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| Titel |
Haar wavelets, fluctuations and structure functions: convenient choices for geophysics |
| VerfasserIn |
S. Lovejoy, D. Schertzer |
| Medientyp |
Artikel
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| Sprache |
Englisch
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| ISSN |
1023-5809
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| Digitales Dokument |
URL |
| Erschienen |
In: Nonlinear Processes in Geophysics ; 19, no. 5 ; Nr. 19, no. 5 (2012-09-13), S.513-527 |
| Datensatznummer |
250014243
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| Publikation (Nr.) |
 copernicus.org/npg-19-513-2012.pdf |
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| Zusammenfassung |
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(x+Δx)-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 x+Δx/2 to x+Δx.
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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