Sensitivity analysis and parameter estimation for distributed hydrological modeling: potential of variational methods

Variational methods are widely used for the analysis and control of computationally intensive spatially distributed systems. In particular, the adjoint state method enables a very efficient calculation of the derivatives of an objective function (response function to be analysed or cost function to be optimised) with respect to model inputs. In this contribution, it is shown that the potential of variational methods for distributed catchment scale hydrology should be considered. A distributed flash flood model, coupling kinematic wave overland flow and Green Ampt infiltration, is applied to a small catchment of the Thoré basin and used as a relatively simple (synthetic observations) but didactic application case. It is shown that forward and adjoint sensitivity analysis provide a local but extensive insight on the relation between the assigned model parameters and the simulated hydrological response. Spatially distributed parameter sensitivities can be obtained for a very modest calculation effort (~6 times the computing time of a single model run) and the singular value decomposition (SVD) of the Jacobian matrix provides an interesting perspective for the analysis of the rainfall-runoff relation. For the estimation of model parameters, adjoint-based derivatives were found exceedingly efficient in driving a bound-constrained quasi-Newton algorithm. The reference parameter set is retrieved independently from the optimization initial condition when the very common dimension reduction strategy (i.e. scalar multipliers) is adopted. Furthermore, the sensitivity analysis results suggest that most of the variability in this high-dimensional parameter space can be captured with a few orthogonal directions. A parametrization based on the SVD leading singular vectors was found very promising but should be combined with another regularization strategy in order to prevent overfitting

On model error in variational data assimilation

[Departement_IRSTEA]Eaux [TR1_IRSTEA]GEUSIInternational audienceThe problem of variational data assimilation for a nonlinear evolution model is formulated as an optimal control problem to find the initial condition. The optimal solution (analysis) error arises due to the errors in the input data (background and observation errors). Under the Gaussian assumption the optimal solution error covariance can be constructed using the Hessian of the auxiliary data assimilation problem. The aim of this paper is to study the evolution of model errors via data assimilation. The optimal solution error covariances are derived in the case of imperfect model and for the weak constraint formulation, when the model euations determine the cost functional

General sensitivity analysis in data assimilation

International audienceThe problem of variational data assimilation for a nonlinear evolution model is formulated as an optimal control problem to nd the initial condition function (analysis). The operator of the model, and hence the optimal solution, depend on the parameters which may contain uncertainties. A response function is considered as a functional of the solution after assimilation. Based on the second-order adjoint techniques, the sensitivity of the response function to the parameters of the model is studied. The gradient of the response function is related to the solution of a non-standard problem involving the coupled system of direct and adjoint equations. The solvability of the non-standard problem is studied. Numerical algorithms for solving the problem are developed. The results are applied for the 2D hydraulic and pollution models. Numerical examples on computation of the gradient of the response function are presented

Posterior Covariance vs. Analysis Error Covariance in Data Assimilation

International audienceThe problem of variational data assimilation for a nonlinear evolution model is formulated as an optimal control problem to find the initial condition function (analysis). The data contain errors (observation and background errors), hence there is an error in the analysis. For mildly nonlinear dynamics, the analysis error covariance can be approximated by the inverse Hessian of the cost functional in the auxiliary data assimilation problem, whereas for stronger nonlinearity - by the 'effective' inverse Hessian. However, it has been noticed that the analysis error covariance is not the posterior covariance from the Bayesian perspective. While these two are equivalent in the linear case, the difference may become significant in practical terms with the nonlinearity level rising. For the proper Bayesian posterior covariance a new approximation via the Hessian is derived and its 'effective' counterpart is introduced. An approach for computing the mentioned estimates in the matrix-free environment using Lanczos method with preconditioning is suggested. Numerical examples which validate the developed theory are presented for the model governed by Burgers equation with a nonlinear viscous term

Vector field regularization by generalized diffusion

Regularization is a common procedure when dealing with inverse problems. Because of the ill-posedness of many inverse problems, one needs to add some constraints as regularization to the problem in order to get a satisfactory solution. A difficulty when using multiple constraints is to properly choose a weighting parameter for each constraint. We propose here a vector field regularization method that combines in a single constraint the two well-known regularization methods namely Tikhonov regularization and smoothing regularization. The particularity of this new method is that one have only one balance parameter to determine. We also suggest a robust implementation of the proposed method based on the equivalent generalized diffusion equation in some particular cases. This implementation is illustrated on a set of vector fields of fluid motio

