# Seismogram Rearrangement as a Perspective Basis for Defining New Attributes
## 2 Materials and Methods  
### Elementary Fluctuation--Formal and Real Meaning  
The term \"fluctuation\" (to fluctuate) is irregular rising and falling (change) of a physical quantity in an alternating pattern, although random, which is consisted of \"ups\" and \"downs\", separated by crossings of a certain axis (mostly the \(t\)-axis), or some constant value, or even some monotonic trend curve (e.g., change of stock price or currency exchange rate). So, one can see some regularity in such irregular behavior: alternating positive-negative pattern, in brief, random cyclic behavior. In this research, the term \"elementary fluctuation\" (EF) is defined, and the seismogram is considered a sequence of EFs. Further, the EF is composed of two subsequent alternate random impulses, which are called single fluctuations (SF), as in [10]. So, the EF, or the SF, is the basic structural element of a seismogram. When the fault slips, the two sides of the fault move in opposite directions, which creates a different polarity of the first motion P in different directions (see [34] \"p. 217, Figure 7.3\"), so one should distinguish two conventions in defining the EF composition: (1) the leading impulse is positive. Hence the other is negative, and vice versa (Figure 1). To put it more clearly [10],  
\[\text{EF}^{+}=\text{SF}^{+}+\text{SF}^{-}\qquad\text{EF}^{-}=\text{SF}^{-}+ \text{SF}^{+} \tag{8}\]  
The EFs are separated by zero-crossings (Figure 1), the points where the random waveform crosses the actual axis, \"the \(t\)-axis, or some constant, or even some monotonic trend curve\". Within repetitive fluctuational patterns one can distinguish regular and irregular fluctuations. Regular EFs can mean quasi-periodicity, while irregular EFs mean just repeating a general pattern [35]. Figure 1: A few examples of EFs (depicted as pairs of positive and negative SFs) were extracted from a fragment of RSN 1616 seismogram (**a**). The cases (**b**,**d**) are of positive polarity (the leading SF is positive), while the case (**c**) is inverted, i.e., of negative polarity (the leading SF is negative). Positive SFs are in red solid line while negative SFs are in blue solid line. The generation of a seismogram is caused by processes during fault rupture in the hypocenter, which picture is conveyed by seismic waves to the seismograph location. In-between, seismic waves propagate through a (spherically) multilayered, inhomogeneous, anisotropic medium, which yields intricate combination of many reverberations of initial waves [8] \"Ch. 2\", [36]. Even with the seismic kinematics (ray formalism), the picture at the site remains pretty entangled. Standard Nomenclature of Seismic Phases speaks for itself, and the distribution of seismic sources as well [37]. Global Earth models are just statistical pictures of real mantle structure for study of teleseismic events. Top-down mantle convection brings much more complicated models and, hence, propagation pictures [7; 38] \"article 1.16, Figure 12, p. 555/576\". At closer distances, propagation conditions can be considered as even more complex, caused by the usually very irregular structure of lithosphere, since the main propagation in these cases takes place mainly in the crust [8] \"Ch. 2; article 1.08.7, Figures 5-12, pp. 260-281\". It is not only a matter of internal but also of external irregularities since there are effects of the Earth's surface topography [7] \"article 4.18.6.13, Figure 15, p. 2453\". The events observed at closer distances also send seismic waves toward all seismographs around the world, where usually are recorded only the basic transients. For teleseismic observations, subsequent reverberations are also recorded. This research is restricted only to close-range earthquakes. The lithosphere is a layered structure, often corrugated (due to tectonic movements), where basins (bedrocks) are infilled with stratified deposits (soil). Bedrock-soil boundaries and soil surfaces affect the propagation of seismic waves in several ways, causing reflections, refractions, and scatterings.