Third-Order Polynomial Normal Transform Applied to Multivariate Hydrologic Extremes

Yeou-Koung Tung,Lingwan You,Chulsang Yoo

doi:10.3390/w11030490

Yeou-Koung Tung, Lingwan You + Show 1 more

Open Access

https://doi.org/10.3390/w11030490

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Journal: Water	Publication Date: Mar 8, 2019
Citations: 3	License type: CC BY 4.0

Affiliation: National Yang Ming Chiao Tung University, Korea University

Abstract

Hydro-infrastructural systems (e.g., flood control dams, stormwater detention basins, and seawalls) are designed to protect the public against the adverse impacts of various hydrologic extremes (e.g., floods, droughts, and storm surges). In their design and safety evaluation, the characteristics of concerned hydrologic extremes affecting the hydrosystem performance often are described by several interrelated random variables—not just one—that need to be considered simultaneously. These multiple random variables, in practical problems, have a mixture of non-normal distributions of which the joint distribution function is difficult to establish. To tackle problems involving multivariate non-normal variables, one frequently adopted approach is to transform non-normal variables from their original domain to multivariate normal space under which a large wealth of established theories can be utilized. This study presents a framework for practical normal transform based on the third-order polynomial in the context of a multivariate setting. Especially, the study focuses on multivariate third-order polynomial normal transform (TPNT) with explicit consideration of sampling errors in sample L-moments and correlation coefficients. For illustration, the modeling framework is applied to establish an at-site rainfall intensity–duration-frequency (IDF) relationship. Annual maximum rainfall data analyzed contain seven durations (1–72 h) with 27 years of useable records. Numerical application shows that the proposed modeling framework can produce reasonable rainfall IDF relationships by simultaneously treating several correlated rainfall data series and is a viable tool in dealing with multivariate data with a mixture of non-normal distributions.

Highlights

In hydrosystem design, performance evaluation, and simulation, the problems often involve multiple random variables that are correlated with a mixture of non-normal marginal distributions.Under this condition, it is generally difficult, if not impossible, to establish an analytical joint probability distribution for these variables
By varying the value of zT for different return periods in Equation (55), in conjunction with the optimal third-order polynomial normal transform (TPNT) coefficients listed in Table 6a–d, one can establish IDF curves as shown in Figures 2 and 3
Focus is placed on the third-order polynomial transform (TPNT) procedure, which relies on the preservation of marginal L-moments and correlations among variables

Summary

Introduction

Performance evaluation, and simulation, the problems often involve multiple random variables that are correlated with a mixture of non-normal marginal distributions. Under this condition, it is generally difficult, if not impossible, to establish an analytical joint probability distribution for these variables. Examples of using analytical multivariate distributions in hydrology are bivariate Gamma distribution [3] and bivariate generalized extreme distribution [4]. Their use is somewhat limited to many practical problems because of different marginal distributions.

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Conclusion