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: