Comparative Analysis of Regression Methods Used for Seismic Magnitude Conversions

The magnitude conversion problem, usually concerning the estimation of proxies of moment magnitude Mw from local (ML) and teleseismic (Ms and mb) 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 Mw. In the last two years, researchers proposed a modified GOR (MGOR) procedure that claimed to achieve unbiased estimates of Mw using an estimator of the true value of the independent variable (xt) in place of the observed value (Xobs) in regressed equations. In this article, we demonstrate both theoretically and experimentally that (a) xt coincides on average with Xobs, (b) the proposed procedure to calculate xt is biased by the choice of using the OLS regression method to fit the linear relationship between xt and Xobs, 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.