Geophysical Journal International

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1 Geophysical Journal International Geophys. J. Int. (2012) 190, doi: /j X x A comparison of moment magnitude estimates for the European Mediterranean and Italian regions Paolo Gasperini, 1 Barbara Lolli, 2 Gianfranco Vannucci 2 and Enzo Boschi 1 1 Dipartimento di Fisica, Università di Bologna, Viale Berti-Pichat 8, Bologna, Italy. paolo.gasperini@unibo.it 2 Istituto Nazionale di Geofisica e Vulcanologia, Sezione di Bologna, Via Donato Creti 12, Bologna, Italy Accepted 2012 June 12. Received 2012 June 12; in original form 2012 February 13 INTRODUCTION The size of an earthquake is physically defined by the scalar seismic moment M 0 but, for historical reasons, it is most commonly expressed in terms of the moment magnitude M w. The conversion formula from M 0 to M w, as proposed by Hanks & Kanamori (1979), is a mere definition rather than an empirical relationship. Its most recent formulation according to international standards is M w = 2 ( log10 M ), (1) 3 where M 0 is measured in Nm. SUMMARY With the goal of constructing a homogeneous data set of moment magnitudes (M w )tobeused for seismic hazard assessment, we compared M w estimates from moment tensor catalogues available online. We found an apparent scaling disagreement between M w estimates from the National Earthquake Information Center (NEIC) of the US Geological Survey and from the Global Centroid Moment Tensor (GCMT) project. We suspect that this is the effect of an underestimation of M w > 7.0 (M 0 > Nm) computed by NEIC owing to the limitations of their computational approach. We also found an apparent scaling disagreement between GCMT and two regional moment tensor catalogues provided by the Eidgenössische Technische Hochschule Zürich (ETHZ) and by the European Mediterranean Regional Centroid Moment Tensor (RCMT) project of the Italian Istituto Nazionale di Geofisica e Vulcanologia (INGV). This is probably the effect of the overestimation of M w < 5.5 (M 0 < Nm), up to year 2002, and of M w < 5.0 (M 0 < Nm), since year 2003, owing to the physical limitations of the standard CMT inversion method used by GCMT for the earthquakes of relatively low magnitude. If the discrepant data are excluded from the comparisons, the scaling disagreements become insignificant in all cases. We observed instead small absolute offsets ( 0.1 units) for NEIC and ETHZ catalogues with respect to GCMT whereas there is an almost perfect correspondence between RCMT and GCMT. Finally, we found a clear underestimation of about 0.2 units of M w magnitudes computed at the INGV using the time-domain moment tensor (TDMT) method with respect to those reported by GCMT and RCMT. According to our results, we suggest appropriate offset corrections to be applied to M w estimates from NEIC, ETHZ and TDMT catalogues before merging their data with GCMT and RCMT catalogues. We suggest as well to discard the probably discrepant data from NEIC and GCMT if other M w estimates from different sources are available for the same earthquakes. We also estimate approximately the average uncertainty of individual M w estimates to be about 0.07 magnitude units for the GCMT, NEIC, RCMT and ETHZ catalogues and about 0.13 for the TDMT catalogue. Key words: Earthquake source observations; Statistical seismology. The most reliable M w estimates come from moment tensor inversions of broad-band waveforms that are feasible only above certain magnitude thresholds, depending on the inversion method. Generally, there is no arbitrary calibration involved in estimating the seismic moment M 0 but different assumptions about properties of the Earth, and different tools used to simulate realistic wave propagation by different computational approaches may result in offsets and/or scaling disagreements between different data sources that need to be tested before various data sources are merged in a unique M w data set. We compare here M w estimates available for the European Mediterranean and the Italian areas to build a GJI Seismology C 2012 The Authors 1733

2 1734 P. Gasperini et al. homogeneous M w catalogue, spanning over a range of magnitudes as wider as possible, to be used in seismic hazard assessment. In particular, this work contributes to the European Union research project SHARE (Seismic Hazard Harmonization in Europe), within the ambit of Task 3.1 European Earthquake Database and to the internal project of the Italian Istituto Nazionale di Geofisica e Vulcanologia (INGV), for the revision of the Italian seismic catalogue used in Italian seismic hazard analyses: the Catalogo Parametrico dei Terremoti Italiani (CPTI; CPTI Working Group 1999, 2004, 2011). At the global scale, the most complete collection of M w magnitudes is the catalogue of the Global Centroid Moment Tensor (GCMT) project (formerly known as Harvard CMT ). It spans from 1976 to few months before the present (Dziewonski et al. 1981; Ekström et al. 2005) and is available online at The moment tensors are computed at GCMT by inverting mantle waves in the band s and body waves in the band s recorded at teleseismic distances and, starting from 2003, even teleseismic surface waves with period s (Hjörleifsdóttir & Ekström 2010). For small events for which the amplitudes of mantle waves are below the noise level for most stations, the solutions before 2003 were constrained only