Optimal solution error covariance in highly nonlinear problems of variational data assimilation

The problem of variational data assimilation for a nonlinear evolution model is formulated as an optimal control problem (see, e.g.[1]) to find the initial condition, boundary conditions or model parameters. The input data contain observation and background errors, hence there is an error in the optimal solution. For mildly nonlinear dynamics, the covariance matrix of the optimal solution error can be approximated by the inverse Hessian of the cost functional of an auxiliary data assimilation problem ([2], [3]). The relationship between the optimal solution error covariance matrix and the Hessian of the auxiliary control problem is discussed for different degrees of validity of the tangent linear hypothesis. For problems with strongly nonlinear dynamics a new statistical method based on computation of a sample of inverse Hessians is suggested. This method relies on the efficient computation of the inverse Hessian by means of iterative methods (Lanczos and quasi-Newton BFGS) with preconditioning. The method allows us to get a sensible approximation of the posterior covariance matrix with a small sample size. Numerical examples are presented for the model governed by Burgers equation with a nonlinear viscous term. The first author acknowledges the funding through the project 09-01-00284 of the Russian Foundation for Basic Research, and the FCP program "Kadry"

On optimal solution error covariances in variational data assimilation problems

The problem of variational data assimilation for a nonlinear evolution model is formulated as an optimal control problem to find unknown parameters such as distributed model coefficients or boundary conditions. The equation for the optimal solution error is derived through the errors of the input data (background and observation errors), and the optimal solution error covariance operator through the input data error covariance operators, respectively. The quasi-Newton BFGS algorithm is adapted to construct the covariance matrix of the optimal solution error using the inverse Hessian of an auxiliary data assimilation problem based on the tangent linear model constraints. Preconditioning is applied to reduce the number of iterations required by the BFGS algorithm to build a quasi-Newton approximation of the inverse Hessian. Numerical examples are presented for the one-dimensional convection-diffusion model

A posteriori error covariances in variational data assimilation

The problem of variational data assimilation for a nonlinear evolution model is formulated as an optimal control problem to find some unknown parameters of the model. The equation for the error of the optimal solution is derived through the statistical errors of the input data (background, observation, and model errors). A numerical algorithm is developed to construct an a posteriori covariance operator of the analysis error using the Hessian of an auxiliary control problem based on tangent linear model constraints

Divergence-Free Motion Estimation

International audienceThis paper describes an innovative approach to estimate motion from image observations of divergence-free flows. Unlike most state-of-the-art methods, which only minimize the divergence of the motion field, our approach utilizes the vorticity-velocity formalism in order to construct a motion field in the subspace of divergence free functions. A 4DVAR-like image assimilation method is used to generate an estimate of the vorticity field given image observations. Given that vorticity estimate, the motion is obtained solving the Poisson equation. Results are illustrated on synthetic image observations and compared to those obtained with state-of-the-art methods, in order to quantify the improvements brought by the presented approach. The method is then applied to ocean satellite data to demonstrate its performance on the real images

Utilisation de contraintes spatiales dans un schéma d'assimilation d'images de télédétection. Exemple sur un modèle de culture simple, Bonsaï

