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A Practical Guide to Discrete Wavelet Decomposition of Hydrologic Time Series

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Abstract

Discrete wavelet transform (DWT) is commonly used for wavelet threshold de-noising, wavelet decomposition, wavelet aided hydrologic series simulation and prediction, as well as many other hydrologic time series analyses. However, its effectiveness in practice is influenced by many key factors. In this paper the “reference energy function” was firstly established by operating Monte-Carlo simulation to diverse noise types; then, energy function of hydrologic series was compared with the reference energy function, and four key issues on discrete wavelet decomposition were studied and the methods for solving them were proposed, namely wavelet choice, decomposition level choice, wavelet threshold de-noising and significance testing of DWT, based on which a step-by-step guide to discrete wavelet decomposition of hydrologic series was provided finally. The specific guide is described as: choose appropriate wavelet from the recommended wavelets and according to the statistical characters relations among original series, de-noised series and removed noise; choose proper decomposition levels by analyzing the difference between energy function of the analyzed series and reference energy function; then, use the chosen wavelet and decomposition level, estimate threshold according to series’ complexity and set the same threshold under each level, and use the mid-thresholding rule to remove noise; finally, conduct significance testing of DWT by comparing energy function of the de-noised series with the reference energy function. Analyses of both synthetic and observed series indicated the better performance and easier operability of the proposed guide compared with those methods used presently. Following the guide step by step, noise and different deterministic components in hydrologic series can be accurately separated, and uncertainty can also be quantitatively estimated, thus the discrete wavelet decomposition result of series can be improved.

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Acknowledgements

The author gratefully acknowledged the helpful comments and suggestions on earlier version of the manuscript given by the Editor-in-Chief, George P. Tsakirisand, and anonymous reviewers. The RS2 discharge data are kindly provided by Dr. Chengpeng Ling. The author also thanked Ms. Fei-Fei Liu and Die Zhu for their assistances in preparation of the manuscript. This project was supported by the National Key Basic Research Development Program of China (No. 2009CB421305), the National Natural Science Foundation of China (NSFC) (No. 40971023), and the Water Resources Public-Welfare Projects (No. 200901042 and 201201072).

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  1. Key Laboratory of Water Cycle & Related Land Surface Processes, Institute of Geographic Sciences and Natural Resources Research, Chinese Academy of Sciences, Beijing, 100101, People’s Republic of China

    Yan-Fang Sang

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Sang, YF. A Practical Guide to Discrete Wavelet Decomposition of Hydrologic Time Series. Water Resour Manage 26, 3345–3365 (2012). https://doi.org/10.1007/s11269-012-0075-4

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