## 3 Materials and Methods
### Data Correlation
This is a crucial part of the process--setting an appropriate high-frequency cut-off threshold. The most commonly used thresholding methods are \"hard\" and \"soft\" thresholding methods. Hard thresholding involves zeroing out the high-frequency components while retaining the rest of the signal (19). \[a_{m}(x)= \begin{cases}1&\text{if }|x|\geq T\\ 0&\text{if }|x|<T\end{cases} . \tag{19}\]  
Figure 13: Schematic diagram showing discrete signal decomposition Legend: A—approximations at a set numerical deconstruction level, D—detail of the signal at a set numerical deconstruction level. Soft thresholding involves removing high-frequency components below a defined threshold, reducing the energy of intermediate frequencies, and retaining the low-frequency components (20). \[\rho_{T}(x)=\left\{ \begin{array}{ll}x-T&\text{if }x\geq T\\ x+T&\text{if }x\leq-T\\ 0&\text{if }|x|\leq T\end{array} \right.. \tag{20}\]  
Universal thresholding involves determining an appropriate threshold, which is given by \(T=\sigma\sqrt{2\log_{\text{c}}N}\), where \(\sigma\) is the estimated standard deviation of the noise, and \(N\) is the number of samples in the signal [63; 64]. In the case of signals from radon probes, the db8 wavelet was selected with the \"universal\" cut-off threshold and the \"hard\" rule. The db8 wavelet was continuous and allowed approximations similar to 4 th degree polynomials. Level 5 of the decomposition provided access to useful high frequencies without cutting the useful low frequencies (i.e., at a maximum of 24 h). According to [PERSON] and [PERSON] [65], the threshold was assumed as the \"universal threshold.\" The selected hard programming rule did not affect the amplitudes during decomposition at the individual levels (Figure 14). A comparison of the original signal (Figure 15A) and denoised signal (Figure 15B) indicated that the signal smoothed using wavelet decomposition was continuous at every point, even in the locations of sharp peaks from the radon probe. In order to obtain an appropriate sensitivity of the filter to the bends occurring in the signal, the operation of the filter (Figure 16) in the case of radon sequences was based on a 3 rd-degree polynomial. The 2 nd degree would be insufficiently sensitive, and the 4 th degree--oversensitive, causing either excessively long frequencies to be filtered or excessively short frequencies to be preserved. The frame length was set at 49 h, which corresponds to two days (the filter required an odd frame length, and therefore one hour was added), thus allowing the smoothing of periods below one day (Figure 17). Figure 14: Coefficients of discrete wavelet analysis on five levels. Explanation: D1-D5—details of the signal after decomposition using the discrete wavelet transform. A5 indicates approximation of the smoothed signal. Figure 16: Operation of the SGolay filter [66] with an example of fitting the quadratic function (red line) into the local data frame (blue points). Figure 15: Signal from probe SRDN-3 No. 2: original and denoised with wavelet decomposition (**A**), zoom on the area indicated with the frame (**B**). Explanations: blue line indicates input signal. Red line indicates smoothing.