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Title
Abstract
Introduction: Which Chaos in Geophysics?
Cascades and Data Analyses
Conclusions
References
All Topics
Mathematics
Geometry

Scale, scaling and multifractals in geophysics: twenty years on

Profile image of Shaun LovejoyShaun Lovejoy

2007

https://doi.org/10.1007/978-0-387-34918-3_18
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28 pages

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Abstract

We consider three developments in high number of degrees of freedom approaches to nonlinear geophysics: a) the transition from fractal geometry to multifractal processes, b) the Twenty Years on 18

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In turbulence, the simplest phenomenological models of the energy cascade are multiplicative processes constructed on a regular grid (in short, M.P.G). They have been used mostly for their simplicity, allowing many of their properties to be derived analytically, and their capacity to reproduce the scale invariance properties of various geophysical fields. However, these M.P.G.'s suffer from the drawback of lacking translation invariance in their spatial statistics (spatial homogeneity), and therefore they cannot be fully satisfactory models for geophysical fields. In this paper, we are interested in finding new construction methods for spatially homogeneous random multifractals. We investigate the scaling properties of a new family of gridless models of multifractals. PACS: 47.25Cg; 42.20Tg; 02.50. +s; 05.40. +j; 64.60.Ak 0378-4371/96/$15.00 ~2) 1996 Elsevier Science B.V. All rights reserved SSDI 0378-4371(95)00338-X

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Existing fractal studies dealing with subsurface heterogeneity treat the logarithm of the permeability K as the variable of concern. We treat K as a multifractal and investigate its scaling and fractality using measured horizontal K data from two locations in the United States. The first data set was from a shoreline sandstone near Coalinga, California, and the second was from an eolian sandstone . By applying spectral analyses and computing the scaling of moments of various orders (using the double trace moment method ), we found that K is multiscaling (i.e., scaling and multifractal). We also found that the so-called universal multifractal (UM) model (essentially a log-Levy multifractal), was able to reproduce the multiscaling behavior reasonably well. The UM model has three parameters: ␣, , and H, representing the multifractality index, the codimension of the mean field, and the "distance" to stationary multifractal, respectively. We found (␣ ϭ 1.7, ϭ 0.23, H ϭ 0.22) and (␣ ϭ 1.6, ϭ 0.11, H ϭ 0.075) for the shoreline and eolian data sets, respectively. The fact that ␣ values were less than 2 indicates that the underlying statistics are non-Gaussian. We generated stationary and nonstationary multifractals and illustrated the role of the UM parameters on simulated fields. Studies that treated Log K as the variable of concern have pointed out the necessity for large data records, especially when the underlying distribution is Levy-stable. Our investigation revealed that even larger data records are required when treating K as a multifractal, because Log K is less intermittent (or irregular) than K.

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    I. IntroductIon Un mode classique de représentation, en relation avec la notion de cascade ([1] pour une revue récente) conduit à analyser et simuler l'extrême variabilité des processus géophysiques sur de grandes gammes d'échelle, en particulier la pluie, dans le cadre des multifractals universels, à l'aide d'un nombre réduit de paramètres [2,3]. De telles analyses menées sur les débits et/ou les précipitations, ont permis d'éclairer la compréhension de la relation pluie-débit en comparant leurs paramètres multifractals respectifs [4,5,6,7] ainsi que le comportement de leurs extrêmes [8]. Par ailleurs, de nombreuses études d'hydrologie plus classiques [9,10,11,12,13] suggèrent un impact significatif de la variabilité spatio-temporelle sur les débits même si des nuances sont à apporter en fonction du type d'événements pluvieux, de la taille du bassin versant et de ses caractéristiques. En hydrologie urbaine, avec des bassins versants plus petits et une proportion plus grande de pluie efficace en raison de coefficients d'imperméabilisation supérieurs, les effets semblent plus importants [14,15]. Cet article met en oeuvre les méthodes multifractales en hydrologie urbaine afin de mieux comprendre comment l'incertitude sur la variabilité spatio-temporelle à très haute résolution de la pluie (i.e. au-delà de la résolution de 1 km*1 km*5min classiquement disponible avec les réseaux radars à bande C des services météorologiques) se reflète dans les résultats de simulation d'un modèle semi-distribué d'hydrologie urbaine. Le cas d'étude est celui du bassin versant de Cranbrook une zone urbaine de 900 hectares située

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    Development and analysis of a simple model to represent the zero rainfall in a universal multifractal framework
    Shaun Lovejoy

    Nonlinear Processes in Geophysics, 2013

    High-resolution rainfall fields contain numerous zeros (i.e. pixels or time steps with no rain) which are either real or artificial -that is to say associated with the limit of detection of the rainfall measurement device. In this paper we revisit the enduring discussion on the source of this intermittency, e.g. whether it requires specific modelling. We first review the framework of universal multifractals (UM), which are commonly used to analyse and simulate geophysical fields exhibiting extreme variability over a wide range of scales with the help of a reduced number of parameters. However, this framework does not enable properly taking into account these numerous zeros. For example, it has been shown that performing a standard UM analysis directly on the field can lead to low observed quality of scaling and severe bias in the estimates of UM parameters. In this paper we propose a new simple model to deal with this issue. It is a UM discrete cascade process, where at each step if the simulated intensity is below a given level (defined in a scale invariant manner), it only has a predetermined probability to survive and is otherwise set to zero. A threshold can then be implemented at the maximum resolution to mimic the limit of detection of the rainfall measurement device. While also imperfect, this simple model enables explanation of most of the observed behaviour, e.g. the presence of scaling breaks, or the difference between statistics computed for single events or longer periods.

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