## 2 Materials and Methods
### Spectral Normalized Entropy \(En_{s}\) and \"Usual\" Spectral Index \(\beta_{s}\)  
The normalized entropy (1) refers to the entropy of the energy distribution on different time scales (because of the use of finite orthogonal basis functions). At the same time, the entropy can be estimated in the frequency domain. Let \(S_{xx}(\omega_{j})\) be an estimate of the signal \(x(t)\) power spectrum, \(\omega_{j}\) be a sequence of discrete frequency values with a step \(\Delta\omega\), \(\omega_{j}\ =\ (j-1)\cdot\Delta\omega\), \(\Delta\omega\ =\ 2\pi/(N\cdot\Delta s)\), \(j\ =\ 1,\ldots,N\). Quantities  
\[p_{j}^{(S)}\ =\ S_{xx}(\omega_{j})/\sum_{k\ =\ 1}^{N/2}S_{xx}(\omega_{k}),\ j \ =\ 1,\ldots,N/2, \tag{6}\]  
can be considered as the probabilities of energy distribution in a sequence of non-overlapping frequency intervals of length \(\Delta\omega\). Here, \(N\) is the minimum length, which is greater than or equal to the sample length \(L\): \(N\ =\ 2^{n}\geq L\), the index \(j\) in Formula (6) varies up to \(N/2\) due to the symmetry of the power spectrum of real signals with respect to the Nyquist frequency \(\omega_{N/2+1}\): \(S_{xx}(\omega_{N/2+1-k})\ =\ S_{xx}(\omega_{N/2+1+k})\), \(k\ =\ 1,\ldots,N/2-1\). Similarly to the values (1) and (5), we introduce the spectral normalized entropy \(En_{S}\) and spectral index \(\beta_{S}\) according to formulae:  
\[\begin{array}{l}En_{S}\ =\ -\sum\limits_{j\ =\ 1}^{N/2}p_{j}^{(S)}\ln(p_{j}^{(S )})/\ln(N/2),\ 0\leq En_{S}\leq 1;\\ \ln(S_{xx}(T_{j}))\ =\ \beta_{S}\cdot\ln(T_{j})+d+\eta_{j},\ j\ =\ 1,\ldots,N/2 \end{array} \tag{7}\]where period \(T_{j}~{}=~{}2\pi/\omega_{j}\), the value of \(\beta_{S}\) is defined from the least squares approach: \(\sum_{j=1}^{N/2}\eta_{j}^{2}\rightarrow\min\limits_{d,\beta_{S}}\eta_{j}\) is a sequence of independent random variables with zero mean.  
The values of \(En_{W}\), \(En_{S}\), \(\beta_{W}\), and \(\beta_{S}\) were estimated in successive time windows consisting of 288 adjacent 5-min readings, which is one day. Before their calculation, the operation of detrending by a polynomial of the fourth order was performed. Thus, for each station, time series were obtained with a time step of 1 day. The maximum entropy method [28] with autoregression order 28 (10% of the sample length in the time window) was used to calculate the power spectrum \(S_{xx}(\omega_{j})\) estimate. The proximity of the normalized spectral entropy \(En_{S}\) to the maximum value of 1 means that the energy distribution over frequencies is close to uniform, i.e., the power spectrum is close to the white noise spectrum.  
Since displacements of the Earth's surface in three orthogonal directions are measured at each point of the network, then, when necessary, we will distinguish the values of the analyzed tremor statistics for different directions of surface displacements by the superscript, for example, for entropy (1) \(En_{W}^{(E)}\), \(En_{W}^{(N)}\), \(En_{W}^{(U)}\), where the superscripts correspond to measurements in the E, N, and Up directions. Similar notation will be used for the quantities, \(En_{S}\), \(\beta_{W}\), and \(\beta_{S}\).  
For an auxiliary network of 16 reference points (Figure 1a) 12 time series were determined with a time step of 1 day, the values of which were calculated as medians of the values \((En_{W}^{(E)},En_{W}^{(N)},En_{W}^{(U)})\), \((En_{S}^{(E)},En_{S}^{(N)},En_{S}^{(U)})\), \((\beta_{W}^{(E)},\beta_{W}^{(N)},\beta_{W}^{(U)})\), and \((\beta_{S}^{(E)},\beta_{S}^{(N)},\beta_{S}^{(U)})\) from the ten nearest operational GPS measurement points. Figure 2 shows graphs of such time series for central reference point number 8. These graphs show a strong seasonal periodicity of tremor properties.  
Figure 2: Plots of daily values for four GPS ground tremor properties for reference point number 8 for three ground displacement components E, N, and Up. The property values are calculated as medians from the ten nearest operable stations.