# Seismogram Rearrangement as a Perspective Basis for Defining New Attributes
## 2 Materials and Methods
### Envelope of Rearranged Individual Fluctuations  
A plain (peak) envelope, drawn over a seismogram's peak amplitudes of fluctuations, clearly shows general variation along the waveform (Figure 2). The segmentation of the seismogram into single fluctuations (SEs) results in sequences of random shape impulses, both positive and negative ones. Recomposition of these sets of SEs, separately for positive and negative SEs, by increasing or decreasing peak amplitudes, will produce another rearranged waveform. Thus, one gets two ordered \"halves\" of the original seismogram, one composed of positive SFs and the other of negative SFs (see Figure 3). The rearrangement processes are schematically represented in [10] \"Figures 3, 5 and 11\". Figure 2: RSN 1616 seismogram is composed of miscellaneous shapes of EFs. It is divided into five segments for easier referencing. The plain envelopes, positive \(A^{+}(t)\) (red solid line) and negative \(A^{-}(t)\) (blue solid line), are drawn over the waveform only within the 3 rd and 4 th segments. The durations of SF\({}^{+}\) and SF\({}^{-}\) within the 1 st segment are measured and drawn by red (positive SFs) and blue (negative SFs) lines and shown downwards with a zoomed-in time scale. The insert showing an EF (green solid line) designates durations of corresponding SF\({}^{+}\) and SF\({}^{-}\), where arrows (dashed red and blue lines) graphically explain the procedure. Now, it is routine to draw the R-envelope along arranged SFs, both over positive and over negative SF sequences [28,52] \"Figure 3\", which contains new information about seismogram. Also, the previous correlative content in the original seismogram is heavily disrupted by recomposition, but new information appeared, telling us something new about fluctuating content in the seismogram, which is much more easily perceived than from the raw one. Therefore, the R-envelope can be designated as a new functional attribute of the actual seismogram, which also provides a basis for defining other new parametric attributes. The (peak) envelope \"hovers\" over amplitude peaks along a waveform, making the shape of fluctuations irrelevant. This means it is sufficient to replace whatever complex form of the SF by only two numbers, one for peak amplitude and another for SF duration, which in geometrical terms means a triangle replacing (or approximating) the respective SF (see Figure 3c). The replacement of a SF by the approximating triangle is explained in more detail in Figure 4. Similar idea is applied to the source time function \(S(t)\) for many earthquakes having body-wave magnitudes less than 6.5, which often can be adequately represented by a simple triangle-shaped pulse, where the triangle width is related to the corner frequency and fault length [7] \"article 1.05.6, Figure 20, pp. 190-182\". In the case of approximation by triangles, one can use symmetric and asymmetric forms (Figure 4) [10]. There are also other approximating shapes, but no further elaboration of this issue will be given here. The triangle approximation of any shape of SF, whatever complex, evidently facilitates numerics about the R-envelope. This is equivalent to a sort of non-uniform sampling of seismograms, where samples are peak amplitudes of SFs. To put it more formally, an elementary fluctuation EF, within a seismogram \(s(t)\), is decomposed into two single fluctuations, SF\({}^{+}\) and SF\({}^{-}\), and their waveforms \(sf_{m}(t)\) are approximated by corresponding triangles \(\varphi_{m}(t)\), which are derived from generative (isosceles) triangular impulse \(i(t)\), scaled by respective parameters \(p/n\) (amplitude), \(b\) (duration factor), and \(t_{m}\) (time shift):  
\[