by the body waves. Hjörleifsdóttir & Ekström (2010) demonstrated that in this case the GCMT might overestimate seismic moments where the crust is thicker than average (like in Tibet) and underestimate them where the crust is thinner than average (like in ocean ridges). Starting from 1980, the National Earthquake Information Center (NEIC) of the US Geological Survey (USGS) also computes routinely global moment tensor solutions (Sipkin 1982, 1994) of most significant earthquakes (M w > 5.0) that are provided at Sipkin (1994) noted however that the low-pass filter employed by the USGS inversion technique can lead to a bias in the estimated scalar moments larger than approximately Nm (M w > 7.27) and actually Kagan (2002) found the average difference between GCMT and NEIC catalogues, from 1982 to 1999, being close to zero for 5.4 M w 7.0 but of the order of units for M w > 7.0. NEIC web pages on significant earthquakes ( also provide solutions computed according to a different centroid moment tensor (CMT) technique, since about September 2007, as well as WPhase Moment tensor solutions (Kanamori & Rivera 2008), since about the beginning of Unfortunately, such solutions are not reported in the revised catalogue then we did not consider them in this paper. The use of long-period body waves recorded at teleseismic distances prevents the general applicability of the standard CMT method to earthquakes of moderate magnitude owing to the low signal-to-noise ratio (Pondrelli et al. 2002). In the practice, the GCMT catalogue includes earthquakes with M w > but is about complete only for M w > , depending on the time and the hypocentre depth (Kagan 2003). A slightly different technique, consisting of the inversion of intermediate period ( s) surface waves recorded at regional distance (Arvidsson & Ekström 1998; Ekström et al. 1998), is used to compute moment tensors also for smaller events with similar reliability to the standard CMT analysis. For the European Mediterranean area, this technique was systematically applied by the INGV that maintains and updates the European Mediterranean Regional Centroid Moment Tensor (RCMT) catalogue at The database provides definitive solutions from 1997 to 2008 as well as quick (but manually reviewed) solutions from 2009 to present and is about complete for M w > (Pondrelli et al. 2011). Pondrelli et al. (1999, 2001, 2004, 2006) computed other RCMT solutions for the Italian area from 1976 to 1996 that are included in the Italy Dataset (Pondrelli et al. 2006) available at (events with codes R, S and T). Since 1999, a regional moment tensor catalogue for a wider European Mediterranean area was also being maintained by the Eidgenössische Technische Hochschule Zürich (ETHZ) at (Braunmiller et al. 2002) but it was discontinued after year ETHZ still provides, at the same website, automatic moment tensor determinations of strong earthquakes of the European Mediterranean area (Bernardi et al. 2004) but the solutions are not reviewed manually. Hence, we did not consider them in our analysis. For the period , Braunmiller et al. (2005) found the average M w difference between ETHZ and GCMT sets being only 0.02 (±0.12) magnitude units with symmetric scatter around 0. They detected as well no systematic variation of the differences with the size of the earthquake. More recently, Konstantinou & Rontogianni (2011) compared the seismic moment estimates made by GCMT with those by RCMT and ETHZ and found that about 20 per cent of events exhibit differences of more than 1.5 times (about 0.2 in magnitude units). The differences were always positive for RCMT (GCMT overestimates) and both positive and negative for ETHZ. In agreement with Hjörleifsdóttir & Ekström (2010), they correlated the positive residuals with large crustal thickness (>40 km) in the source region and the use of only body waves (before 2003) in the GCMT inversions. After the Time-Domain INVerse Code (TD-INVC) became available (Dreger 2003), time-domain moment tensor (TDMT) solutions of earthquakes with M w > are routinely determined by several regional observatories worldwide using the method originally proposed by Dreger & Helmberger (1993). TDMT solutions of Italian earthquakes with M w 3.5, using broad-band (T = 40 s) recordings from the Rete Sismica Nazionale Centralizzata of the INGV, are available at (Scognamiglio et al. 2009). The calibration of M w computed by the TMDT technique, using data of the Southern California Seismic Network (SCSN), has been analysed by Clinton et al. (2006) and showed a general good correlation with a set of GCMT solutions. In summary, we considered the following sources of M w magnitudes. (1) Global CMT catalogue from 1976 to 2010 (GCMT). (2) USGS NEIC MT catalogue from 1980 to 2010 February (NEIC). (3) INGV RCMT catalogue from 1997 to 2008, integrated with the quick RCMT from 2009 to 2010 and with the Italy Dataset (RCMT). (4) ETHZ Regional MT catalogue from 1999 to 2006 (ETHZ). (5) INGV TDMT catalogue from 2006 to 2010 (TDMT). The GCMT catalogue is the most complete among the various sources and, in general, it can be considered the most authoritative for moderate to large earthquakes, hence, we will assume its M w estimates as response variable for calibrating other sources. This does not imply necessarily that estimates by GCMT are the most accurate for all of the earthquakes and particularly for the smallest ones.

