Analysis of Geophysical Time Series Using Discrete Wavelet Transforms: An Overview
Discrete wavelet transforms (DWTs) are mathematical tools that are useful for analyzing geophysical time series. The basic idea is to transform a time series into coefficients describing how the series varies over particular scales. One version of the DWT
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Abstract. Discrete wavelet transforms (DWTs) are mathematical tools that are useful for analyzing geophysical time series. The basic idea is to transform a time series into coefficients describing how the series varies over particular scales. One version of the DWT is the maximal overlap DWT (MODWT). The MODWT leads to two basic decompositions. The first is a scale-based analysis of variance known as the wavelet variance, and the second is a multiresolution analysis that reexpresses a time series as the sum of several new series, each of which is associated with a particular scale. Both decompositions are illustrated through examples involving Arctic sea ice and an Antarctic ice core. A second version of the DWT is the orthonormal DWT (ODWT), which can be extracted from the MODWT by subsampling. The relative strengths and weaknesses of the MODWT, the ODWT and the continuous wavelet transform are discussed.
Keywords: Arctic sea ice, Haar wavelet, Ice cores, Maximal overlap discrete wavelet transform, Multiresolution analysis, Wavelet spectrum, Wavelet variance
1 Introduction The wide-spread use of wavelets to analyze data in the geosciences can be traced back to work by Morlet and coworkers [1, 2] in the early 1980s. Their efforts were motivated by signal analysis in oil and gas exploration and resulted in the continuous wavelet transform (CWT). Work in the late 1980s by Daubechies, Mallat and others [3, 4, 5, 6] led to various discrete wavelet transforms (DWTs), which are the focus of this article. While CWTs and DWTs are closely related, DWTs are more amenable to certain types of statistical analysis, making them the transform of choice for tackling certain – but not all – problems of interest in geophysical data analysis. The intent of this article is to give an overview of how DWTs can be used in the analysis
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of geophysical time series, i.e., a sequence of observations recorded over time (usually at regularly spaced intervals such as once per second). The remainder of this article is structured as follows. In Sect. 2 we review the important notion of scale and the basic ideas behind the maximal overlap DWT (MODWT). The MODWT leads to two basic decompositions. The first (the subject of Sect. 3) is a scale-based analysis of variance known as the wavelet variance (or wavelet spectrum). The second (Sect. 4) is an additive decomposition known as a multiresolution analysis, in which a time series is reexpressed as the sum of several new series, each associated with a particular physical scale. In Sect. 5 we discuss another form of the DWT known as the orthonormal DWT (ODWT) that can be extracted from the MODWT and that has certain strengths and weaknesses in comparison to the MODWT. Our overview concentrates on the so-called Haar wavelet, but we note the existence of other wavelets in Sect. 6 and discuss why they might be preferred over the Haar wavelet for certain types of analyses. Finally we make some concluding comments in Sect. 7, including a comparison of the strengths and weaknesses of DWTs and
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