# Seismogram Rearrangement as a Perspective Basis for Defining New Attributes
## 1 Introduction
### Envelope of Seismogram--Issue of Complexity  
If one draws the peak envelope over positive elementary fluctuations (EF) of a seismogram, one gets better insight into peak amplitude variation, which would not be that much visible through mere spikes (see, [14; 28]). It has something to do with gestalt principles of perception--if spikes along a seismogram are not so dense as necessary, one's vision cannot gather them into the shape to perceive a variation. Just the fact that the envelope is drawn as a line over the peaks, making visual continuation, helps in the perceptive grouping of impulses [29].  
The envelope is a widely used attribute of seismograms or other signals. Besides the plain (peak) envelope, there is also the envelope defined as a mixture of a signal itself and its Hilbert transform [11]  
\[E(t)=\sqrt{s^{2}(t)+s_{H}^{2}(t)}\quad s_{H}(t)=H[s(t)]\ \{=Hs(t)\}\ \{=H(s)(t)\} \tag{5}\]  
where various labels of the operator's action are quoted and direct and smoothed envelopes as well.  
Apart from the information about P- and S-phase, seismologists often focus on the characteristics of seismogram envelopes (rms envelope in [30]). Extensive use of envelopes can be found in many papers [29; 31]. Another interesting kind of envelope is the signal-to-noise ratio:  
\[STA(t)=\frac{1}{T_{STA}}\int\limits_{t}^{t+T_{STA}}SNR(\tau)\,d\tau\quad LTA(t )=\frac{1}{T_{LT}A}\int\limits_{t-T_{LT}A}^{t}SNR(\tau)\,d\tau \tag{6}\]  
which is calculated for short- and long-term averages (\(STA\), \(LTA\)) to detect phases as [32]  
\[[STA-LTA]_{C}(t)=\int\limits_{0}^{t}\left[STA(\tau)-LTA(\tau)\right]d\tau \tag{7}\]  
Seismograms are recorded for various physical quantities (acceleration, velocity displacement), orientations (e.g., Hor-1, Hor-2, and Vert, in PEER database nomenclature, or equivalent ones), and filtered (low-, band, or high-frequency) [33].  
It is obvious that the envelope, as an attribute of seismogram, gains in importance with waveform complexity or fluctuational density. Different recordings (acceleration, velocity, or displacement) of the same seismic event at the same site show different waveform densities [33], and the same record is different with different filtering ([7] \"article 1.01, p. 22, Figure 13\").  
The seismogram gains in complexity with propagation distance, which is documented in many references [29], but the distance is not the only cause. Just to mention processes during fault rupture, the size, and geometry of the fault, emitted (delivered) energy, etc.  
Seismic processes, in the whole range from the fault to the site, are 'buried', so they are difficult for open study. In Section 2, a brief analysis of the specific open 'fault' mechanism of the Chelyabinsk event is given, and the following consequences, are as an illustration of said complexity.  
Rise of seismogram complexity with the distance has its coin reverse. Comparison of accelerograms for a range of event sizes, chosen to have a hypocentral distance of approximately 25 km, shows an increase of complexity with magnitude ([7] \"article 4.18, Figure 4, p. 2437\").  
In one word, the envelope seeks a critical \"mass\" of a waveform to justify its applicability. Otherwise, if a seismogram waveform is heavily and randomly indented (contains only a few fluctuations), its envelope is of little use (e.g., displacement record).  
With all these various types of envelopes, it seems there is no room left for a new one. Nevertheless, in this article it is proposed a new envelope, better to say a new type of envelope, which is neither the result of some complicated mathematical transformation, such as, Fourier or Hilbert, nor it is that simple to stem from the raw waveform. The proposed new envelope is mainly based on the rearrangement (recomposition) of single fluctuations (SFs) from a raw seismogram into a new sequence, ordered by increasing or decreasing peak amplitudes. Because of that, the new envelope is named as the R-envelope (from \"Rearranged\", \"Recomposed\").