Abstract
An increased focus on seismic hazard related to induced seismicity means that stateoftheart approaches for earthquake monitoring and hazard estimation associated to tectonic earthquakes are now being applied at smaller and smaller scales. This chapter focuses on a specific issue related to this shift of focus to relatively small earthquakes in close proximity to urban areas. In tectonic earthquake hazard analyses we typically rely on a simple powerlaw scaling relating earthquake magnitude and recurrence. It is known, however, that for smaller earthquakes, the scaling between different magnitude types is not necessarily linear – meaning that a power law cannot be maintained over all magnitude types. Extrapolation to estimate the recurrence of earthquakes not yet recorded at the study site is therefore nontrivial. For earthquake hazard, the moment magnitude is typically used as input as it is easy to relate to ground motion through empirical equations or simulation approaches. However, for earthquake monitoring, maintaining a complete catalogue including moment magnitude of small events is technically difficult. Instead, a pointmeasure based magnitude, such as the local magnitude is usually determined. In the following the impact of the nonlinear scaling between the magnitude of choice for local monitoring – the local magnitude – and that used for hazard analysis – the moment magnitude – is explored.
Keywords
 Seismic Hazard
 Stress Drop
 Earthquake Magnitude
 Moment Magnitude
 Seismic Hazard Analysis
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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18.1 Introduction
Ground shaking from induced seismicity associated with stimulation and exploitation of the nearsurface, for example, related to geothermal reservoirs, shale oil or gas extraction and CO_{2} storage increases the risk exposure of the local population. For instance, an enhanced deepgeothermal project in Basel, Switzerland, triggered an M_{L}3.4 (M_{w}3.2) earthquake at a depth of less than 5 km below the city (Baer et al. 2007). Along with thousands of aftershocks (Deichmann and Giardini 2009), the event led to insurance claims relating to nonstructural damage (e.g., hairline cracks) of more than $7.5 M with total costs of $9 M (pers. Comm. Geo Explorers Ltd., 2012; Giardini 2009).
A growing body of evidence, while still inconclusive, suggests that seismicity related to injection induced earthquakes is increasing. For example, Ellsworth (2013) showed that “within the central and eastern United States, the earthquake count has increased dramatically over the past few years”. This, in turn, means that “regions where the rate increased may be more hazardous than forecast by the 2008 version of the U.S. National Seismic Hazard Map” (Ellsworth 2013). Although direct causality between increasing seismicity and projects related to the exploitation of the shallow crust is not clear in all cases, what is important is that the potential for increased hazard related to induced seismicity (and consequently risk) should be assessed prior to, and during, such operations. Two primary components drive estimates of seismic hazard (and its uncertainty): seismicity rates and groundmotion prediction. In the following chapter, the issues surrounding the determination of seismicity rates are discussed with a focus on the influence of earthquake magnitude assessment, as routinely carried out during seismic monitoring.
Seismicity is typically modelled in probabilistic seismic hazard analyses (PSHAs) using the GutenbergRichter (1944, hereafter GR) relation, with the cumulative number of events (with magnitude greater than M), N, given by:
with a maximum magnitude M _{ Max } (Smith 1976) defined by a probability density function. In practice the truncated exponential GR relation is used (Cornell and Vanmarcke 1969). This relation is used to characterize faults or source zones based on observed seismicity in terms of overall activity rate (a) and the proportion of large to small events (b) (e.g., Wiemer and Wyss 2002). While the avalue can be thought of as a measure of the overall seismicity, the bvalue has been previously linked to factors such as changes in differentialstress, for example, due to: asperities (Wiemer and Wyss 1997), different faulting regimes (Schorlemmer et al. 2005) or due to source depth (Spada et al. 2013). a and bvalues can be directly estimated for a given source zone or fault based on observed (and historical) seismicity above a timedependent completeness threshold M _{ c }. Maximumlikelihood approaches (Aki 1965; Utsu 1965; Bender 1983) are normally used to determine a and bvalues, along with their confidence intervals. While based on simple observational statistics, the estimation of GR parameters is subject to uncertainties due to determined magnitudes (Kijko 1985; Tinti and Mulargia 1985) and due to catalogue completeness (Lee and Brillinger 1979; Weichert 1980).
