows good agreement with the east-west plane, but the north-south plane is less well constrained. However, the approach may provide some constraint in the high noise case if there are enough
receivers sampling the focal sphere. Using an arrival time PDF from a well calibrated onset picker (Section 5.2) would be a much larger improvement in the source constraints.
Figure 5.21: Lower hemisphere equal area fault-plane plots and marginalised full moment tensor
lune plots showing the comparison of results from automated polarity using a manual prior of 0.85 (first two columns) and manual polarity (second two columns) for the synthetic (A) and real (B) events shown in Fig. 5.18. See also Fig. 5.18.
Using manual polarity picks as a prior probability further constrains the source PDF, leading to a sharper solution than the manual polarities (Fig. 5.21). The full moment tensor solutions are also more constrained by the prior, compared to the equivalent solutions in Fig. 5.18.
5.6
Summary and Discussion
The Bayesian approach to automated polarity estimation proposed here allows a quantitative estimate of the measurement uncertainties. This method surmounts the inconsistencies in manual polarity picking, as well as producing a quantitative estimate of the uncertainty on the source.
The polarity probabilities have a clear dependence on the time pick accuracy and the noise level of the trace, requiring an accurate arrival time pick approach. When an automated arrival picking approach is used, it should produce accurate onset picks, with results comparable to a human, otherwise the resultant PDF is usually too large. However, many of these automated pickers require parameters that often that are dependent on the arrival characteristics, such as the frequency and onset characteristics, to produce accurate picks, requiring human input to correctly set the values.
The choice of arrival time PDF can be adjusted depending on the perceived quality of the arrival time picking approach. The arrival time PDF can also be adjusted using the pick
quality estimate, such as the pick weight (0-4 range from HYPO71 (Lee and Lahr, 1975)), although a range with higher discretisation would prove more accurate.
There are few differences between the results obtained from manual and automated picking for the events from the Upptyppingar and Krafla volcanic systems in Iceland. The estimation of the probability of correct first motions produces a quantitative estimate of the uncertainty of the polarities which carries through to the calculation of the resulting source mechanism PDF. In the cases where the automated and manual picks seem to disagree, some of that can be attributed to human error in the arrival time and manual polarity estimation.
The time required for calculating the PDFs is less than one second per arrival, which is much shorter than that required for manual polarity picking. Consequently, this approach adds little time to an automated processing work-flow, and can easily be included into near real-time event detection, unlike the much slower manual polarity picking. Additionally, the approach is useful for estimating polarities of phases measured on location-dependent seismogram components, such as the SH phase. These are often ignored in source inversion due to the requirement to return and pick the polarities after the event has been located.
Polarity probabilities can be incorporated into the source inversion approach of Chapter 4, producing results that are similar to inversions from the manual polarity picks. This quantitative approach for estimating and including the uncertainties in the source inversion is better than the qualitative approach for manual polarity observation described in Chapter 4, as well as the inclusion of an arbitrary probability of a mistaken pick.
The results from this method will never be better than those from manual polarities with no uncertainties, as the manual polarities have massively underestimated the uncertainties on the measurements, so produce artificially sharp results.
6 The Effects of Uncertainties on Source
Inversion
Source inversion is usually carried out as the final part of a processing workflow involving detecting, picking and locating the events, and at each step in the chain uncertainties are introduced and assumptions made.
The uncertainties in the steps preceding the source inversion have been investigated in the literature (e.g. Lomax and Michelini, 2001; Husen and Hardebeck, 2010; Drew et al., 2013), and will only be reviewed in this chapter, which is focussed on the effects of different uncertainties on the resultant source PDF, using synthetic tests controlled for different un- certainties. The synthetic data are generated using a finite difference approach (Bernth and
Chapman, 2011), for different source types and station distributions, and the posterior model
probabilities (Section 4.2.6) are calculated for the constrained and unconstrained solutions.
6.1
Event Detection and Arrival Picking
Event detectors are used to decide whether a signal is due to an earthquake or not. This can be easy for larger seismic events, which can sometimes even be detected in space (Garcia
et al., 2013). However, microseismic events are often small in magnitude and frequent,
making manual detection difficult. Additionally, microseismic events are often separated by only a few seconds, which complicates the assignment of arrivals to individual events. Consequently, automated monitoring approaches, such as those used in early-warning systems or microseismic monitoring, often try to locate an event as part of the detection step.
Picking the arrival times is sensitive to the random background noise at the station. The noise may be larger than the first arrival, leading to emergent rather than impulsive arrival characteristics, which can prove difficult for automated approaches, increasing the arrival time uncertainty. Arrival picking can occur concurrently with event detection, but the picks can be refined, either manually, helping to account for false event detection, or automatically using a more complex picking algorithm. Section 5.2 introduces several of these automated approaches. Many of the approaches in common usage essentially compare the time evolution of the signal and whether it differs from the long-term noise environment in such a way that could be classed as an event.
Modern seismograms are mostly digital, so some interpolation is done between the points to produce the observed waveform. However, sample rates are usually sufficiently large that the associated uncertainties from the digitisation are small, compared with the arrival time uncertainty.
Classifying the arrival phase often uses known characteristics to reduce the uncertainty, such as the expected orientation of the arrival phases. The phase of the arrival can also be estimated using polarisation analysis (Baillard et al., 2014; Ross and Ben-Zion, 2014), requiring a prior estimate of the expected polarisation directions for the P and S phases. This estimate can be verified after determining the hypocentre location.
The detection step itself does not introduce many uncertainties into the source inversion workflow, apart from the risk of a false event detection. If the detection approaches are used to pick the arrival and locate the event, there are corresponding uncertainties for these steps, as discussed below.
Arrival picking is also when manual or automated polarity determination is carried out. This depends on an accurate arrival time pick and phase classification to correctly identify the onset and first arrival polarity. Many automated workflows do not measure arrival polarities at all, due to the difficulties in determining it, but Chapter 5 describes a method to include this in an automatic workflow.