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Nonlinear Processes in Geophysics An interactive open-access journal of the European Geosciences Union
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Volume 19, issue 5
Nonlin. Processes Geophys., 19, 513–527, 2012
© Author(s) 2012. This work is distributed under
the Creative Commons Attribution 3.0 License.
Nonlin. Processes Geophys., 19, 513–527, 2012
© Author(s) 2012. This work is distributed under
the Creative Commons Attribution 3.0 License.

Research article 13 Sep 2012

Research article | 13 Sep 2012

Haar wavelets, fluctuations and structure functions: convenient choices for geophysics

S. Lovejoy1 and D. Schertzer2 S. Lovejoy and D. Schertzer
  • 1Physics, McGill University, 3600 University st., Montreal, Que. H3A 2T8, Canada
  • 2LEESU, École des Ponts ParisTech, Université Paris-Est, 6-8 Av. B. Pascal, 77455 Marne-la-Vallée Cedex 2, France

Abstract. Geophysical processes are typically variable over huge ranges of space-time scales. This has lead to the development of many techniques for decomposing series and fields into fluctuations Δv at well-defined scales. Classically, one defines fluctuations as differences: (Δvdiff = v(xx)-v(x) and this is adequate for many applications (Δx is the "lag"). However, if over a range one has scaling Δv ∝ ΔxH, these difference fluctuations are only adequate when 0 < H < 1. Hence, there is the need for other types of fluctuations. In particular, atmospheric processes in the "macroweather" range ≈10 days to 10–30 yr generally have −1 < H < 0, so that a definition valid over the range −1 < H < 1 would be very useful for atmospheric applications. A general framework for defining fluctuations is wavelets. However, the generality of wavelets often leads to fairly arbitrary choices of "mother wavelet" and the resulting wavelet coefficients may be difficult to interpret. In this paper we argue that a good choice is provided by the (historically) first wavelet, the Haar wavelet (Haar, 1910), which is easy to interpret and – if needed – to generalize, yet has rarely been used in geophysics. It is also easy to implement numerically: the Haar fluctuation (ΔvHaar at lag Δx is simply equal to the difference of the mean from x to x+ Δx/2 and from xx/2 to xx. Indeed, we shall see that the interest of the Haar wavelet is this relation to the integrated process rather than its wavelet nature per se.

Using numerical multifractal simulations, we show that it is quite accurate, and we compare and contrast it with another similar technique, detrended fluctuation analysis. We find that, for estimating scaling exponents, the two methods are very similar, yet Haar-based methods have the advantage of being numerically faster, theoretically simpler and physically easier to interpret.

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