# Spatial Correlations of Global Seismic Noise Properties
## 4 Probability Densities of Extreme Values of Seismic-Noise Properties  
Further analysis of the properties of seismic noise was based on the identification of regions that are characterized by the most frequent occurrences of extreme values for certain seismic-noise statistics. To do this, it was necessary to estimate the probability densities of the distribution over the space of extreme values by their values in the network of reference points. The following process was used. Denote by \(\zeta_{k},\ k=1,\ldots,m=50\) coordinates of the reference points. Let \(Z_{k}\) be the values of any property of the noise at the reference point with the number \(k\). We will be interested in maps of the distribution of probability densities of extreme values (minimums or maxima) of values \(Z\) over the Earth's surface. To construct such maps, consider a regular grid of 50 nodes in latitude and 100 nodes in longitude covering the entire earth's surface. Next, maps will be built for various seismic-noise statistics \(Z\), the values of which will be estimated in time windows of various lengths. For each such window, we define the time index \(t\), which corresponds to its right end. For each time index \(t\), find the reference point with the number \(k_{t}^{*}\), at which the extremum of one or another property \(Z\) is realized, and let \(\zeta_{t}^{*}\) be the coordinate of the reference point with the number \(k_{t}^{*}\). Let \(\rho(\zeta_{k},r_{ij})\) be the distance between the points \(\zeta_{k}\) and the coordinates of the nodes of the regular grid \(r_{ij}\). Then, the value of the probability density of extreme values for the quantity \(Z\) at the node \(r_{ij}\) of the regular grid can be calculated according to Gaussian distribution:  
\[p(r_{ij}|t)=\exp(-\rho^{2}(\zeta_{t}^{*},r_{ij})/(2h^{2}))/P(t) \tag{3}\]  
Here, \(h\) is the distance, which corresponds to an arc of 15 angular degrees, which is approximately equal to 1700 km, and \(P(t)\) is a divisor that provides the normalization condition so that the numerical integral of function (3) over the Earth's surface is equal to 1. Formula (3) gives an \"elementary map\" of the probability density of the property \(Z\) extrema distribution for each current window with a time index \(t\). This makes it possible to calculate the average probability density for a given time interval from \(t_{0}\) to \(t_{1}\):  
\[\overline{p}(r_{ij}|t_{0},t_{1})=\sum_{t=t_{0}}^{t_{1}}p(r_{ij}|t)/n(t_{0},t_ {1}) \tag{4}\]  
where \(n(t_{0},t_{1})\) is the number of time windows labeled from \(t_{0}\) to \(t_{1}\). Figure 3: Graph of the average daily values of the DJ-index for all 50 reference points. Red lines present linear trends for 3 time intervals: 1997–2003, 2004–2015 and 2016–2022. Another way to estimate the variability of the probability density of extreme properties of seismic noise is to construct histograms of the distribution of the numbers of control points, in which the statistics extrema are realized. The construction of such histograms in a moving time window, the length of which should include a sufficiently large number of initial time windows within which the noise properties are estimated, makes it possible to determine the reference points at which the minima or maxima of the seismic-noise properties \(Z\) are realized most often. This method has the disadvantage that it does not give a direct distribution of the places of the most probable realization of the noise-property extrema in the form of a geographical map, although each reference point has geographical coordinates. On the other hand, this method makes it possible to compactly visualize the temporal dynamics and, in particular, to determine the time intervals when the places of concentration of extreme values for the noise statistics change sharply. To solve a similar problem of visualizing temporal dynamics using averaged probability densities (4), one would have to build a laborious and long sequence of maps for time fragments consisting of the same number of time windows.