Comparison of First-Order and Second-Order Derived Moment Approaches in Estimating Annual Runoff Distribution

The first-order second-moment approximation (FOSM) has been proposed to estimate the annual runoff ($Q$) distribution from the statistical information of annual precipitation ($P$), annual potential evapotranspiration ($Ep$), and interannual water storage change ($\mathrm{\Delta }S$). However, FOSM can only estimate two statistical parameters for theoretical distributions, thus is unable to estimate the annual runoff distribution that is fitted by three-parameter probability distribution. Hence, this paper extends FOSM to second-order third-moment approximation (SOTM). This study first establishes an annual rainfall-runoff model. Then the moments of $Q$ are derived from the data of $P$, $Ep$, and $\mathrm{\Delta }S$ in the annual rainfall-runoff model by FOSM and SOTM. Because the observed data of $\mathrm{\Delta }S$ are difficult to obtain, $\mathrm{\Delta }S$ is simulated by two monthly hydrological models. Both FOSM and SOTM are applied to the Gan River basin (GRB) and the Tongtian River basin (TRB). The results show that the determination coefficients $R_{QQ}^2$ of curve fitting by FOSM and SOTM are both larger than 0.80, indicating their capability to estimate the annual runoff distribution. The availability of $\mathrm{\Delta }S$ has a certain impact on the performance of FOSM and SOTM, particularly in the TRB, where $R_{QQ}^2$ with the use of $\mathrm{\Delta }S$ information can increase by 2–4% compared with that without using $\mathrm{\Delta }S$. In practice, however, the impact of $\mathrm{\Delta }S$ can be neglected in the ungauged basins. It is also found that although the $R_{QQ}^2$ values estimated by SOTM for both Pearson Type III and three-parameter lognormal distributions are not greater than the $R_{QQ}^2$ values of gamma distribution estimated by FOSM, SOTM is capable of presenting more statistical information with the $R_{QQ}^2$ values all larger than 0.80.