## 3 Results
The solid and dashed lines represent the regression type as indicated in the graph. N is the number of data points used for regression. Data by [PERSON] et al. (2015) from a larger earthquake compilation related to \(\rm M_{w}\) versus \(\rm m_{b}\) is shown in gray symbols in the background for comparison [19]. Figure 3 shows TLS regressions on NEIC, BJI, MOS, and IDC data to be steeper relative to OLS regressions most likely due to the 2005 March 20 M\({}_{\rm W}\) 6.6 Fukuoka earthquake. Figure 4 plots M\({}_{\rm W}\) and M\({}_{\rm L}\) pairs from multiple agencies. Data from KMA, JMA, BJI, KEA, and IDC are shown in Figure 4a-e, respectively. In the ISC Bulletin, magnitudes listed as M should be treated equivalent to M\({}_{\rm L}\)[16]. Since the local magnitude in the JMA seismological bulletin is listed as M\({}_{\rm JMA}\), the local magnitude used by JMA will be termed M\({}_{\rm JMA}\) herein for consistency. Data from JMA, BJI, and KEA appear linear with decent scatter having R\({}^{2}\) = 0.895, 0.838, and 0.930 for JMA, BJI, and KEA, respectively. The KMA data appear relatively scattered with R\({}^{2}\) = 0.780 while the IDC data appear the most scattered with R\({}^{2}\) = 0.408. The figures show there is a lack of M\({}_{\rm L}\) data pairs for BJI and KEA at M\({}_{\rm L}\) \(<\) 4, while KMA, JMA, and IDC are showing a good amount of data for M\({}_{\rm L}\) \(<\) 4. Interestingly, Figure 4a appears to show a type of magnitude saturation where, as M\({}_{\rm L}\) decreases, M\({}_{\rm W}\) stays within a small range. Considering this, a bilinear relationship is proposed where an OLS and TLS regression is applied on magnitude pair data above M\({}_{\rm L}\) = 3.6, but kept constant for M\({}_{\rm L}\) \(<\) 3.6. In this approach, R\({}^{2}\) for data where M\({}_{\rm L}\) \(<\) 3.6 is 0.810, a moderate increase from 0.780, but still suggesting a linear relationship. Figure 3.5 Relationships between M\({}_{\rm w}\) against M\({}_{\rm L}\) from (**a**) KMA, (**b**) JMA, (**c**) BJI, (**d**) KEA, and (**e**) IDC. The solid and dashed lines represent the regression type as indicated in the graph. N is the number of data points used for regression. OLS regressions on the \(\mathrm{M_{L}}\) data result in the following relationships:  
Figure 3.6 Figure 3.7 Figure 3.8 (7b) Figure 3.9 (7c) Figure 2.0 (7d) Figure 2.1  
Additionally, TLS regressions on the \(\mathrm{M_{L}}\) data result in the following relationships:  
Figure 2.2 Figure 2.3 Figure 2.4 Figure 2.5 (8c) Figure 2.6 (8d) Figure 2.7  
In addition to \(\mathrm{M_{W}}\) and \(\mathrm{M_{JMA}}\) also uses a displacement magnitude, \(\mathrm{M_{D,JMA}}\), and a velocity magnitude, \(\mathrm{M_{V,JMA}}\)[32; 33; 34]. Earthquakes with these JMA displacement and velocity-based magnitudes and a corresponding \(\mathrm{M_{W}}\) are plotted in Figure 2.8 The data for \(\mathrm{M_{D,JMA}}\) is not as plentiful, ranging from \(\mathrm{M_{D,JMA}}\)-4.0 to 6.0. On the other end, \(\mathrm{M_{V,JMA}}\) appears to be associated with events at the lower magnitude range, with \(\mathrm{M_{V,JMA}}\)-3.5 to 5.0. Both magnitudes show linear behavior and little scatter. OLS regressions on these magnitude types result in the following relationships:  
Figure 2.9 Figure 2:0  
Figure 2:1 Relationships between \(\mathrm{M_{w}}\) and (**a**) \(\mathrm{M_{V}}\) and (**b**) \(\mathrm{M_{D}}\) by JMA. The solid and dashed lines represent the regression type as indicated in the graph. \(\mathrm{N}\) is the number of data points used for regression. Additionally, TLS regressions on these magnitude types result in the following relationships:  
Figure 2:2 Figure 2:3  
As stated previously, the \(\sigma\)'s will be used to prioritize magnitude relations. Note that \(\sigma_{\text{TLS}}\) treats the residual as the shortest distance from data points to the regressed line. Table 4 compares the ranking of each relationship in ascending order for OLS and TLS.