Evaluation of statistical methods for quantifying fractal scaling in water-quality time series with irregular sampling

Abstract

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 because (1) fractal scaling has the potential to lead to false inference about the statistical significance of trends and (2) the abundance of irregularly spaced data in water-quality monitoring networks complicates efforts to quantify fractal scaling. Traditional methods for estimating fractal scalingin the form of 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 autocorrelation, as the interpolation methods consistently underestimate or overestimate β under a wide range of prescribed β values and gap distributions. Second, the widely used Lomb-Scargle spectral method also consistently underestimates β. A previously published 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-meansquared 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.

Detecting dominant changes in irregularly sampled multivariate water quality data sets

“…That would have required an assessment of the lag-1 autocorrelation which was hampered by the irregular sampling. Neither did we consider long-term memory and its effects on the statistical significance of the trends (Cohn and Lins, 2005;Zhang et al, 2018). Consequently, we did not consider the possible effects of the irregular sampling on the long-term memory (fractal scaling) of the water quality series either (Zhang et al, 2018).…”

“…Neither did we consider long-term memory and its effects on the statistical significance of the trends (Cohn and Lins, 2005;Zhang et al, 2018). Consequently, we did not consider the possible effects of the irregular sampling on the long-term memory (fractal scaling) of the water quality series either (Zhang et al, 2018). Due to the limited number of samples per year and non-equidistant sampling, the seasonal Mann-Kendall test was not applicable (Fig.…”

“…Thus, in environmental monitoring practice, data sets with gaps and periods with corrupted measurements are more the rule rather than the exception (see e.g. Zhang et al, 2018, for river quality data). Lischeid et al (2010) suggested a combination of exploratory data analysis methods to detect and analyse dominant processes and their temporal development in multivariate water quality data sets that is capable of dealing with irregular time series.…”

Response to referees #1 and #2

“…That would have required an assessment of the lag-1 autocorrelation which was hampered by the irregular sampling. Neither did we consider long-term memory and its effects on the statistical significance of the trends (Cohn and Lins, 2005;Zhang et al, 2018). Consequently, we did not consider the possible effects of the irregular sampling on the long-term memory (fractal scaling) of the water quality series either (Zhang et al, 2018).…”

“…Neither did we consider long-term memory and its effects on the statistical significance of the trends (Cohn and Lins, 2005;Zhang et al, 2018). Consequently, we did not consider the possible effects of the irregular sampling on the long-term memory (fractal scaling) of the water quality series either (Zhang et al, 2018). Due to the limited number of samples per year and non-equidistant sampling, the seasonal Mann-Kendall test was not applicable (Fig.…”