General Orthogonal Regression Relations between Body-Wave and Moment Magnitudes

Abstract

For determining the size of earthquakes occurring in different environs, several magnitude scales are in use. One or more magnitude scales may be appropriate for estimating the size of an earthquake; therefore, earthquake catalogs invariably report different magnitude types. Toward preparation of a homogenized earthquake catalog, an important input parameter for seismic hazard estimation and other seismological applications, it is generally required to express the different magnitude types into a preferred magnitude type. Most studies prefer moment magnitude ( M w), as M w does not saturate for higher magnitude earthquakes unlike other magnitude scales, and also, it is derived directly from seismic moment ( M ), a basic physical property of the source. The process of estimating earthquake magnitude using different magnitude scales is subject to measurement errors. Standard linear regression (SR) assumes that the independent variable is either error free or the order of its error is very small compared with the measurement error of the dependent variable. Hence, use of SR is not appropriate for magnitude conversion as both magnitude estimates contain errors. General orthogonal regression (GOR) relation takes into account the errors on both the magnitude types (Fuller, 1987; Castellaro et al. , 2006; Thingbaijam et al. , 2008; Ristau, 2009). GOR is obtained using minimization of the squares of the orthogonal distances to the best‐fit line, whereas SR is derived by minimizing the squares of the vertical offsets. GOR relations between different magnitude types have been reported in several studies (Castellaro et al. , 2006; Ristau, 2009; Thingbaijam et al. , 2009; Das et al. , 2011). As the derivation of GOR is based on the orthogonal projection of observed magnitude data pairs on the GOR line, the calculation of estimates of a preferred magnitude type using these GOR relations in their existing form is liable …