## 3 Materials and Methods
### Time Series Decomposition
\begin{table}
\begin{tabular}{c c c c c c} \hline  & **SRDN-3 No. 2** & **SRDN-3 No. 3** & **SRDN-3 No. 4** & **SRDN-3 No. 5** & **SRDN-3 No. 6** \\ \hline \(p\) Value & \(0.0001\) & \(0.0001\) & \(0.0001\) & \(0.0001\) & \(0.0001\) \\ c Value & \(-1.9416\) & \(-1.9416\) & \(-1.9416\) & \(-1.9416\) & \(-1.9416\) \\ statistic & \(-39.2701\) & \(-35.7463\) & \(-37.8563\) & \(-40.8262\) & \(-39.0372\) \\ Null hypothesis & rejected & rejected & rejected & rejected & rejected \\ \hline \end{tabular}
\end{table} 
Table 2: Dickey–Fuller test, data after removing the periodic component. Figure 6: Residues for radon probe SRDN-3 No. 2.  
which provides a global frequency distribution over the entire signal \(x(t)\). Nevertheless, there exists a function that enables the inverse operation, i.e., with knowledge of the above-mentioned distribution \(X_{FT}(f)\), it is possible to reconstruct the original signal (8). \[x(t)=\int_{-\infty}^{\infty}X_{FT}(f)e^{j2\pi ft}df \tag{8}\]  
In order for the calculations to be feasible, the analyzed signal must satisfy the Dirichlet conditions, meaning, among other things, that it must be at least piecewise continuous. This condition is not met by the signals we analyze, as they are sampled at discrete time intervals. Therefore, the Discrete Fourier Transform (DFT) is applied (9),  
\[X_{DFT}(f_{n})=\frac{1}{N}\sum_{k=0}^{N-1}x(k)e^{-j2\pi k\Delta T} \tag{9}\]  
which also has its inverse form (10):  
\[x(k)=\frac{1}{\Delta T}\sum_{f_{n}=0}^{\frac{N-1}{t}}X_{DFT}(f_{n})e^{j2\pi f_ {n}k\Delta T} \tag{10}\]  
For each frequency \(k\), the result of the transform \(X(f)\) is a complex number, from which the following can be calculated (11) and (12),  
\[A(t)=\frac{2}{N}\cdot|X(f)| \tag{11}\]  
\[\phi(t)=arg(X(f)) \tag{12}\]  
where:  
\(A(k)\)--the amplitude of the component at frequency \((k)\),  
\(\phi(k)\)--phase of this component,  
\(|X(f)|\)--the magnitude of the complex number \((X(f))\),  
\(arg(X(f))\)--argument (phase angle of complex number \((X(f))\). The amplitude indicates the strength (energy) of a given frequency in the signal, while the phase describes the temporal shift of that component [59; 60]. The signal from the radon sensors contained 40,531 samples with an hourly resolution, making it a large dataset. Fourier Transform for such large data volumes is computationally expensive, which is why the Fast Fourier Transform (FFT) is used. The FFT requires the number of samples to be adjusted to N = 2 m. This can be performed in two ways: by trimming the signal to the required number of samples or by zero-padding the data up to that number. However, this leads to a problem known as spectral leakage, where energy from a signal component spreads into neighboring frequencies (causing reduced amplitude). The solution is to apply a window function, which sharpens the frequency components within the window but also introduces additional periodicity into the signal [59]. Another issue is aliasing, which may result in fundamental frequencies being mis-represented as different ones. The solution is to sample the signal properly--according to [PERSON]'s sampling theorem--at a rate at least twice the highest frequency of interest. In the case of radon sensor data, the sampling frequency far exceeds that necessary for detecting a primary frequency of one year. Wavelet analysis is an alternative method for signal analysis. This approach offers significantly more capabilities, including local and multiscale analyses. Sampled signals from \"geo\" sources often contain frequencies arising from different interrelated processes. Therefore, continuous wavelet analysis was selected as the most appropriate method because its key advantage is that it operates in both the frequency and time domains. This is particularly important due to the large amount of data and the likelihood of frequency components appearing and disappearing over time.