Appendix I Determination of M W and calibration of M L (SED) M W regression

Transcription

1 Appendix I Determination of M W and calibration of M L (SED M W regression Bettina Allmann, Benjamin Edwards, Falko Bethmann, and Nicolas Deichmann Internal report of the Swiss Seismological Service Institute of Geophysics, ETH Zürich March, Executive Summary We estimate moment magnitudes M W for earthquakes in Switzerland recorded between 998 and 9. Compared to previous studies the M W determination in Switzerland could be extended to lower magnitudes and scaling relations can be investigated from M L = to.. Three different spectral methods are applied to estimate M W and the scaling relations resulting from the different methods are compared with one another. Above M L =, the obtained estimates are consistent with the previously used scaling relation of M W = M L.. Below M L =, all three methods indicate that a :-type relationship is inappropriate. Therefore, we propose a new empirical piecewise M L to M W scaling relation for earthquakes in Switzerland. The scaling is linear below M L = and above M L =. To obtain a smooth transition between the two linear scales we fit a quadratic relation in between ( M L <. This scaling relation is also consistent with M W estimates from moment tensor solutions based on broadband waveform fitting of local earthquakes with M L >.. Introduction The routine determination of seismic moment, and hence moment magnitude M W, at the Swiss Seismological Service (SED is made using moment tensor solutions, which require the availability of long-period data. Estimates of M W are typically only available for local events above a

2 threshold of M W =., which is imposed by the signal-to-noise (STN ratio at long periods of ground-motion. The estimation of M W below this threshold is typically performed by applying an empirical scaling relation between local magnitude M L and M W. In the following report we aim to improve the scaling between M L and M W for earthquakes in Switzerland by extending the M W estimates to lower magnitudes using three different methods. All methods attempt to obtain the frequency-independent long-period spectral level below the corner frequency of a displacement spectrum (Brune, 97, 97. Placing long-period in quota indicates that this is a relative term that changes with the magnitude and stress drop of the analyzed event. The three methods are briefly explained and the scaling relations resulting from the different methods are compared with one another. The results are also compared to previous M W determinations by Clinton (pers. com. and Braunmiller et al. (. Estimates of moment magnitude between different authors (Allmann, Edwards and Bethmann differ by up to. units. Possible causes for this variation are discussed. An improved scaling relation for Switzerland is proposed. However, it is noted that the new scaling relation is still purely empirical and does not intend to reflect any physical relationship between M L and M W. Moment magnitude estimation. Method of Edwards Data processing follows the methodology of Edwards et al. (, which is based on the previous work in Edwards et al. (8 and Edwards and Rietbrock (9. In brief, the multi-taper FFT algorithms of Park et al. (987 and Lees and Park (99 are applied to an analysis window whose position and duration is based on the S-wave arrival and the arias intensity of the recorded timeseries. The aim is to encapsulate the main duration of shaking defined by to 7 % of the energy from the earthquake. The resulting Frequency spectra have differing minimum, maximum and delta frequency, depending on the length of the time-window used. However, the minimum frequency available to the subsequent inversion is typically between. and Hz, with the maximum ranging from to Hz. Noise estimates are taken from the recording before the P-phase arrival and are carefully compared with the signal in the frequency domain in order to retain only the highest quality data. All data are converted to velocity spectra and the instrument response is removed. Following the selection and processing of data, the absolute values of M W for events with more than recordings are extracted using a two-stage regression that deconvolves the source, path and site effects as detailed in Edwards et al. (8. In this case, Q-tomography was not used as the M W is stable without it. The earthquake source is based on the Brune (97, 97 model, with geometrical attenuation described by a simple model with decay of /R in the first km, leading to decay of /R. at greater distances. In Edwards and Rietbrock (9, it was shown that the M W computed using the spectral fitting method are consistent with M W computed using a moment tensor solution based on broadband waveform fitting of local Swiss earthquakes of M W =.8 to M W =. with a negligible offset and standard deviation of less than.. This difference is further discussed in section. M W computed using the method of Edwards depend on a good azimuthal coverage of recordings. For smaller events, the relatively large inter-station spacing results in few recordings passing

