## 2 Materials and Methods
### Classical Spectral Methods  
Of the groups of mathematical methods listed above, the group 'time series analysis', especially spectral analysis, is important for solving cyclostratigraphy problems. A brief description of the spectral methods most commonly used in cyclostratigraphy is provided below.  
Any time series can be analyzed in terms of its description in the frequency domain. The classical way to detect frequency components in time series is Fourier spectral analysis. The importance of each frequency component in the time series is established using paired sine and cosine waves. Cyclostratigraphic data are discrete. For this reason, a discrete Fourier transform is applied. In a discrete Fourier transform, a time series is multiplied, point by point, by a cosine wave of a specific frequency. The results are summed and multiplied by a constant (\(2/N\), where \(N\) is the number of points in the data series). This calculation gives the average amplitude of the cosine frequency component. The calculations continue assuming half the spectrum exists as a mirror image of the actual spectrum at 'negative frequencies'. Since negative frequencies have no physical meaning for time series observations, the average amplitude must be doubled using a constant. The time series is then multiplied by a sine wave of the same frequency. The results are again summed and multiplied by a constant. For each frequency component investigated, the relative size of the average amplitude of the sine and cosine waves determines the average phase of the time series oscillations. Fourier transform can be considered as reorganizing time series data to a different location based on frequency.  
To summarize, the Fourier transform is a mathematical operation that takes any waveform and breaks it down into separate sinusoidal components with different frequencies and amplitudes. The components are then presented as peaks in the frequency spectrum.  
Spectral analysis of data series can be performed using the periodogram method. The periodogram is the square of the modulus of the amplitude of the Fourier spectrum. The Lomb periodogram is a frequency analysis technique for non-uniform data series. It is more suitable for cyclostratigraphic data than the Fourier transform. The Lomb periodogram is invariant concerning the time scale shift, and its statistical properties for non-uniform samples are equivalent to the Fourier transform properties for uniform samples. Frequency analysis by the Lomb method solves the problem of detecting oscillatory processes in data series. However, to study the evolution of the observed phenomena a time-frequency analysis is required. In this analysis, a subset of the sample is selected by a sliding observation window. When using an observation window, the data series is multiplied by a specific weight function [15].  
Time series spectra are often characterized by a continuous decrease in spectral amplitude with increasing frequency (red noise). Additionally, time series are often unevenly distributed in time, making it difficult to estimate their red noise spectra. The REDFIT mathematical method [16] is an advanced version of the simple periodogram that solves this problem by directly fitting the first-order autoregressive process to unevenly distributed time series. In this way, the method avoids interpolation in the time domain and its inevitable shift. REDFIT is used to test if peaks in the spectrum of a time series are significant against a background of red noise from a first-order autoregressive process.