## 2 Materials and Methods  
### Seismogram as a Sequence of Random Impulses  
As stated in the Introduction, a seismogram is the result of many complex processes taking place in the so-called \"information chain\" of seismology, from the earthquake source to the recording place.  
Considered only by its appearance, the waveform, a seismogram is a fluctuational transient of certain mechanical quantities (acceleration, velocity, displacement), where the waveform looks as if it is composed of random impulses [2] \"Vol. 1, paper 1.03.1.3, pp. 73, 131\", with random amplitudes, durations, and shapes. Although the term \"fluctuation\" (to fluctuate) refers to the irregular rising and falling (change) of a physical quantity, we will further formalize this term and introduce the concept of elementary fluctuation (EF), consisting of \"ups\" and \"downs\", separated by crossings of a certain axis (usually the \(t\)-axis). Hence, the seismogram is considered a sequence of EFs since an alternating pattern, although random, occurs in its waveform.  
In this sense, the EF is usually composed of two subsequent alternate random impulses, which we will call single fluctuations (SFs).  
To put it more clearly:  
\[\text{EF}^{(+)}=\text{SF}^{(+)}+\text{SF}^{(-)}\qquad\text{EF}^{(-)}=\text{SF}^{ (-)}+\text{SF}^{(+)} \tag{1}\]  
Therefore, the EF, or the SF, is the basic structural element of the seismogram (see Figure 1).  
For numerical simplification, we express the complex waveform of SFs with two parameters: the maximum amplitude of the SF and its duration (Figure 1), which, geometrically, is equivalent to the approximation of SFs by respective triangles, either symmetrical or asymmetrical (Figure 1). In other words, a seismogram is represented by a non-equidistant sequence of amplitude samples and is a sort of non-uniform sampling.  
When we speak of geometry, we will distinguish intrinsic SFs and approximated (by triangles) SFs, which are labeled as SF and SF(A), respectively.  
The concept of the EF or SF enables the parsing of a given seismogram into a sequence of SFs, which is a very important operation. Notice that, from the point of view of this research, it is the same as if the whole seismogram waveform is parsed directly into SFs or, after an interstep of approximation, into respective triangles.  
After segmentation, the seismogram is actually divided (separated) into two sequences of SFs (or respective triangles SF(A)s), one comprising only SFs with positive amplitudes and another with negative amplitudes. This can be described in terms of the set theory and is a nice example of set theory application.  
The original set of SFs is divided into two disjoint subsets:  
\[\{\text{SF}\}=\left\{\text{SF}^{(+)}\right\}\,\cup\,\left\{\text{SF}^{(-)} \right\}, \tag{2}\]  
\[\left\{\text{SF}^{(+)}\right\}\,\cap\left\{\text{SF}^{(-)}\right\}=\varnothing, \tag{3}\]  
Figure 1: **(left)** A fragment from the seismogram of the RSN 1616 event, Düzce, Turkey, shows the complex random behavior of the seismic process. Units on axes scales are in g and in s. The trace in red color indicates the concept of elementary fluctuation (EF) introduced in this paper. **(right)** The EF can be parsed into two single fluctuations (SFs) as a pair of positive (red line) and negative (blue line) random impulses. **(far right)** Geometrical approximation of SFs by symmetric (sym) and asymmetric (asym) triangles.  
As a next step of the procedure, elements of each of these sequences are assigned with integers:  
\[\left\{\mathrm{SF}^{(+)}\right\}\,=\left\{\mathrm{SF}_{i}^{(+)}:i\in I\right\}, \tag{4}\]  
where \(I\) is the index set (Figure 2) (see, e.g., [36]).  
Similarly, these relations can be easily rewritten for triangles approximating the SFs.