3 Comparison of moment magnitude estimates 1735 T E S T I N G T H E C A L I B R AT I O N O F M w D ATA S E T S We compared common M w estimates for all the earthquakes reported by global catalogues (GBL set) as well as for the earthquakes occurred within the European Mediterranean (MED set: Latitude 25N 60N, Longitude 20W 50E) and Italian (ITA set: Latitude 35E 48E, Longitude 6N 20N) areas (Fig. 1). The association of earthquakes between different catalogues was done based on the origin time difference (within 1 min) and the epicentre distance (within 30 km). The thresholds were chosen, after some trials, to avoid estimates of different earthquakes are merged together. For all catalogues, we recomputed M w from the scalar moment M 0 according to eq. (1). For the ETHZ data, such computation gives M w values that are about 0.03 units lower than those reported by the catalogue. This is probably due to the use of a slightly different conversion formula between M 0 and M w at the ETHZ. As the NEIC catalogue gives M 0 (and M w ) with only one decimal (two significant digits), for homogeneity with other sources, we recomputed M 0 from moment tensor components, as the average of the modulus of the two largest eigenvalues of the moment tensor (which components are actually given with two decimals in the MT format of the USGS site), according to standard procedures (see Gasperini & Vannucci 2003 for details). We verified for NEIC as well as for other catalogues that the difference between the reported M 0 values and those computed from moment tensor eigenvalues are negligible (of the order of 0.01 units or less) in almost all cases. For only one earthquake of the NEIC catalogue (1995 September 6 22:48, M 0 = Nm), we found a discrepancy between the M 0 and the moment tensor components. The comparison with the scalar moment of the same earthquake given by the GCMT catalogue (M 0 = dyn cm) indicates that the M 0 (and the M w ) reported by NEIC are correct whereas the tensor components are overestimated of a factor of 10. We also found a discrepancy of one order C 2012 The Authors, GJI, 190, C 2012 RAS Geophysical Journal International of magnitude between the scalar moment and the moment tensor eigenvalues reported in the FM format on the USGS site, for several tens of earthquakes but this does not affect our computations that are based on moment tensor components provided by the USGS MT format. We evaluate the agreement between pairs of catalogues by computing the coefficients of the linear regression between corresponding M w estimates. We determine if their scaling is the same by testing if the slope coefficient of the straight line is not significantly different from 1 (in the statistical sense). If this occurs, there is no need of applying a scaling correction between the two catalogues and then we evaluate if the average difference (offset) between corresponding M w estimates is significantly different from 0. In case the average difference is significant, it can be applied back to the M w data to homogenize the different sources. For computing the straight-line coefficients, the ordinary least square (OLS) method is not appropriate because it assumes that the error of the independent variable is negligible with respect to that of the dependent variable whereas the errors of M w estimates coming from different catalogues are plausibly of the same order of magnitude. When the independent variable is affected by errors, the OLS underestimates the slope coefficient with respect to the true value (Fuller 1987). Moreover, the computed regression line differs from that obtained by inverting the two variables (i.e. the OLS regression is not invertible). Castellaro et al. (2006) demonstrated that the application of empirical relationships computed using OLS may produce systematic magnitude errors in the resulting catalogues as well as an heavy bias in estimates of the slope of the frequency magnitude distributions (Gutenberg & Richter 1944). To compute unbiased regression estimates, they suggested the use of the general orthogonal regression (GOR) method (Fuller 1987) that requires that only the ratio η between the variances of the two variables is known (and not necessarily the variances of the two variables or the variances of each Figure 1. Map of the MED and ITA areas considered in this work and of the epicentres of the earthquakes used for the analysis.

