Untenable nonstationarity: An assessment of the fitness for purpose of trend tests in hydrology

The detection and attribution of long-term patterns in hydrological time series have been important research topics for decades. A significant portion of the literature regards such patterns as ‘deterministic components’ or ‘trends’ even though the complexity of hydrological systems does not allow easy deterministic explanations and attributions. Consequently, trend estimation techniques have been developed to make and justify statements about tendencies in the historical data, which are often used to predict future events. Testing trend hypothesis on observed time series is widespread in the hydro-meteorological literature mainly due to the interest in detecting consequences of human activities on the hydrological cycle. This analysis usually relies on the application of some null hypothesis significance tests (NHSTs) for slowly-varying and/or abrupt changes, such as Mann-Kendall, Pettitt, or similar, to summary statistics of hydrological time series (e.g., annual averages, maxima, minima, etc.). However, the reliability of this application has seldom been explored in detail. This paper discusses misuse, misinterpretation, and logical flaws of NHST for trends in the analysis of hydrological data from three different points of view: historic-logical, semantic-epistemological, and practical. Based on a review of NHST rationale, and basic statistical definitions of stationarity, nonstationarity, and ergodicity, we show that even if the empirical estimation of trends in hydrological time series is always feasible from a numerical point of view, it is uninformative and does not allow the inference of nonstationarity without assuming a priori additional information on the underlying stochastic process, according to deductive reasoning. This prevents the use of trend NHST outcomes to support nonstationary frequency analysis and modeling. We also show that the correlation structures characterizing hydrological time series might easily be underestimated, further compromising the attempt to draw conclusions about trends spanning the period of records. Moreover, even though adjusting procedures accounting for correlation have been developed, some of them are insufficient or are applied only to some tests, while some others are theoretically flawed but still widely applied. In particular, using 250 unimpacted stream flow time series across the conterminous United States (CONUS), we show that the test results can dramatically change if the sequences of annual values are reproduced starting from daily stream flow records, whose larger sizes enable a more reliable assessment of the correlation structures

Multifractality, imperfect scaling and hydrological properties of rainfall time series simulated by continuous universal multifractal and discrete random cascade models

Discrete multiplicative random cascade (MRC) models were extensively studied and applied to disaggregate rainfall data, thanks to their formal simplicity and the small number of involved parameters. Focusing on temporal disaggregation, the rationale of these models is based on multiplying the value assumed by a physical attribute (e.g., rainfall intensity) at a given time scale <I>L</I>, by a suitable number <I>b</I> of random weights, to obtain <I>b</I> attribute values corresponding to statistically plausible observations at a smaller <I>L/b</I> time resolution. In the original formulation of the MRC models, the random weights were assumed to be independent and identically distributed. However, for several studies this hypothesis did not appear to be realistic for the observed rainfall series as the distribution of the weights was shown to depend on the space-time scale and rainfall intensity. Since these findings contrast with the scale invariance assumption behind the MRC models and impact on the applicability of these models, it is worth studying their nature. This study explores the possible presence of dependence of the parameters of two discrete MRC models on rainfall intensity and time scale, by analyzing point rainfall series with 5-min time resolution. Taking into account a discrete microcanonical (MC) model based on beta distribution and a discrete canonical beta-logstable (BLS), the analysis points out that the relations between the parameters and rainfall intensity across the time scales are detectable and can be modeled by a set of simple functions accounting for the parameter-rainfall intensity relationship, and another set describing the link between the parameters and the time scale. Therefore, MC and BLS models were modified to explicitly account for these relationships and compared with the continuous in scale universal multifractal (CUM) model, which is used as a physically based benchmark model. Monte Carlo simulations point out that the dependence of MC and BLS parameters on rainfall intensity and cascade scales can be recognized also in CUM series, meaning that these relations cannot be considered as a definitive sign of departure from multifractality. Even though the modified MC model is not properly a scaling model (parameters depend on rainfall intensity and scale), it reproduces the empirical traces of the moments and moment exponent function as effective as the CUM model. Moreover, the MC model is able to reproduce some rainfall properties of hydrological interest, such as the distribution of event rainfall amount, wet/dry spell length, and the autocorrelation function, better than its competitors owing to its strong, albeit unrealistic, conservative nature. Therefore, even though the CUM model represents the most parsimonious and the only physically/theoretically consistent model, results provided by MC model motivate, to some extent, the interest recognized in the literature for this type of discrete models

