# Expanding Moment Magnitude Pools for Earthquake Magnitude Homogenization
## 2 Materials and Methods
### Magnitude Pool Expansion  
The target scale for homogenization herein is M\({}_{\rm W}\) from GCMT, (M\({}_{\rm W,GCMT}\)) due to reasons stated previously. The magnitude homogenization regressions using the pool of M\({}_{\rm W,GCMT}\) records from the compiled catalogs are considered the controls in this study. From this, two approaches are used to expand the M\({}_{\rm W,GCMT}\) pool for magnitude homogenization. One is to include all relevant sources related to GCMT. These include HRVD and ISC-GEM. However, only ISC-GEM records with a listed M\({}_{\rm W,GCMT}\) or a M\({}_{\rm W}\) from a bibliographic source are included. This excludes records where ISC homogenization was applied. Moreover, since bibliographic sources are included, earthquakes from the ISC Event Bibliography (EVBIB) with a recorded M\({}_{\rm W}\) are also included. Most of the records in EVBIB are in ISC-GEM. This extended M\({}_{\rm W}\) pool is called M\({}_{\rm W,GCMT}\)-related. Another approach is to use statistical hypothesis testing [4; 8]. The assumption is if one moment magnitude type from any source is significantly similar enough to the control pool M\({}_{\rm W,GCMT}\), then the remaining magnitudes of the same magnitude type from the same source would also be similar to M\({}_{\rm W,GCMT}\). Hypothesis testing herein involves applying \(t\)-tests to regression parameters \(\beta_{1}\) and \(\beta_{0}\) derived from magnitude homogenization with other M\({}_{\rm W}\) formats. For this study, \(t\)-tests are conducted when a magnitude type from a specific agency or project has at least 5 data pairs due to regression calculation issues. For each comparison, a calculated t-statistic, t, is used to estimate a \(p\)-value from Student's t-distribution. This \(p\)-value represents the probability that, given a model, results as extreme as the observed results could occur. \(p\)-values are compared to a pre-determined threshold, \(\alpha\), where \(p\)-values less than or equal to \(\alpha\) suggest a rejection of the null hypothesis, H\({}_{0}\), and that the differences are statistically significant. Using an \(\alpha\) = 0.05, the null hypotheses are H\({}_{0}\):\(\beta_{1}\) = 1 and H\({}_{0}\):\(\beta_{0}\) = 0, which is a one-to-one line in magnitude homogenization space. The t-statistic for H\({}_{0}\):\(\beta_{1}\) = 1 is:  
\[{\rm t}=\frac{\beta_{1}-1}{SE(\beta_{1})} \tag{2}\]where \(SE(\beta_{1})\) is the standard error of the estimate of \(\beta_{1}\) and is taken as:  
\[SE(\beta_{1})=\sqrt{\frac{\text{MSE}}{\sum_{i=1}^{n}(x_{i}-\overline{x})^{2}}} \tag{3}\]  
where \(i\) is the index, \(n\) is the number of data points, \(x_{i}\) are the dependent variable, \(x^{-}\) is the mean of the dependent variable, and MSE is the mean squared error taken as:  
\[\text{MSE}=\frac{\sum_{i=1}^{n}(y_{i}-\hat{y_{i}})^{2}}{n-2} \tag{4}\]  
where \(y_{i}\) is the dependent variable and \(\hat{y_{i}}\) is the appropriate regression model. However, when applying hypothesis testing for the equivalence of two regressed slopes, that is H0:\(\beta_{1}\) = \(\beta_{\text{a}}\), the t-statistic becomes:  
\[\text{t}=\frac{\beta_{1}-\beta_{\text{a}}}{\sqrt{SE(\beta_{1})^{2}+SE(\beta_{ \text{a}})^{2}}} \tag{5}\]  
where \(\beta_{\text{a}}\) is the target regressed slope for comparison. If the null hypothesis for the slope is not rejected, then another \(t\)-test is applied to the regressed intercept. The t-statistic for H0:\(\beta_{0}\) = 0 is:  
\[\text{t}=\frac{\beta_{0}-0}{SE(\beta_{0})} \tag{6}\]  
where \(SE(\beta_{0})\) is the standard error of the estimate of \(\beta_{0}\) and is taken as:  
\[SE(\beta_{0})=\sqrt{\text{MSE}}\sqrt{\frac{1}{n}+\frac{\overline{x}^{2}}{\sum _{i=1}^{n}(x_{i}-\overline{x})^{2}}} \tag{7}\]  
Similar to the \(t\)-test for two regressed slopes, hypothesis testing between two regressed intercepts has a similar t-statistic as calculated in Equation (5) with the substitution of Equation (7) into Equation (5) for the standard error of the estimate of \(\beta_{0}\).