Magnitude conversion problem using general orthogonal regression

Abstract

In this study, a procedure for the application of general orthogonal regression (GOR) towards conversion of different magnitude types is described. Through minimization of the squares of orthogonal residuals, GOR relation is obtained in terms of the abscissas (M x*) of the projected points corresponding to the observed data pairs (M x, obs, M y, obs). In many studies, M x* is replaced by M x, obs in the GOR relation for convenience of obtaining the estimates of a preferred magnitude type for given magnitude values. Such forms of GOR, however, lead to biased estimates of the dependent variable. To represent the GOR relation correctly in terms of M x, obs, a linear relation has been obtained between M x* and M x, obs using given points and the corresponding projected points on the GOR line. Based on events data for the whole globe during the period 1976-2007, GOR relations have been derived for conversion of m b to M w,m b to M s,m b to M e and M s to M w following the proposed procedure and using specific error variance ratio (η) values. The superiority of the GOR relations obtained following the proposed procedure over the commonly used forms has been shown by computing the absolute average difference and standard deviation between the observed and the estimated values using events data not used in the derivation. It is observed that the proposed GOR relations yield better estimates compared to the commonly used GOR forms. This procedure has been further tested for a wide range of η values between 0.1 and 7.0. The procedure proposed in this study can be used for the purpose of catalogue homogenization where GOR relations are applicable for conversion of different magnitude types. © 2012 The Authors Geophysical Journal International © 2012 RAS.