Gumbel–Hougaard copula-based tetravariate flood frequency analysis for the Hirakud reservoir catchment

ABSTRACT

The floods are averted depending upon the effective operations of gate keeping downstream conditions in mind. In this regard inflow hydrographs have a major role to play. The tetravariate flood frequency analysis has been done in this study by taking four variables of a hydrograph i.e. peak (Qp), volume (V), duration (D) and time to peak (Tp). The variable time to peak has given more emphasis in this study, as it characterizes the inflow more effectively and its severity gives less time for operation of dam. In this study, the dependence parameters (${\theta }_1,{\theta }_2,{\theta }_3$) of the proposed copula model are determined, finally the values of best fit marginal for all the four variables and dependence parameters (${\theta }_1,{\theta }_2,{\theta }_3$) are applied in four-dimensional asymmetric Gumbel–Hougaard copula. The main advantage of using this model is that, it relaxes the restriction of using a similar type of marginal distributions for all the four basic variables. The three hourly inflow data of Hirakud reservoir has been taken to fit the proposed model. Finally, this model is validated using the observed tetravariate probability plotting position model. The result of the copula model is in better agreement with the observed tetravariate probability, ultimately using this copula model, conditional probability is determined.

KEYWORDS:

• Four-dimensional symmetric Gumbel–Hougaard copula
• trial and error method
• marginals
• Hirakud reservoir

Notations

PDF(f) = Probability Density Function

CDF(F) = Cumulative Probability Distribution Function

MLE = Maximum Likelihood Estimation Method.

EV1 = Extreme Value Type 1 probability distribution

AIC = Akaike Information Criteria

MSE = Mean Square Error

RMSE = Root Mean Square Error

TCM = Thousand Cubic Meter

${\mu }_{\mathrm{y}}$= mean of y

${\mathrm{s}}_{\mathrm{y}}$ = standard deviation of y

θ = dependence parameter

a = marginal CDF for peak discharge

b = marginal CDF for volume

c = marginal CDF for duration

d = marginal CDF for time to peak

n = number of observations.

P = non-exceedance probability

q = value of the random variable Q

v = value of random variable V

d = value of random variable D

t_p = value of the random variable TP

$Q_p$ = Peak discharge

V = Flood volume

D = Duration

T_P = Time to peak

Th = Theoritical tetravariate CDF

Ob = Observed tetravariate CDF

Acknowledgments

Authors are highly appreciative of the Department of Water Resources for providing the inflow data for making this research work a success.

Data Availability Statement

The inflow data used in this study are from Department of Water Resources, Government of Odisha. Requests for data should be directed to the provider indicated in the Acknowledgements.https://www.dowrodisha.gov.in/

Disclosure statement

No potential conflict of interest was reported by the author(s).