3 7 6 # events:9 y=.6 M w (Edwards : 6 relative Mw Figure : Moment magnitude M W from Edwards versus relative moment magnitude for 9 events. The black dashed line shows a : relation. The red line shows the least-squares fit. the quality control procedure. Consequently there also exists a minimum magnitude for which the method is stable. This is highly dependent on the location of the earthquake and the level of background noise. However, solutions for earthquakes with M L <. are rarely available after quality control and signal processing procedures.. Method of Allmann Allmann computes displacement spectra of about earthquakes between M L =. and. in Switzerland using the multi taper method of Park et al. (987 for. s long time window following the S-wave arrival. The events are recorded between September 998 and March 9 at broadband and s seismometer stations of the Swiss Digital Seismic Network (SDSNet in addition to selected stations from neighboring countries. The maximum epicentral distance used is km. All traces are corrected for the instrument response and equally resampled to Hz. A minimum signal-to-noise (STN ratio of at least three between and Hz is required. The noise is computed over a. s long window before the P-wave arrival. The STN ratio is estimated in the frequency domain using a moving window average over two neighboring sample points. Each event has to be recorded by at least three different stations in an attempt to average out directivity effects. The data processing closely follows the approach of Shearer et al. (6 and Allmann and Shearer (7 to which the reader is referred for more detailed explanations. The spectra are separated into relative source, propagation path, and near-receiver terms such as site effects by using an iterative least-squares stacking approach to obtain an unbiased estimate of the relative source spectrum of each event. The relative seismic moment is proportional to the long-period amplitude (Ω of the source spectrum. Ω of each event is computed from the average spectral amplitudes between. Hz and.6 Hz. However, the spectral deconvolution method as parameterized by Allmann only allows for the extraction of relative moments, which have to be calibrated in a subsequent step to absolute

4 x 6868 station OTTER S pick original processed displacement in nm time in s 6 Log log plot: 6868 station OTTER Lin lin plot Amplitude density in nms fourier transform spectral fit corner frequency fc =9.7 Hz Ω =67.6 [nms] n = γ = fit =grid frequency in Hz Amplitude density in nms 8 6 frequency in Hz Figure : Example of cutting a single sided pulse, shifting it to the baseline and performing a spectral fit for the determination of Ω. values for M W. On the other hand, the spectral deconvolution into relative source terms can be applied to a larger dataset without restriction, and ultimately allows for the determination of M W down to smaller events. The calibration can be performed with a much smaller dataset of a few independently obtained absolute M W estimates. To calibrate relative M W estimates (equals / Ω to an absolute M W, Allmann uses M W estimates from Edwards et al. ( for selected events (Figure. The bestfitting y-intercept of.6 is used as a constant shift to scale the relative M W to absolute values without changing the difference in the slope between Edwards and Allmann s M W.. Method of Bethmann We manually select uni-polar displacement pulses for our analysis under the assumption that all energy for a specific radiation angle is captured within the first arrival. To better sample our spectra in the lower frequency part we zero pad the seismogram to 8 samples. Instead of using a taper at the end of the seismogram we remove the trend between the beginning and the end of the pulse. The pulse will then start and end with a value of. We calculate the spectral amplitude at low frequency (Ω by a fit in the frequency domain using the expression of Abercrombie (99. A grid search with manually picked starting values for Ω and f c is used for fitting. An example of how a pulse is selected and the resulting fit is shown in Figure.

