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

Abstract

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.

Streamflow simulation: A nonparametric approach

“…By allowing the noise term Wt, • to be from a skewed distribution [Lettenmaier and Burges, 1977;Todini, 1980], streamflow from a skewed distribution can be approximated. Thus this model reproduces the mean, variance, and correlations between monthly streamflows and approximates the skewness.…”

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

“…We remark that this was acknowledged by the authors [3] (pp. [53][54][55][56][57] as well as later remarked by other researchers ( [26,37,38], (p. 66)); (2) In order to reproduce the skewness of the underlying process it is required (due to central limit theorem) to use white noise with higher skewness [11,23,38,39], which can cause, in some cases, failure of the random number generator itself; (3) This simulation scheme generates time series that can have negative values, which is not consistent with many physical processes (e.g., rainfall, wind, streamflow, etc.). This is attributed to the fact that the lower bound of the white noise distribution may be negative in order to match the target statistics (as estimated from observed time series); (4) Finally, we prove and demonstrate in the next sections that this scheme leads to bounded and thus unrealistic dependence patterns that are not observed in natural processes (such as those depicted in Figure 1).…”

“…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].…”

Adapting California’s water system to warm vs. dry climates

“…This is in line with classical reservoir operations theory that reservoirs with over-year storage capability are affected much less by seasonal changes in flows (Hazen 1914). Many approaches are available to quantify the over-year storage character of a reservoir (Hoshi et al 1978;Lettenmaier and Burges 1977;Vogel et al 1999). All are imperfect.…”