## 2 Materials and Methods
### New Type of Bipartite Graph as a Transform of Seismogram  
#### 2.3.1 Twin Bipartite Graph  
Twofold Result of Transform  
The previous section considered representation for only positive SFs (or SF(A)s) by corresponding bipartite graph (Figures 3 and 4), mainly for the sake of procedure illustration. This means that a bipartite graph is created only for half of the seismogram. The same should be performed for negative SFs. Ordering and mapping (Figure 5) produce an additional bigraph, which means that the (whole) seismogram is represented by a pair of bipartite graphs (Figure 6). Indexing SFs reduces their usually complex waveforms to integers, which means that the entire seismogram's intricate waveform is described by two sequences of integers. In such a reduction of random impulse shapes, only the maximum amplitudes of SFs remain important as a factor for ordering purposes. The seismogram can be considered as digitized in a twofold manner:  
* As a sequence of integers;
* As a sequence of real numbers (maximum amplitudes of SFs, or of approximating triangles SF(A)s), whose positions within the sequence are implicitly marked by integers, i.e., indexes. Splitting the whole seismogram into positive and negative SFs resulted in two separate bigraphs, one unique result of this procedure. Figure 5: Complete rearrangement of (approximated) seismogram from Figure 2 involves both positive and negative SF(A)s, whose mappings are clearly indicated by arrows. Notice that the positive and negative SF(A)s are indexed in the same manner, while the respective mappings produce different random sequences of integers. Summed durations of positive and negative SF(A)s are different. To overcome this inconsistency, one simply can consider them as united and transpose them into 'the twin bigraph' (Figure 6). There may be the impression that this is only semantic because they remain separated. However, the two bigraphs are not as separated as it may seem from Figure 6. In the background, they are quite correlated since they are parts of a singular seismogram. Therefore, one can treat the 'double' result of these operations as a whole, as a singular mathematical entity, even though it comprises two actually separated bigraphs. Those two, to some extent, mirror each other so that all operations applied to the seismogram can be treated as a kind of transform. \[\text{seismogram waveform}=s(t)\xrightarrow{\mathcal{F}}\ (BG_{+}\oplus BG_{-}), \tag{8}\]  
The operator labeled by \(\mathcal{F}\) stands for this transformation. The unity of two bigraphs, i.e., the twin bigraph, is labeled by the symbol \(\oplus\), lent from XOR operation (addition modulo 2). Using the symbol \(\oplus\) for such a purpose does not mean it executes this operation between two bigraphs strictly, but symbolizes the unity of these two mathematical objects and points out the fact that the two bigraphs should be considered as one. In addition, the operation \(\oplus\) brings with it some combinatorial unifying of SF\({}^{+}\)s and SF\({}^{-}\)s, defined by its truth table, particularly for meaning \"one or the other but not both\", which additionally vindicates the handy use of strictly defined operation.