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Hydrology and Earth System Sciences An interactive open-access journal of the European Geosciences Union
https://doi.org/10.5194/hess-2017-315
© Author(s) 2017. This work is distributed under
the Creative Commons Attribution 3.0 License.
Research article
21 Jun 2017
Review status
This discussion paper is a preprint. It is a manuscript under review for the journal Hydrology and Earth System Sciences (HESS).
Evaluation of statistical methods for quantifying fractal scaling in water quality time series with irregular sampling
Qian Zhang1,a, Ciaran J. Harman2, and James W. Kirchner3,4,5 1University of Maryland Center for Environmental Science at the US Environmental Protection Agency Chesapeake Bay Program Office, 410 Severn Avenue, Suite 112, Annapolis, MD 21403
2Department of Environmental Health and Engineering, Johns Hopkins University, 3400 North Charles Street, Baltimore, Maryland 21218
3Department of Environmental System Sciences, ETH Zurich, Universitätstrasse 16, CH-8092 Zurich, Switzerland
4Swiss Federal Research Institute WSL, Zürcherstrasse 111, CH-8903 Birmensdorf, Switzerland
5Department of Earth and Planetary Science, University of California, Berkeley, California 94720
aformerly at: Department of Geography and Environmental Engineering, Johns Hopkins University, 3400 North Charles Street, Baltimore, Maryland 21218
Abstract. River water-quality time series often exhibit fractal scaling, which here refers to autocorrelation that decays as a power law over some range of scales. Fractal scaling presents challenges to the identification of deterministic trends, but traditional methods for estimating spectral slope (β) or other equivalent scaling parameters (e.g., Hurst exponent) are generally inapplicable to irregularly sampled data. Here we consider two types of estimation approaches for irregularly sampled data and evaluate their performance using synthetic time series. These time series were generated such that (1) they exhibit a wide range of prescribed fractal scaling behaviors, ranging from white noise (β = 0) to Brown noise (β = 2), and (2) their sampling gap intervals mimic the sampling irregularity (as quantified by both the skewness and mean of gap-interval lengths) in real water-quality data. The results suggest that none of the existing methods fully account for the effects of sampling irregularity on β estimation. First, the results illustrate the danger of using interpolation for gap filling when examining auto-correlation, as the interpolation methods consistently under-estimate or over-estimate β under a wide range of prescribed β values and gap distributions. Second, the long-established Lomb-Scargle spectral method also consistently under-estimates β. A modified form, using only the lowest 5 % of the frequencies for spectral slope estimation, has very poor precision, although the overall bias is small. Third, a recent wavelet-based method, coupled with an aliasing filter, generally has the smallest bias and root-mean-squared error among all methods for a wide range of prescribed β values and gap distributions. The aliasing method, however, does not itself account for sampling irregularity, and this introduces some bias in the result. Nonetheless, the wavelet method is recommended for estimating β in irregular time series until improved methods are developed. Finally, all methods' performances depend strongly on the sampling irregularity, highlighting that the accuracy and precision of each method are data-specific. Accurately quantifying the strength of fractal scaling in irregular water-quality time series remains an unresolved challenge for the hydrologic community and for other disciplines that must grapple with irregular sampling.

Citation: Zhang, Q., Harman, C. J., and Kirchner, J. W.: Evaluation of statistical methods for quantifying fractal scaling in water quality time series with irregular sampling, Hydrol. Earth Syst. Sci. Discuss., https://doi.org/10.5194/hess-2017-315, in review, 2017.
Qian Zhang et al.
Qian Zhang et al.

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Short summary
River water-quality time series often exhibit fractal scaling, which here refers to autocorrelation that decays as a power law over some range of scales. This paper provides a comprehensive overview of the various approaches for quantifying fractal scaling in irregularly sampled data and provides new understanding and quantification of the methods’ performances. More generally, the findings and approaches may be broadly applicable to irregularly sampled data in other scientific disciplines.
River water-quality time series often exhibit fractal scaling, which here refers to...
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