# Integrated Earthquake Catalog of the Eastern Sector of the Russian Arctic
## 2 Materials and Methods
We consider the events of the additional Catalog 2, with the value of the neighborhood function as less than the threshold one, as duplicates. The rest of the Catalog 2 events are declared unique and added to Catalog 1. Further, any number of catalogs can be sequentially added. The choice of the proximity function is based on a probabilistic model. We assume that the difference in earthquake detection by different networks is a random variable with a normal distribution and zero mean for each of the parameters:  
\[f(DT)=\frac{1}{\sigma_{T}\sqrt{2\pi}}\exp\biggl{(}-\frac{DT^{2}}{2\sigma_{T} ^{2}}\biggr{)},\]  
\[f(DX)=\frac{1}{\sigma_{X}\sqrt{2\pi}}\exp\biggl{(}-\frac{DX^{2}}{2\sigma_{X}^{ 2}}\biggr{)},\]  
\[f(DY)=\frac{1}{\sigma_{Y}\sqrt{2\pi}}\exp\biggl{(}-\frac{DY^{2}}{2\sigma_{Y}^{ 2}}\biggr{)}.\]  
Here \(DT\), \(DX\), and \(DY\) are the differences in time, longitude, and latitude, respectively, between different determinations of a seismic event, and \(\sigma_{T}\), \(\sigma_{X}\), and \(\sigma_{Y}\) are the corresponding standard deviations. If we assume that all errors are independent, then the duplicate probability density will be the product of the error probabilities for all parameters. This will be the multivariate normal distribution, as follows:  
\[f(DT,DX,DY)=\frac{1}{\sigma_{T}\sigma_{X}\sigma_{Y}(2\pi)^{\frac{3}{2}}}\cdot \exp\biggl{(}-\biggl{(}\frac{DT^{2}}{2\sigma_{T}^{2}}+\frac{DX^{2}}{2\sigma_{X} ^{2}}+\frac{DY^{2}}{2\sigma_{Y}^{2}}\biggr{)}\biggr{)}.\]Thus, we naturally arrive at the Euclidean metric:  
\[Ro=\sqrt{\frac{DT^{2}}{{\sigma_{T}}^{2}}+\frac{DX^{2}}{{\sigma_{X}}^{2}}+\frac{ DY^{2}}{{\sigma_{Y}}^{2}}} \tag{1}\]  
The preliminary identification of duplicates is done with the standard metric parameters in (1): \(\sigma_{0T}=0.05\) min and \(\sigma_{0X}=\sigma_{0Y}=15\) km. The initial values of the parameters have little effect on the identification of duplicates; however, they significantly affect the value of the duplicate probability and the estimate of the percentage of errors. At this stage, we check that each of the parameters follows a normal distribution, and we refine the values of the standard deviations \(\sigma_{T}\), \(\sigma_{X}\), and \(\sigma_{Y}\). After that, the final identification of duplicates is performed. The choice of the optimal metric threshold for identifying duplicates and the estimation of the percentage of errors will be explained in detail below. Before proceeding with the merging process of the four catalogs, we checked each of them for internal duplicates. For this reason, we built the distribution of metric (1) for the nearest events within each catalog (Figure 2). The analysis was performed with the metric parameters \(\sigma_{T}=0.05\) min and \(\sigma_{X}=\sigma_{Y}=15\) km. As a result of the analysis, no events with the same time and epicenter coordinates were found in any of the four catalogs. For such events, \(Ro=0\), and, thereafter, we will call them absolute duplicates. Statistical analysis also did not reveal anomalous groups of nearest events. The duplicates are characterized by the value \(Ro<10\). The value \(Ro=10\) corresponds to a distance of 150 km or a time interval of 0.5 min. From our experience, we know that duplicates have smaller differences in instrumental catalogs, and the number of such nearest events within each of the catalogs is very small. These are mainly the early aftershocks of the Olyutorsk earthquake, with \(M=7.6\), on 20 April 2006, and the Ilin-Tas earthquake, with \(M=6.6\), on 14 February 2013. There is no reason to consider such events as duplicates since the early aftershocks can occur at very small distances and time intervals. Thus, the necessary condition for applying the method in [34] is met. Figure 2: Distribution of metric (1) for events within the source catalogs (Tables 1 and 2). The catalogs are indicated on the histograms.