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Authors: Gasperini, P.* 
Lolli, B.* 
Vannucci, G.* 
Title: Body-wave magnitude mb is a good proxy of moment magnitude Mw for small earthquakes
Issue Date: Nov-2013
Series/Report no.: 6/84 (2013)
DOI: 10.1785/0220130105
Keywords: body-wave magnitude
orthogonal regression
moment magnitude
Subject Classification05. General::05.02. Data dissemination::05.02.02. Seismological data 
Abstract: Body-wave magnitude mb is usually considered a poor proxy of moment magnitude Mw because it saturates for moderate and large earthquakes (M w > 5:5–6) and generally shows a poor correlation with Mw. On the other hand, the observed distri- bution of data at the global scale also seems to indicate an in- verse saturation at low magnitudes (M w < 4:5–5:0) in which M w appears to be almost uncorrelated with mb . We show here that the latter is an artifact of the incompleteness of the global M w datasets for M w < 4:5–5:0 and that disappears considering lower Mw estimates available from regional centroid moment tensor (CMT) catalogs and/or using general orthogonal regres- sion methods. In these cases we show that mb well corresponds to M w < 4:5–5:0 and hence can confidently be used for approximating the Mw of small earthquakes. Conversion relations between the band-limited short- period body-wave magnitude mb (Gutenberg and Richter, 1956) and moment magnitude Mw (Hanks and Kanamori, 1979) have been obtained in the past by several authors using ordinary least-squares (OLS) regression methods (e.g., Heaton et al., 1986; Johnston, 1996; Scordilis, 2006). Such a computa- tional approach, however, is inappropriate when the error in the independent variable (predictor) is not negligible compared with that of the dependent variable (response). Castellaro et al. (2006) have shown that the use of OLS in conversion relations produces a bias of the frequency–magnitude distribution law (Gutenberg and Richter, 1944), which can be avoided using the general orthogonal regression (GOR) method described by Fuller (1987). The latter method has been used in numerous studies of this type, both at the global and regional scale (Ristau, 2009; Wang et al., 2009; Deniz and Yucemen, 2010; Das et al., 2011, 2012a; Baruah et al., 2012; Gasperini et al., 2012). Other gen- eral orthogonal regression methods have been proposed in recent literature: the chi-square (CSQ) regression described by Stromeyer et al. (2004), which was used for conversions between magnitudes even by Grünthal and Wahlström (2003), Grünthal et al. (2009), and Gasperini et al. (2013); and the total weighted least-squares (WLS) method (Krystek and Anton, 2007), which was used by Bethmann et al. (2011). Gutdeutsch et al. (2011) showed that, if the ratio between the variances of the dependent and independent variables η is constant, the coefficients computed by the GOR and the CSQ methods have the same formulations. Under the same condi- tion (η const:), Lolli and Gasperini (2012) demonstrated that all three general orthogonal regression methods (CSQ, GOR, and WLS) provide virtually identical regression coeffi- cients and very similar uncertainties. Some recent works (Das et al., 2012b, 2013; Wason et al., 2012) proposed a modification to the GOR method that was intended by the authors to correct an alleged bias due to the use of observed values, affected by errors, in place of the true values (actually unknown) of the independent variable. Unfortu- nately, as argued by Gasperini and Lolli (2013), the new method is based on some incorrect assumptions. In particular, to demonstrate the superiority of their approach with respect to the original one by Fuller (1987), such authors use as goodness-of-fit estimator the simple standard deviation (s.d.) between observed and calculated values, which by definition does not consider the error of the independent variable. For this reason, the new method simply has to be rejected as well as all the regression relations formed thereby.
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