4 1736 P. Gasperini et al. individual observation). Other GOR methods have been proposed by the recent literature on magnitude conversions: the chi-square regression (Stromeyer et al. 2004) and the weighted total least squares (Krystek & Anton 2007; Bethmann et al. 2011). Lolli & Gasperini (2012) showed that they are both equivalent to the GOR when only the ratio η between the variances of the two variables is known. In this work, we adopted the GOR because it does not require numerical minimizations and is also the most widely used in the recent literature (e.g. Deniz & Yucemen 2010; Faenza & Michelini 2010; Das et al. 2011; Gutdeutsch et al. 2011). The uncertainties of M w estimates are usually not reported by moment tensor catalogues. They were estimated to range from 0.05 to 0.15 magnitude units and to be very similar for both the GCMT and NEIC catalogues (Helffrich 1997; Kagan 2002, 2003). In this work,wemaketheaprioriassumption that the variances are equal (η = 1) for all of the catalogues and will show in the following that such assumption is quite appropriate in most cases. On the other hand, Castellaro & Borman (2007) demonstrated that the GOR with η = 1 performs anyhow better than the OLS if the ratio between the uncertainties of the two variables lies within the range from 0.7 to 1.8. They also showed that in case the true variances are not equal, the slope of the regression line computed by the GOR with η = 1 tends to be rotated away from the true slope towards the axis with the larger variance. This means that if the variance of the independent variable (x) is larger than that of the dependent variable (y) then the true slope will be steeper than that computed by the GOR with η = 1 and will be less steep in the opposed case. Ideally, the mean magnitude difference should be 0 and the regression slope coefficient should be 1, but they might differ owing to the different computational approaches adopted by the different catalogues. Deviations might reflect effective calibration biases as well as random fluctuations without statistical meaning. Hence, for each pair of catalogues, we estimated the significance level (s.l.) at which the H 0 hypotheses that the mean difference is 0 and the slope is 1 can be rejected, using the Student s t-test (Press et al. 2003). According to the common statistical practice, we can confidently reject the H 0 hypothesis if the s.l. is lower than 0.01 and tentatively reject such hypothesis if the s.l. is larger than 0.01 but lower than We also tested, using the Student s t-test, if the mean differences and the slope coefficients estimated for the MED and ITA sets represent real peculiar properties of the two areas by comparing them with the corresponding estimates made for independent complementary sets (MED C and ITA C sets) obtained from the GBL set by excluding the earthquakes occurred in the European Mediterranean and Italian areas, respectively. To simplify the reading, we will use in the following common terms in place of rigorous statistical sentences. Hence, when the H 0 hypothesis the slope is equal to 1 ( the average difference is equal to 0 or is equal for two data sets ) can be rejected with s.l. <0.01 we will simply state that the slope is significantly different from 1 ( the average difference is significantly different from 0 or is significantly different for the two data sets ). When 0.01 < s.l. < 0.05, we will say that that the slope is possibly different from 1 and when H 0 cannot be rejected confidently (s.l. > 0.05), we will say that the slope is not significantly different from 1. GCMT versus NEIC We have in all 6401 common estimates (Fig. 2), 171 of which belong to the MED area and 18 to the ITA area. For the GBL set, the mean M w difference between GCMT and NEIC is (NEIC slightly underestimates) and the standard deviation of the mean is According to Student s t-test, we can assert that such difference (albeit very small) is significantly different from 0 (Table 1). For the MED and ITA sets, the average differences are ± and 0.07 ± 0.03, respectively, which also both differ significantly from 0. The comparison of average differences of the MED and ITA sets with those of the corresponding complementary set MED C and ITA C indicates that for the MED set they are significantly different and that for the ITA set they are possibly different from the global value. For the ITA set, the small number of data (18) suggests caution in assuming such difference as really representative of a regional deviation from the global average. The slope of the GOR regression is ± for the GBL set, 1.02 ± 0.02 for the MED set and 1.05 ± 0.09 for the ITA set. The Student s t-test indicates that the slope coefficient is significantly different from 1 at the global scale only but not for the MED and ITA areas. As well, the comparison between the MED and ITA sets with their respective complementary sets does not show significant differences with respect to the global scale. One might argue that the value of the regression slope significantly larger than 1 at the global scale is due to the incorrect assumption of the apriorivariance ratio (η = 1). According to the theory of GOR regression outlined earlier, this implies that the (true) variance of NEIC M w would be lower than that of GCMT M w but