Application of the Thomas and Fiering Model to Skewed Hydrologic Data

McMahon, Thomas A.; Miller, A. J.

doi:10.1029/wr007i005p01338

Cited by 29 publications

(11 citation statements)

References 3 publications

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“…In hydrology, log-normal and gamma distributions have been the most popular; the gamma variable being generated by a X 2 variable and the Wilson and Hilferty (1931) transformation. However, McMahon and Miller (1971) showed that this was poor for skewness in excess of two, as it often is, and Kirby (1972) suggested a modified transformation for Pearson Type III distributions. However, Bernier (1970), Weiss (1973a, b) and Lewis (1975, private communication) have obtained the exact distribution of {St} which ensures {Vi} has a gamma distribution; solutions in other cases are awaited.…”

Section: A T+1 Atmentioning

confidence: 99%

Stochastic Modelling of Riverflow Time Series

Lawrance¹,

Kottegoda²

1977

Journal of the Royal Statistical Society. Series A (General)

202

100

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Statistical work in hydrology on the topic of riverflow time series is discussed. The theme is the "data-generation method" which in the present situation aims to produce simulations corresponding to a chosen historical riverflow series; they are required to resemble the historical data in respect of hydrologically important properties, such as runs of low flows. In practice the simulations are then used as input to a projected water resources system, but this aspect is not treated here. Capture of statistical resemblence entails a thorough statistical analysis of the historical series, and the paper describes the distinctive non-Gaussian and periodic features of typical riverflow series; it emphasizes the hydrological importance of seasonality, marginal distribution, dependence and crossing properties. The paper then describes hydrological adaptions and use of some of the traditional time series and shot noise models. It discusses the hydrologically motivated long memory fractional noise and broken line models; here the aim is to bring them to the attention of statisticians for appraisal while at the same time making their theory more accessible to hydrologists. Places where the presently used methodology needs consolidating are mentioned, as are areas where further statistical and mathematical work would be of value.

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Section: A T+1 Atmentioning

confidence: 99%

Stochastic Modelling of Riverflow Time Series

Lawrance¹,

Kottegoda²

1977

Journal of the Royal Statistical Society. Series A (General)

202

100

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“…Historically, most of the questions raised regarding the TF approach have concerned the case of the AR(1) model and the range of attainable skewness coefficients [20,38,43]. This was mainly due to the use of Wilson-Hilferty transformation which was used for generating Gamma or Pearson type-III RVs [44].…”

Section: Discussionmentioning

confidence: 99%

“…Nowadays, this technical issue is out of interest, since such RVs can be easily generated with high accuracy by modern random number generators which are available in almost every programming language (e.g., R, MATLAB, etc.). Additionally, we note that McMahon and Miller [20] reported that Thomas and Burden [18] and Fiering [5] tested their approach for skewness values ranging in (−0.5, 1.0). This work focused on the effect of using Pearson type-III white noise in AR(1) models and we show that this approach leads to unrealistic dependence patterns.…”

Section: Discussionmentioning

confidence: 99%

A Cautionary Note on the Reproduction of Dependencies through Linear Stochastic Models with Non-Gaussian White Noise

Tsoukalas

Papalexiou

Efstratiadis

et al. 2018

Water

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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.

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“…The general gamma distribution is a three-pai'ameter distri- where G, is the desired skew coefficient of •t and where r/t is a normal independent process of mean zero and variance one, resulting in an independent process with mean zero, variance one, and skew coefficient G•. McMahon and Miller [1971] showed that this transformation is inaccurate for Gx >• 2. Kirby [1972] provided a modification of the transformation which gives good results for Gx •< 10.…”

Section: Gamma Distributionmentioning

confidence: 99%

An operational approach to preserving skew in hydrologic models of long‐term persistence

Lettenmaier¹,

Burges²

1977

Water Resources Research

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Formulation of two models of long‐term persistence, fast fractional Gaussian noise (ffGn) and the first‐ order autoregressive‐first‐order moving average (Arma (1, 1)) process for a three‐parameter log normal and three‐parameter gamma distribution are given. For the three‐parameter log normal distribution the marginal probability distribution is generated exactly for both models, but the desired autocorrelation functions are distorted. Use of the three‐parameter log normal distribution requires a transformation of the lag one correlation coefficient and the Hurst coefficient for the ffGn model and of the parameters ϕ and θ for the Arma (1, 1) model. Because of the nonlinearity of the three‐parameter log normal transformation, sequences with slightly higher long‐term persistence (larger H or ϕ) must be generated in the logarithmic domain to achieve the desired model properties in the skewed domain. For the gamma distribution, the autocorrelation function is preserved exactly, but the desired marginal probability distribution is distorted. Monte Carlo tests showed that this distortion is evidenced primarily in the loss of the theoretical lower bound of the gamma distribution. This distortion is most extreme for time series having high long‐term persistence (H or ϕ) and large coefficients of variation. For operational applications, there appeared to be no clear advantage to use of either of the skewed distributions.

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Application of the Thomas and Fiering Model to Skewed Hydrologic Data

Abstract: An inconsistency is noted in the transformation to modify random normal variates to random skewed variates used in the Thomas and Fiering hydrologic generation model.

Cited by 29 publications

References 3 publications

Stochastic Modelling of Riverflow Time Series

Stochastic Modelling of Riverflow Time Series

A Cautionary Note on the Reproduction of Dependencies through Linear Stochastic Models with Non-Gaussian White Noise

An operational approach to preserving skew in hydrologic models of long‐term persistence

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