Abstract
Motivated by the necessity of developing new multifractal analysis techniques to characterize empirical fields by scale invariant (sensor resolution independent) codimension functions, we introduce a new method, PDMS, to directly estimate the codimension of the singularity spectrum c(γ) and we also indicate the theoretical (or practical) limits of this method as well as its consequences for the determination of highest values of c(γ). These properties also have implications for the behaviour of K(h) — related to c(γ) by a Legendre transformation — in particular for large h. The characteristic behaviour of c(γ) and K(h) are illustrated respectively by the estimation of the scaling properties of the probability distribution and of statistical moments of simulated fields, obtained with multiplicative self similar cascade processes.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Bialas, A., R. Peschanski, 1986: Moments of rapidity distributions as a measure of short-range fluctuations in high-energy collisions. Nucl. Phys. B. B273. 703–718.
Bender, C.M., S.A. Orszag, 1978: Advanced mathematical methods for scientists and engineers. McGraw-Hill, New York, NY.
Frisch, U., G. Parisi, 1985: A multifractal model of intermittency. in Turbulence and predictability in geophysical fluid dynamics and climate dynamics. Eds. Ghil, Benzi, Parisi, North-Holland, 84–88.
Gabriel, P., S. Lovejoy, D. Schertzer, and G.L. Austin, 1988: Multifractal analysis of resolution dependence in satellite in satellite imagery. Geophvs. Res. Lett., 15, 1373–1376.
Grassberger, P., 1983: Generalized dimensions of strange attractors. Phys. Lett., A 97.227.
Halsey, T.C., M.H. Jensen, L.P. Kadanoff, I. Procaccia, B. Shraiman, 1986: Fractal measures and their singularities: the characterization of strange sets. Phys. Rev., A 33.1141–1151.
Hentschel, H.G.E., I. Proccacia, 1983: The infinite number of generalized dimensions of fractals and strange attractors. Physica. 8D. 435–444.
Hubert, P., J. P. Carbonnel, 1988: Caractérisation fractale de la variabilité et de l’anisotropie des prestations tropicales. C. R. Acad. Sci. Paris. 2–307.909–914.
Kolmogorov, A. N., 1962: A refinement of previous hypothesis concerning the local structure of turbulence in viscous incompressible fluid at high Reynolds number. J. Fluid Mech., 13, 82–85
Lavallée, D., 1990: Ph. D. Thesis, University McGill, Montrdal, Canada.
Lovejoy, S., D. Schertzer, A.A. Tsonis, 1987a: Functional box-counting and multiple elliptical dimensions in rain. Science. 235.1036–1038.
Lovejoy, S., D. Schertzer, 1990: Multifractal analysis techniques and the rain and cloud fields from 10−3 to 10−6m. (this volume).
Levitch, E., I. Shtilman, 1990: Helicity fluctuations and coherence in developed turbulence, (this volume).
Mandelbrot, B., 1974: Intermittent turbulence in self-similar cascades: Divergence of high moments and dimension of the carrier. J. Fluid Mech., 62,331–350.
Obukhov, A., 1962: Some specific features of atmospheric turbulence. J. Geophvs. Res. 67, 3011–3014
Meneveau, C., K. R. Sreenivasan, 1987: Simple multifractal cascade model for fully developed turbulence. Phys. Rev. Lett., 59(13) 1424–1427.
Schertzer, D., S. Lovejoy, 1983: On the dimension of atmospheric motions. Preprint Vol., IUTAM Svmp. on Turbulence and Chaotic phenomena in Fluids. Kyoto, Japan, IUTAM, 141–144.
Schertzer, D., S. Lovejoy, 1984: On the dimension of atmospheric motions. Turbulence and chaotic phenomena in fluids. Ed. Tatsumi, Elsevier North-Holland, New York,.505–508.
Schertzer, D., S. Lovejoy, 1987a: Physical modelling and analysis of rain and clouds by anisotropic, scaling multiplicative processes. J. Geophvs. Res., 92(D8). 9693–9714.
Schertzer, D., S. Lovejoy, 1987b: Singularity anisotropes, divergences des moments en turbulence: invariance d’echelle generalise et processus multiplicatifs. Annales Math. du Oué. 11.139–181
Schertzer and Lovejoy 1988: Multifractal simulation and analysis of clouds by multiplicative process. Atmospheric Research. 21,337–361
Schertzer, D., S. Lovejoy, 1990a: Scaling nonlinear variability in geodynamics: Multiple singularities, observables and universality classes, (this volume).
Schertzer, D., S. Lovejoy, 1990b: Nonlinear variability in geophysics: multifractal simulations and analysis, in Fractals: physical origins and properties, edited by L. Pietronero, Plenum, New York, NY, pp.49–79.
Wilson, J., D. Schertzer, S. Lovejoy, 1990: Physically based cloud modelling by multiplicative cascade processes, (this volume).
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1991 Kluwer Academic Publishers
About this chapter
Cite this chapter
Lavallée, D., Schertzer, D., Lovejoy, S. (1991). On the Determination of the Codimension Function. In: Schertzer, D., Lovejoy, S. (eds) Non-Linear Variability in Geophysics. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-2147-4_7
Download citation
DOI: https://doi.org/10.1007/978-94-009-2147-4_7
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-010-7466-7
Online ISBN: 978-94-009-2147-4
eBook Packages: Springer Book Archive