Given a source model, PSHAs estimate the probability of exceedance for a given groundmotion using a Ground Motion Prediction Equation (GMPE) (Cornell 1968). Moment magnitude (M_{w}) is the magnitude of choice for seismic hazard based on tectonic seismicity: it does not saturate and is compatible with palaeoseismological and geological estimates of ancient earthquakes and maximum earthquake magnitude. This is because it is based on the seismic moment (M_{0}), of which the contributing factors are fault area and slip (assuming constant rigidity). In addition, and of importance to GMPEs developed for induced seismicity is the possibility to base timeseries simulations directly on M_{w}. For instance, Douglas et al. (2013) presented GMPEs using 36 simulation models based on various source (M_{w}, stressdrop), path (regional Q) and local site conditions (local attenuation κ_{0}). As a result of these advantages, GMPEs in the literature are predominantly developed using M_{w} and I will use it as the ‘reference’ magnitude for this discussion.
Clearly, in order to maintain consistency in PSHA, magnitudes used in the source model and the GMPE should be the same. If a GR based source model was developed using, for example, local magnitude (Richter 1935) – typically routinely determined at seismological observatories – then a magnitude conversion would be required between estimating the rate (based on M_{L}) and computing the associated groundmotion (based on M_{w}). In the ideal case, a GR source model can be developed entirely using M_{w} (i.e., for tectonic activity in seismically active areas). However, M_{w} cannot always be calculated for small events due to ambient noise. Furthermore, depending on the frequency content analysed, estimation of M_{w} for small events may introduce systematic bias due to high frequency effects such as siteamplification and attenuation (Stork et al. 2014). In order to obtain complete earthquake catalogues (critical for measuring the cumulative number of events in Eq. 18.1) local monitoring network operators therefore typically estimate magnitudes based on simpletomeasure parameters such as peakamplitude (M_{L}) or signal duration (M_{d}). These catalogues are then supplemented with M_{w} in the case that it is available (e.g., Fäh et al. 2011) and conversion equations (e.g., M_{w} = f(M_{L})) are used to estimate M_{w} of small events.
It is logical to reason that an earthquake has a single ‘magnitude’, and that while some scatter may be apparent, each measure (M_{w}, M_{d}, M_{L}…) should lead to the same broadly consistent value for properly calibrated scales. However, this is not the case: independent estimates of different earthquake properties can lead to systematic differences between different scales, particularly at extremes of magnitudes (either very small or very large) relative to where the initial magnitude scale calibration was made. For instance, Hanks and Boore (1984) showed that the observed curvature of M_{L} versus M_{W} data over an extended magnitude range of Californian earthquakes (0 ≤ M_{L} ≤ 7) could be explained by a frequencydependent interaction of the earthquake source, attenuation and instrument response of the WoodAnderson Seismometer. In this case, determination of a linear scaling between M_{L} and M_{w} would lead to a systematic underestimation of M_{w} (Hanks and Boore 1984). Edwards et al. (2010) used the same simulation method to model and explain, in terms of source, path, site, and instrumenteffects, the observed curvilinear scaling of Swiss earthquake magnitudes over a range 1.4 ≤ M_{L} ≤ 5.5. Subsequently, GoertzAllmann et al. (2011) developed a M_{L} to M_{w} scaling relation for Switzerland by collecting independent estimates of M_{w} and M_{L} over a range of 0.2 ≤ M_{L} ≤ 5.5 and found similar scaling to a model developed based on a Europewide dataset (Grünthal et al. 2009). In the range 3 ≤ M_{L} ≤ 6 the models tend to show that M_{w} ≈ M_{L} – 0.3. However, below M_{L} 3 the models deviate from 1:1 scaling systematically: with a decrease of 1 unit in M_{L} corresponding to only a ~0.6–0.7 unit decrease in M_{w}.
There is no guarantee that different magnitudes scale 1:1. Indeed, such comparisons extend to many different magnitude scales (e.g., Bormann et al. 2009). Careful initial calibrations nevertheless ensure that over a broad region of interest, earthquake magnitudes using different scales are consistent. For instance, Choy and Boatwright (1995) defined the energy magnitude M_{E} to be consistent to the surface wave magnitude (M_{S}) in the range 5.5 < Ms ≤ 8.2. In the past, earthquakes of ‘interest’ have focussed on those easily recordable on national networks (e.g., M ≳ 3) or teleseismic networks (e.g., M ≳ 5). This then corresponds to where magnitude scales tend to be broadly consistent (i.e., M ≳ 3–5). In terms of monitoring induced seismicity, and the estimation of seismic hazard based on these observations, we must therefore fully consider not only the influence of measured earthquake magnitude, but also the magnitude scale itself.
