# Seismogram Rearrangement as a Perspective Basis for Defining New Attributes
## 3 Results and Analysis
### Attributes from Rearranged and Ordered Fluctuations
#### 3.2.1 R-Envelope  
Real Cases  
Combining the principle, shown in Figure 3b,c, and a real case, seismogram RSN 1616 shown in Figure 2, the result of the whole procedure consisting of parsing into SFs, approximating the SFs by (asymmetric) triangles, ordering the SFs by peak amplitude, is represented on Figure 14.  
The R-envelope is a new characteristic of the seismogram. One could also call it the new waveform or the profile waveform of seismogram. It is a kind of seismogram's transform, although not so strictly defined as, e.g., the Fourier transform. Considered as a \"kind of transform\", it can also be considered as a \"kind of spectrum\", relying this claim on its arranged structure. Speaking in terms of attributes, it can be called the functional attribute of the seismogram. The R-envelope, in the notation \(R(\tau)\), is formally a function of (relative) time since it is created over structural elements of the seismogram, the SFs, which brought the time with, either in raw form, the intrinsic SFs, or in approximated form, labeled by SF\({}^{(\rm A)}\)s. If the time is quantized, as in Figures 8 and 9, it becomes irrelevant, and the \(R(\tau)\) becomes a discrete function \(R(n)\), with a neutral indexed variable, labeled by \(R_{\rm Q}(n)\) and called \(R_{\rm Q}\)-envelope with the letter Q to assign the quantization (see [10] for further details). Figure 13: (**left**) The EF consists of SF\({}^{+}\) (red solid line) and SF\({}^{-}\) (blue solid line). The period of its basic sinusoid 1, dark red dashed line, is \(T_{\rm E}=t_{3}{-}t_{1}\). With the SF\({}^{-}\) rectified (modulo operation), the two SFs from the EF are one next to the other, having the basic sinusoid 2 (orange dashed line) with the period \(T_{\rm S}=t_{2}{-}t_{1}\). The \(T_{\rm E}\) and \(T_{\rm S}\) differ in frequency (period) approximately by the factor 2. It is an additional reason to consider EFs in assessments of local features of the seismogram spectrum. (**right**) The spectrum of RSN 1616, taken from the PEER database, shows characteristic peaks. Both axes are with linear scales, the abscissa for the period (in s), and the ordinate is for pSa (in g), with scale increments Dx = 0.5 s and Dy = 0.02 g.  
Following the reference [10], the R\({}_{\rm Q}\)-envelopes of several earthquakes are calculated as a family of functions presented on the same diagram (Figure 15). Figure 14: (**a**) The raw seismogram RSN 1616, shown in Figure 2, is segmented into positive and negative SFs, which are approximated by asymmetric triangles (see Figure 4). (**b**) After rearrangement (ordering) of SFs, separately for positive and negative SFs, positive (red solid line) and negative (blue solid line) peak envelopes are drawn labeled by \(R^{+}(\tau)\) and \(R^{-}(\tau)\), respectively. By the black point line it is depicted the modulo of \(R^{-}(\tau)\) for better comparison between the two. The diagram (**b**) is depicted with 200% stretched scale than the (**a**), which is indicated by the green dotted vertical lines. Figure 15: For the list of seismograms given between diagrams (**a**,**b**), an ensemble of R\({}_{\rm Q}\)-envelopes, both positive and negative, is represented as the family of functions. Colors of event names are correspondent to lines colors. All seismograms are taken from the PEER seismic database. Individual diagrams are normalized in certain aspects and overlapped. The colors of curves on diagrams correspond with the colors of event names in the list. (**b**) The R\({}_{\rm Q}\)-envelopes from indicated seismograms are only overlapped and justified by maximum amplitudes without normalization. (**a**) The R\({}_{\rm Q}\)s are normalized by maximum amplitudes individually, where tails are additionally normalized by the smallest tail. The similarity between RQs in Figure 15a strongly suggests the existence of invariant in the population of seismograms (see about this issue in [10]). This similarity does not end with the RQ-envelope.