# Spatial Correlations of Global Seismic Noise Properties
## 3 Wavelet-Based Measure of Time-Series Non-Stationarity  
Let us consider a random signal \(x(t)\) where \(t=1,\ldots,N\) is an integer discrete-time index. For a finite sample of the signal, the entropy [20] could be defined as:  
\[En=-\sum_{k=1}^{N}p_{k}\cdot\log(p_{k}),\ p_{k}=c_{k}^{2}/\sum_{j=1}^{N}c_{j}^ {2} \tag{1}\]  
Here, \(c_{k}\) are the coefficients of orthogonal wavelet decomposition. Within a finite set of Daubechies wavelet bases [20] with the number of vanishing moments from 1 to 10, the optimal basis is chosen from the minimum of (1). The DJ-index was introduced in the problem of noise reduction and compression of information by setting all \"small\" wavelet coefficients to zero and performing an inverse wavelet transform [14; 20]. This operation of noise reduction is performed for the optimal wavelet basis which was found from the minimum of entropy. The problem is how to define the threshold which separates \"large\" wavelet coefficients from \"small\" ones. In [14], this problem was solved using a formula of the asymptotic probability of the maximum deviations of Gaussian white noise. The DJ threshold is given by the expression \(T_{DI}=\sigma\sqrt{2\cdot\ln N}\), where \(\sigma\) is the standard deviation of wavelet coefficients of the first detail level of wavelet decomposition. The first level is the most high frequency and is considered as the most noisy. Thus, in order to estimate the noise standard deviation, it is  
Figure 1: Positions of 229 broadband seismic stations (blue circles) and a network of 50 reference points (numbered red circles). possible to calculate the standard deviation of the first-level wavelet coefficients \(c_{k}^{(1)}\). The robust estimate was proposed:  
\[\sigma=med\Big{\{}\left|c_{k}^{(1)}\right|\,,\;k=1,\ldots,N/2\Big{\}}/0.6745, \tag{2}\]  
where \(N/2\) is the number of wavelet coefficients on the first level, and \(med\) is the median. Formula (2) gives the relationship between the standard deviation and the median for the Gaussian random variable. Finally, it is possible to define the dimensionless signal characteristic \(\gamma\), \(0<\gamma<1\), as the ratio of the number of the most informative wavelet coefficients, for which the inequality \(|c_{k}|>T_{DJ}\) is satisfied, to the total number \(N\) of all wavelet coefficients. For stationary Gaussian white noise, which is a kind of ideal stationary random signal, the index \(\gamma=0\). That is why the DJ-index could be called a measure of non-stationarity. For each reference point, the index value \(\gamma\) is calculated daily as a median of the values in the five nearest operational stations. Thus, continuous time series are obtained with a time sampling of one day at 50 reference points. Figure 2 shows the graphs of the index \(\gamma\) for 9 reference points (point numbers are indicated next to each graph). Figure 3 shows a graph of average daily values of index \(\gamma\) calculated for all 50 reference points. The red lines represent the linear trends of the average values calculated for the time fragments 1997-2003, 2004-2015 and 2016-2022. The boundaries of time fragments were chosen so that the continuity of the broken line at the union of linear trends was approximately observed. For the time intervals 1997-2003 and 2016-2022, linear trends coincided with the average values and it can be seen that during the intermediate time interval 2004-2015 there had been a significant decrease in the global average \(\gamma\). According to the results published in articles [15; 17; 18], a decrease in the average value of the index \(\gamma\), which is a sign of the simplification of the statistical structure of noise (approximation of  
Figure 2: Graphs of daily index \(\gamma\) values for 9 reference points. The green lines represent the moving averages in a 57-day window. its properties to the properties of white noise), was associated with an increase in seismic hazard.