18.2 Influence of M_{L} on GR a and bValues
In the following the focus is placed on M_{L} as it is the most commonly routinely determined magnitude at seismological observatories or local monitoring networks. The computation of M_{L} at small scale monitoring networks can be usually considered complete down to even M_{L} ≈ 1 or even lower. For example, a small scale monitoring network (consisting of seven stations with interstation spacing on the order of a few km) related to a deep geothermal project in St. Gallen, Switzerland, had a magnitude of completeness M_{L} ≈ −1. Catalogue completeness levels at such low magnitude levels was achieved by implementing crosscorrelation techniques and single boreholestation magnitude determination (Edwards et al. 2015). While the GR relation was initially determined based on the analysis of M_{L} data in California, common usage tends to make the assumption that it is M_{w} (and consequently log(M_{0}) and its constituents: fault area and slip) that scales as a power law with the cumulative number of events. As an example of the impact of curvilinear scaling between M_{L} and M_{w} on seismic hazard estimation I therefore show in the following section its impact on GutenbergRichter (GR) a – and bvalues using simulated earthquake catalogues.
18.2.1 Simulation Method
I generate a synthetic earthquake catalogue consisting of events with seven different moment magnitudes (M_{w} 1–7), with each event recorded at seven locations (10, 20, 30, 50, 70, 100 and 200 km). The occurrence of events follows an arbitrary GR relation with a = 3.0 and b = 1.0. For each recording a stochastic seismogram is generated (Hanks and Boore 1984; Boore 2003), which can in turn be used to determine M_{L} of the event by measuring the peak amplitude on a synthesized WoodAnderson Seismometer.
The synthetic stochastic seismograms are generated from a simple model of the Fourier acceleration spectrum (FAS), comprising of the farfield representation of the displacement source (Brune 1970, 1971):
with M _{0} the seismic moment (in SI units), and C a constant which accounts for the freesurface, average radiation pattern, slip velocity and density. Geometrical spreading is accounted for using the hypocentral distance, R. The moment magnitude scale is defined by M _{0}:
(Hanks and Kanamori 1979). Assuming a circular crack model (Eshelby 1957) the source cornerfrequency (f _{ c }) can be calculated using:
where Δσ is the stress drop of the earthquake (SI units) and β is the shearwave velocity (β = 3,500 m/s). In order to account for anelastic attenuation along the sourcereceiver path an exponential decay function is used:
(Knopoff 1964) with Q the quality factor and R the hypocentral distance (in m). The full synthetic stochastic acceleration timeseries can be calculated based on the FAS (E(f)B(f)) and a given duration model. For the duration model we adopt a simple model based on source magnitude and distance from the source:
(Herrmann 1985). The Local Magnitude is calculated using an adapted from of the original relation used by the Swiss Seismological Service:
with R in km and A the peak displacement (in mm) of the groundmotion convolved with the response of a WoodAnderson Seismograph. In practice, since M_{L} is based on a peak measure (A) of the WoodAnderson seismogram with natural period 0.8 s and damping of 0.69, randomvibration theory [RVT, (Cartwright and LonguetHiggins 1956)] can be used to directly calculate A using the duration model adapted to account for the WoodAnderson oscillator response (Liu and Pezeshk 1999) and the FAS (Hanks and Boore 1984).
18.3 Results: GR a and bValues from M_{L}
Across the various simulation models with different source (Δσ, M_{w}) and attenuation terms (Q) it is apparent, as expected, that the a and bvalues obtained using M_{L} are different from those obtained using M_{w}. This fact is of course, consistent with the fact that the M_{L} to M_{w} scaling relation is curvilinear. The exact difference is driven by the source properties (Δσ), the attenuation (Q), and the interaction of the earthquake spectrum and the WoodAnderson Seismometer used to compute M_{L}. This was originally shown by Hanks and Boore (1984) to be the driving force behind the M_{L} to M_{w} scaling behaviour and therefore directly propagates into the GR relation. As a result of the curvilinear scaling, the magnitude range over which the GR relation is calculated also has a significant impact on the differences found by using different magnitude scales.
