# Seismogram Rearrangement as a Perspective Basis for Defining New Attributes
## 4 Discussion
### R-Envelope as Functional Attribute or Transform of Seismogram  
It is said that the R/R\({}_{\text{Q}}\)-envelope is a functional attribute. This claim has the reverse. The R/R\({}_{\text{Q}}\)-envelope, represented by a polygonal continuous line drawn over SFs peaks, really seems to have a continuous function. However, each segment of a polygonal line is drawn (interpolated) between adjacent peaks. Thus, the segment does not carry any additional information which is not already contained in peak coordinates. It serves only for visual impressions to human vision. In that sense, the R/R\({}_{\text{Q}}\)-envelope actually represents a group-parametric attribute of the original seismogram, not the functional one, unless polygonal line segments are not taken as a fact of continuity. The same holds for the seismogram: a string of several thousand amplitudes (samples) expressed as real numbers, which peaks the plotting software connects by linear segments, creating a polygonal line just for the needs of the software user (human). In plain words, a human sees the polygonal line where the computer'sees' a string of numbers. The R/R\({}_{\text{Q}}\)-envelope actually does not exist as a function, nor the other 'functions' derived from the R/R\({}_{\text{Q}}\)-envelope, but it is very useful to treat it formally as a function, not only for the sake of human vision, but for mathematical reasons because in such case it is a single object to manipulate with, and not a string of numbers, i.e., a collection of separate mathematical objects. It is now quite clear that there is no necessity for any R/R\({}_{\text{Q}}\)-envelope in defining the seismogram duration. R-Envelope and Spectrum Issues  
There is an analogy between the spectrum of the raw seismogram and the rearranged seismogram since both are ordered sequences of elements (SFs and spectral components), although the order is the result of different operations (plain sorting in size and the Fourier transform). In that sense, the R/R\({}_{\text{Q}}\)-envelope, as the main product of seismogram rearrangement), can be considered as a kind of its transform. This issue is also considered in [10]. In the case of analytic function, the FT transforms one form into the exact another, and vice versa, showing its cyclic ability. For non-analytic cases, usually expressed as numerical sets, the FT (i.e., DFT, FFT) transforms one set into strictly another one. With digital computers, even analytic cases are processed as numerical sets. It means that the FT transform operates between two numerical domains, which are actually one and the same domain. This further means that this is not a transformation since it circles within one domain. The same situation then occurs with the DFT: it transfers one set of numbers into another one. The human is the factor that prescribes the meaning to bare numbers prior to and after the transform action. If one would ask the computer, could it tell the difference? When parsed into SFs (raw or approximated by triangles), the seismogram becomes a finite string of real numbers (peak amplitudes and durations of SFs). When indexed,it becomes a string of integers. The above expression 'vice versa' designates the cyclic ability of a transform, to equally transfer mathematical objects between two domains. The R/R\({}_{\text{Q}}\)-transform shows limits of such ability. At the integer level, it works completely. At the real-number level (maximum amplitudes or SFs' peaks), it also works, with an addition of another set of real numbers--positions of peaks along the \(\tau\)-axis. On the SF level, it does not work since it cannot return SFs shapes from R/R\({}_{\text{Q}}\)-envelope.