5 The seismic moment is calculated following Aki and Richards (. M = πρ ξ ρ x β ξ β x R ξx Ω ( F θφ S Equation includes station and location specific information such as density and velocity information at the source (ρ ξ, β ξ and the receiver (ρ x, β x, radiation characteristics represented by the radiation coefficient (F θφ, dependent on azimuth (Φ and takeoff angle (θ, amplification at the free surface (S, hypocentral distance (R ξx, and long period spectral amplitude Ω. The moment magnitude is derived from the seismic moment M in [Nm] using the relation of Hanks and Kanamori (979 M w = log M 6.. (.. Simplification of scaling relations for series of similar earthquakes. One advantage of investigating sequences of similar earthquakes is that we can assume an identical propagation path and use identical velocity density information. Also, the radiation pattern and surface amplification can assumed to be identical for each station in the sequence. The scaling relation of M L vs. M w can then be reduced to a simple relation of Wood-Anderson (WA amplitude vs. spectral amplitude. We identified seven sequences of similar earthquakes in Switzerland in the past 8 years. Those are ideally suited to determine a relationship of magnitudes below magnitude. A : relation of WA-amplitude and spectral amplitude corresponds to a relation of M L =.M W (Deichmann, 6. Scaling relation and comparison of different studies Figure shows a compilation of M W residuals of matching events from different authors. Note that M W estimates from Allmann are generally lower for larger magnitude events (M L >. Bethmann obtains lower M W estimates compared to Allmann or Edwards with a mean shift of about.7. The mean M W of Allmann is.6 lower compared to Clinton and the variance is also high with.8. The relatively high variance is mainly due to the much lower M W estimates of Allmann for the largest events. Using the newly derived M W, a new M L to M W scaling relation is computed for all three datasets (Figure. In all cases, the quantity of data for this regression is substantially larger than the original regression of Braunmiller et al. (. The minimum magnitude is ML =. We first use an orthogonal quadratic fit to estimate a scaling relation. This approach follows the work of Grünthal and Wahlström ( and Grünthal et al. (9, who found a quadratic relation using a chi-square fit to fit well to a European catalog of M L and M W values. Since events follow a power-law magnitude-frequency distribution (Gutenberg and Richter, 9 there are a decreasing number of events with increasing magnitude, which may bias the quadratic fit. To avoid this bias the data are split into bins of equal M L, and subsequently apply bootstrap resampling whilst enforcing an equal quantity of data in each M L bin. This helps to evaluate the robustness of the solution and avoid bias due to data selection and magnitude. The regression

6 a b c d e M W (E. - M W (A. M W (B. - M W (A. M W (C. - M W (A. M W (B. - M W (E. M W (C. - M W (E.... µ =. σ = events µ = -.7 σ =.8 events 6 µ =.6 σ =.8 events 6 µ = -.8 σ =.7 6 events 6 µ = -. σ =. events 6 M L Figure : M W residuals versus M L for matching events from a Edwards and Allmann b Bethmann and Allmann c Clinton and Allmann d Bethmann and Edwards e Clinton and Edwards. Mean µ and variance σ of the residuals are given for each distribution. Note that positive residuals mean lower M W estimates of the second author, respectively. 6

7 .. Allmann: 986 events Edwards: 677 events Clinton: 6 events Braunmiller: events Bethmann: 8 events Grünthal: 8 events M W - M L M W =.9+. M L +. M L (Allmann M W =.+.7 M L +. M L (Edwards M W = M L +. M L (Grünthal et al. ( M W =.+.66 M L +.76 M L (Grünthal et al. (9 6 M L Figure : Difference between M L and M W versus M L of different datasets from various authors (different shaped and colored symbols. The colored lines show the best quadratic fit to some datasets described by the equations. is an orthogonal L solution assuming equal variance in both magnitudes. A final regression is performed over all possible scaling relations given by the bootstrap analysis, in order to find the orthogonal L solution of the bootstrap possibilities. As an example, Figure shows the range of fits found by the bootstrap approach for the dataset of Edwards. The scaling laws of Grünthal and Wahlström ( and Grünthal et al. (9 are included for comparison. The scaling relation from bootstrap resampling (solid line matches those of Grünthal and Wahlström ( and Grünthal et al. (9 very well for magnitudes greater than ML =.. For lower magnitudes the scaling relation of Edwards gives higher M W estimates for respective M L values. It is important to note that the validity of the derived relation does not extend beyond the limits for which data is available as there is no theoretical basis for a purely quadratic relation between M L and M W. This is apparent in Figure, where it can be observed that the variance of the scaling relation grows significantly outside of the magnitude range where data is available (in this case for M L <. and M L >.. The resulting relations for each dataset are given in Figure. For comparison, we have also added the data of Clinton (pers. com., Braunmiller et al. (, and Grünthal and Wahlström (. The scaling relations of Allmann and Edwards agree in the magnitude range between and but show strong deviations at larger magnitudes. However, this is expected due to the quadratic term being ill-constrained outside this range. The scaling relation obtained from Bethmann s dataset is more linear with an almost negligible quadratic term. His relation is also shifted compared to Allmann and Edwards but seems to agree better with the relation obtained by Grünthal and Wahlström ( and Grünthal et al. (9 for small magnitude events (<.. For events below magnitude the relation of M L vs. M W can be fitted by a linear relation. An attempt to fit a quadratic relation to the data leads to a quadratic term of ±. for most of the stations. Figure 6 7

