# Advanced Unified Earthquake Catalog for North East India
## Appendix A  
Let \(M_{x}\) and \(M_{y}\) be the true values, and \(m_{x}\) and \(m_{y}\) be the derived values with \(\delta\) and \(\varepsilon\) as magnitudes of errors in measuring for the independent and dependent variables. Then we can write:  
\[m_{x}=M_{x}+\delta, \tag{1}\]  
\[m_{y}=M_{y}+\varepsilon, \tag{2}\]  
as well as the regression-like model:  
\[m_{y}=\alpha+\beta M_{x}+\varepsilon, \tag{3}\]  
with \(M_{y}=\alpha+\beta M_{x}\), where \(\alpha\) and \(\beta\) are the slopes and intercepts of a linear relationship between genuine values  
\(M_{x}\), \(\varepsilon\), and \(\delta\) are assumed to be distributed in the relationships on a regular and independent basis.  
The observed value covariances \(\sigma_{m_{y}}^{2}\), \(\sigma_{m_{x}m_{y}}\) and \(\sigma_{m_{x}}^{2}\) are given by:  
\[\sigma_{m_{y}}^{2}=\beta^{2}\sigma_{M_{x}}^{2}+\sigma_{\varepsilon}^{2}, \tag{4}\]  
\[\sigma_{m_{x}m_{y}}=\beta\sigma_{M_{x}}^{2}, \tag{5}\]  
\[\sigma_{m_{x}}^{2}=\sigma_{M_{x}}^{2}+\sigma_{\delta}^{2}, \tag{6}\]  
where the error variance ratio:  
\[\eta=\frac{\sigma_{\varepsilon}^{2}}{\sigma_{\delta}^{2}} \tag{7}\]  
If, \(s_{m_{y}}^{2}\), \(s_{m_{x}}^{2}\) and \(s_{m_{x}m_{y}}\) are the sample covariances of \(m_{y}\), \(m_{x}\) and between \(m_{y}\) and \(m_{x}\), then:  
\[s_{m_{y}}^{2}=\beta^{2}\partial_{M_{x}}^{2}+\eta\phi_{\delta}^{2}, \tag{8}\]  
\[s_{m_{x}m_{y}}=\hat{\beta}\partial_{M_{x}}^{2}, \tag{9}\]  
\[s_{m_{x}}^{2}=\hat{\sigma}_{M_{x}}^{2}+\hat{\sigma}_{\delta}^{2} \tag{10}\]  
The estimators \(\hat{\beta}^{2}\), \(\hat{\sigma}_{M_{x}}^{2}\) and \(\hat{\sigma}_{\delta}^{2}\) maybe simply determined using the above simultaneous Equations (8)-(10). For example, the elimination of \(\hat{\sigma}_{M_{x}}^{2}\) and \(\hat{\sigma}_{\delta}^{2}\), we get the quadratic equation:  
\[\hat{\beta}^{2}s_{m_{x}m_{y}}-\hat{\beta}\Big{(}s_{m_{y}}^{2}-\eta s_{m_{x}}^{ 2}\Big{)}-\eta s_{m_{x}m_{y}}, \tag{11}\]Which yields  
\[\hat{\beta}=\frac{s_{m_{y}}^{2}-\eta s_{m_{x}}^{2}+\sqrt{\left(s_{m_{y}}^{2}-\eta s _{m_{x}}^{2}\right)^{2}+4\eta s_{m_{x}m_{y}}^{2}}}{2s_{m_{x}m_{y}}} \tag{12}\]  
The \(\partial_{M_{x}}^{2}\) and \(\partial_{\delta}^{2}\) are similarly derived as follows:  
\[\partial_{M_{x}}^{2}=\frac{\sqrt{\left(s_{m_{y}}^{2}-\eta s_{m_{x}}^{2}\right) ^{2}+4\eta s_{m_{x}m_{y}}^{2}-\left(s_{m_{y}}^{2}-\eta s_{m_{x}}^{2}\right)}}{2 \eta}, \tag{13}\]  
\[\hat{\sigma}_{\delta}^{2}=\frac{\left(s_{m_{y}}^{2}+\eta s_{m_{x}}^{2}\right) -\sqrt{\left(s_{m_{y}}^{2}-\eta s_{m_{x}}^{2}\right)^{2}+4\eta s_{m_{x}m_{y}} ^{2}}}{2\eta} \tag{14}\]  
The estimator for \(\alpha\) can be obtained from the relation:  
\[\hat{\alpha}=\overline{m}_{y}-\hat{\beta}\overline{m}_{x} \tag{15}\]  
where \(\overline{m}_{x}\) and \(\overline{m}_{y}\) are the averages of the observed values.  
Using Ref. [29] equations, the estimated variances of regression parameters \(\hat{\beta}\) and \(\hat{\alpha}\), may be expressed as follows:  
\[\hat{\sigma}_{\beta}^{2}=\frac{\partial_{M_{x}}^{2}(n-1)\big{(}\eta+\hat{ \beta}^{2}\big{)}\hat{\sigma}_{\delta}^{2}+\big{(}\hat{\sigma}_{\delta}^{2} \big{)}^{2}(n-1)\big{(}\eta+\hat{\beta}^{2}\big{)}-(n-2)\big{(}-\hat{\beta} \hat{\sigma}_{\delta}^{2}\big{)}^{2}}{(n-2)(n-1)\Big{(}\hat{\sigma}_{M_{x}}^{2 }\Big{)}^{2}} \tag{16}\]  
and  
\[\hat{\sigma}_{\alpha}^{2}=\frac{(n-1)\big{(}\eta+\hat{\beta}^{2}\big{)}\hat{ \sigma}_{\delta}^{2}}{n(n-2)}+\overline{m}_{x}^{2}\hat{\sigma}_{\beta}^{2}, \tag{17}\]  
where n is the sample size.