The Role of "Equation Error" in Empirical Regressions for Seismic Magnitude Conversions

In the last two decades, empirical regressions for the conversions of traditional magnitudes (e.g., M-L, M-s, and mb) to moment magnitude M-w were mostly computed using error-invariable (EIV) method, also known as general orthogonal regression, and not the ordinary least-squares (OLS) method because the uncertainties of traditional magnitudes are not negligible when compared to that of M-w. However, a few recent articles criticized such approaches based on the hypothesis that an error in the equation, that is, the epistemic uncertainty associated with the form of the regression equation, is present, and this is better accounted for by the OLS rather than by the EIV. In this work, we first discuss the problem theoretically and then make some analysis on real datasets. We find that the regression coefficients are strongly influenced by the assumption made about the sizes of the errors of traditional magnitudes. In particular, the role of equation error may be significant if the assumed errors of the traditional magnitudes are small ( <= 0.1 magnitude units) or may even be almost negligible if the errors are only slightly larger ( > 0.12-0.15). In general, the EIV method has always to be preferred with respect to OLS, but it might require a small correction for equation error. These findings have significant implications for statistical forecasting and seismic hazard assessment.