this can be excluded quite reasonably based on previous efforts (e.g. Kagan 2003). The different scaling between the two catalogues indicates that their relative calibration depends on magnitude. To better understand this behaviour, we plotted in Fig. 3 the mean M w differences between GCMT and NEIC catalogues (solid line), computed over bins of 0.2 units, as a function of the central M w of the bin. NEIC M w underestimates GCMT M w of about units for M w 5.4, of about units for 5.4 < M w 7.0 and of more than 0.10 units for larger magnitudes (with the only exception of the bin 7.7 ± 0.1). The clear underestimation of large magnitudes by NEIC with respect to GCMT, was already noted by Kagan (2002), and might be related to the computational limitations of the MT inversion method adopted by NEIC cited earlier (Sipkin 1994). As well, we might argue that the larger offsets observed for M w 5.4 with respect to the range 5.4 < M w 7.0 might be due to a poor accuracy of the Global CMT inversion method for earthquakes close or below the limit of full applicability of such technique (approximately corresponding to the completeness threshold of the GCMT catalogue). Actually, if we only consider the earthquakes in the range 5.4 < M w 7.0, the slope coefficient between GCMT and NEIC becomes ± (Table 2) that means that it is not significantly different from 1 anymore. As well, the regression slope is not significantly different from 1 even for the MED and ITA areas and in the MED area the (significant) mean difference between GCMT and NEIC is slightly larger (about 0.05 units) than that computed at the global scale. For the ITA area, the mean difference is similar (about 0.06) and not significantly different that of the GBL set (Table 2). We can argue that it is reasonable to discard NEIC magnitudes if other estimates of M w > 7.0 for the same earthquake are available from GCMT or other catalogues. For smaller earthquakes, the only corrections needed for the GBL area would be a positive shift of about 0.02 units that, however, is substantially negligible (albeit statistically significant). For the MED area, the positive offset correction needed is about 0.05 units. Due to the small number of common M w estimates and the small difference with

5 Comparison of moment magnitude estimates 1737 Figure 2. Data distribution and regression lines for the comparison between GCMT and NEIC M w catalogues for the GBL (top panel) and MED (bottom panel) sets. OLS direct and OLS inverse refer to regression lines computed by the Ordinary Least Squares method using NEIC M w and GCMT M w as (error-free) independent variable, respectively. respect to the MED area, we do not consider the offset for the ITA area being representative of a real deviation from the MED offset. The long time interval covered by the two catalogues (about 30 yr) and the large number of data available allow to analysing the behaviour of the mean difference and of the slope coefficient as a function of time. In Fig. 4, we see that, for 5.4 < M w 7.0, the two catalogues were almost equivalent from 1980 to about 1994 with a slightly negative (NEIC overestimates) mean difference (Fig. 4 top panel) ranging from 0.01 to 0.04 units and an orthogonal

6 1738 P. Gasperini et al. Table 1. Mean differences and GOR coefficients for the GCMT-NEIC data set. Significance levels (s.l.) indicate the probabilities of rejecting the corresponding H 0 hypotheses (within parentheses) when they are true (in boldface when <0.05). Values after ± indicate one-sigma uncertainties. GCMT-NEIC Data set N Mean difference (d) s.l. (d = 0) s.l. (d = d C ) Intercept b Slope a s.l. (a = 1) s.l. (a = a C ) GBL ± < ± ± <0.01 MED ± <0.01 < ± ± MED C ± < ± ± <0.01 ITA ± ± ± ITA C ± < ± ± <0.01 Figure 3. Mean M w differences between GCMT and NEIC, GCMT and ETHZ catalogues, within bins of 0.2 magnitude units as a function of NEIC, GCMT and ETHZ M w. Error bars indicate 95 per cent confidence intervals. regression slope (Fig. 4 middle panel) rather close to 1. Starting from about 1995, the average difference becomes positive (NEIC underestimates) of the order of units and the slope coefficient becomes slightly larger than 1. Finally, starting from about 2007, the positive offset further increases to about units and the slope coefficient becomes in most years significantly higher than 1. We can also note how the abrupt increase of mean difference in 1995 closely corresponds to a strong increase of the number of earthquakes (Fig. 4 bottom panel) per year (from about 100 before to more than 200 after). We are not aware whether some changes of the moment tensor inversion procedure took place in 1995 and in 2007 but the observed discrepancies might deserve investigations by the compilers of the NEIC catalogue. GCMT versus RCMT We have in all 294 common estimates for the whole set (Fig. 5) that in this case is limited to the MED area covered by the RCMT catalogue (51 of them belong to ITA area). At the MED scale, the mean difference between GCMT and RCMT M w is ± (RCMT underestimates), whereas for the Italian area it is ± (Table 3). In both cases, the average difference is significantly different from 0 and for the ITA set it is possibly different from that of the complementary set. The slope of the GOR regression for the MED set (0.936 ± 0.012) is significantly different from 1, whereas for the ITA set (0.94 ± 0.03) it is not. As well, the ITA estimate of slope is not significantly different from that of the complementary set. We can exclude that the significant deviation from 1 of the slope coefficient might be due to the underestimation of the error of the independent variable (RCMT M w ) because the inverse of the slope computed by the OLS regression of RCMT versus GCMT, which corresponds to assuming a negligible error for GCMT with respect to RCMT, is also lower than 1 (see Fig. 5). We can note in Fig. 5 several large outliers at M w < 5.0 confirming the tendency of GCMT to overestimate low magnitudes that we have argued in the case the previous comparison with respect to NEIC. In Fig. 3, we can see how the average difference between GCMT and RCMT (dotted line) is about 0.15 units for magnitudes M w = 4.5 ± 0.1 and decrease monotonically to 0 or slightly less for M w > 5.4. Even in this case, we may argue that GCMT solutions of earthquakes with true M w magnitude close or below the limit of full applicability of the standard CMT technique might be biased high and might be the cause of the observed deviation from the 1:1 scaling for the entire catalogues. Actually, in Fig. 6 bottom panel, we can note how the slope coefficient monotonically increases for increasing minimum M w and becomes not significantly different from 1 for all thresholds larger or equal to 5.0 (Table 4). The mean difference decreases as well from about 0.03 for minimum M w = 4.6 to nearly 0 for minimum M w 5.0 and is not significantly different from 0, for all but one (5.4) threshold larger or equal to 5.0 (Table 4). Following Hjörleifsdóttir &Ekström (2010), which hypothesized that the seismic moment of small events occurred before 2003 are not well constrained due to the inversion made using only the body waves, we computed the average differences for the two time intervals before and after 2003 January 1. In Fig. 7 (top panel), we can see that the decreasing trend of the average difference between GCMT and RCMT as a function of magnitude is present even for the most recent period but the magnitude threshold below which Table 2. AsTable1butfor5.4< M w 7.0. GCMT-NEIC (with 5.4 < M w 7.0) Data set N Mean difference (d) s.l.(d = 0) s.l. (d B = d A ) Intercept b Slope a s.l. (a = 1) s.l. (a B = a A ) GBL ± < ± ± MED ± <0.01 < ± ± MED C ± < ± ± ITA ± ± ± ITA C ± < ± ±

7 Comparison of moment magnitude estimates 1739 We can conclude that within the range of full applicability of the standard CMT inversion technique (M w > ) the M w estimates from the RCMT data set do not significantly differ from those of the GCMT data set and that neither a scaling nor an offset correction are needed to merging them into a unique data set. This also indicates that we can assume the RCMT catalogue as a reference data set to calibrate other catalogues in the European Mediterranean area. Figure 4. Mean M w differences (upper panel), slope coefficients (middle panel) and numbers of earthquakes (bottom panel) per year, for the comparison between GCMT and NEIC catalogues with 5.4 < M w 7.0. Error bars indicate 95 per cent confidence intervals. a significant overestimation is observed is M w < 5.5 before 2003 and M w < 5.0 since This indicates that the use of surface waves actually improved the accuracy of seismic moment estimates by GCMT but that a residual bias is still present at very low magnitudes. GCMT versus ETHZ There are in all 300 common estimates (Fig. 8), 201 of which belong to the MED area and 20 to the ITA area. At the GBL scale, the mean difference is ± (ETHZ overestimates), whereas for the MED and ITA areas it is ± (ETHZ overestimates) and 0.08 ± 0.04 (ETHZ underestimates), respectively (Table 5). For the GBL and MED sets, the mean difference is significantly different from 0, whereas for the ITA it is only possibly different. As well, for the MED set the average difference is not significantly different to that of the complementary set whereas for the ITA set it is significantly different. This might indicate a real peculiar property of the ITA set, although the small number of data suggests caution even in this case. The slopes for the GBL and MED areas (0.932 ± 0.013, ± 0.015) are significantly different from 1 whereas that for the ITA area (0.82 ± 0.08) is not. For both the MED and ITA sets, the slopes are not significantly different from those of their respective complementary sets. For the same reason described earlier in the case of RCMT (even the slope of the inverse OLS regression is lower than 1 in Fig. 8), we can exclude that a substantial underestimation of the slope coefficient might be due to the incorrect assumption of the ratio of the variances of the two variables. As for the RCMT data set, we can note in Fig. 8 some outliers at relatively low magnitudes (M w < 5.5) that might be the effect of an overestimation of GCMT M w. The analysis as a function of the minimum M w (Fig. 6 and Table 6) shows that the mean difference decreases from about 0.02 units, for low thresholds, to about 0.06 for thresholds larger than 5.0 (significantly different from 0 in all cases). As well, the slope coefficient increases monotonically for increasing minimum magnitude and is not significantly