# Seismogram Rearrangement as a Perspective Basis for Defining New Attributes
## 2 Materials and Methods
### Attributes Based on Rearrangement of EFs/SFs
#### 2.6.1 Issue of Seismogram Symmetry  
Formally considered, the seismogram fluctuates around the \(t\)-axis. Physically, the \(t\)-axis is a reference, around which seismic waves compress or extend the medium. It would be interesting to examine seismogram symmetry with respect to the \(t\)-axis. The files used for this purpose are recordings of acceleration. The simplest method to do that is to compare peak envelopes, \(A^{+}(t)\) and \(A^{-}(t)\), drawn over positive and negative peaks of seismogram waveform, i.e., over peaks of EFs (Figure 2). The difference between these two plain envelopes apparently exists, but they are very indented for a reliable visual assessment. A more regular picture of symmetry is attained by the R-envelope, both for positive and negative SFs, see Figure 3, and see [10] \"Figure 20\". As can be seen, there are small differences between the envelopes \(R_{+}(\tau)\) and \(R_{-}(\tau)\) so that one can speak of an invariant on the population of seismograms [10]. Additionally, one can complain about the small variation range to be useful for qualifying and quantifying the difference between the two envelopes in order to describe the seismogram symmetry. The cause is simple: rearrangement of SFs equalizes differences, putting similar SFs face-to-face. The seismogram conveys the influence right by its waveform, not by arranged SFs, but by a sequence of EFs. Finally, the chief factor by which the earthquake acts on the environment are transitions, which means that a more proper structural element would be the EF than the SF. It seems that it would be more proper to create the R-envelope by arranging the EFs, but it is not simple as in the case of SFs, since the EF is ambiguous, having two randomly different maximum amplitudes, and makes the ordering by peak amplitude uncertain. Then, the better criterion for ordering would be the EF energy: firstly, to calculate the energy of each EF, then to order the EFs by increasing/decreasing energy, and then to draw the R-envelope over maximum amplitudes of respective positive and negative peak amplitudes. This approach is the square-linear combination. Of course, such an envelope would not be labeled by R, but by, e.g., \(\mathrm{R_{E}}\) (with E for energy). A linear-linear version of this energy approach is described in Figure 8. The EFs are ordered twice: once by positive peak amplitudes of EFs (not SFs) and then by negative ones. Each time, the correspondent R-envelope is drawn over the peaks by which the sorting is performed (Figure 9). The quantized durations \(\Delta\)s make the time variable \(\tau\) irrelevant, so a new index variable \(n\) can be introduced, and all the envelopes from Figure 9 are rewritten as \(R_{\mathrm{Q}}{}^{+}(n)\), \(N_{\mathrm{Q}}{}^{+}(n)\), etc. Two key elements should be noted in Figures 8 and 9. Instead of the structural axis of the EF, already introduced in Figure 5, here, one deals with the skeleton of EF, graphically explained in Figure 8. The other one is the quantizing operation over EF skeleton durations (Figure 9a), and its role is apparent--to minimize the influence of EF shape variability on the result. Consequently, after the quantization, the creation of the R-envelope is meaningless, but not the creation of the quantized envelope labeled by \(R_{Q}\). The positive \(R_{Q}^{+}(\tau)\) is monotonic and topologically identical to the corresponding \(R^{+}(\tau)\) envelope, i.e., the R\({}^{+}\)-envelope. The reverse of the \(R_{Q}^{+}(\tau)\) is the \(N_{Q}{}^{+}(\tau)\), which clearly shows the asymmetry of ordered EFs. The similar holds for \(R_{Q}{}^{-}(\tau)\) and \(N_{Q}{}^{-}(\tau)\). Figure 8: (**up**) Several EFs are taken from Figure 3, and for each one, peak amplitudes of belonging EF\({}^{+}\) and EF\({}^{-}\) are drawn. (**down**) The base points of peak amplitudes of each EF are connected by a horizontal line to make a skeleton of EF, in which the horizontal line represents an effective duration of EF. Notice their variability. Dim gray solid lines just indicate the EFs shapes from upper figure.