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.