

The EOLDAS Prototype 
The EOLDAS software allows for various forms of EO and other data to be combined to provide an optimal estimate of state variables. It requires that the uncertainties of the input data are quantified. At present, this is limited to an assumption that Gaussian statistics can be used to describe the uncertainty, i.e. input data are represented by their mean value and standard deviation (or more fully a covariance matrix, although some of the input data formats currently accepted only consider standard deviation). This can cause problems for highly nonlinear systems, especially if the uncertainties are large, so this can be partially overcome by transforming the representation of the state variable. Arbirtary transformations are allowed, but a typical one would be to work with a state variable that is an exponential transformation of the desired state variable. The DA system requires ‘Observation Operators’ to map from EO data (radiance or reflectance data) to the state variables of interest (e.g. LAI). A few example Observation Operators are available in this release of EOLDAS, including the semidiscrete radiative transfer model of Gobron et al. 1997, for which an adjoint has been developed here to allow more rapid state estimation. In addition, the linear Kernel models used in the MODIS BRDF/Albedo product are included and an interface to the 6S atmospheric model is provided. The EO data that can be used is then essentially limited to observations in the optical domain (~4502500 nm), if these operators are used. No ‘biophysical’ process models are included in this release, only a form of regularisation model.
The EOLDAS prototype uses a family of observation operators to fit the observations (contaminated by normal error), tempered by prior knowledge of the parameter distributions (assumed to be normal, and hence characterised by a mean vector and a covariance matrix) and prior knowledge of the parameter trajectory (either through smoothness conditions or through the deployment of a model of the parameters' trajectory). This latter term is the “dynamic model''. Mathematically, this can be expressed as a cost function which is the sum of three terms:
(2.1) Or more explicitly:
(2.1)
In the light of this, the best estimate of the state of the land surface is obtained at the minimum value of J(x).The uncertainty can be calculated by first estimating the Hessian around the minimum of J(x) and then inverting it to obtain the posterior covariance matrix. Note that in this formalism, there are a number of parameters that need to be specified:
· the observation noise through its covariance matrix C_{obs}, · the prior information through its mean vector x_{prior }and covariance matrix C_{prior} · the dynamic model M(x) · and the confidence we have in it, C_{model}
Note that ignoring J_{model }(2.1) is just a typical 4DVar DA system. The addition of J_{model}turns it into a 4DVar system with a ''weak constraint'': the solution ought to fit the model predictions, but this fit is tempered by the associated covariance structure, resulting in a solution that may diverge from the space of solutions spanned by the model.
The role of the prototype software is to minimise equation (2.1), find the state at the minimum and estimate the posterior covariance by calculating the Hessian. Where possible, this is aided by the provision of both J(x), as well as its derivative .
The latter can be used to aid the minimisation of the cost function. In practice, more components can be added to the above formulation. If the assumptions of normality still hold, the prototype will be able to take these new components into account.
