Multifractal large number of drops limit in rain

Lilley, M.; Lovejoy, S.; Desaulniers-Soucy, N.; Schertzer, D.

doi:10.1016/j.jhydrol.2005.11.063

Cited by 32 publications

(28 citation statements)

References 60 publications

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“…The most appropriate manner to describe the discrete nature of rainfall is still the topic of ongoing debates (e.g., Kostinski et al, 2006;Lilley et al, 2006;Jameson, 2007). However, that the seemingly conflicting approaches of generalized Poisson process models versus multi-fractal models do not necessarily have to exclude each other has been argued by Fabry (1996) and more recently theoretically investigated by Schertzer (2006, 2008).…”

Section: Discussionmentioning

confidence: 99%

“…The hypothesis supporting this reconciliation is that gravitational mixing (associated with the differential fall speeds of the raindrops) tends to lead to homogeneous rainfall at the smallest spatial scales, whereas (fractal) scaling regimes exist at larger scales. Indeed, spatial distributions of raindrops measured using stereo-photography indicate that the multi-fractal character of rain would be more clearly detectable in the spatial distribution of the liquid (rain)water content than in the spatial distribution of the drops per se -which has been found to be relatively homogeneous at small scales (DesaulniersSoucy et al, 2001;Lovejoy et al, 2003;Lilley et al, 2006).…”

Section: Discussionmentioning

confidence: 99%

“…Later, Zawadzki (1995), Lavergnat and Golé (1998) presented fractal analyses of the (one-dimensional) temporal distribution of raindrop arrivals at the ground (such analyses were recently challenged by Larsen et al, 2005). Desaulniers-Soucy et al (2001), Lovejoy et al (2003) and Lilley et al (2006) reported on multi-fractal analyses of (three-dimensional) spatial distributions of raindrops measured using stereo-photography. Here we concentrate on the blotting paper experiment of LS, which has lead to a lively debate in the scientific literature concerning the statistical significance of the supposed fractal nature of the microstructure of rain (Jameson and Kostinski, 1998;Jameson and Kostinski, 2001b;).…”

Section: Introductionmentioning

confidence: 99%

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Edge effect causes apparent fractal correlation dimension of uniform spatial raindrop distribution

Uijlenhoet

Porrà

Sempere‐Torres

et al. 2009

Nonlin. Processes Geophys.

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Abstract. Lovejoy and Schertzer (1990a) presented a statistical analysis of blotting paper observations of the (twodimensional) spatial distribution of raindrop stains. They found empirical evidence for the fractal scaling behavior of raindrops in space, with potentially far-reaching implications for rainfall microphysics and radar meteorology. In particular, the fractal correlation dimensions determined from their blotting paper observations led them to conclude that "drops are (hierarchically) clustered" and that "inhomogeneity in rain is likely to extend down to millimeter scales". Confirming previously reported Monte Carlo simulations, we demonstrate analytically that the claims based on this analysis need to be reconsidered, as fractal correlation dimensions similar to the ones reported (i.e. smaller than the value of two expected for uniformly distributed raindrops) can result from instrumental artifacts (edge effects) in otherwise homogeneous Poissonian rainfall. Hence, the results of the blotting paper experiment are not statistically significant enough to reject the Poisson homogeneity hypothesis in favor of a fractal description of the discrete nature of rainfall. Our analysis is based on an analytical expression for the expected overlap area between a circle and a square, when the circle center is randomly (uniformly) distributed inside the square. The derived expression (πr 2 −8r 3 /3+r 4 /2, where r denotes the ratio between the circle radius and the side of the square) can be used as a reference curve against which to test the statistical significance of fractal correlation dimensions determined from spatial point patterns, such as those of raindrops and rainfall cells.

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Section: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Edge effect causes apparent fractal correlation dimension of uniform spatial raindrop distribution

Uijlenhoet

Porrà

Sempere‐Torres

et al. 2009

Nonlin. Processes Geophys.

