2.3.4 Calibration process and uncertainty estimation
After the specification of a prior parameter pdf, the uncertain parameters in crop model and residual error model can be calibrated by conditioning on the data through the combined log likelihood function of Eqn 11. In this study, the prior pdf of uncertain parameters was treated as a normal distribution. The prior means of crop model parameters were defined as midpoints of their identified range, while their prior STDs were derived from the fixed coefficient of variation (CV) at 0.25. Regarding residual error parameters for multivariate observations, the prior means of \(\lambda\), \(\sigma_{1}\) and \(\xi\) were set as one, zero and one, respectively, and their prior STDs were 0.25. The DREAM-zs (DiffeRential Evolution Adaptive Metropolis) algorithm was adopted to generate a representative sample from the posterior distribution. DREAM-zs is designed to accelerate convergence for high-dimensional problems, by sampling from an archive of past parameter candidates. Furthermore, DREAM-zs increases the diversity of candidate points by generating jumps beyond parallel direction updates (Schoups and Vrugt, 2010). Although the original DREAM-zs does not require outlier detection, the step of detecting an outlier chain (Vrugt et al., 2009a) was included in this study, as outlier chains can significantly deteriorate the performance of the MCMC sampler, especially when there is more than one type of observation.
The number of chains was set as two times the number of calibrated parameters following the suggestion of Vrugt et al. (2009b) and in total, 100 000 evaluations were conducted in the process of MCMC sampling. The outlier chain was detected until 60% evaluations were finished. The last 10% evaluations of each chain were compiled to calculate the mean and STD of the posterior pdf of each parameter.
As for the uncertainty estimation in the crop model, 2000 parameter sets were sampled from the posterior pdf to generate the corresponding simulations \(\hat{\mathbf{Y}}\) and residual errors\(\mathbf{\varepsilon}\). By calculating the 2.5% and 97.5% percentiles of simulations with or without \(\mathbf{\varepsilon}\) for each type of observation, the uncertainties of the crop model caused by uncertain crop model parameters with or without model structural error were obtained. The estimation of \(\mathbf{\varepsilon}\) involves generating independent samples from a SEP distribution. The followed sampling algorithm is described completely in Schoups and Vrugt (2010). With sampled residual error parameters, the corresponding residual errors \(\mathbf{\varepsilon}\) were calculated using Eqn 8.