Measurement of the dimension spectrum ƒ(α): Fixed-mass approach

Badii, R.; Broggi, G.

doi:10.1016/0375-9601(88)90784-0

Cited by 49 publications

(21 citation statements)

References 21 publications

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“…For more details see BADII andBROGGI (1988) andHIRABAYASHI et al (1992). According to BADII and BROGGI (1988), the fixed-radius method works better for q > 1, while the fixed-mass method gives better results for q £ 1.…”

Section: General Features Of Vrancea (Romania) Region Seismicitymentioning

confidence: 99%

Multifractal and Chaotic Analysis of Vrancea (Romania) Intermediate-depth Earthquakes: Investigation of the Temporal Distribution of Events

Enescu

Itō

Rădulian

et al. 2005

Pure appl. geophys.

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The Vrancea seismic region contains an isolated cluster of events beneath the Carpathian Arc Bend in Romania, dipping to about 200 km depth. Seismic activity mainly occurs at intermediate depths (h > 60 km). The main goal of the paper is to perform an in-depth, complex analysis of the occurrence times of these intermediate-depth events. We also try to show the versatility of the methods used to characterize different aspects of the seismicity evolution and to offer a user-friendly software toolbox to do most of the related computations. The earthquake catalog used in this study spans from 1974 to 2002 and includes only the intermediate-depth events. In the first part of the paper, we analyze the multifractal characteristics of the temporal distribution of earthquakes. The study reveals two distinct scaling regimes. At small scales we found a clear nonhomogeneous, multifractal pattern, while at large scales the temporal distribution of events shows a monofractal, and close to Poissonian (random), behavior. The multifractal behavior at small scales (minutes-hours) is shown to be clearly an effect of the ''short'' aftershock sequences that occurred after some major Vrancea earthquakes. In the second part of the paper we analyze whether our temporal series shows a persistent (or anti-persistent) long-term behavior, by using the Detrended Fluctuation Analysis (DFA) method. The results suggest that the analyzed temporal series of Vrancea earthquakes is a non-correlated process. In part three of the paper we seek to determine whether the dynamics of our earthquake system (described by the occurrence time of Vrancea earthquakes) is deterministically chaotic, deriving from a rather simple evolution law, or whether it is stochastic and is generated by a system that possesses many degrees of freedom. The results suggest that our signal is stochastic (probably does not possess an attractor). The limited time-span of the catalog and the analysis performed in this paper cannot rule out the emergence of an interesting, quasideterministic and low-dimensional structure in the case of major Vrancea earthquakes.

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Section: General Features Of Vrancea (Romania) Region Seismicitymentioning

confidence: 99%

Multifractal and Chaotic Analysis of Vrancea (Romania) Intermediate-depth Earthquakes: Investigation of the Temporal Distribution of Events

Enescu

Itō

Rădulian

et al. 2005

Pure appl. geophys.

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“…For example, for 5% error the minimum number of points needed is Nmin = 5. (73) If one were to use a value of r for which C(N, r) were of the order of 1/N2 and if one employed the Takens best estimator instead of the local slope, the minimum number of points needed for 5% error would be Nmin (1.25)m. It i possible to reduce this value even further if one's model for the strange attractor is, instead of an m-dimensional hypercube, …”

Section: Discretizationmentioning

confidence: 99%

“…This value is quite a bit different from that given in Eq. (73), which points to the sensitivity of these kinds of estimate to the assumptions in the model. It is not at all clear that the edge effect is a good model for making a priori estimates of errors that arise from dimension-estimation methods.…”

Section: Discretizationmentioning

confidence: 99%

Estimating fractal dimension

1990

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Fractals arise from a variety of sources and have been observed in nature and on computer screens. One of the exceptional characteristics of fractals is that they can be described by a noninteger dimension. The geometry of fractals and the mathematics of fractal dimension have provided useful tools for a variety of scientific disciplines, among which is chaos. Chaotic dynamical systems exhibit trajectories in their phase space that converge to a strange attractor. The fractal dimension of this attractor counts the effective number of degrees of freedom in the dynamical system and thus quantifies its complexity. In recent years, numerical methods have been developed for estimating the dimension directly from the observed behavior of the physical system. The purpose of this paper is to survey briefly the kinds of fractals that appear in scientific research, to discuss the application of fractals to nonlinear dynamical systems, and finally to review more comprehensively the state of the art in numerical methods for estimating the fractal dimension of a strange attractor.Confusion is a word we have invented for an order which is not understood.

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“…The box-counting method, the fixed-radius method and the fixed-mass method. The fixedradius method is a generalization of the correlation method and reported to performed better for determining D q for q > 0, but is unstable for negative q when the number of points is limited (Badii and Broggi, 1988;Hirabayashi et al, 1992). The fixed-mass method which is also a generalization of the correlation method is essentially akin to the fixed-radius one.…”

Section: Methodsmentioning

confidence: 97%

“…The fixed-mass method which is also a generalization of the correlation method is essentially akin to the fixed-radius one. According to Badii and Broggi (1988) and Hirabayashi et al (1992) the fixed-mass method is superior for determining D q for q ≤ 1 rather than the fixed-radius one. The main idea in the fixed-mass method is to determine the minimum radius R from a point i, that contains m points, R i (<m):…”

Section: Methodsmentioning

confidence: 98%