Abstract
Multivariate hydrologic frequency analysis has been widely studied using: (1) commonly known joint distributions or copula functions with the assumption of univariate variables being independently identically distributed (I.I.D.) random variables; or (2) directly applying the entropy theory-based framework. However, for the I.I.D. univariate random variable assumption, the univariate variable may be considered as independently distributed, but it may not be identically distributed; and secondly, the commonly applied Pearson’s coefficient of correlation (g) is not able to capture the nonlinear dependence structure that usually exists. Thus, this study attempts to combine the copula theory with the entropy theory for bivariate rainfall and runoff analysis. The entropy theory is applied to derive the univariate rainfall and runoff distributions. It permits the incorporation of given or known information, codified in the form of constraints and results in a universal solution of univariate probability distributions. The copula theory is applied to determine the joint rainfall-runoff distribution. Application of the copula theory results in: (i) the detection of the nonlinear dependence between the correlated random variables-rainfall and runoff, and (ii) capturing the tail dependence for risk analysis through joint return period and conditional return period of rainfall and runoff. The methodology is validated using annual daily maximum rainfall and the corresponding daily runoff (discharge) data collected from watersheds near Riesel, Texas (small agricultural experimental watersheds) and Cuyahoga River watershed, Ohio.
Highlights
Comparing to the existing frameworks, the proposed framework has the following advantages: (i) the universal probability distribution can be obtained from appropriately defined constraints; (ii) the multi-mode can be captured using the maximum entropy theory if the data show the multi-mode structure which may result in better estimation of multivariate/conditional return periods of given events; and (iii) the nonlinear dependence can be captured among the correlated random variables by applying the copula theory rather than applying the known or entropy-based multivariate probability distribution with the dependence captured by linear covariance
Using X as rainfall random variable and Y as runoff random variable, the conditional return period of runoff events of given rainfall events can be written in two cases: Case I: Return period of runoff events conditioned on rainfall events greater than the given rainfall values: Applying the copula theory, the exceedance conditional distribution is written as:
This study investigates the relationship between annual maximum daily rainfall amount and the corresponding daily runoff using maximum entropy and copula theories to address the questions arising from the assumptions in the commonly applied approaches and to better estimate risk
Summary
In multivariate hydrological frequency analysis, studies have been extensively carried out along three lines: (I) application of the covariance structure (i.e., Pearson’s linear covariance/correlation matrix) with known multivariate and univariate probability distributions [1,2,3,4,5]; (II) application of copula theory to the pseudo-observations (i.e., empirical probability distribution function) first and study the risk with fitted univariate distributions [6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24]; and (III) application of linear covariance with the maximum entropy framework [25,26,27,28,29]. Comparing to the existing frameworks, the proposed framework has the following advantages: (i) the universal probability distribution can be obtained from appropriately defined constraints; (ii) the multi-mode can be captured using the maximum entropy theory if the data show the multi-mode structure which may result in better estimation of multivariate/conditional return periods of given events; and (iii) the nonlinear dependence can be captured among the correlated random variables by applying the copula theory rather than applying the known or entropy-based multivariate probability distribution with the dependence captured by linear covariance.
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