## 3 Results
### Seismogram Conversion--R\({}_{Q}\)-Envelope
#### 3.2.2 Examples of R\({}_{\rm Q}\)-Envelope  
In the following, several seismograms are represented by the R\({}_{\rm Q}\)-envelope, which provides more complete insight into its main characteristics.  
In addition to Figure 14, all Figures 15-19, given below, confirm the previous impression that the R\({}_{\rm Q}\)-envelope, and the R-envelope as well, is of similar shape over the whole population of seismograms.  
Moreover, it can be said that an invariant exists across the seismogram population. The authors expected such a conclusion, but not to that extent. If a certain, large level of seismogram variability remains in the head of the R\({}_{\rm Q}\)-envelope, the tail is completely invariant. To further verify this inference, which might be an imp  
Figure 14: The R\({}_{\rm Q}\)-envelope of RSN70 SFERN (file taken from the PEER seismic database), drawn over only positive peaks. Two parts of the envelope are easily distinguished: the ‘head’, as non-stationary, and the ‘tail’, as gradually decreasing, almost monotonic. The gray bar indicates an arbitrarily positioned border between the two parts. The weighting function (window) \(w(n)\) is depicted by a dashed line to give the idea of tail trimming.  
Figure 15: R\({}_{\rm Q}\)-envelope of RSN57 (San Fernando), taken from the PEER database, focal mechanism: Reverse, Horizontal-1 Component, contains both positive (red line) and negative (blue line) R\({}_{\rm Q}\)s. The modulo of R\({}_{\rm Q}\) negative is depicted with a green line. At the right, just for quick visual reference, a small picture of seismogram waveform is inserted, with scale divisions: Dx=10s; Dy=0.2g (acceleration).  
Having in mind all preceding diagrams and in the context of R(\(\tau\)) symmetry, a general remark, even a recommendation, can be given; it suffices to analyze and process only the positive half of the seismogram, i.e., only positive SFs, thus saving of 50% of all necessary calculations in this approach.  
Figure 19: For the list of seismograms, given between diagrams (**a**,**b**), an ensemble of R\({}_{\text{Q}}\)-envelopes, both positive (solid lines) and negative (dashed lines), is represented as the families of diagrams. All seismograms are taken from the PEER seismic database. Individual diagrams are normalized in certain aspects and overlapped. Colors of curves on diagrams correspond with colors of event names in the list. (**c**) The R\({}_{\text{Q}}\)-envelopes from indicated seismograms are only overlapped and justified by maximum amplitudes without normalization. (**b**) The four R\({}_{\text{Q}}\)s from (**c**) are normalized by maximum amplitudes individually. (**a**) As in (**b**), where tails are additionally normalized by the smallest tail.  
Figure 20: Diagram \(\varepsilon(\tau)\), magnified 500% for better clearness, shows the difference between positive (red solid line) and negative (blue solid line) envelopes of a seismogram, where the negative one is overlapped (green solid line) to be compared with the positive one. The change in overlapping measure along the \(t\)-axis gives the idea of where to put the separation point between the head and the tail of the R-envelope.  
Also, it makes the previous consideration about attaining a singular result in converting a seismogram into its bigraph slightly irrelevant. However, this remark can be regarded only in practical value, while the mathematical significance of seeking the singular product in the transformation process of a seismogram remains in full significance.