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

Abstract

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 v at well-defined scales. Classically, one defines fluctuations as differences:and this is adequate for many applications ( x is the "lag"). However, if over a range one has scaling v ∝ x H , 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 ( v) Haar at lag x is simply equal to the difference of the mean from x to x + x/2 and from x + x/2 to x + x. 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.

What Is Climate?

“…Although this distinction may seem arcane, starting in the 1980s, analyses using differences and spectra were not sufficiently clear. The failure to define fluctuations in this way is at least partly responsible for the lack of awareness of macroweather [ Lovejoy and Schertzer , 2012b].…”

Universal multifractal Martian topography

“…The fluctuation at scale L is defined by the absolute difference | B − A |, L being the spherical distance between the first point and the last point. On synthetic multifractal series obtained by simulation (Lovejoy and Schertzer, 2012), Haar fluctuations have proven to be strong estimators of the input multifractal parameters. However, Haar fluctuations unlike first differences are more complicated to implement, particularly in the case of one-dimensional irregular series (see Appendix A).…”

Intermittent particle dynamics in marine coastal waters

“…If ζ (q) is linear, the statistical behaviour is mono-scaling; if ζ (q) is nonlinear and concave/convex, the behaviour is defined as multiscaling, corresponding to a multifractal process. The concavity of this function is a characteristic of the intermittency: the more concave the curve is, the more intermittent the process is (Frisch, 1995;Schertzer et al, 1997;Vulpiani and Livi, 2003;Lovejoy and Schertzer, 2012). The slight curvature which is found here for all size classes (Fig.…”

“…First a negative Hurst exponent is found: H U = −0.30 ± 0.02 and H V = −0.20 ± 0.02. Such a negative sign for H values indicates that small scales show larger fluctuations than the larger scales in a scaling way (Lovejoy and Schertzer, 2012). Both curves become quite different for larger moments: the U curve is more nonlinear, associated with larger intermittency (Fig.…”