8 Figure : Left: plot of possible polynomial fits found during the bootstrap process for the dataset of Edwards. Darker colors (excluding the background indicate a higher density of solutions in one area. Right: Same plot showing data (circles and best model (solid line of Edwards (see Figure in addition to the scaling relation of Grünthal and Wahlström ( (dashed line. shows an example of a linear fit for two borhole stations of Basel. Figure 7 shows plots of Wood- Anderson amplitude vs. spectral amplitude for the Martigny sequence of similar earthquakes in. Note that a slope of in these plots corresponds to a M L =.M W relation. Discussion Results from Allmann, Edwards, and Bethmann do not disagree with a quadratic scaling relation such as that proposed by Grünthal and Wahlström ( and the updated scaling relation by Grünthal et al. (9. Grünthal and Wahlström ( and Grünthal et al. (9 obtained a scaling relation of earthquakes from central, northern, and northwestern Europe, including selected Swiss earthquakes. Therefore, all results are first compared with respect to a Grünthal-type quadratic fit. The scatter is generally large and the quadratic term is not well constrained. It should be noted that a linear fit would, for all three authors, result in similar misfits. We emphasize again that any scaling relation obtained in this manner is purely empirically motivated and strictly valid only for the magnitude range in which it is determined. When plotting the data against M W M L (Figure, we notice that the quadratic relation of Grünthal and Wahlström ( does not appear to fit very well to his own data, which casts further doubt on the applicability of a quadratic fit in this case. Leaving the quadratic fits aside and focusing on the data displayed in Figure, we observe two main features: For local magnitudes above, M W estimates of Allmann are lower than estimates of Edwards and Clinton, and generally show less scatter. The magnitude estimates of Bethmann are generally lower than the estimates from others. For M L >, the magnitude estimates of Allmann could be biased by saturation due to the fixed frequency range (. -.6 Hz in which the relative moment Ω is estimated. The frequency range used for the Ω estimation is constrained by the choice of the window length (. s and the 8

9 Ω vs. max. WA amplitude at station JOHAN Ω vs. max. WA amplitude at station MATTE log max(a WA log max(a WA slope:.+/.6 equivalent to M L =.M w slope:.+/.6 equivalent to M L =.M w 6 log Ω 6 log Ω Figure 6: Comparison of fits for a sequence of similar earthquakes in the Basel sequence at two different stations. A linear fit is best suited to describe the relation between Wood-Anderson Amplitude vs. Ω (resp. M L vs. M W in the lower magnitude range. spectral estimation method. Note that the STN ratio also decreases rapidly below Hz. To test the influence of saturation on the estimates, Allmann compared relative magnitude estimates between two different frequency bands ( Hz vs Hz and found no differences in the estimated magnitudes, which suggests that saturation is not a dominant effect. However, this needs to be explored further by a comparison with longer window lengths to allow an estimation at even longer periods for the larger events. Edwards uses a variable window length of up to s for the largest events, and therefore these data are unlikely to be affected by saturation at higher magnitudes. The generally lower M W estimates of Bethmann may be explained by the difference in spectral fit and accounting for impedance contrasts near the surface. While Edwards assumes a homogeneous half space where the source velocity enters with an exponent of (equation, Bethmann uses an exponent of.. The surface velocity included by Bethmann is determined by a quarter wavelength method (Joyner et. al, 98 and uses an exponent of.. The different assumptions made by Bethmann and Edwards can lead to differences of up to. magnitude units for surface stations sitting on sediment. The difference in seismogram-length used for spectral analysis differs significantly between the methods of Edwards ( s to s Allmann (. s and Bethmann (pulse widths in the order of.s to.s. This may also contribute to the observed offset in absolute values between Bethmann and Edwards. Bethmann assumes that the first arriving pulse contains all the energy of event coming from the source as theoretically expected. By cutting the first pulse only he intends to get a high resolution estimate of M W. Edwards and Allmann also include parts of the coda to account for scattered energy along the ray path. Further tests, such as a detailed synthetic test including reflected phases and scatter, are needed to better understand the effect of different window length on M W estimates. The magnitude estimates of Allmann and Edwards are in good agreement from M L = to.. The results of Edwards and/or Clinton are preferred above M L =. due to the unresolved possible 9