different from 1 for thresholds larger than 5.0. Apart from the offset, which in this case is always significant, the general behaviour resembles that found for RCMT, thus confirming the hypothesis that the scaling disagreement might be due to some sort of inadequacy of the inversion method used by GCMT for earthquakes with M w < producing a slight overestimation of such magnitudes. In Fig. 3, we can note how the average differences between GCMT and ETHZ over bins of 0.2 units (dashed line) closely follow the path of those computed between GCMT and RCMT from bins with average M w = 4.5, where the offset is largely positive, to about M w = 5.1 where the offset is close to 0. Starting from the bin with M w = 5.3, the offset of ETHZ becomes negative and decreases down to about 0.08 units for M w > 5.5. Even in this case, Table 3. As Table 1 for the GCMT-RCMT data set. GCMT-RCMT Data set N Mean difference (d) s.l.(d = 0) s.l. (d = d C ) Intercept b Slope a s.l. (a = 1) s.l. (a = a C ) MED ± < ± ± <0.01 ITA ± < ± ± > ITA C ± < ± ± <0.01

8 1740 P. Gasperini et al. Figure 5. Data distribution and regression lines for the comparison between GCMT and RCMT M w catalogues. OLS direct and OLS inverse refer to regression lines computed by the Ordinary Least Squares method using NEIC M w and GCMT M w as (error-free) independent variable, respectively. we computed the average differences as a function of magnitude for the time intervals before and after 2003 January 1 and found (Fig. 7 bottom panel) that similarly to RCMT the magnitude threshold below which the overestimation of GCMT is significant decreases from about M w < 5.5 for the earlier period to about M w < 5.0 for the most recent one. Our results are somehow at odds with Braunmiller et al. (2005) that found a substantial equality of M w estimates from GCMT and ETHZ. This discrepancy can be explained by the shorter time interval considered by such authors (from 1999 to 2001) and by the effect of the possible overestimation of GCMT at low magnitudes that might have masked the overestimation of ETHZ. We can conclude that a scaling correction is not needed and that only an offset correction ranging between 0.05 to 0.08 units has to be applied to ETHZ to make it compatible with GCMT. GCMT versus TDMT We have only 19 common estimates for the MED area and 11 for the ITA area, hence the comparison is scarcely significant. We can note however (Table 7) a clear mean offset of about 0.2 units (TDMT underestimates) that is significantly different from 0 both for the MED and the ITA set and that is not significantly different between the ITA and ITA C sets. Regarding the empirical slopes, the Student s t-test does not reject any of the hypotheses, hence we can assert that they are not different from 1 and that they are not different between the ITA and ITA C sets. As the small number of data suggests caution in assuming such results as representative of the real calibration of TDMT catalogue, we proceeded to a further analysis comparing TMDT with RCMT that includes much more M w estimates in common with TDMT. RCMT versus TMDT The number of common data is now 60 (Fig. 9) and most of them (52) belong the Italian region. We can confirm that the average difference, of the order of 0.20 units, is significantly different from 0 both for the MED and the ITA sets and we cannot exclude that the offsets for the two sets are equal (Table 8). As well, the scaling coefficient between RCMT and TDMT is not significantly different from 1 for both the MED and the ITA sets. In summary, we can conclude that for both the MED and ITA areas a scaling correction is not necessary for merging TDMT with GCMT and RCMT data sets but a positive offset correction of about 0.20 units is certainly required. ESTIMATION OF UNCERTAINTIES FOR INDIVIDUAL MAGNITUDE ESTIMATES FOR THE MED AREA After the application of the offset corrections indicated earlier for NEIC (+0.05 units), ETHZ ( 0.05 units) and TDMT catalogues (+0.20 units), the standard deviations σ m of magnitude difference between GCMT, NEIC, RCMT and ETHZ result of the order of 0.10 units and between TDMT and other catalogues of the order 0.15 units. Being the calibration bias removed by the application of corrections, the variance of the magnitude difference is only the result of random errors σ m1 and σ m2 of individual observations from both catalogues (e.g. Kagan 2003) σ 2 m = σ 2 m 1 + σ 2 m 2. (2) The approximate equality of the standard deviations for GCMT, NEIC, RCMT and ETHZ catalogues suggests that magnitude

