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"

Discharge estimation under uncertainty using variational methods with application to the full Saint-Venant hydraulic network model

International audienceEstimating river discharge from in situ and/or remote sensing data is a key issue for evaluation of water balance at local and global scales and for water management. Variational data assimilation (DA) is a powerful approach used in operational weather and ocean forecasting, which can also be used in this context. A distinctive feature of the river discharge estimation problem is the likely presence of significant uncertainty in principal parameters of a hydraulic model, such as bathymetry and friction, which have to be included into the control vector alongside the discharge. However, the conventional variational DA method being used for solving such extended problems often fails. This happens because the control vector iterates (i.e., approximations arising in the course of minimization) result into hydraulic states not supported by the model. In this paper, we suggest a novel version of the variational DA method specially designed for solving estimation-under-uncertainty problems, which is based on the ideas of iterative regularization. The method is implemented with SIC2, which is a full Saint-Venant based 1D-network model. The SIC2 software is widely used by research, consultant and industrial communities for modeling river, irrigation canal, and drainage network behavior. The adjoint model required for variational DA is obtained by means of automatic differentiation. This is likely to be the first stable consistent adjoint of the 1D-network model of a commercial status in existence. The DA problems considered in this paper are offtake/tributary estimation under uncertainty in the cross-device parameters and inflow discharge estimation under uncertainty in the bathymetry defining parameters and the friction coefficient. Numerical tests have been designed to understand identifiability of discharge given uncertainty in bathymetry and friction. The developed methodology, and software seems useful in the context of the future Surface Water and Ocean Topography satellite mission

On computation of the design function gradient for the sensor-location problem in variational data assimilation

The optimal sensor-location problem is considered in the framework of variational data assimilation for a large-scale dynamical model governed by partial differential equations. This problem is formulated as an optimization problem for the design function defined on the limited memory approximation of the inverse Hessian of the data assimilation cost function. The expression for the gradient of the design function with respect to the sensor-location coordinates is derived via the adjoint to the Hessian derivative. An efficient algorithm for the gradient evaluation suitable for large-scale applications is suggested. This algorithm exploits the special structure of the limited memory inverse Hessian defined by a small number of Ritz pairs obtained by the Lanczos method. If additional memory is allocated and certain data are stored during the computation of the Ritz pairs, no additional runs of the tangent linear model are required to evaluate the gradient. The accuracy of the gradients is checked in the numerical experiments. These gradients can be used for the gradient-based optimization of the design function within the chosen global optimization procedure

Implicit treatment of model error using inflated observation-error covariance

International audienceData assimilation involving imperfect dynamical models is an important topic in meteorology, oceanography and other geophysical applications. In filtering methods, the model error is compensated for by inflation. In variational data assimilation, authors usually try to estimate it, which means that all uncertainty-loaded model inputs are included into the control vector. However, this approach suffers from implementation difficulties. In this paper we suggest an alternative method, motivated by the 'nuisance parameter' concept known in statistics. This method allows the model error to be treated implicitly by inflating the observation-error covariance. The equivalency theorem substantiating the method has here been proved. We also consider a case with a biased model error. In the corresponding mixed formulation, the spatially distributed mean error is included into the control vector, whereas the time-dependent fluctuations around the mean are subjected to the proposed implicit treatment. Numerical experiments for the 1D generalized Burgers' equation illustrate the presented theory. In these experiments the model error related to uncertainty in the advection coefficient has been considered

Adjoint to the Hessian derivative and 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 the initial condition function. The optimal solution error is considered through the errors of input data (background and observation errors). The optimal solution error covariance operator is approximated by the inverse Hessian of the auxiliary (linearized) data assimilation problem, which involves the tangent linear model constraints. We show that the derivative of the inverse Hessian with respect to the exact solution may be treated as the measure of nonlinearity for analysis error covariances in variational data assimilation problems

A multilevel approach for computing the limited-memory Hessian and its inverse in variational data assimilation

[Departement_IRSTEA]Eaux [TR1_IRSTEA]GEUSIInternational audienceUse of data assimilation techniques is becoming increasingly common across many application areas. The inverse Hessian (and its square root) plays an important role in several different aspects of these processes. In geophysical and engineering applications, the Hessian-vector product is typically defined by sequential solution of a tangent linear and adjoint problem; for the inverse Hessian, however, no such definition is possible. Frequently, the requirement to work in a matrix-free environment means that compact representation schemes are employed. In this paper, we propose an enhanced approach based on a new algorithm for constructing a multilevel eigenvalue decomposition of a given operator, which results in a much more efficient compact representation of the inverse Hessian (and its square root). After introducing these multilevel approximations, we investigate their accuracy and demonstrate their efficiency (in terms of reducing memory requirements and/or computational time) using the example of preconditioning a Gauss-Newton minimization procedure