## 2 Materials and Methods
### Probability Densities of Extreme Values  
Further, estimates of 12 parameters of the Earth's surface tremors will be considered in the sequence of time windows 182 days long (half a year), taken with a shift of 3 days. In this case, in each window, we will consider only operable GPS stations, by which we mean those stations for which the total number of missing data in the current window does not exceed 0.1 of the window length. Consider a regular grid of 30 \(\times\) 30 nodes covering an area with a latitude from 28\({}^{\circ}\) N to 46\({}^{\circ}\) N and longitude from 128\({}^{\circ}\) E to 146\({}^{\circ}\) E. Let \(U\) be any value \(En_{W}\), \(En_{S}\), \(\beta_{W}\), or \(\beta_{S}\), whose estimates are obtained in a sliding time window. For each grid node \((i,j)\) and for each time window labeled \(t\) at the right end of the window, find the 10 nearest seismic stations, which gives 10 values of \(U\). Let's take their median value \(U_{ij}^{(t)}\). The values \(U_{ij}^{(t)}\) form an \"elementary\" map corresponding to a time window of 182 days. Consider the values of the parameter as a function of two-dimensional vectors \(z_{ij}~{}=~{}(x_{i},y_{j})\) of longitudes and latitudes of nodes \((i,j)\) in an explicit form: \(U_{ij}^{(t)}\equiv U^{(t)}(z_{ij})\). For each \"elementary map\" with a discrete time index \(t\) we find the coordinates of the nodes \(z_{mn}^{(t)}~{}=~{}(x_{m}^{(t)},y_{n}^{(t)})\) at which the extreme value of \(U\) is reached with respect to all other nodes of the regular grid. If \(U~{}=~{}En_{W}\) or \(U~{}=~{}En_{S}\), then they will look for the minimum values. Otherwise, for \(U~{}=~{}\beta_{W}\) or \(U~{}=~{}\beta_{S}\), the maximum values will be searched. As noted above, the choice of a minimum for entropies and a maximum for spectral indexes is due to considerations that anomalous tremor regions should be characterized by the values of statistics that reflect the maximum deviation from the properties of white noise. A cloud of two-dimensional vectors \(z_{mn}^{(t)}\) considered within a certain time interval \(t\in[t_{0},t_{1}]\) forms a random set. Let us estimate their two-dimensional probability distribution function for each node of the regular grid. For this, the Parzen-Rosenblatt estimate with the Gaussian kernel function [26] will be applied:  
\[p(z_{ij}|t_{0},t_{1})~{}=~{}\frac{1}{2\pi\hbar^{2}(t_{1}-t_{0}+1)}\sum_{t~{}= ~{}t_{0}}^{t_{1}}\sum_{m~{}=~{}t_{0}}^{t_{2}}\sum_{m~{}=~{}t_{0}}\exp\left(- \frac{\left|z_{ij}-z_{mn}^{(t)}\right|^{2}}{2\hbar^{2}}\right) \tag{8}\]  
Here, \(h\) is the kernel averaging radius and \(t_{0},t_{1}\) are integer indices that enumerate the \"elementary\" maps of each time window. Thus, \((t_{1}-t_{0}+1)\) is the number of time windows in the considered time interval. The width of the smoothing band was used \(h~{}=~{}0.5^{\circ}\) which is approximately equal to 56 km. It should be noticed that the maximum distance from the nearest 10 th stations to the reference points equals 47.5 km, the minimum equals 25.4 km, and the median maximum distance equals 37 km. Figures 3 and 4 show the average distribution maps of the extreme values of the four tremor statistics under consideration for all three displacement components. Even a purely visual perception of these maps suggests that they have a strong similarity to each other. This visual impression is confirmed by the estimate of the matrix of pairwise correlation coefficients presented in Table 1. The minimum value of the correlation coefficient is 0.7044 for the pair, while for the pair the correlation coefficient reaches 0.9903.  
spectral index exceeds the level of 98% quantile. These regions are considered as regions of anomalous GPS-measured Earth tremors. Figure 5b is a plot of the squares of the eigenvector components of the correlation matrix corresponding to the maximum eigenvalue after sorting in descending order. From this plot, it can be seen that the GPS noise statistics corresponding to measurements in the N direction contribute the most to the averaging weights.