[1] Distributional analysis of river discharge time series is an important task in many areas of hydrological engineering, including optimal design of water storage and drainage networks, management of extreme events, risk assessment for water supply, and environmental flow management, among many others. Having diverging moments, heavy-tailed power law distributions have attracted widespread attention, especially for the modeling of the likelihood of extreme events such as floods and droughts. However, straightforward distributional analysis does not connect well with the complicated dynamics of river flows, including fractal and multifractal behavior, chaos-like dynamics, and seasonality. To better reflect river flow dynamics, we propose to carry out distributional analysis of river flow time series according to three ''flow seasons'': dry, wet, and transitional. We present a concrete statistical procedure to partition river flow data into three such seasons and fit data in these seasons using two types of distributions, power law and lognormal. The latter distribution is a salient property of the cascade multiplicative multifractal model, which is among the best models for turbulence and rainfall. We show that while both power law and lognormal distributions are relevant to dry seasons, river flow data in wet seasons are typically better fitted by lognormal distributions than by power law distributions.

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Tree ring width data are among the best proxies for reconstructing past temperature and precipitation records. The discovery of fractal scaling and long‐memory in meteorological and hydrological signals motivates us to investigate such properties in tree ring chronologies. Detrended fluctuation analysis and adaptive fractal analysis are utilized to estimate the Hurst parameter values of 697 tree ring chronologies from the continental United States. We find significant differences in the Hurst parameter values across the 10 species studied in the work. The long‐range scaling relations found here suggest that the behavior of tree ring growth observed in a short calibration period may be similar to the general behavior of tree ring growth in a much longer period, and therefore, the limited calibration period may be more useful than originally thought. The variations of the long‐range correlations within and across species may be further explored in future to better reconstruct paleoclimatic records.

This paper presents an adaptive procedure for estimating the variability and determining error bars as confidence intervals for climate mean states by accounting for both short- and long-range dependence. While the prevailing methods for quantifying the variability of climate means account for short-range dependence, they ignore long memory, which is demonstrated to lead to underestimated variability and hence artificially narrow confidence intervals. To capture both short- and long-range correlation structures, climate data are modeled as fractionally integrated autoregressive moving-average processes. The preferred model can be selected adaptively via an information criterion and a diagnostic visualization, and the estimated variability of the climate mean state can be computed directly from the chosen model. The procedure was demonstrated by determining error bars for four 30-yr means of surface temperatures observed at Potsdam, Germany, from 1896 to 2015. These error bars are roughly twice the width as those obtained using prevailing methods, which disregard long memory, leading to a substantive reinterpretation of differences among mean states of this particular dataset. Despite their increased width, the new error bars still suggest that a significant increase occurred in the mean temperature state of Potsdam from the 1896–1925 period to the most recent period, 1986–2015. The new wider error bars, therefore, communicate greater uncertainty in the mean state yet present even stronger evidence of a significant temperature increase. These results corroborate a need for more meticulous consideration of the correlation structures of climate data—especially of their long-memory properties—in assessing the variability and determining confidence intervals for their mean states.