ALBERT

Description: The magnitude conversion problem, usually concerning the estimation of proxies of moment magnitude M w from local ( M L ) and teleseismic ( M s and m b ) magnitude estimates, was addressed by a number of recent articles. In most of them, the general orthogonal regression (GOR) method is employed; in place of the ordinary least squares (OLS), owing to the fact that the errors of local and teleseismic magnitudes are not negligible with respect to those of M w . In the last two years, researchers proposed a modified GOR (MGOR) procedure that claimed to achieve unbiased estimates of M w using an estimator of the true value of the independent variable ( x t ) in place of the observed value ( X obs ) in regressed equations. In this article, we demonstrate both theoretically and experimentally that (a)  x t coincides on average with X obs , (b) the proposed procedure to calculate x t is biased by the choice of using the OLS regression method to fit the linear relationship between x t and X obs , and (c) the claimed better fit of MGOR proxies with observed data is due to the use of goodness-of-fit estimators that neglect the error of the independent variable. In particular, we show the regression method that best minimizes such estimators is the OLS, which assumes that the error of the independent variable is negligible. This, however, does not mean that the OLS is the better approach for computing conversion equations between different magnitudes; because, owing to the presence of errors of the independent variable, it is simply not applicable to magnitude conversion as well as the MGOR procedure.

Description: We analysed a catalogue of Italian earthquakes, covering 55 yr of data from 1960 to 2014 with magnitudes homogeneously converted to M w , to compute the time-dependent relative frequencies with which strong seismic shocks (4.0 ≤ M w 〈 5.0), widely felt by the population, have been followed by main shocks ( M w ≥ 5.0) that threatened the health and the properties of the persons living in the epicentral area. Assuming the stationarity of the seismic release properties, such frequencies are estimates of the probabilities of potentially destructive shocks after the occurrence of future strong shocks. We compared them with the time-independent relative frequencies of random occurrence in terms of the frequency gain that is the ratio between the time-dependent and time-independent relative frequencies. The time-dependent relative frequencies vary from less than 1 per cent to about 20 per cent, depending on the magnitudes of the shocks and the time windows considered (ranging from minutes to years). They remain almost constant for a few hours after the strong shock and then decrease with time logarithmically. Strong earthquakes (with M w ≥ 6.0) mainly occurred within two or three months of the strong shock. The frequency gains vary from about 10 000 for very short time intervals to less than 10 for a time interval of 2 yr. Only about 1/3 of main shocks were preceded by at least a strong shock in the previous day and about 1/2 in the previous month.

Description: We analysed the conversion problem between teleseismic magnitudes ( M s and m b ) provided by the Seismological Bulletin of the International Seismological Centre and moment magnitudes ( M w ) provided by online moment tensor (MT) catalogues using the chi-square general orthogonal regression method (CSQ) that, differently from the ordinary least-square regression method (OLS), accounts for the measurement errors of both the predictor and response variables. To account for the non-linearity of the relationships, we used two types of curvilinear models: (i) the exponential model (EXP), recently proposed by the authors of the Global Catalogue sponsored by the Global Earthquake Model (GEM) Foundation and (ii) a connected bilinear (CBL) model, similar to that proposed by Ekström & Dziewonski, where two different linear trends at low and high magnitudes are connected by an arc of circle that preserves the continuity of the function and of its first derivative at the connecting points. For M s , we found that the regression curves computed for a global data set (GBL) are likely to be biased by the incompleteness of global MT catalogues for M w 〈5.0–5.5. In fact, the GBL curves deviate significantly from a similar regression curve computed for a Euro-Mediterranean data set (MED) integrated with the data provided by two regional MT catalogues including many more events with M w 〈 5.0–5.5. The GLB regression curves overestimate the M w proxies computed from M s up to 0.5 magnitude units. Hence for computing M w proxies at the global scale of M s ≤ 5.5, we suggest to adopt the coefficients obtained from the MED regression. The analysis of the frequency–magnitude relationship of the resulting M w proxy catalogues confirms the validity of this choice as the behaviour of b­ -value as a function of cut-off magnitude of the GBL data set is much more stable using such approach. The incompleteness of M w 's provided from MT global catalogues also affects the m b GBL data set but in this case the use of the CSQ regression method, in place of the OLS, mitigates the bias and then, at low magnitudes, the EXP regression curve computed from the more complete MED data set almost coincides with that computed from the GBL data set. Our results also indicate that the slope at low magnitudes of the M w – M s relationship is substantially consistent with the hypothesized theoretical value of 2/3 for M s 〈 5.0 while the slope of the M w – m b relationship at high magnitudes probably reaches the theoretically expected value of 2 only in the proximity of the upper limit of m b determinations in our data set ( m b  = 7.2).

