2.5 Evaluation metrics
With respect to the corresponding observations \(y_{i}\), the performance of \({\hat{y}}_{i}\), simulations in model calibration and validation or updated simulations in DA, was evaluated by the coefficient of determination (R 2), the root mean square error (RMSE ) and the normalized RMSE(NRMSE ):
\(R^{2}=1-\frac{\sum_{i\ =\ 1}^{n}{(y_{i}-{\hat{y}}_{i})}^{2}}{\sum_{i\ =\ 1}^{n}{(y_{i}-\overset{\overline{}}{y})}^{2}}\)(18)
\(RMSE=\sqrt{\sum_{i\ =\ 1}^{n}{(y_{i}-{\hat{y}}_{i})}^{2}/\left(n-1\right)}\)(19)
\(NRMSE=\frac{\text{RMSE}}{\overset{\overline{}}{y}}\) (20)
where \(n\) represents the number of evaluated data points, and\(\overset{\overline{}}{y}\) is the mean value of observations across the whole growing season.
Moreover, to assess the filter behavior of EnKF, we introducedf rc, which represents the relative change of the states after and before updating compared with the measured \(y_{i}\),
\(f_{\text{rc}}=\frac{{\overset{\overline{}}{Y}}_{i}^{a}-y_{i}}{{\overset{\overline{}}{Y}}_{i}^{f}-y_{i}}\)(21)
in which \({\overset{\overline{}}{Y}}_{i}^{a}\) is calculated as\(\frac{\left(\sum_{j=1}^{N_{\text{ens}}}Y_{t,j}^{a}\right)}{N_{\text{ens}}}\). The closer f rc is to one, the more likely does filter divergence occur.