[Departement_IRSTEA]Eaux [TR1_IRSTEA]ARCEAUInternational audienceAssimilation of remote sensing data into crop models is generally applied pixel by pixel, while the whole image data is available. Estimating the crop model parameters of a pixel independently from the neighboring ones generates at least four issues: (1) the method does not take into account the possible spatial structures although there are not necessarily existing or easy to quantify (2) the spatial properties of images, and even more of time series of images constitute an additional source of information to use, which is neglected in this case (3) the inverse problem is generally ill-posed when applied at the pixel level and (4) repeating the same action a great number of times is not computer efficient and provides sub-optimal solutions. When inverting models of significant size, computational cost becomes a clear limitation. Some constraints are sometimes added to ensure better spatial consistency through the regularization of the behavior of a pixel by that the neighboring ones. However, this ignores the spatial dependencies of the parameters to be estimated. We propose here to exploit some spatial structures of the parameters to reduce the size of the problem and make its inversion manageable. It is applied to process concurrently a set of pixels using variational assimilation using the adjoint model. This method was applied on a simple model of plant growth, the BONSAI model, which simulates LAI (Leaf Area Index) as a function of 6 parameters. The method proposed here assumes that the parameters are governed by spatial structures depending on several levels: the cultivar, the field, and the pixel, while some of them are assumed to be stable over the whole image. For example, cultivar parameters govern phenological stages. At a lower level, some parameters depend on the field, such as agricultural practices. Other parameters depend on the pixel level, such as soil parameters. Finally, to improve the robustness of the method and reduce the space of realization, the parameters to which the model is not sensitive were considered fixed to a default value on all plots and all cultivars. The constrained variational method has been tested on twin experiments (with virtual data) and on actual observations from the ADAM experiement and evaluated on (1) the quality of the estimation of the model input parameters, (2) the quality of LAI simulation and (3) its sensitivity on the frequency of observations, i.e. satellite revisit time. If it is not always relevant to assert that space constraints allow a better reproduction of the trajectory of LAI as compared to a conventional method when a lot of observations are available, the method allows obtaining an equally satisfactory result for a much lower computational cost. However, the spatial constraints allow more stable and robust results when the frequency of observations is relaxed as compared to the assimilation by pixel. In addition, this new method requires the minimization of a single cost function easier to control when there are divergence or local minimum. Finally, estimate of the model input parameters, which is generally the main objective of data assimilation, is much less dependent on the number of observations assimilated by using these spatial constraints

On optimal solution error covariances in variational data assimilation

The problem of variational data assimilation for a nonlinear evolution model is formulated as an optimal control problem to find some unknown parameters of the model. The equation for the error of the optimal solution is derived through the statistical errors of the input data. The covariance operator of the optimal solution error is obtained using the Hessian of an auxiliary data assimilation problem based on the tangent linear model constraints

Computation of the analysis error covariance in variational data assimilation problems with nonlinear dynamics

The problem of variational data assimilation for a nonlinear evolution model is formulated as an optimal control problem to find the initial condition function. The data contain errors (observation and background errors), hence there will be errors in the optimal solution. For mildly nonlinear dynamics, the covariance matrix of the optimal solution error can often be approximated by the inverse Hessian of the cost functional. Here we focus on highly nonlinear dynamics, in which case this approximation may not be valid. The equation relating the optimal solution error and the errors of the input data is used to construct an approximation of the optimal solution error covariance. Two new methods for computing this covariance are presented: the fully nonlinear ensemble method with sampling error compensation and the ‘effective inverse Hessian’ method. The second method relies on the efficient computation of the inverse Hessian by the quasi-Newton BFGS method with preconditioning. Numerical examples are presented for the model governed by Burgers equation with a nonlinear viscous term

A Water Supply Optimization Problem for Plant Growth Based on GreenLab Model

Disponible en ligne ArimaGreenLab est un modèle structure-fonction de croissance des plantes. Son formalisme mathématique permet la simulation dynamique de la croissance et l'analyse du modèle. Dans cet article est introduit une équation bilan de l'eau dans le sol afin de décrire les interactions entre la croissance de la plante et les ressources en eau disponibles. Un problème d'optimisation des apports d'eau au cours de la croissance est présenté et résolu par la méthode de recherche intuitive et par les algorithmes génétiques: le poids du fruit de tournesol est maximisé en fonction de différentes stratégies d'apports d'eau, pour une quantité d'eau totale identique. Le formalisme présenté est intéressant en ce qu'il ouvre la voie à d'importantes applications en agronomie. ABSTRACT. GreenLab is a structural-functional model for plant growth based on multidisciplinary knowledge. Its mathematical formalism allows dynamic simulation of plant growth and model analysis. A simplified soil water balance equation is introduced to illustrate the interactions and feedbacks between the plant functioning and water resources. A water supply optimization problem is then described and solved: the sunflower fruit weight is optimized with respect to different water supply strategies in a theoretical case. Intuitive searching method and genetic algorithms are used to solve this mixed integer nonlinear problem. The optimization results are analyzed and reveal possible agronomic applications

On error sensitivity analysis in variational data assimilation

Theme 4 - Simulation et optimisation de systemes complexes - Projet IdoptSIGLEAvailable from INIST (FR), Document Supply Service, under shelf-number : 14802 E, issue : a.2001 n.4220 / INIST-CNRS - Institut de l'Information Scientifique et TechniqueFRFranc