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“…All Rights Reserved. [e.g., Schertzer and Lovejoy, 1987;Schmitt et al, 1992;Tessier et al, 1993Tessier et al, , 1996Olsson and Niemzcynowicz, 1996;Harris et al, 1996;Lovejoy et al, 1996Lovejoy et al, , 2008Deidda, 2000;Lilley et al, 2006;Kantelhardt et al, 2006;Veneziano et al, 2006;Venugopal et al, 2006;Lovejoy and Schertzer, 2006, 2010a, 2010bGarcia-Marin et al, 2008;Serinaldi, 2010;. In particular, Lui [2002] Anh et al [2007Anh et al [ , 2008 and Yu et al [2010Yu et al [ , 2012 used MFA and fractional stochastic differential equations to study the AE and geomagnetic field data.…”

Section: Journal Of Geophysical Research: Space Physicsmentioning

confidence: 99%

Underlying scaling relationships between solar activity and geomagnetic activity revealed by multifractal analyses

Anh

Eastes

2014

JGR Space Physics

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This paper identifies some scaling relationships between solar activity and geomagnetic activity. We examine the scaling properties of hourly data for two geomagnetic indices (a p and AE), two solar indices (solar X-rays X l and solar flux F10.7), and two inner heliospheric indices (ion density Ni and flow speed Vs) over the period 1995-2001 by the universal multifractal approach and the traditional multifractal analysis. We found that the universal multifractal model (UMM) provides a good fit to the empirical K(q) and (q) curves of these time series. The estimated values of the Lévy index in the UMM indicate that multifractality exists in the time series for a p , AE, X l , and Ni, while those for F10.7 and Vs are monofractal.The estimated values of the nonconservation parameter H of this model confirm that these time series are conservative which indicate that the mean value of the process is constant for varying resolution. Additionally, the multifractal K(q) and (q) curves, and the estimated values of the sparseness parameter C 1 of the UMM indicate that there are three pairs of indices displaying similar scaling properties, namely a p and X l , AE and Ni, and F10.7 and Vs. The similarity in the scaling properties of pairs (a p , X l ) and (AE, Ni) suggests that a p and X l , AE and Ni are better correlated-in terms of scaling-than previous thought, respectively. But our results still cannot be used to advance forecasting of a p and AE by X l and Ni, respectively, due to some reasons.

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“…Multifractal analysis was initially proposed to treat turbulence data and is a useful way to characterise the spatial heterogeneity of both theoretical and experimental fractal patterns (Grassberger and Procaccia, 1983;Halsy et al, 1986). It has been applied successfully in many different fields including financial modelling (e.g., Anh et al, 2000;Canessa, 2000), biological systems (e.g., Yu et al, 2001Yu et al, , 2003Yu et al, , 2004Yu et al, , 2006Anh et al, 2001Anh et al, , 2002Zhou et al, 2005), geophysical systems (e.g., Schertzer and Lovejoy, 1987;Schmitt et al, 1992;Tessier et al, 1993Tessier et al, , 1996Olsson, 1995;Olsson and Niemczynowicz, 1996;Harris et al, 1996;Lovejoy et al, 1996;Deidda, 2000;Lilley et al, 2006;Kantelhardt et al, 2006;Veneziano et al, 2006;Venugopal et al, 2006;Schertzer, 2006, 2010a, b;Garcia-Marin et al, 2008;Serinaldi, 2010) and high energy physics (e.g., Ratti et al, 1994). Fractal and multifractal approaches have been quite successful in extracting salient features of physical processes responsible for the near-Earth magnetospheric phenomena (Lui, 2002).…”

Section: Introductionmentioning

confidence: 99%

Multifractal analysis of solar flare indices and their horizontal visibility graphs

Anh

Eastes

et al. 2012

Nonlin. Processes Geophys.

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Abstract. The multifractal properties of the daily solar Xray brightness, X l and X s , during the period from 1 January 1986 to 31 December 2007 which includes two solar cycles are examined using the universal multifractal approach and multifractal detrended fluctuation analysis. Then we convert these time series into networks using the horizontal visibility graph technique. Multifractal analysis of the resulting networks is performed using an algorithm proposed by us. The results from the multifractal analysis show that multifractality exists in both raw daily time series of X-ray brightness and their horizontal visibility graphs. It is also found that the empirical K(q) curves of raw time series can be fitted by the universal multifractal model. The numerical results on the raw data show that the Solar Cycle 23 is weaker than the Solar Cycle 22 in multifractality. The values of h(2) from multifractal detrended fluctuation analysis for these time series indicate that they are stationary and persistent, and the correlations in the time series of Solar Cycle 23 are stronger than those for Solar Cycle 22. Furthermore, the multifractal scaling for the networks of the time series can reflect some properties which cannot be picked up by using the same analysis on the original time series. This suggests a potentially useful method to explore geophysical data.

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Multifractal large number of drops limit in rain

Cited by 32 publications

References 60 publications

Edge effect causes apparent fractal correlation dimension of uniform spatial raindrop distribution

Edge effect causes apparent fractal correlation dimension of uniform spatial raindrop distribution

Underlying scaling relationships between solar activity and geomagnetic activity revealed by multifractal analyses

Multifractal analysis of solar flare indices and their horizontal visibility graphs

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