9 Comparison of moment magnitude estimates 1741 Figure 6. Mean M w difference (upper panel) and GOR slope coefficient (lower panel) between GCMT and RCMT and ETHZ catalogues as a function of minimum M w magnitude. Error bars indicate 95 per cent confidence intervals. uncertainties for such catalogues are approximately equal (Kagan 2003). Then, from eq. (2), we can write σ m = 2σm 2 (3) from which we can estimate the uncertainty for an individual magnitude measurement as σ m = σ m. (4) 2 For GCMT, NEIC, RCMT and ETHZ catalogues this gives σ m = 0.07 units. We can even compute the uncertainty of individual Figure 7. Mean M w differences between GCMT and GCMT (top panel) and between GCMT and ETHZ (bottom panel) catalogues, within bins of 0.2 magnitude units as a function of NEIC M w, for the time intervals before (dotted) and after (solid) 2003 January 1. Error bars indicate 95 per cent confidence intervals. TDMT M w estimates from eq. (2) as σ TDMT = σ m 2 σ other 2. (5) that gives σ TDMT = CONCLUSIONS We have analysed the relative calibration of M w estimates of moment tensor catalogues available, at the global scale, from the Table 4. Mean differences and GOR coefficients as a function of minimum M w magnitude for the GCMT-RCMT data set. Significance levels (s.l.) indicate the probabilities of rejecting the corresponding H 0 hypotheses (within parentheses) when they are true (in boldface when <0.05). Values after ± indicate one-sigma uncertainties. GCMT-RCMT Minimum M w N Mean difference d s.l. (d = 0) Intercept b Slope a s.l. (a = 1) ± < ± ± < ± < ± ± < ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ±

10 1742 P. Gasperini et al. Figure 8. As Fig. 5 for the comparison between GCMT and ETHZ M w catalogues. GCMT) project and the NEIC of the US Geological Survey, at the European Mediterranean scale, from the ETHZ and the RCMT project of the INGV and at the Italian scale from the TDMT data set of the INGV. We showed that, within the ranges of full applicability of the inversion techniques adopted by various catalogues, the slope coefficients of the GOR between common M w estimates from pairs of catalogues do not significantly differ from 1. Hence, there is no need to apply scaling corrections when such data are merged in a unique data set. The only observed deviations from the 1:1 scaling can be attributed indeed to the physical limitations of the computational methods used by global catalogues NEIC and GCMT within certain magnitude ranges. In particular, the computational approach adopted by NEIC tends to underestimate the magnitudes of earthquakes with M w > 7.0 (M 0 > Nm). Hence, we suggest to discard the data from NEIC when M w estimates for the same earthquake from other catalogues are larger than 7.0. As well, the standard CMT method used by GCMT tends to overestimate the magnitude of earthquakes with M w < Even in this case, we suggest to discard the data from GCMT when M w estimates for the same earthquake from other catalogues are smaller than 5.5 (M 0 < Nm) before 2003 (when only the body waves are used for the moment tensor inversion) and smaller than 5.0 (M 0 < Nm) Table 5. As Table 1 for the GCMT-ETHZ data set. GCMT-ETHZ Data set N Mean difference (d) s.l. (d = 0) s.l. (d = d C ) Intercept b Slope a s.l. (a = 1) s.l. (a = a C ) GBL ± < ± ± <0.01 MED ± > ± ± < MED C ± < ± ± ITA ± ± ± > ITA C ± < ± ± <0.01 Table 6. As Table 4 for the GCMT-ETHZ data set. GCMT-ETHZ Minimum M w N Mean difference d s.l. (d = 0) Intercept b Slope a s.l. (a = 1) ± < ± ± < ± < ± ± < ± < ± ± ± < ± ± ± < ± ± ± < ± ± ± < ± ± ± ± ±

11 Table 7. As Table 1 for the GCMT-TDMT data set. Comparison of moment magnitude estimates 1743 GCMT-TDMT Data set N Mean difference (d) s.l. (d = 0) s.l. (d = d C ) Intercept b Slope a s.l. (a = 1) s.l. (a = a C ) MED ± < ± ± ITA ± <0.01 > ± ± ITA C ± ± ± Figure 9. As Fig. 5 for the comparison between RCMT and TDMT M w catalogues. Table 8. As Table 1 for the RCMT-TDMT data set. RCMT-TDMT Data set N Mean difference (d) s.l. (d = 0) s.l. (d = d C ) Intercept b Slope a s.l. (a = 1) s.l. (a = a C ) MED ± < ± ± ITA ± < ± ± ITA C ± < ± ± since 2003 (when even the surface waves are considered in the computations). For M w > 5.4, we observed a close correspondence between the M w estimates of GCMT and RCMT that allows to consider the latter as a reference set for the European Mediterranean area. We found instead slight but significant mean offsets for NEIC, ETHZ and TDMT catalogues with respect to both GCMT and RCMT. Hence, when merging their data in a unique data set for the MED area we suggest to apply corrections of units to NEIC, of 0.05 units to ETHZ and of units to TDMT. In general, the scarce number of data available prevents a reliable assessment of specific offset corrections for the ITA area. Hence, even for reason of geographical contiguity, we suggest the application to ITA area of the same corrections deduced for the MED area. At the global scale, the correction to be applied to NEIC to make it equivalent to GCMT would be units, although the latter is substantially negligible when compared with the uncertainties of each single M w determination. However, we observed an increase of the positive offset between GCMT and NEIC starting from about year 2007 that would require further investigations. More generally, the offsets we found, which might be due to the different assumptions made about the properties of the Earth, and the different tools used to simulate realistic wave propagation by different catalogues, deserve future investigation into their possible causes. After the applications of the proposed corrections, the standard deviations of magnitude differences between the M w estimates of GCMT, NEIC, RCMT and ETHZ catalogues are of the order of 0.10 units. Based on the error propagation law, this indicates that the uncertainties of individual M w estimates are almost equal for all catalogues. The standard deviations between such catalogues and TDMT are instead of the order of 0.15 units. We can

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