## 2 Materials and Methods
### Ordering of SFs  
Having indexed sets of SFs, or SF\({}^{(\mathrm{A})}\)s, both positive and negative, the next operation is applied: the ordering of each SF\({}^{(\mathrm{A})}\) sequence by maximum amplitude. For clarity, the ordering is only applied to positive SFs, which is presented in Figure 3.  
A random sequence of SFs is converted into an ordered sequence of SFs, while the initial sequence of naturally ordered integers is turned into randomized indexes. This is obviously a kind of random mapping, which can be written as:  
\[\left\{\mathrm{SF}^{(+)}\right\}\,=\left\{\mathrm{SF}_{i}^{(+)}:i\in I\right\} \,\stackrel{{ F_{0}}}{{\rightarrow}}\,\left\{\mathrm{SF}_{j}^{(+) }:j=F_{0}(i)\in I\right\}\,, \tag{5}\]  
A similar relation can be written for triangles \(\left\{\mathrm{SF}^{(\mathrm{A})}\right\}\)  
\[\left\{\mathrm{SF}^{(A+)}\right\}\,=\left\{\mathrm{SF}_{i}^{(\mathrm{A+})}:i \in I\right\}, \tag{6}\]  
It is important to note that the approximation goes from the complex waveform of an SF down to an integer only, which is the ultimate level of approximation (simplification).  
Thus, the ordering maps one sequence of integers into another one. This function is another form of mapping. It seems to be a dead end. Reducing either SFs or SF\({}^{(\mathrm{A})}\)s to integers, one gets simple, bare numbers and loses information for their mapping contained in their max amplitudes. How, then, to obtain an ordered set?  
The answer is obvious, but it should be explicated: the amplitude information is hidden in the function that performs the mapping.  
Figure 2: (**a**) Seismogram waveform \(s(t)\) is composed of random impulses. (**b**) Approximation by (symmetric) triangles is geometrically equivalent to the numerical reduction of each random impulse to its maximum amplitude and duration. Further reduction involves assigning positive integers to positive and negative triangles. Zoomed-in examples of approximated and indexed SFs (positive: 1, 10, 19, and negative: 4, 20) are inserted around approximated waveform. The original SFs are depicted by dash line.  
There, one finds the meaning of twofold digitizing of the seismogram: behind the sequence of integers is hidden a corresponding sequence of real numbers (SFs maximum amplitudes), which serve for the ordering operation.  
Moreover, the mapping can be considered a sequential operation consisting of a pair of instantaneous acts: taking an input integer, applying the function, and obtaining the result--the output integer. Therefore, it is possible to represent the mapping, which includes input and output and the beginning and ending stages of the mapping process. This is the basis for representing such mapping with the bipartite graph, which shows all acting stages of function.  
If reduced to the level of pure integers, the mapping (function) can be written as:  
\[\forall i\in I:\;\;i\;\stackrel{{ F_{+}}}{{\rightarrow}}\;\;F_{+}(i)=j\in I, \tag{7}\]  
for positive SFs (or SF(A)s).  
This kind of mapping can be represented with bipartite graphs (shown in Figure 4).  
Figure 4: A bipartite graph (vertical configuration), by its edges, connects input and output nodes assigned by integers so that the input ordered sequence shuffles into a random one. The bigraph structure schematically follows the mapping indicated on Figure 3. Such random mapping relies on SFs in the original seismogram and on their random maximum amplitudes.  
Figure 3: Rearrangement (ordering) of positive SF(A)s, from approximated waveform (b), retaken from Figure 2, by maximum amplitude, yields the waveform (c\({}_{1}\)). This means the ordered sequence of integers is mapped into a random sequence of integers, or a random sequence of triangles is mapped into an ordered sequence of triangles. The scale on the \(\tau\)-axis of (c\({}_{1}\)) is zoomed in by 150% with respect to diagram (b).