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

2012 ◽  
Vol 19 (2) ◽  
pp. 177-184 ◽  
Author(s):  
V. Shutyaev ◽  
I. Gejadze ◽  
G. J. M. Copeland ◽  
F.-X. Le Dimet
Keyword(s):  
Nonlinear Dynamics ◽  
Data Assimilation ◽  
Optimal Solution ◽  
Nonlinear Problems ◽  
Variational Data Assimilation ◽  
Model Parameters ◽  
Efficient Computation ◽  
Viscous Term ◽  
Highly Nonlinear ◽  
The Cost

Abstract. The problem of variational data assimilation (DA) for a nonlinear evolution model is formulated as an optimal control problem to find the initial condition, boundary conditions and/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 function. For problems with strongly nonlinear dynamics, a new statistical method based on the 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. Numerical examples are presented for the model governed by the Burgers equation with a nonlinear viscous term.

scholarly journals On model error in variational data assimilation

2016 ◽  
Vol 31 (2) ◽  
Author(s):  
Victor Shutyaev ◽  
Arthur Vidard ◽  
François-Xavier Le Dimet ◽  
Igor Gejadze
Keyword(s):  
Data Assimilation ◽  
Nonlinear Evolution ◽  
Optimal Solution ◽  
Variational Data Assimilation ◽  
Initial Condition ◽  
Model Errors ◽  
Auxiliary Data ