4.1 Impacts of residual error assumption for crop model calibration and data assimilation
The residual error assumption heavily affects the performance of the formal Bayesian approach (Beven et al., 2008). In our study,\(\sigma_{1}\) denoted the heteroscedastic residual error, assuming that the residual error linearly increased with the crop model simulations, as the STDs in field measurements tended to vary with the averages (Fig. 3). Similarly, due to the increase of both the averages and the STDs of the W above throughout the seasons, the likelihood function was revised with the observational variances by Dumont et al. (2014) for the non-stationary residual errors. In line with their results, our study showed that \(\sigma_{1}\) depicted the heteroscedasticity well for the crop model simulations (Fig. 3), although the calibration of the uncertain parameters tended to be sensitive to the range of \(\sigma_{1}\) (Fig. 2). Instead of the heteroscedastic residual error, Zhang et al. (2021a) used time-series variance across the growing season to investigate the heteroscedasticity of the uncertain parameters in the crop model AquaCrop, and showed that it significantly improved the effectiveness of the particle filter as well, when assimilating remotely sensed canopy cover into the crop model. In addition to the hypothesized heteroscedasticity, non-Gaussian errors were also introduced in our study and it was shown that the residual errors were more likely to be negatively skewed (\(\xi<1\)), except for those of simulated W grain(\(\xi_{W_{\text{grain}}}=2.86\)) and N grain(\(\xi_{N_{\text{grain}}}=1.75\)) (Table 5).
Compared with the complex residual error assumptions, the i.i.d. Gaussian errors worked well in the MCMC approach for the parameter calibration of the crop model (Dumont et al., 2014; Kang and Özdoğan, 2019). However, while further assimilating crop model simulated and remotely sensed crop traits by EnKF, instead of the pre-assumed i.i.d. Gaussian errors, the inflation factor is introduced for accounting for model uncertainties (Kang and Özdoğan, 2019). Our results showed that the estimated uncertainties resulted in a better performance of the DA system, compared with the ones when the pre-assumed uncertainties was used alone or when it was combined with the inflation factor being introduced into EnKF (Fig. 8). Moreover, the inflation factor tends to not only be sensitive to the filter performance (Kang and Özdoğan, 2019; Whitaker and Hamill, 2012), but also differ between updated traits of crop and soil (Ines et al., 2013). With the rapid development of various satellites and unmanned aerial vehicles, a wide range of crop and soil information will be accessible to be incorporated into crop models (Jin et al., 2018). However, due to the uncertainties of crop and soil traits, rigorous determination of inflation factors in DA is difficult. Rather than the obscure adjustment of inflation factors, quantifying crop model uncertainties by the MCMC process together with the adapted residual error assumption in this study has great potential. With the Bayes’ multiplication method, we calibrated the crop model parameters well by exploiting multiple crop traits (Table 6, Fig. 3), in line with the results of He et al. (2010). The uncertainties of multiple crop traits in crop model simulation were estimated simultaneously (Table 5). The estimated crop model uncertainties worked well in EnKF while assimilating remotely sensed leaf traits into the crop model GECROS, while the NRMSE of updated W grain andN grain increased slightly, compared with simulations using mean values of uncertain parameters (Table 6, Fig. 7). Recently, next to inflation factors and residual error models, the parameter and model structural errors of the simulation of rice phenology were determined by a multi-model ensemble method with assumed i.i.d. Gaussian errors, in which the different simulations generated from the different models were treated as the samples from the distribution of the unknown true model (Gao et al., 2021). Thus, more research is needed regarding the exploration of quantified crop model uncertainty and of their performance comparison in DA.