On error sensitivity analysis in variationnal data assimilation

The problem of data assimilation for a nonlinear evolution model is considered with the aim to identify the initial condition. The equation for the error of the optimal solution through the errors of the input data is derived, based on the Hessian of the misfit functional. The solvability of the error equation is studied. The fundamental control functions are used for error analysis. The error sensitivity coefficients are obtained using the singular vectors of the specific response operators in the error equation. An application to the data assimilation problem in hydrology is given. Numerical results are presented

Posterior Covariance vs. Analysis Error Covariance in Data Assimilation

International audienceThe problem of variational data assimilation for a nonlinear evolution model is formulated as an optimal control problem to find the initial condition function (analysis). The data contain errors (observation and background errors), hence there is an error in the analysis. For mildly nonlinear dynamics, the analysis error covariance can be approximated by the inverse Hessian of the cost functional in the auxiliary data assimilation problem, whereas for stronger nonlinearity - by the 'effective' inverse Hessian. However, it has been noticed that the analysis error covariance is not the posterior covariance from the Bayesian perspective. While these two are equivalent in the linear case, the difference may become significant in practical terms with the nonlinearity level rising. For the proper Bayesian posterior covariance a new approximation via the Hessian is derived and its 'effective' counterpart is introduced. An approach for computing the mentioned estimates in the matrix-free environment using Lanczos method with preconditioning is suggested. Numerical examples which validate the developed theory are presented for the model governed by Burgers equation with a nonlinear viscous term

Generalised diffusion based regularization for inverse problem in image processing

International audienceDue to the ill-posedness of inverse problems, it is important to make use of most of the \textit{a priori} informations while solving such a problem. These informations are generally used as constraints to get the appropriate solution. In usual cases, constrains are turned into penalization of some characteristics of the solution. A common constraint is the regularity of the solution leading to regularization techniques for inverse problems. Regularization by penalization is affected by two principal problems: - as the cost function is composite, the convergence rate of minimization algorithms decreases - when adequate regularization functions are defined, one has to define weighting parameters between regularization functions and the objective function to minimize. It is very difficult to get optimal weighting parameters since they are strongly dependant on the observed data and the truth solution of the problem. There is a third problem that affects regularization based on the penalization of spatial variation. Although the penalization of spatial variation is known to give best results (gradient penalization and second order regularization), there is no physical underlying foundation. Penalization of spatial variations lead to smooth solution that is an equilibrium between good and bad characteristics. Here, we introduce a new approach for regularization of ill-posed inverse problems. Penalization of spatial variations is weighted by an observation based trust function. The result is a generalized diffusion operator that turns regularization into pseudo covariance operators. All the regularization informations are then embedded into a preconditioning operator. On one hand, this method do not need any extra terms in the cost function, and of course is affected neither by the ill-convergence due to composite cost function, nor by the choice of weighting parameters. On the other hand, The trust function introduced here allows to take into account the observation based a priori knowledges on the problem. We suggest a simple definition of the trust function for inverse problems in image processing. Preliminary results show a promising method for regularization of inverse problems

De la prévision météo à la prévision environnementale

National audienceL'augmentation des capacités de calcul et la généralisation des satellites pour l'observation de l'atmosphère, de la terre et des océans ont permis le formidable essor de la prévision en météorologie et son extension à d'autres domaines : océanographie, prévision des crues, etc..

Topological data assimilation using Wasserstein distance

International audienceThis work combines a level-set approach and the optimal transport-based Wasserstein distance in a data assimilation framework. The primary motivation of this work is to reduce assimilation artifacts resulting from the position and observation error in the tracking and forecast of pollutants present on the surface of oceans or lakes. Both errors lead to spurious effect on the forecast that need to be corrected. In general, the geometric contour of such pollution can be retrieved from observation while more detailed characteristics such as concentration remain unknown. Herein, level sets are tools of choice to model such contours and the dynamical evolution of their topology structures. They are compared with contours extracted from observation using the Wasserstein distance. This allows to better capture position mismatches between both sources compared with the more classical Euclidean distance. Finally, the viability of this approach is demonstrated through academic test cases and its numerical performance is discussed