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 (Yt) and independent variable (Xt) based on observed data (Xobs, Yobs). Recent investigations have shown that the conventional GOR procedure for obtaining Yt by substituting Xobs in the GOR relation is incorrect, because the error in Xobs introduces some bias in Yt. One way to reduce this bias in the Yt estimate is to use Xt (instead of Xobs) given by a standard linear regression (SLR) relation between Xt and Xobs. 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 Yt and Xobs to estimate Yt for a given Xobs, thereby eliminating the need to estimate and substitute Xt.
We verify the supremacy of the proposed GOR procedure over the conventional procedure (Xt replaced by Xobs) and the SLR procedure by using observed and synthetic datasets. We observe that the proposed GOR procedure provides an improved estimate of Yt 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 (RXY) and standard error values as compared with the other two approaches. The newly developed GOR procedure with η 1 provides the highest accuracy in Yt 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.