## 4 Discussion
However, this may be due in part to the relatively higher number of earthquakes assigned this magnitude type. Figure 8e shows the M\({}_{\rm L,IDC}\) residuals that appear the most scattered across the magnitude range, resulting in relatively large \(\sigma\). Even with a large \(\sigma\), there are four and three events beyond 2\(\sigma\) for OLS and TLS regressions, respectively. Moreover, the decreasing behavior with M\({}_{\rm L,IDC}\) suggest a linear model for regression may not have been appropriate. As such, the proposed regression relationship herein is not recommended for homogenization purposes but is presented for completeness. Figure 7: Regression residuals for m\({}_{\rm b}\) from (**a**) ISC, (**b**) NEIC, (**c**) BJI, (**d**) MOS, and (**e**) IDC. The shaded regions represent \(\pm 1\sigma\) from OLS regression. Interestingly, the residuals for \(\rm M_{V,JMA}\) and \(\rm M_{D,JMA}\) appear the most linear of all the regressions presented herein as shown in Figure 9. This may be perhaps due to the limited number of earthquake events. However, the reason why there are so few events is that JMA started using \(\rm M_{D,JMA}\) and \(\rm M_{V,JMA}\) after discontinuing the use of \(\rm M_{JMA}\) in 2016. Nonetheless, all the data points fall within 2\(\sigma\) and suggest a linear model to be appropriate to the data. Overall, these findings reinforce the notion that \(\rm M_{W}\) estimates from magnitude type regressions are dependent on regression type. A linear trend model appears to serve the data well with perhaps the exceptions of \(\rm M_{L,KMA}\) and \(\rm M_{L,IDC}\). For magnitude types that exhibited more linear behavior, either OLS or TLS regression would yield similar results. Figure 8: Regression residuals for \(\rm M_{L}\) from (**a**) KMA, (**b**) JMA, (**c**) BJI, (**d**) KEA, and (**e**) IDC. The shaded regions represent \(\pm 1\sigma\) from OLS regression. Figure 9: Regression residuals for (**a**) \(\rm M_{V,JMA}\) and (**b**) \(\rm M_{D,JMA}\). The shaded regions represent \(\pm 1\sigma\) from OLS regression. Differences between OLS and TLS relationships appear when the data show events that seem to be significant outliers, relative to a linear trend. An additional comparison is made between the proposed homogenization regressions herein against regressions from previous studies in Figure 10. As previously mentioned, a study based on global ISC data provided regressions for \(\mathrm{M_{S,ISC}}\) and \(\mathrm{m_{b,ISC}}\)[19]. Another study focused on the continent of Africa also offered homogenization relationships \(\mathrm{M_{S,ISC}}\), \(\mathrm{m_{b,ISC}}\), \(\mathrm{M_{S,N,NEC}}\), and \(\mathrm{m_{b,NEC}}\)[20]. Both studies derived their relationships using orthogonal regression, which is comparable to the TLS approach used here. Figure 10a,b show the residuals to both global and African relationships for \(\mathrm{M_{S,ISC}}\) and \(\mathrm{m_{b,ISC}}\), respectively. The global relationship appears to overestimate \(\mathrm{M_{S,ISC}}\) for the compiled data and shows a linearly increasing trend for the \(\mathrm{m_{b,ISC}}\) residuals, suggesting an inappropriate model. Figure 10c,d show the residuals from the African relationships for \(\mathrm{M_{S,N,NEC}}\) and \(\mathrm{m_{b,NEC}}\), respectively. The residuals from the African model plot slightly above and away from the TLS \(\mathrm{M_{S,NEC}}\) residuals, while the residuals from the African model plot mostly lower and away from the TLS \(\mathrm{m_{b,NEC}}\) residuals. A South Korean study presenting a local magnitude calibration also showed magnitude type relationships between \(\mathrm{M_{W}}\), \(\mathrm{M_{L}}\), and \(\mathrm{M_{L,KMA}}\)[21]. They separated between horizontal and vertical \(\mathrm{M_{L}}\) derivations, but each one showed similar results. The residuals from the horizontal model are presented in Figure 10e. These residuals show significantly more scatter relative to the TLS residuals shown herein and seem to suggest bias, especially at \(\mathrm{M_{L}}<3.6\).