Estimation of the distribution of annual runoff from climatic variables using copulas

An approach of deriving the annual runoff distribution using copulas from an annual rainfall-runoff model is proposed to provide an alternative annual runoff frequency analysis method in case of changing climatic variables. The annual rainfall-runoff model is established on the basis of the Budyko formula to estimate annual runoff, with annual precipitation and potential evapotranspiration as input variables. The model contains one single parameter k that guarantees that annual water balance is satisfied. In the derivation of the annual runoff distribution, annual precipitation, annual potential evapotranspiration, and parameter k are treated as three random variables, while the annual runoff distribution is obtained by integrating the joint probability density function of the three random variables over the domain constrained by the annual rainfall-runoff model using the canonical vine copula. This copula-based derivation approach is tested for 40 watersheds in two large basins in China. The estimated annual runoff distribution performs well in most watersheds. The performance is mainly related to the accuracy of the marginal distribution of precipitation. The copula-based derivation approach can also be used in ungauged watersheds where the distribution of k at the local site is estimated from the regional information of the k variable, and it also has acceptable performance in most watersheds, while poor performance is observed in a few watersheds with low accuracy in the Budyko formula.