# Seismogram Rearrangement as a Perspective Basis for Defining New Attributes
## 3 Results and Analysis
### Attributes from Rearranged and Ordered Fluctuations
#### 3.2.2 R-envelope Difference Function and Seismogram Symmetry  
A plain envelope is well known from amplitude modulation (AM), with a continuous high-frequency sinusoid carrier, in which positive and negative impulses are symmetrical in respect to abscissa (\(t\)-axis) so that it is sufficient to draw only an envelope over positive impulses, knowing that its negative counterpart is symmetrical. Moreover, circuits in AM receivers deal only with positive envelopes.  
Randomly alternating processes, such as speech and seismic signals, do not have symmetry in respect to abscissa, so it is natural to define a negative envelope as well. Asymmetry is preserved during seismogram rearrangement hence the envelopes \(R_{+}(\tau)\) and \(R_{-}(\tau)\) are not equal (see Figures 3, 14 and 15), and it is quite logical to define the difference function (Figure 16)  
\[\varepsilon(\tau)=R_{+}(\tau)-R_{-}(\tau) \tag{26}\]  
as a measure of seismogram asymmetry. The \(\varepsilon(\tau)\) is a functional attribute and provides an asymmetry profile of the seismogram.  
One-parametric measure (attribute) of asymmetry is defined simply as  
\[A_{\varepsilon}=\int_{0}^{T_{0}}\lvert\varepsilon(\tau)\rvert\,d\tau \tag{27}\]  
Figure 16: (**a**) The R-envelopes, positive \(R^{+}(\tau)\) (red solid line), and negative \(R^{-}(\tau)\) (blue dashed line), of seismogram RSN 1616 are drawn on the same diagram. For comparison, the modulo of \(R^{-}(\tau)\) is drawn by a blue solid line. (**b**) The difference function of two R-envelopes is drawn with a 500% extended ordinate scale for better insight in details.  
while cumulative functional attribute, analog to the Husid diagram  
\[A_{\varepsilon}(\tau)=\frac{\int_{0}^{\tau}\lvert\varepsilon(\tau)\rvert\,d\tau}{ \int_{0}^{T_{0}}\lvert\varepsilon(\tau)\rvert\,d\tau} \tag{28}\]  
provides a better picture of asymmetry distribution along seismograms. Note that the modulus is used instead of the square to avoid mutual canceling of positive and negative amplitudes in \(\varepsilon(\tau)\).  
Additional examples of the difference function is represented in Figure 17. In these cases, the R\({}_{\text{Q}}\)-envelopes are compared, giving \(\varepsilon_{\text{Q}}\)-functions.  
\[\varepsilon_{\text{Q}}(n)=R_{\text{Q}}^{+}(n)-R_{\text{Q}}^{-}(n) \tag{29}\]  
The three cases in Figure 17, RSN57, RSN70 RSN88, belong to the same event, San Fernando 1971, magnitude 6.61, recorded at the sites Castaic--Old Ridge Route, Lake Hughes #1, and Santa Felita Dam, respectively (further details see in [33]), show certain similarities. On the other hand, the similarities between RSN57, RSN79, and RSN286, with polarity excluded, seem to be closer than those between RSN57, RSN70, and RSN88. This suggests the necessity for further research before inferring any more founded claims. Additional cases for study can be found in [10]. In [10], the terms 'head' and 'tail' are used to describe different variability in the diagram of the R/R\({}_{\text{Q}}\)-envelope. It is much more transferred to the diagram of \(\varepsilon_{\text{Q}}(\tau)\).  
Another attribute that addresses the issue of seismogram symmetry proposed in Section 2.6.1 (Figures 8 and 9), applied on RSN1616, is presented in Figure 18, and defined by the difference function of random envelopes \(N_{\text{Q}^{+}}(n)\) and \(N_{\text{Q}^{-}}(n)\), called the random difference function:  
\[\eta_{\text{Q}}(n)=N_{\text{Q}}^{+}(n)-N_{\text{Q}}^{-}(n) \tag{30}\]