A New Type of Bipartite Random Graph as a Transform of Seismogram and Its Potential for Organizing Seismic Databases

This paper proposes a method to reduce seismogram variability as a determining factor in its interpretation, processing, and clustering. By introducing the concept of single fluctuations (SFs), the seismogram can be parsed into a sequence of random impulses with subsequent ordering. This rearrangement of SFs, if they are assigned by positive integers, represents the formal mapping of a regular string of integers into a random one, which can be represented with a bipartite random graph (bigraph). Due to its specific randomness, such a bigraph is considered a new type of random balanced bigraph. The R-envelope and RQ-envelope, its equidistant version, are defined by tracing the peak envelope over ordered SFs. The equivalence and complementariness of the RQ and bigraph are considered and discussed, forming a combined characteristic of the seismogram. The R/RQ provided a considerable reduction in seismogram variability, which was confirmed by creating and analyzing an ensemble of RQ from several seismograms. In the RQ domain, distance is defined as a possible basis for metrics and clustering, but the ensemble variability was quite narrow and not as suitable for this purpose. Otherwise, the ensemble shows high redundancy hidden in the seismogram population. As for the bigraph, the mesh of its edges is structuralized in bundles, forming a skeleton, which reflects the internal structural content of the seismogram. The distance over the domain of bigraphs is proposed to show the possibility of clustering. This means that only a combined RQ and bigraph provides a suitable frame for seismogram representation with reduced variability and, thus, the potential for more effectively organizing seismic databases and a deeper interpretation of seismograms; therefore, RQs and bigraphs can be considered as a transform of a seismogram. Many aspects of these concepts are thoroughly discussed. The similarity between concepts of SF and wavelets is briefly considered. This very complex theme is new and promises broad further research. All issues considered in the paper are abundantly illustrated.