# Seismogram Rearrangement as a Perspective Basis for Defining New Attributes
## 2 Materials and Methods
### Order--Key Feature of the R-Envelope and of the Spectrum  
The seismogram, labeled with \(s(t)\), is a raw information about an earthquake, while its spectrum is a chief seismogram's functional attribute, usually given in earthquake databases, e.g., [33]:  
\[S(\omega)=\int\limits_{-\infty}^{\infty}s(t)\,e^{-i\omega t}dt=F[s(t)]\quad s (t)=\frac{1}{2\pi}\int\limits_{-\infty}^{\infty}S(\omega)\,e^{i\omega t}d \omega=F^{-1}[S(\omega)] \tag{21}\]Quite a decent survey of FT, elaborate but brief, can be found in [54]. Being the Fourier transform of seismic signal \(s(t)\), the spectrum \(S(\omega)\) is fully equivalent but not equal to \(s(t)\). Another notation, \(s\) and \(\hat{s}\), emphasizes their equivalence:  
\[s(t)\rightarrow\hat{s}(\omega)=s(t)\ \xrightarrow{F}\ \hat{s}(\omega)=(Fs)( \omega)=\int\limits_{-\infty}^{\infty}\left[e^{-i\omega t}dt\!\cdot\!s(t)\right] \tag{22}\]  
Since \(s(t)\) and \(s(\omega)\) are equivalent but not equal, they do not contain the same quantity of information (\(Q\))  
\[Q[s(t)]>Q[\hat{s}(\omega)] \tag{23}\]  
To get \(s(t)\) from \(S(\omega)\), one must add sinusoids (through the \(F^{-1}\)) to the spectrum values:  
\[s(t)=\frac{1}{2\pi}\!\int\limits_{-\infty}^{\infty}S(\omega)\,e^{i\omega t}d \omega=\int\limits_{-\infty}^{\infty}\left[\frac{1}{2\pi}|S(\omega)|d\omega \right]e^{i[\omega t+\theta(\omega)]}=\int\limits_{-\infty}^{\infty}\left[ \frac{1}{2\pi}|S(\omega)|d\omega\right]\cdot\cos[\omega t+\theta(\omega)] \tag{24}\]  
which are previously removed time components in operation (21). Whatever the \"raw\" domain, where one originally observes certain physical phenomena (time, length, angle, etc. ), for all of them, the transformed domain will always be the frequency, or the \(\omega\)-domain, because the Fourier operator itself contains sinusoids, either in complex (\(e^{i\omega t}\)) or in plain (cos\(\omega t\)) forms (see Equation (24)), which the most characteristic parameter is the frequency [55]. It might seem pretty obvious, but the \(\omega\)-domain (\(\omega\)-axis) is arranged by frequency. The same stands for the \(t\)-domain (the \(t\)-axis). In both cases, there is a continuous increase of numeric values along both axes. What makes the important difference between these two arrangements is the fact that a point on the \(\omega\)-axis is not a simple numerical quantity, as on the \(t\)-axis--it is the basic parameter of a sinusoid component--its frequency. Figuratively, each point at the \(\omega\)-axis hides a sinusoid waveform of assigned frequency since the term frequency corresponds only to sinusoid shape so that one can ascribe arrangement of sinusoid shapes along the \(\omega\)-axis, with respective frequencies, and make a parallel to the arrangement in the R-envelope, which is, however, simpler and more obvious. Moreover, there is a general tendency in the spectrum (amplitude or power, not in phase), nevertheless, its complexity, to steadily decrease with frequency, see, e.g., [54], so that one can treat the \(\omega\)-axis as the spectrum's horizontal asymptote. In similar way, the R-envelope not only steadily, but monotonically approaches the \(t\)-axis. This issue of R-envelope depends on several, mostly practical factors, and will be considered in Section 4. Although the R-envelope proposed in this paper, seems to be a continuous function (functional attribute), it is actually a group-parametric attribute (set of parameters) since it abstracts real shapes of SFs, and relies on peak amplitudes and durations of SFs, i.e., \"equivalent triangles\". So, it is not possible to reconstruct the original waveform of a seismogram just from this envelope (see Figure 3). A similar fact holds for the spectrum, which parameters need adding sinusoid waveforms to create the original signal. Also the spectrum can be represented as a group-parametric attribute, as the line spectrum, given by the Fourier series. If a seismogram is formally periodized [56], the result is a line (discrete) spectrum.