18.3.1 Influence of StressDrop
The reference GR relation using M_{w} with b = 1.0 is shown in Fig. 18.1 along with four simulated catalogues for which M_{L} is used instead of M_{w}. In this case only geometrical (not anelastic) attenuation was applied. For each of the four catalogues a different stress drop is used: 0.1, 1, 5 and 10 MPa. The resulting differences show the influence of the stressdrop on M_{L} and consequently a and bvalues. In Fig. 18.1a it can be immediately seen that the largest deviation from the GR relation is apparent at higher magnitudes. This is due to the saturation of the M_{L} scale: at some point, despite increasing the moment of the earthquake in equal steps, the increase in M_{L} slows (and eventually stops). This is a wellknown phenomenon typically considered to occur around M67, however, an interesting aspect to consider is that the point at which saturation begins is controlled by the stressdrop. For low values (e.g., Δσ = 0.1 MPa to 1 MPa), even M_{L} > 4 events show the beginning of saturation: an effect that artificially increases the bvalue. As a general observation, systematically low stressdrop events tend to increase the apparent bvalue when using M_{L} in the high magnitude range. This effect should not be particularly problematic since we can observe a departure from the linear GR law, and consequently limit the fitting range to below where the effect starts. Furthermore, for the larger events susceptible to saturation, it is likely that direct computations of M_{w} will be available.
Avoiding the saturation effect, which is not particularly relevant for induced seismicity, and instead focusing on the lower magnitudes (Fig. 18.1b) we nevertheless still see a systematic variation of the bvalue depending on the stressdrop used in the simulations. Generally (apart from the lowest stressdrop catalogue), the bvalue is artificially decreased. For the Δσ = 10 MPa catalogue, the bvalue is 0.86; for Δσ = 5 MPa, b = 0.89; for Δσ = 1 MPa, b = 0.94 and for Δσ = 0.1 MPa b = 1.05. These values are computed over the range 1 ≤ M_{w} ≤ 5, limiting the upper magnitude to 4 may slightly decrease the values further.
18.3.2 Influence of Attenuation
In the previous analysis the influence of the stressdrop was isolated by neglecting anelastic attenuation. However, in reality a complex interaction between the stressdrop, attenuation and the instrument response all play a role in the M_{L} assigned to a particular event (Hanks and Boore 1984). A further reason that anelastic attenuation is important is apparent if we consider that the M_{L} relation (Eq. 18.7) only accounts for frequency independent attenuation (i.e., the + αR term).
Taking the catalogue with Δσ = 5 MPa I now apply anelastic attenuation (Fig. 18.2). For Q = 1,000 the bvalue is further reduced from b = 0.89 without attenuation to b = 0.82, and using Q = 500 I obtain b = 0.79. Such variations in the bvalue seem quite strong, however, if we look only in the range 3 ≤ M_{w} ≤ 5, where such statistics are often calculated (e.g., for seismic hazard), the smallest effect on the bvalue is observed (Fig. 18.3). Interestingly, the avalue is increased in this example, consistent with the aforementioned difference between M_{w} and M_{L} of 0.3 in this range (GoertzAllmann et al. 2011; Grünthal et al. 2009). As a result, if using magnitudes of interest for tectonic seismic hazard (3 ≤ M_{w} ≤ 5), and applying a simple linear conversion (e.g., M_{w} = M_{L}0.3), one would observe similar a and bvalues to if one had been able to use M_{w} directly. The effects discussed here may also be further pronounced if considering the influence of siteamplification on M_{L} values. Whilst site amplification tends not to affect M_{w} due to its longperiod basis, M_{L} is computed over a period range where it is common for strong amplification effects to be present.
18.4 Regional Variability Between M_{L} Values
So far this chapter has focused on the impact of using M_{L} in the standard GR relation without accounting for the curvilinear scaling between M_{w} and M_{L}. However, a major problem to address in the prediction of ground motion for induced seismicity is the significant variability of reported earthquake magnitude from agency to agency (Fäh et al. 2011). Edwards and Douglas (2014) homogeneously computed earthquake moment and localmagnitude for events related to Enhanced Geothermal Systems (EGSs) in Basel (Switzerland), Soultz (France) and Cooper Basin (Australia); natural geothermal fields in Geysers (California) and Hengill (Iceland), and a gasfield in Roswinkel (Netherlands). As shown in previous studies, published catalogue (M_{L}) magnitudes differed widely with respect to a common M_{L}–M_{w} scaling relation, with up to a unit of magnitude difference. Using nonspecific conversions from catalogue magnitudes (e.g., M_{L}) to M_{w} for use in GMPEs would subsequently lead to significant bias. On the other hand, Edwards and Douglas (2014) showed that given a common magnitude definition (and corresponding attenuation corrections), the scaling between moment and localmagnitude of small induced earthquakes follows a secondorder polynomial (Fig. 18.4) consistent with previous studies of natural seismicity (GoertzAllmann et al. 2011; Grünthal et al. 2009). Using both the SouthernCalifornia M_{L} scale and M_{equiv} (Bommer et al. 2006) Edwards and Douglas (2014) found that the analysed datasets fell into two subsets offset by 0.5 magnitude units, with welldefined relation to M_{w} (Fig. 18.4a, b). M_{equiv} was shown to correlate 1:1 with M_{L}, albeit with regionspecific offsets.
