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.