Haar wavelets, fluctuations and structure functions: convenient choices for geophysics
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 &Delta;<i>v</i> at well-defined scales. Classically, one defines fluctuation...
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Copernicus Publications
2012-09-01
|
| Series: | Nonlinear Processes in Geophysics |
| Online Access: | http://www.nonlin-processes-geophys.net/19/513/2012/npg-19-513-2012.pdf |
| _version_ | 1818564152862441472 |
|---|---|
| author | S. Lovejoy D. Schertzer |
| author_facet | S. Lovejoy D. Schertzer |
| author_sort | S. Lovejoy |
| collection | DOAJ |
| description | 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 &Delta;<i>v</i> at well-defined scales. Classically, one defines fluctuations as differences: (&Delta;<i>v</i><sub>diff</sub> = <i>v(x</i>+&Delta;<i>x</i>)-<i>v(x)</i> and this is adequate for many applications (&Delta;<i>x</i> is the "lag"). However, if over a range one has scaling &Delta;<i>v</i> &propto; &Delta;<i>x</i><sup><i>H</i></sup>, these difference fluctuations are only adequate when 0 < <i>H</i> < 1. Hence, there is the need for other types of fluctuations. In particular, atmospheric processes in the "macroweather" range &approx;10 days to 10–30 yr generally have −1 < <i>H</i> < 0, so that a definition valid over the range −1 < <i>H</i> < 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 (&Delta;<i>v</i><sub>Haar</sub> at lag &Delta;<i>x</i> is simply equal to the difference of the mean from <i>x</i> to <i>x</i>+ &Delta;<i>x</i>/2 and from <i>x</i>+&Delta;<i>x</i>/2 to <i>x</i>+&Delta;<i>x</i>. 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. <br><br> 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. |
| first_indexed | 2024-12-14T01:25:36Z |
| format | Article |
| id | doaj.art-8726f3cc2c094e34912f13e0351d4f32 |
| institution | Directory Open Access Journal |
| issn | 1023-5809 1607-7946 |
| language | English |
| last_indexed | 2024-12-14T01:25:36Z |
| publishDate | 2012-09-01 |
| publisher | Copernicus Publications |
| record_format | Article |
| series | Nonlinear Processes in Geophysics |
| spelling | doaj.art-8726f3cc2c094e34912f13e0351d4f322022-12-21T23:22:12ZengCopernicus PublicationsNonlinear Processes in Geophysics1023-58091607-79462012-09-0119551352710.5194/npg-19-513-2012Haar wavelets, fluctuations and structure functions: convenient choices for geophysicsS. LovejoyD. SchertzerGeophysical 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 &Delta;<i>v</i> at well-defined scales. Classically, one defines fluctuations as differences: (&Delta;<i>v</i><sub>diff</sub> = <i>v(x</i>+&Delta;<i>x</i>)-<i>v(x)</i> and this is adequate for many applications (&Delta;<i>x</i> is the "lag"). However, if over a range one has scaling &Delta;<i>v</i> &propto; &Delta;<i>x</i><sup><i>H</i></sup>, these difference fluctuations are only adequate when 0 < <i>H</i> < 1. Hence, there is the need for other types of fluctuations. In particular, atmospheric processes in the "macroweather" range &approx;10 days to 10–30 yr generally have −1 < <i>H</i> < 0, so that a definition valid over the range −1 < <i>H</i> < 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 (&Delta;<i>v</i><sub>Haar</sub> at lag &Delta;<i>x</i> is simply equal to the difference of the mean from <i>x</i> to <i>x</i>+ &Delta;<i>x</i>/2 and from <i>x</i>+&Delta;<i>x</i>/2 to <i>x</i>+&Delta;<i>x</i>. 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. <br><br> 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.http://www.nonlin-processes-geophys.net/19/513/2012/npg-19-513-2012.pdf |
| spellingShingle | S. Lovejoy D. Schertzer Haar wavelets, fluctuations and structure functions: convenient choices for geophysics Nonlinear Processes in Geophysics |
| title | Haar wavelets, fluctuations and structure functions: convenient choices for geophysics |
| title_full | Haar wavelets, fluctuations and structure functions: convenient choices for geophysics |
| title_fullStr | Haar wavelets, fluctuations and structure functions: convenient choices for geophysics |
| title_full_unstemmed | Haar wavelets, fluctuations and structure functions: convenient choices for geophysics |
| title_short | Haar wavelets, fluctuations and structure functions: convenient choices for geophysics |
| title_sort | haar wavelets fluctuations and structure functions convenient choices for geophysics |
| url | http://www.nonlin-processes-geophys.net/19/513/2012/npg-19-513-2012.pdf |
| work_keys_str_mv | AT slovejoy haarwaveletsfluctuationsandstructurefunctionsconvenientchoicesforgeophysics AT dschertzer haarwaveletsfluctuationsandstructurefunctionsconvenientchoicesforgeophysics |