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Beyond Multifractional Brownian Motion: New Stochastic Models for Geophysical Modelling : Volume 20, Issue 5 (11/09/2013)

By Lévy Véhel, J.

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Book Id: WPLBN0003990282
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Reproduction Date: 2015

Title: Beyond Multifractional Brownian Motion: New Stochastic Models for Geophysical Modelling : Volume 20, Issue 5 (11/09/2013)  
Author: Lévy Véhel, J.
Volume: Vol. 20, Issue 5
Language: English
Subject: Science, Nonlinear, Processes
Collections: Periodicals: Journal and Magazine Collection (Contemporary), Copernicus GmbH
Historic
Publication Date:
2013
Publisher: Copernicus Gmbh, Göttingen, Germany
Member Page: Copernicus Publications

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Véhel, J. L. (2013). Beyond Multifractional Brownian Motion: New Stochastic Models for Geophysical Modelling : Volume 20, Issue 5 (11/09/2013). Retrieved from http://hawaiilibrary.net/


Description
Description: Regularity Team, Inria and MAS Laboratory, Ecole Centrale Paris, Grande Voie des Vignes, Chatenay Malabry, France. Multifractional Brownian motion (mBm) has proved to be a useful tool in various areas of geophysical modelling. Although a versatile model, mBm is of course not always an adequate one. We present in this work several other stochastic processes which could potentially be useful in geophysics. The first alternative type is that of self-regulating processes: these are models where the local regularity is a function of the amplitude, in contrast to mBm where it is tuned exogenously. We demonstrate the relevance of such models for digital elevation maps and for temperature records. We also briefly describe two other types of alternative processes, which are the counterparts of mBm and of self-regulating processes when the intensity of local jumps is considered in lieu of local regularity: multistable processes allow one to prescribe the local intensity of jumps in space/time, while this intensity is governed by the amplitude for self-stabilizing processes.

Summary
Beyond multifractional Brownian motion: new stochastic models for geophysical modelling

Excerpt
Echelard, A. and Lévy Véhel, J.: Self-regulating processes-based modelling for arrhythmia characterization, in: ISPHT 2012, International Conference on Imaging and Signal Processing in Health Care and Technology, 14–16 May, Baltimore, 2012.; Echelard, A., Barrière, O., and Lévy Véhel, J.: Terrain modelling with multifractional Brownian motion and self-regulating processes, Springer, Lect. Notes Comput. Sc., 6374, 342–351, 2010.; Echelard, A., Lévy Véhel, J., and Philippe, A.: Statistical estimation for a class of self-regultaing processes, preprint, 2012.; Falconer, K. J.: The local structure of random processes, J. London Math. Soc., 67, 657–672, 2003.; FracLab: a software toolbox for irregular signals analysis, available at http://fraclab.saclay.inria.fr/ (last access: 9 September 2013), 2012.; Gaci, S. and Zaourar, N.: A New Approach for the Investigation of the Multifractality of Borehole Wire-Line Logs, Research Journal of Earth Sciences, 3, 63–70, 2011.; Keylock, C. J.: Characterizing the structure of nonlinear systems using gradual wavelet reconstruction, Nonlin. Processes Geophys., 17, 615–632, doi:10.5194/npg-17-615-2010, 2010.; Kolmogorov, A. N.: Wienersche Spiralen und einige andere interessante Kurven im Hilbertschen Raume, Doklady, 26, 115–118, 1940.; Le Guével, R. and Lévy Véhel, J.: A Ferguson-Klass-LePage series representation of multistable multifractional processes and related processes, Bernoulli, 18, 1099–1127, 2012.; Li, M., Zhao, W., and Chen, S.: mBm-Based Scalings of Traffic Propagated in Internet, Math. Probl. Eng., 2011, 389803, doi:10.1155/2011/389803, 2011.; Lovejoy, R. and Schertzer, D.: The Weather and Climate: emergent laws and multifractal cascades, Cambridge University Press, Cambridge, 2013.; Mandelbrot, B. B. and Van Ness, J. W.: Fractional Brownian motions, fractional noises and applications, SIAM Rev., 10, 422–437, 1968.; Peltier, R. F. and Lévy Véhel, J.: Multifractional Brownian motion: definition and preliminary results, Rapport de recherche de l'INRIA, No. 2645, available at: http://hal.inria.fr/docs/00/07/40/45/PDF/RR-2645.pdf (last access: 9 September 2013), 1995.; Samorodnitsky, G. and Taqqu, M.: Stable Non-Gaussian Random Process, Chapman and Hall, New York, 1994. \bibitem[Wanliss(2005)] Wanliss Wanliss, J.: Fractal properties of SYM-H during quiet and active times, J. Geophys. Res., 110, A03202, doi:10.1029/2004JA010544, 2005.; Wei, H. L., Billings, S. A., and Balikhin, M.: Analysis of the geomagnetic activity of the Dst index and self-affine fractals using wavelet transforms, Nonlin. Processes Geophys., 11, 303–312, doi:10.5194/npg-11-303-2004, 2004.

 

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