Where TWBM represents the two-parameter monthly water balance model; \(w_{k+1}\) represents the observation error, following a Gaussian distribution with zero mean; \(P_{k+1}\) is the precipitation at time k +1; \(\text{EP}_{k+1}\) is the potential evapotranspiration at time k +1; \(Q_{k+1}\) is the observed runoff at time k +1.
In this study, the uncertainties in output and input (i.e.  v, \(\xi\), \(\eta\) and w in the Equation 8, Equation 9, Equation 10 and Equation 11, respectively) are important to the performance of Particle filter in assimilating the parameters (i.e. SC and C ). The uncertainties (i.e.  v, \(\xi\), \(\eta\) and w in the Equation 8, Equation 9, Equation 10 and Equation 11, respectively) are specified empirically to follow a Gaussian distribution with zero mean and specified deviation following previous studies(Moradkhani et al., 2005; Wang, Chen, & Cai, 2009; Xie & Zhang, 2010; Deng et al., 2016). The deviation of C is set as 0.05. Both the deviation of SC andS is assumed to be proportional to the forecasted SC and S(i.e. obtained from state-transition equation), respectively, at each time. The proportional factors are both set as 0.05. The deviation of observed error is assumed to be proportional to the observed runoff. The proportional factor is set as 0.15. The Particle filter is run 100 times account the possible uncertainty in assimilated parameters and the ensemble mean of the 100 times estimated parameters is considered as the final estimated parameters.

Pettitt’s test for step change detection

In this paper, the Pettitt’s test (Pettitt, 1979) is used to detect significant change in the average value for the assimilated times series of SC and C. The test employs a statistic \(U_{t,\ \ N}\), to verify whether there is a single change-point between two samples \(x_{1}\),…\(x_{t}\) and\(x_{t+1}\),…,\(x_{N}\). The test statistic for each timet (t= 1, 2 …, N) is calculated by: