## 4 Discussion  
In terms of performance, residuals from \(\text{M}_{\text{S}}\) regressions are plotted in Figure 6. These plots show the OLS and TLS residual biases for each regression are essentially zero. Additionally, TLS residuals also show less variability relative to OLS residuals. The regression residuals for \(\text{M}_{\text{S,IDC}}\) appear more scattered especially at \(\text{M}_{\text{S,IDC}}\) < 4, with three events beyond 2\(\sigma\) for both OLS and TLS regressions. Even so, these plots suggest a linear model can be appropriate for \(\text{M}_{\text{S}}\) regressions. \begin{table}
\begin{tabular}{c c c c} \hline
**Priority** & **Magnitude Type** & \(\sigma_{\text{OLS}}\) & **Magnitude Type** & \(\sigma_{\text{TLS}}\) \\ \hline
1 & \(\text{M}_{\text{V,JMA}}\) & 0.075 & \(\text{M}_{\text{V,JMA}}\) & 0.062 \\
2 & \(\text{M}_{\text{D,JMA}}\) & 0.109 & \(\text{M}_{\text{D,JMA}}\) & 0.080 \\
3 & \(\text{M}_{\text{S,MOS}}\) & 0.149 & \(\text{M}_{\text{S,MOS}}\) & 0.122 \\
4 & \(\text{M}_{\text{S,BII}}\) & 0.158 & \(\text{M}_{\text{S,BII}}\) & 0.122 \\
5 & \(\text{M}_{\text{S,ISC}}\) & 0.184 & \(\text{M}_{\text{JMA}}\) & 0.138 \\
6 & \(\text{M}_{\text{JMA}}\) & 0.185 & \(\text{M}_{\text{L,KEA}}\) & 0.139 \\
7 & \(\text{M}_{\text{L,KEA}}\) & 0.188 & \(\text{M}_{\text{L,BII}}\) & 0.143 \\
8 & \(\text{M}_{\text{S,IDC}}\) & 0.198 & \(\text{m}_{\text{b,BII}}\) & 0.147 \\
9 & \(\text{m}_{\text{b,ISC}}\) & 0.198 & \(\text{M}_{\text{S,ISC}}\) & 0.149 \\
10 & \(\text{M}_{\text{L,BII}}\) & 0.210 & \(\text{m}_{\text{b,MOS}}\) & 0.150 \\
11 & \(\text{m}_{\text{b,MOS}}\) & 0.224 & \(\text{M}_{\text{S,IDC}}\) & 0.155 \\
12 & \(\text{M}_{\text{S,NEiC}}\) & 0.237 & \(\text{m}_{\text{b,ISC}}\) & 0.157 \\
13 & \(\text{M}_{\text{L,KMA}}\)\({}^{1}\) & 0.257 & \(\text{m}_{\text{b,NEiC}}\) & 0.174 \\
14 & \(\text{m}_{\text{b,BII}}\) & 0.260 & \(\text{m}_{\text{b,IDC}}\) & 0.183 \\
15 & \(\text{m}_{\text{b,NEiC}}\) & 0.263 & \(\text{M}_{\text{L,KMA}}\)\({}^{1}\) & 0.190 \\
16 & \(\text{m}_{\text{b,IDC}}\) & 0.281 & \(\text{M}_{\text{S,NEiC}}\) & 0.206 \\
17 & \(\text{M}_{\text{L,IDC}}\) & 0.501 & \(\text{M}_{\text{L,IDC}}\) & 0.341 \\ \hline \end{tabular} \({}^{1}\) Regression is bilinear.  
\end{table} 
Table 4: Comparison of the different rankings of magnitude type regressions based on \(\sigma_{\text{OLS}}\) and \(\sigma_{\text{TLS}}\). Residuals from \(\rm m_{b}\) regressions are plotted in Figure 7. These plots show the OLS and TLS residual biases for each regression are close to zero, however there appears to be considerable scatter within the residuals for each regression, with each regression having about three events beyond \(2\sigma\) for OLS regressions. Interestingly, the TLS regression for \(\rm m_{b,NEIC}\) in Figure 7b had no events outside of \(2\sigma\). Although the figure shows decent scatter across magnitudes, there does not appear to be any trend, suggesting a linear model can be appropriate for \(\rm m_{b}\) regression. Figure 6: Regression residuals for \(\rm M_{S}\) from (**a**) ISC, (**b**) NEIC, (**c**) BJI, (**d**) MOS, and (**e**) IDC. The shaded regions represent \(\pm 1\sigma\) from OLS regression. Residuals from M\({}_{\rm L}\) regressions are plotted in Figure 8. Residual bias from OLS and TLS regressions are zero with TLS residuals showing less variability relative to OLS residuals. The proposed regression for M\({}_{\rm L,KMA}\) was bilinear to account for the stagnation at lower magnitudes and the residuals appear to agree in Figure 8a. It is not known if this saturation effect is a characteristic of lower magnitudes or an artifact of M\({}_{\rm L}\) calibrated for Japan [21,35]. Interestingly, there are several events between 4.5 \(<\) M\({}_{\rm L,KMA}\) < 5 where the bilinear model appears to underestimate magnitudes. Residuals for M\({}_{\rm JMA}\) also appear to show events at M\({}_{\rm JMA}\)\(\sim\)3.7 and 5.2 to be underestimated as shown in Figure 8b, with five events beyond 2\(\sigma\) for both OLS and TLS regressions, the highest amongst all models.