10 Ω vs. max. WA amplitude at station SALAN Ω vs. max. WA amplitude at station EMV log max(a WA log max(a WA slope:.+/. slope:.6+/.9 6 log Ω 6 log Ω Ω vs. max. WA amplitude at station AIGLE Ω vs. max. WA amplitude at station DIX log max(a WA log max(a WA slope:.+/.8 slope:.+/.8 6 log Ω 6 log Ω Ω vs. max. WA amplitude at station LKBD AIGLE DIX EMV LKBD SALAN Ω vs. max. Amplitude log max(a WA log max(a WA slope: +/.9 6 log Ω 6 log(ω Figure 7: Plots of Wood-Anderson amplitude vs. spectral amplitude at low frequency for the sequence of similar earthquakes in Martigny.

11 a b c Mw(JC vs. Mw(JB: number of events = 8 Ml(SED vs. Mw(JB: number of events = Ml(SED vs. Mw(JC: number of events = 8 Mw(JC =.98 Mw(JB.6 mean(mw(jc Mw(JB =.68+/.77 median(mw(jc Mw(JB =.... Mw(JC Ml(SED Ml(SED... median(ml Mw =. mean(ml Mw =.+/. Ml =. Mw.8 median(ml Mw =. mean(ml Mw =.+/.7 Ml =. Mw... Mw(JB.. Mw(JB.. Mw(JC Figure 8: a Comparison of moment magnitude estimates from Clinton and Braunmiller. b Local magnitude versus M W from Braunmiller. c Local magnitude versus M W from Clinton. Note the constant -. mean offset between M L and Clinton s M W estimates. bias due to saturation of Allmann s result in this magnitude range. Fitting a quadratic relation to the data of Allmann and Edwards only for M L < yields very similar fitting parameter as before, indicating that the high magnitude events, whether biased or not, have only little influence on the quadratic scaling relation. The scaling relations of Edwards and Allmann are within its respective error margins and therefore in good agreement. The quadratic coefficients of Grünthal fit the data of Edwards and Clinton reasonably well at magnitudes above M L =, but appear to be inapplicable for lower magnitudes, even though their data also includes some events of this catalogue. 6 New scaling relation for Switzerland Previous M W vs. M L scaling relations for Switzerland were derived based on the M W estimates of Clinton and Braunmiller for events larger than about magnitude. In this magnitude range, a : scaling between M L and M W was found, consistent with theoretical considerations by Deichmann (6. However, both scaling relations showed a constant offset between M L and M W (Figure 8. The : scaling with an offset of -. is currently used by the SED for standard estimation of M W. The data of Figure suggest that a linear : scaling between M L and M W is incompatible with the observation for magnitudes below M L =. However, an obtained quadratic relation is only valid for the magnitude range in which it is determined and cannot be extended to smaller or larger magnitudes. We therefore propose a new scaling relationship with different slopes in different magnitude ranges (Figure 9 where we fit a quadratic relation in between two linear scales at smaller and larger magnitudes. Above M L =, we enforce a linear : relationship with a constant offset of -., as previously estimated by Clinton (see Figure 8 c. The orthogonal quadratic fit in the center is estimated from the data of Allmann and Edwards below magnitude with a constraint imposed on the slope and intercept at M L = to obtain a smooth transition. The data of Bethmann (Figure 6 and Figure 7 suggest a consistent slope of /, albeit with different offsets for different clusters. Therefore, we enforce a / slope extending to lower magnitudes starting at the / gradient of the quadratic fit at M L =.. Noting that the / slope below M L = does not fit the data of Edwards and Allmann very well we also propose to fit the data