18.5 Discussion
From the analysis presented here it is clearly important for consistent and transparent magnitude determination at various stages of seismic hazard analysis. A key conclusion is that if one assumes that the moment magnitude M_{w} follows the GR relationship, then the M_{L} scale does not. The most significant deviation is for M_{L} > 5, where we begin to see the effect of magnitude scale saturation, which leads to locally increased bvalues. The magnitude at which this saturation effect occurs depends on the stressdrop. Very low stressdrop events (e.g., 0.1 MPa) led to magnitude saturation effects impacting the bvalue estimation at M ≈ 4–5. However, since this effect is most significant in the magnitude range where direct computation of M_{w} is typically available, it is not an issue in PSHA. More importantly for induced seismicity is the fact that at lower magnitudes we noticed that the bvalue is typically reduced (for all but the lowest stressdrop catalogue). This is related to the fact that for M_{w} ≈ 3–5, M_{L} values tend to be greater than M_{w} for the simulated catalogues (and empirical catalogues), while at lower magnitudes, the difference reverses (M_{w} tends to be greater than M_{L}). Attenuation tends to systematically decrease the bvalue computed with M_{L} because it affects increasingly smaller events (with proportionally more high frequency energy) more strongly. For instance, attenuation can be considered as a lowpass filter: for large events with little high frequency energy (relative to the lowfrequency energy), this filter has little effect, whereas for smaller events a significant portion of the energy is cut from the signal. Nevertheless, it is evident that these effects offset one another, or are minimised to a certain degree when choosing particular magnitude ranges (e.g., when using 3 ≤ M_{w} ≤ 5, as often the case in hazard studies related to tectonic seismicity).
Routine computation of M_{L} is often a requirement in order to have a complete data catalogue for computing the GR relation at small magnitudes. Best practice for recovering bvalues should be to convert M_{L} using a quadratic polynomial (Grünthal et al. 2009) or curvilinear function (GoertzAllmann et al. 2011) (Fig. 18.5). The form of this polynomial will depend on:

(a)
the local conditions – including source properties (e.g., stress drop or slip velocity), path attenuation (Q and geometrical spreading) and site effects (amplification and attenuation) (Edwards et al. 2010);

(b)
the form of the M_{L} equation used (Edwards and Douglas 2014).
Since data M_{w} – M_{L} pairs are rarely available down to small enough magnitudes; one option to consider is the creation of synthetic catalogues, as used in this chapter. Different source, path and site modelling terms can then also be used to cover the epistemic uncertainty of the M_{L} to M_{w} conversion.
Finally, even when we have a homogenous moment magnitude scale and consistent GR source models, analysis by Douglas et al. (2013) highlighted considerable variation in source and path parameters (e.g., stressdrops, local attenuation) among different regions and sites. Both M_{L} to M_{w} conversions and ground motion prediction may therefore be region dependent. While the focus here has been on the computation of magnitude for defining GR models, consistency should be ensured between the conversion used for M_{w}(M_{L}) and the model terms used for ground motion prediction in PSHA (or deterministic/scenario based hazard). Clearly, when conducting seismic hazard assessment for a given geothermal project it is not known a priori which source, path and site terms (or equivalent GMPEs) are most applicable. However, Edwards and Douglas (2013) showed that as seismograms are recorded at a site, the applicability of particular models becomes quickly evident using either spectral or residual analysis approaches. It is therefore important to establish and maintain an optimised and effective monitoring network, with broadband (and if possible, borehole) instrumentation. While region specific wavepropagation behaviour is initially difficult to define, it has the advantage that significantly lower values of uncertainty can be observed compared to previous studies combining small earthquakes from different regions.
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Edwards, B. (2015). The Influence of Earthquake Magnitude on Hazard Related to Induced Seismicity. In: Ansal, A. (eds) Perspectives on European Earthquake Engineering and Seismology. Geotechnical, Geological and Earthquake Engineering, vol 39. Springer, Cham. https://doi.org/10.1007/9783319169644_18
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