Comment on "Multivariate non-normally distributed random variables in climate research – introduction to the copula approach" by C. Schölzel and P. Friederichs, Nonlin. Processes Geophys., 15, 761–772, 2008

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Multivariate linear parametric models applied to daily rainfall time series

International audienceThe aim of this paper is to test the Multivariate Linear Parametric Models applied to daily rainfall series. These simple models allow to generate synthetic series preserving both the time correlation (autocorrelation) and the space correlation (crosscorrelation). To have synthetic daily series, in such a way realistic and usable, it is necessary the application of a corrective procedure, removing negative values and enforcing the no-rain probability. The following study compares some linear models each other and points out the roles of autoregressive (AR) and moving average (MA) components as well as parameter orders and mixed parameters

Spatial And Temporal Modeling Of Radar Rainfall Uncertainties

It is widely acknowledged that radar-based estimates of rainfall are affected by uncertainties (e.g., mis-calibration, beam blockage, anomalous propagation, and ground clutter) which are both systematic and random in nature. Improving the characterization of these errors would yield better understanding and interpretations of results from studies in which these estimates are used as inputs (e.g., hydrologic modeling) or initial conditions (e.g., rainfall forecasting). Building on earlier efforts, the authors apply a data-driven multiplicative model in which the relationship between true rainfall and radar rainfall can be described in terms of the product of a systematic and random component. The systematic component accounts for conditional biases. The conditional bias is approximated by a power-law function. The random component, which represents the random fluctuations remaining after correcting for systematic uncertainties, is characterized in terms of its probability distribution as well as its spatial and temporal dependencies. The space-time dependencies are computed using the non-parametric Kendall\u27s τ measure. For the first time, the authors present a methodology based on conditional copulas to generate ensembles of random error fields with the prescribed marginal probability distribution and spatio-temporal dependencies. The methodology is illustrated using data from Clear Creek, which is a densely instrumented experimental watershed in eastern Iowa. Results are based on three years of radar data from the Davenport Weather Surveillance Radar 88 Doppler (WSR-88D) radar that were processed through the Hydro-NEXRAD system. The spatial and temporal resolutions are 0.5. km and hourly, respectively, and the radar data are complemented by rainfall measurements from 11 rain gages, located within the catchment, which are used to approximate true ground rainfall. © 2013 Elsevier B.V

Complexity–entropy analysis of daily stream flow time series in the continental United States

Complexity–entropy causality plane (CECP) is a diagnostic diagram plotting normalized Shannon entropy HS versus Jensen–Shannon complexity CJS that has been introduced in nonlinear dynamics analysis to classify signals according to their degrees of randomness and complexity. In this study, we explore the applicability of CECP in hydrological studies by analyzing 80 daily stream flow time series recorded in the continental United States during a period of 75 years, surrogate sequences simulated by autoregressive models (with independent or long-range memory innovations), Theiler amplitude adjusted Fourier transform and Theiler phase randomization, and a set of signals drawn from nonlinear dynamic systems. The effect of seasonality, and the relationships between the CECP quantifiers and several physical and statistical properties of the observed time series are also studied. The results point out that: (1) the CECP can discriminate chaotic and stochastic signals in presence of moderate observational noise; (2) the signal classification depends on the sampling frequency and aggregation time scales; (3) both chaotic and stochastic systems can be compatible with the daily stream flow dynamics, when the focus is on the information content, thus setting these results in the context of the debate on observational equivalence; (4) the empirical relationships between HS and CJS and Hurst parameter H, base flow index, basin drainage area and stream flow quantiles highlight that the CECP quantifiers can be considered as proxies of the long-term low-frequency groundwater processes rather than proxies of the short-term high-frequency surface processes; (6) the joint application of linear and nonlinear diagnostics allows for a more comprehensive characterization of the stream flow time series.Centro de Investigaciones Óptica