12 .. Allmann: 986 events Edwards: 677 events Clinton: 6 events Braunmiller: events Bethmann (Basel: 9 events Bethmann: 6 events M W - M L M W =.9 M L +.98 M W =.667 M L +.8 M W =.7+. M L +.8 M L M W = M L -. 6 M L Figure 9: Difference between M L and M W versus M L of different datasets from various authors (different shaped and colored symbols. The black solid line shows the combined scaling relation from three different segments as follows: M L > follows a linear : relation with an offset of -.; between M L < the relation is quadratic; M L < follows a linear relation with slope determined from the quadratic at M L =. The black dashed line shows a linear relation below M L =. with a slope of /. below M L = using a linear scale with the slope determined by the gradient of the quadratics at M L = (Figure 9. Figure shows the three-step fit to the data whereas Bethmann s data were shifted up by the mean difference of.8 between his and Edwards dataset for visual comparison. All datasets follow the proposed relation aside from the discussed uncertainties. 7 Error analysis An estimate of the error in moment magnitude (σ MW is obtained through a combination of error propagation from the uncertainty of the regression coefficients and an assumed uncertainty in M L along with residual analysis of the ML SED - M W dataset. First, we apply the bootstrap method with resamples of the original data distribution and compute a regression for each realization. We again enforce a linear : relationship with a constant offset above M L =. However, for each resample the offset can vary randomly around -. with a standard deviation of.. Below M L = we use a linear scale with the slope determined by the gradient of the quadratics at M L =. We then compute the covariance matrix of the regression coefficients from the bootstrap resamples for the linear fit at M L < and the quadratic fit at < M L, respectively.

13 .. Allmann: 986 events Edwards: 677 events Clinton: 6 events Braunmiller: events Bethmann (Basel: 9 events Bethmann: 6 events M W - M L M W =.9 M L +.98 M W =.667 M L +.8 M W =.7+. M L +.8 M L M W = M L -. 6 M L Figure : Same as Figure 9. The dataset of Bethmann has been shifted up by.8 for a better visual comparison. We obtain the following covariance matrix for the linear fit (M W = a + bm L for M L < : ( σ a σ ab σ ab σ b = (.7e.96e.96e.7788e ( For the quadratic fit (M W = a + bm L + cml we obtain: σ a σ ab σ ac σ ab σ b σ bc σ ac σ bc σ c =.6766e.7e.88e.7e.96e.9797e.88e.9797e.998e A Gaussian propagation of errors is then applied. For the quadratic term this becomes: ( σ M W = σ a ( MW a + σ ab ( MW a + σ b ( MW b ( MW b + σ c + σ ac ( MW a ( MW c ( MW c + σ bc ( MW b ( MW c ( + σm MW L, M L ( where σ a, σ b, σ c, σ ab, σ ab, and σ ac are the coefficients of the covariance matrix. σ M L is the uncertainty in M L.

14 Table : Error estimates of M W from error propagation. σ ML = σ ML =. σ ML =. M L < : ±. ±.69 ±. < M L < : ±. ±.79 ±. M L > : ±. ±. ±. Table shows the mean σ MW for the three segments, assuming different uncertainties in M L. Due to the enforcement of a certain behaviour in the scaling relation below M L = and above M L = the errors derived from the combined bootstrap and error propagation do not completely reflect the actual scatter of the data. Therefore the errors given in Table will underestimate the actual uncertainty in M W. The additional uncertainty in M W can be ascribed to the simplification of our model relative to nature. This is known as epistemic uncertainty and is defined as uncertaintly due to limited data and knowledge. In seismic haszard analysis this is typically tackled using several different models in a logic-tree approach. In this case however, we can simply estimate the contribuion of episemic uncertainty through residual analysis. Dividing up between the different scaling regimes (magnitude ranges, we observe that the standard deviation in M W relative to our model is given by: ±.9 for M W < ±. for < M W < ±.7 for M W > By definition, the error in the M L - M W scaling relation should encapuslate 6% (or one standard deviation of the observed data. Therfore we can define: σ T = σ M W + σ e (6 Where σ T is the total uncertainty in M W from the scaling relation, σ MW is the error computed using the bootstrap resampling method and Gaussian error propagation, and σ e is the epistemic uncertainty. In the case of the SED M L and M W data we can then compute the σ e that is required to give σ T that encapsulates 6% of the data (e.g.,.9 for M W < relative to the predictive equation. This is given by: ±. for M W < ±. for < M W < ±. for M W > The increase of σ e with M W can be explained by the fact that for smaller earthqukes M L is insensitive to stress drop, whereas for larger earthquakes stress drop has an effect on the M L value for a particular M W. As this is not included in our model, it is attributed to epitemic uncertainty. Assuming that this epistemic uncertainty is defined simpy by the simplification of our model relative

