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.