Empirical conversion between teleseismic magnitudes (mb and Ms) and moment magnitude (Mw) at the Global, Euro-Mediterranean and Italian scale

We analysed the conversion problem between teleseismic magnitudes (Ms and mb) provided by the Seismological Bulletin of the International Seismological Centre and moment magni- tudes (Mw) provided by online moment tensor (MT) catalogues using the chi-square general orthogonal regression method (CSQ) that, differently from the ordinary least-square regres- sion 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 curvilin- ear 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 Ekstro ̈m & 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 Ms, 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 Mw <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 Mw < 5.0–5.5. The GLB regression curves overestimate the Mw proxies computed from Ms up to 0.5 magnitude units. Hence for computing Mw proxies at the global scale of Ms ≤ 5.5, we suggest to adopt the coefficients obtained from the MED regression. The analysis of the frequency–magnitude relationship of the resulting Mw 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 Mw’s provided from MT global catalogues also affects the mb 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 Mw–Ms relationship is substantially consistent with the hypothesized theoretical value of 2/3 for Ms < 5.0 while the slope of the Mw–mb relationship at high magnitudes probably reaches the theoretically expected value of 2 only in the proximity of the upper limit of mb determinations in our data set (mb = 7.2).