Abstract
Since the prime days of stochastic hydrology back in 1960s, autoregressive (AR) and moving average (MA) models (as well as their extensions) have been widely used to simulate hydrometeorological processes. Initially, AR(1) or Markovian models with Gaussian noise prevailed due to their conceptual and mathematical simplicity. However, the ubiquitous skewed behavior of most hydrometeorological processes, particularly at fine time scales, necessitated the generation of synthetic time series to also reproduce higher-order moments. In this respect, the former schemes were enhanced to preserve skewness through the use of non-Gaussian white noise— a modification attributed to Thomas and Fiering (TF). Although preserving higher-order moments to approximate a distribution is a limited and potentially risky solution, the TF approach has become a common choice in operational practice. In this study, almost half a century after its introduction, we reveal an important flaw that spans over all popular linear stochastic models that employ non-Gaussian white noise. Focusing on the Markovian case, we prove mathematically that this generating scheme provides bounded dependence patterns, which are both unrealistic and inconsistent with the observed data. This so-called “envelope behavior” is amplified as the skewness and correlation increases, as demonstrated on the basis of real-world and hypothetical simulation examples.
Highlights
A Glimpse of History “Remember that all models are wrong; the practical question is how wrong do they have to be to not be useful.”—George Box and Norman Draper [1]The celebrated Harvard water program and the development of the so-called Thomas-Fiering (TF) model in the early 60s [2,3,4,5] played a historically crucial role in definition and advancement of the scientific discipline of stochastic hydrology—of synthetic hydrology
We mainly focus on AR models with non-Gaussian white noise, which have been widely adopted in hydrology, and briefly discuss three alternative schemes, two of which are based on moving average (MA) models and one based on an autoregressive moving average model (ARMA)
A way to answer this question is through impact assessments of the envelope behavior in real-world applications, e.g., in important engineering studies, and of its effect on decision-making related to water resources management
Summary
A Glimpse of History “Remember that all models are wrong; the practical question is how wrong do they have to be to not be useful.”. The celebrated Harvard water program and the development of the so-called Thomas-Fiering (TF) model in the early 60s [2,3,4,5] played a historically crucial role in definition and advancement of the scientific discipline of stochastic hydrology—of synthetic hydrology. We investigate the effect on the established dependence patterns that arise from the use of P III white noise within stationary univariate and multivariate linear stochastic models for generating synthetic hydrological data via stochastic simulation. It is remarkable that the model reproduces almost perfectly the (often regarded as essential) statistical characteristics of historical data, i.e., the mean, variance, and skewness, as well as the month-to-month linear correlations (Pearson’s), which is the typical measure of statistical dependence that is encountered in all linear stochastic schemes It seems that the preservation of the latter does not ensure the generation of fully consistent dependence patterns. The red line (—) depicts the envelope the TF model
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