Keywords: wavelet, signal, annual time series, autoregressive modeling.

Multiresolution wavelets analysis (MRA) provides a flexible set of tools to solve problems related to science and engineering. In recent years wavelets have been successfully applied to the analysis of signals in various areas such as medicine, electrical engineering, remote sensing and many others. One of the main strengths of MRA is that it allows modeling processes that depend heavily on time and for which their behaviour is not soft. The wavelet transform is particularly effective when extracting information concerning non-periodic finite life signals. On the other hand there are many families of wavelets that allow us to analyze signals of a different nature.

In 2005 Labat [1] provided a review of the most recent wavelet applications in the field of earth sciences, in particular he focused more particularly on the use of wavelet analysis for long-term discharge series. The earlier wavelets applications to discharge time series consisted of an identification and classification of the different regimes of several American catchments. Nevertheless this technique remains unknown to a large majority of land surface hydrologists.

Autoregressive models (AR) models have been extensively used in hydrology and water resources since the early 1960s, for modeling annual and periodic hydrologic time series because of their simplicity and the intuitive type of time dependence. Time series modeling has mainly two uses in hydrology and water resources: for generating synthetic hydrologic time series and for forecasting future hydrologic series. However, these models are basically linear models assuming that data are stationary, and have a limited ability to capture nonstationarities and nonlinearities in hydrologic data. Moreover, AR models do not successfully reproduce the structure of temporal dependence of the series.

One of the most interesting features of the wavelet transform is that the wavelet coefficients from the noise of a given high-frequency random signal f focus on the portion of the fluctuations. So we can generate from f another signal f' and keep its non-random component. We proceed by taking the coefficients of the first fluctuation and generating randomly new ones with the same mean and standard deviation. Then we rebuild the signal using the generated coefficients to obtain f' (the rest of the coefficients of fluctuation and the trend remains).

This contribution provides a new method to generate synthetic time series by using MRA. To this end we use Haar's wavelets which allow a clear and descriptive presentation.

1

D. Labat, "Recent advances in wavelet analyses: Part 1. A review of concepts", Journal of Hydrology, 314, 275-288, 2005. doi:10.1016/j.jhydrol.2005.04.003

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