15 Table : Final M W Error values for various institutions. SED LED ZAMG BCSF CSI Bolletino ISIDE LDG Bolletino (M L (M d ML <= : < ML <= : < ML <= : ML > : to nature, this uncertainty can be applied to other data or datasets. Using the σ MW from Table and the σ e given above along with Equation 6 the total uncertainty in M W was computed for the converted M W from the M L of several institutions. 8 Conclusion The proposed three-step scaling relation (Figure fits all data reasonably well, while avoiding a bias at the smaller and larger magnitudes. Results from previous investigations, namely Grünthal and Wahlström (; Grünthal et al. (9 along with independently derived M W from Clinton are taken into account to the extent possible. We emphasize again that the obtained relation is of an empirical nature. This means in particular that we do not attempt to explain the apparent break in symmetry around M L =. Further work needs to investigate whether the change in scaling is due to residual bias in the M W estimates, due to the method of estimating M L, or indicative of a change in the physical source properties. References Abercrombie, R. (99, Earthquake source scaling relationships from to M L using seismograms recorded at. km depth, J. Geophys. Res.,,,,6. Aki, K., and P. Richards (, Quantitative seismology nd edition, University Science Books. Allmann, B., and P. Shearer (7, Spatial and temporal stress drop variations in small earthquakes near Parkfield California, J. Geophys. Res., (B, doi:.9/6jb9. Braunmiller, J., N. Deichmann, D. Giardini, S. Wiemer, and the SED Magnitude Working Group (, Homogeneous Moment-Magnitude Calibration in Switzerland, Bull. Seismol. Soc. Am., 9(, 8 7. Brune, J. (97, Tectonic Stress and Spectra of Seismic Shear Waves from Earthquakes, J. Geophys. Res., 7, Brune, J. (97, Correction: Tectonic Stress and Spectra of Seismic Shear Waves from Earthquakes, J. Geophys. Res., 76,.

16 Deichmann, N. (6, Local Magnitude, a Moment Revisited, Bull. Seismol. Soc. Am., 96(A, Edwards, B., and A. Rietbrock (9, A comparative study on attenuation and source-scaling relations in the Kanto, Tokai, and Chubu regions of Japan,using data from Hi-Net and Kik-Net, Bull. Seismol. Soc. Am., 99, doi:.78/89. Edwards, B., A. Rietbrock, J. J. Bommer,, and B. Baptie (8, The acquisition of source, path and site effects from micro-earthquake recordings using Q tomography: Application to the UK, Bull. Seismol. Soc. Am., 98, 9 9. Edwards, B., A. Allmann, D. Faeh, and J. Clinton (, Automatic computation of moment magnitudes for small earthquakes and the scaling of local to moment magnitude., Geophys. J. Int., in review. Grünthal, G., and R. Wahlström (, An M w based earthquake catalogue for central, northern and northwestern Europe using a hierarchy of magnitude conversions, J. Seismol., 7, 7. Grünthal, G., R. Wahlström, and D. Stromeyer (9, The unified catalogue of earthquakes in central, northern, and northwestern Europe (CENEC updated and expanded to the last millennium, J. Seismol.,, 7. Gutenberg, B., and C. Richter (9, Frequency of earthquakes in California, Bull. Seismol. Soc. Am.,, Hanks, T., and H. Kanamori (979, A moment magnitude scale, J. Geophys Res., (8, 8. Joyner, W. B., R. E. Warrick, and T. E. Fumal (98, The effect of quaternary alluvium on strong ground motion in the Coyote Lake, California, earthquake of 979, Bull Seismol. Soc. Am., 7(, 9. Lees, J. M., and J. Park (99, Multiple-taper spectral analysis: a stand-alone C subroutine, Computers and Geosciences,, Park, J., C. Lindberg, and F. Vernon (987, Multitaper Spectral Analysis of High Frequency Seismograms, J. Geophys. Res., 9,,67,68. Shearer, P., G. Prieto, and E. Hauksson (6, Comprehensive analysis of earthquake source spectra in southern California, J. Geophys. Res., (B6, doi:.9/jb979. 6