Description: Using general orthogonal regressions (GORs), we calibrated local magnitudes, estimated in Italy using various methods in different periods of time from 1981 to 2010, with a set of homogeneous moment magnitudes ( M w ). Magnitude uncertainties, necessary for the application of GOR methods, are inferred by a trial-and-error procedure based on a priori information and empirical regression results. We found that local magnitudes determined using real or synthesized Wood–Anderson waveforms ( M L ) scale 1:1 with M w in most cases but in general underestimate M w by about 0.1–0.2 magnitude units. The only significant deviation from the 1:1 scaling concerns the most recent data provided by the online ISIDE bulletin of the Istituto Nazionale di Geofisica e Vulcanologia and is probably due to the use of a distance correction table (–log A 0 ) not fully appropriate for the Italian area. Magnitudes computed from the duration of the seismogram coda ( M D ) do not generally scale 1:1 with M w and are also underestimated. The relevant regression coefficients vary significantly from one data set to another depending on the empirical formulas used by different catalogs and bulletins. The derived regression coefficients are used to build a homogenized catalog in terms of M w that also includes a consistent estimate of uncertainty for all reported magnitudes. The analysis of the frequency–magnitude distribution of the resulting catalog, covering 30 years of data, shows a b -value slightly lower than 1, which is reasonably uniform over the different time intervals and data sets. It also shows a progressive decay of the earthquake rates below the best-fit straight line for M w 〉4.5 that might reflect a magnitude distribution truncated or tapered to relatively small maximum magnitudes for some Italian seismic zones with low activity. This behavior also seems to exclude a characteristic earthquake recurrence mechanism for Italy. Online Material: Description of data sets and details of regression and frequency–magnitude distribution analyses.

Description: The argument proposed by Wason et al. that the conversion of magnitudes from a scale (e.g. M s or mb) to another (e.g. M w ), using the coefficients computed by the general orthogonal regression method (Fuller) is biased if the observed values of the predictor (independent) variable are used in the equation as well as the methodology they suggest to estimate the supposedly true values of the predictor variable are wrong for a number of theoretical and empirical reasons. Hence, we advise against the use of such methodology for magnitude conversions.

Description: Using general orthogonal regressions (GORs), we calibrated local magnitudes, estimated in Italy using various methods in different periods of time from 1981 to 2010, with a set of homogeneous moment magnitudes (M (sub w) ). Magnitude uncertainties, necessary for the application of GOR methods, are inferred by a trial-and-error procedure based on a priori information and empirical regression results. We found that local magnitudes determined using real or synthesized Wood-Anderson waveforms (M (sub L) ) scale 1:1 with M (sub w) in most cases but in general underestimate M (sub w) by about 0.1-0.2 magnitude units. The only significant deviation from the 1:1 scaling concerns the most recent data provided by the online ISIDE bulletin of the Istituto Nazionale di Geofisica e Vulcanologia and is probably due to the use of a distance correction table (-logA (sub 0) ) not fully appropriate for the Italian area. Magnitudes computed from the duration of the seismogram coda (M (sub D) ) do not generally scale 1:1 with M (sub w) and are also underestimated. The relevant regression coefficients vary significantly from one data set to another depending on the empirical formulas used by different catalogs and bulletins. The derived regression coefficients are used to build a homogenized catalog in terms of M (sub w) that also includes a consistent estimate of uncertainty for all reported magnitudes. The analysis of the frequency-magnitude distribution of the resulting catalog, covering 30 years of data, shows a b-value slightly lower than 1, which is reasonably uniform over the different time intervals and data sets. It also shows a progressive decay of the earthquake rates below the best-fit straight line for M (sub w) 〉4.5 that might reflect a magnitude distribution truncated or tapered to relatively small maximum magnitudes for some Italian seismic zones with low activity. This behavior also seems to exclude a characteristic earthquake recurrence mechanism for Italy.