Theoretical Background

The different assimilation systems try to estimate selected parameters of the ORCHIDEE vegetation model, possibly including the initial conditions of state variables, with respect to various data sources.

The approach relies on the minimization of a misfit function \( J(x) \) that measures the mismatch between 1) a set of observations \( Y \) and corresponding model outputs \( M(x) \), and 2) the values \( x \) of the parameters (to optimize) and some prior information on them \( x_b \), weighted by the prior error covariance matrices on observations \( R \) and parameters \( P_b \):

\[ J(x) = (Y-M(x))^t R^{-1} (Y-M(x)) + (x-x_b)^t P_b^{-1} (x-x_b) \]

With this Bayesian inversion framework, we thus account for uncertainties associated to the model and the observations (\( R \) matrix), and the prior parameters (\( P_b \) matrix).

For all applications implemented so far, we assumed that the errors on prior parameters and observations follow Gaussian distributions. With this hypothesis, the posterior (optimized) parameter distributions are also Gaussian and their expectation correspond to the minimum of the misfit function. The uncertainties associated to the optimized parameter values are characterized by the posterior error covariance matrix \( P'_b \), that can be estimated assuming model linearity in the vicinity of the solution and Gaussian errors, by:

\[ P'_b = \Big[ H^t R^{-1} H + P_b^{-1} \Big]^{-1} \]

where the \( H \) matrix is the Jacobian matrix of the model \( M \) at the minimum of \( J \) (i.e. the sensitivity of the model outputs with respect to each parameter).

Propagation of the parameter errors (i.e., the variance-covariance matrix \( P'_b \)) to errors on the state variables (\( R' \)) can be done assuming linearity of the model around the optimal parameter set and following the standard rule of error propagation:

\[ R'_{sv} = H P'_b H^t \]

Note that the \( H \) matrix may be different than the one used for the computation of \( P'_b \), given that the state variables of interest may differ from the observations that are assimilated.