## 3 Materials and Methods
### Data Correlation  
The signal from the SRDN-3 radon probes had a very high noise level; therefore, in the subsequent stages of the analysis, it was transformed, i.e., smoothed. The aim of this procedure was to remove high frequencies (below 1 day) that dominated in the radon time series. The signals from the water tube tiltmeters had a very smooth flow and frequencies higher than 1 day. Two methods were used: discrete wavelet analysis and Savitzky-Golay filtering. The first method was based on the decomposition of the signal by passing it through two filters: a high-pass and a low-pass filter. Depending on the n level of decomposition, this operation needs to be repeated n times on the signals generated using the high-pass filter (Figure 13). Figure 12: Scalogram and 24 h signal reconstruction together with significance level Legend: (**A**)—signal having a particular frequency, applied to data from probe SRDN-3 No. 2, (**B**)—power graph for a particular scale together with noise power (red line), (**C**)—scalogram showing frequencies occurring in a time series within a particular range, (**D**)—reconstructed signal with a period of about 24 h. Explanations: top figure (**A**)—green dots indicate the analyzed signal, pink line indicates the recon-structed signal at the selected frequency. The middle figure on the left (**B**), the sum of the energy at each frequency is marked in blue. The red line indicates the power of the red noise. If the peaks on the blue line exceed the power of the noise, this indicates the signal’s significance. The green line indicates the selected scale/frequency. The middle figure on the right (**C**) a scalogram shows the signal energy for each scale/frequency. The line indicates the selected frequency. The green line and the line in the scalogram are the same line, representing the selected frequency. The bottom figure (**D**)—the reconstructed signal itself. High-frequency signals were filtered (smoothed) by eliminating signals with frequencies above a certain signal strength threshold. Each time, the signals with higher frequencies were cut so that the final signal had only lower frequencies than those of the original signal. For this purpose, it was necessary to select an appropriate wavelet, its scale, decomposition level, and the method of selecting the frequency cut-off threshold. The filtering (smoothing) of high-frequency signals is based on eliminating components with frequencies above a certain signal power threshold. In each step, the filtering removed higher-frequency components, such that the final signal retained only lower frequencies compared to the original. Therefore, an appropriate wavelet, its scale, the level of decomposition, and a method for selecting the cutoff threshold must be chosen [62]. Denoising signals using wavelet decomposition requires the selection of an appropriate mother wavelet. It should be characterized by a sufficient number of vanishing moments, depending on the complexity of the signal and the nature of the noise. A commonly used family for such purposes is the Daubechies wavelets (dbN--where N indicates the level). The higher the level, the greater the number of vanishing moments, which enables the approximation of more complex signals and higher order polynomials [61; 62]. For example, the db1 wavelet with one vanishing moment can approximate a polynomial with one term, db2, with two terms, and so on. The more complex the waveform, the higher the degree. In our study, the exact approximated function is unknown; however, based on the signal's behavior, a high polynomial complexity can be assumed. Therefore, the wavelet level was set to db8. Additionally, the higher the order of the dbN wavelet, the smoother its shape is. When denoising a signal using DWT, the signal is decomposed into successive components (a decomposition level of n = 5 was chosen as the optimal), each representing increasingly lower frequencies. To perform denoising, some of these high-frequency components must be removed.