Earthquake Magnitude Conversion Problem

General orthogonal regression (GOR) is a superior regression procedure for conversion of different magnitude types into preferred ones. It yields a linear relation between dependent (Y-t) and independent variable (X-t) based on observed data (X-obs, Y-obs). Recent investigations have shown that the conventional GOR procedure for obtaining Y-t by substituting X-obs in the GOR relation is incorrect, because the error in X-obs introduces some bias in Y-t. One way to reduce this bias in the Y-t estimate is to use X-t (instead of X-obs) given by a standard linear regression (SLR) relation between X-t and X-obs. However, this approach does not have ease of application and does not permit statistical verification. Therefore, we propose a modified procedure in which the derived GOR relationship is used to develop a SLR relation directly between Y-t and X-obs to estimate Y, for a given X-obs, thereby eliminating the need to estimate and substitute X-t. We verify the supremacy of the proposed GOR procedure over the conventional procedure (X-t replaced by X-obs) and the SLR procedure by using observed and synthetic datasets. We observe that the proposed GOR procedure provides an improved estimate of Y-t compared with both the conventional GOR and SLR. The proposed GOR provides lower errors in slope and intercept compared with SLR and yields improved correlation coefficient (R-xy) and standard error values as compared with the other two approaches. The newly developed GOR procedure with eta = 1 provides the highest accuracy in Y-t estimates as compared with the SLR and conventional GOR approaches. Our analysis also concludes that incorrect application of regression procedure can introduce a bias in the Gutenberg-Richter parameter b in the 5%-42% range. Therefore, we recommend the use of the proposed GOR to develop regression relations for earthquake magnitude conversions.