Unbiased Estimation of Moment Magnitude from Body- and Surface-Wave Magnitudes

Abstract

Abstract For regression of variables having measurement errors, general orthogonal regression (GOR) is the most appropriate statistical procedure that yields a linear relation between the true values of the dependent ( y t ) and independent ( x t ) variables. However, the GOR procedure to obtain unbiased estimate of the dependent variable for a given error‐affected value of the predictor variable is not well addressed in the literature. In the conventional GOR approach, the error‐affected value of the predictor variable is substituted as such in the GOR relation, yielding biased estimates of y t . In another approach, the orthogonal projections of the given points on the GOR line are used to first estimate x t and then y t . In this study, a procedure making use of true points on the GOR line is proposed to obtain improved estimates of y t . The proposed GOR procedure is applied to the magnitude conversion problem between m b to M w and M s to M w , using real data set. The absolute average differences of the estimates obtained and their standard deviations are compared, indicating that the proposed GOR procedure provides improved estimates of the dependent variable ( y t ) compared with the conventional GOR approach. The improved unified magnitudes obtained using the proposed GOR procedure will result in more realistic seismic hazard for a given catalog and seismotectonic environment.