## 4 Spatial Correlations of Tremor Properties  
In this section of the article, spatial correlations between tremor properties in the network of 16 reference points presented in Figure 1 will be investigated. Property values are computed at 16 reference points as medians of daily values from the 10 closest opera  
Figure 6: (a1,a2) are the latitudes and longitudes of the nodes of a regular grid of 30 \(\times\) 30 in size, in which the maxima of the average weighted probability density of the distribution of extreme values of the properties of the tremor of the Earth’s surface were realized in sliding time windows 182 days long; (a3) are the corresponding numbers of clusters of maximum points. (b)—division of points of the trajectory of maxima into five clusters. tional stations. Examples of graphs of such time series for reference point #8 are shown in Figure 2.  
Consider the correlation coefficients between the same tremor properties at different reference points. Denote by \(\rho^{(\tau)}_{ij}(P)\) the estimates of the correlation coefficients between the values of the property \(P\) at the reference points with numbers \(i\) and \(j\) for a sequence of time windows 182 days long with time marks \(\tau\) of the right ends of the windows. As \(P\) can be any of the 12 tremor properties: \(En^{(E)}_{W}\), \(En^{(N)}_{W}\), \(En^{(U)}_{W}\), \(En^{(E)}_{S}\), \(En^{(N)}_{S}\), \(En^{(U)}_{S}\), \(\beta^{(E)}_{W}\), \(\beta^{(N)}_{W}\), \(\beta^{(U)}_{S}\), \(\beta^{(E)}_{S}\), \(\beta^{(N)}_{S}\), \(\beta^{(U)}_{S}\). For each property, we determine the average value of the correlation coefficients for all pairs of reference points:  
\[\overline{\rho}^{(\tau)}(P)\ =\ \sum_{(i,j)}\rho^{(\tau)}_{ij}(P)/M \tag{9}\]  
In Formula (9) \(M\) is the number of different pairs of reference points. In our case, for 16 reference points \(M\ =\ 120\). Let us denote by \(n^{(\tau)}(P)\) the number of pairs of reference points for which the corresponding correlation coefficients \(\rho^{(\tau)}_{ij}(P)\) exceed the threshold of 0.7 in the time windows with the label of the right end \(\tau\). These quantities define some of the strong spatial correlations of tremor properties. The long-range action of strong correlations is described by the values \(d^{(\tau)}(P)\) of the maximum distances between those pairs of reference points for which the correlation coefficients \(\rho^{(\tau)}_{ij}(P)\) exceed the threshold of 0.7. The obtained values \(\overline{\rho}^{(\tau)}(P)\), \(n^{(\tau)}(P)\), \(d^{(\tau)}(P)\) average over all 12 properties, that is, consider the values:  
\[\rho^{*}_{\tau}\ =\ \sum_{P}\overline{\rho}^{(\tau)}(P)/12,\ n^{*}_{\tau}\ =\ \sum_{P}n^{(\tau)}(P)/12,\ d^{*}_{\tau}\ =\ \sum_{P}d^{(\tau)}(P)/12 \tag{10}\]  
Dependences (10) are shown in Figure 7. Quantities (10) could be called measures of the activity of the Earth's surface tremor. For each reference point with a number \(j\ =\ 1,\ldots,16\), it is possible to calculate how the average value \(\rho^{(j)}_{\tau}\) of the correlation coefficients of the tremor statistics with the rest of the reference points changes in each time window. When calculating the values \(\rho^{(j)}_{\tau}\), the average is made over all 12 properties of the tremor of the Earth's surface. Graphs of values \(\rho^{(j)}_{\tau}\) are shown in Figure 8.  
Figure 7: (**a**) is a plot of mean correlations \(\rho^{*}_{\tau}\); (**b**) is a graph of the average number \(n^{*}_{\tau}\) of pairs of control points for which the correlations exceeded the threshold of 0.7; (**c**) is a graph of the average maximum distances \(d^{*}_{\tau}\) between those pairs of control points for which the correlation coefficients exceed the threshold of 0.7. Vertical red lines indicate time moments of earthquakes: 1—Tohoku, 11 March 2011, \(M=9.1\); 2—Kumamoto, 16 April 2016, \(M=7\). The presence of values in 16 reference points distributed throughout Japan allows the construction of maps of spatial values.