Symmetropy of earthquake patterns: asymmetry and rotation in a disordered seismic source

Nanjo, Kazuyoshi; Nagahama, Hiroyuki; Yodogawa, Eiji

doi:10.2478/s11600-006-0002-2

Cited by 3 publications

(7 citation statements)

References 18 publications

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“…On the other hand, to estimate symmetry and entropy of fracture process, we use discrete Walsh analysis and the concept of symmetropy. As details of the mathematical procedures were given in a previous paper (Nanjo et al, 2006;Nishiyama et al, 2008), only a brief outline is described below.…”

Section: Methods I: Fractal Analysis and Walsh Analysismentioning

confidence: 99%

“…We regard the AE events as circles with finite diameter. Following previous studies (e.g., Nanjo et al, 2000;2005;2006), the spatial patterns of AE were covered with 2 5 ×2 6 cells. If we find a part of or whole of one or more of circles in a cell of (i, j), then x ij = 1 , otherwise x ij = 0.…”

Section: Methods I: Fractal Analysis and Walsh Analysismentioning

confidence: 99%

“…This symmetropy approach is a useful tool for the quantification of anisotropy, asymmetry and entropic heterogeneity of fracturing patterns (Nanjo et. al., 2005;2006). One of the unique features of the fractal analysis is to enlarge or reduce the scale of observation windows.…”

Section: Introductionmentioning

confidence: 99%

“…Such a critical comparison of a large scale (a whole) versus a small scale (parts) will better enable us to find out an essential feature of a geometrical pattern. In the previous study (e.g., Nanjo et al, 2000;2005;2006), however, only a whole symmetropy of fracturing patterns has been considered. That is, the relationship between a whole symmetropy and a partial symmetropy (PS) has not been discussed.…”

Section: Introductionmentioning

confidence: 99%

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A new mathematical tool for analyzing the fracturing process in rock: Partial symmetropy of microfracturing

Yamasaki

Nanjo

2009

Physics of the Earth and Planetary Interiors

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We have shown that the symmetry of fracturing process in a macro scale can be quantified by using the concept of symmetropy (an entropylike measure of symmetry). Here we extend this approach to examining the symmetries in a range from small (partial) scales to larger (whole) scales: This approach is called partial symmetropy (PS). To check its applicability, we consider one illustrative example, the temporal change of the spatial patterns of acoustic-emission events in a well-documented rock fracture experiment. Our results are summarized as follows: (i) The PS enables us to distinguish the nucleation phases from the other phases such as the pre-nucleation and the propagation phases; (ii) The variation of the PS shows that the fracturing process is a associated with a type of phase transitions from the subcritical state to the critical state; (iii) The scale dependence of the PS reveals the presence of the sandwich structure that consists of order and non-order in the evolution of the fracturing pattern; (iv) Within the framework of nonextensive Tsallis entropy we develop the PS concept, and show that the degree of nonextensivity on a large scale increases immediately after the nucleation. The results shown

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Section: Methods I: Fractal Analysis and Walsh Analysismentioning

confidence: 99%

Section: Methods I: Fractal Analysis and Walsh Analysismentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A new mathematical tool for analyzing the fracturing process in rock: Partial symmetropy of microfracturing

Yamasaki

Nanjo

2009

Physics of the Earth and Planetary Interiors

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“…The entropy and the symmetry of discrete patterns can be estimated by using the discrete Walsh analysis (Yodogawa, 1982). In this paper, we use the results of the computer simulation for the cell sorting (Mochizuki et al, 1996), and the experimental data for spatial self-organization in mussel bed ecosystems (Koppel et al,200), because they are suitable for applying the discrete Walsh analysis used in the previous papers (Nanjo et al, 2006;Nishiyama et al, 2008;Yamasaki and Nanjo, 2009). Details are as given below.…”

Section: Datamentioning

confidence: 99%

Symmetry and entropy of biological patterns: Discrete Walsh functions for 2D image analysis

2011

Self Cite

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To quantify symmetry and entropy inherent in the discrete patterns such as spatial self-organization in cell sorting and mussel bed ecosystems, we introduce the discrete Walsh analysis. This analysis enables us to estimate the degree of the complicated symmetry, and to extract the symmetry from the pattern that seems to be asymmetric. The results obtained in this paper are summarized as follows. (I) The geometrical patterns of the cell sorting become systematic with the predominance of the particular symmetry. This implies that not only the entropy but also the particular symmetry can decrease in the biological process. (II) The magnitude of the symmetry is related to the absolute value of the adhesion, and the type of the symmetry is related to the sign of the adhesion. That is, centro-symmetry dominates in the cell sorting pattern caused by large negative adhesion, and double symmetry dominates in the pattern caused by large positive adhesion. (III) Spatial self-organization in mussel bed is accompanied by the decreasing of the centro-symmetry. This implies that the positive "adhesion" between mussel individuals increases with time. (IV) In the biological process, the Curie symmetry breaking occurs at intervals.

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Geometrical minimum units of fracture patterns in two-dimensional space: Lattice and discrete Walsh functions

Nishiyama

Nanjo²,

Yamasaki

2008

Physica A: Statistical Mechanics and its Applications

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Symmetropy of earthquake patterns: asymmetry and rotation in a disordered seismic source

Cited by 3 publications

References 18 publications

A new mathematical tool for analyzing the fracturing process in rock: Partial symmetropy of microfracturing

A new mathematical tool for analyzing the fracturing process in rock: Partial symmetropy of microfracturing

Symmetry and entropy of biological patterns: Discrete Walsh functions for 2D image analysis

Geometrical minimum units of fracture patterns in two-dimensional space: Lattice and discrete Walsh functions

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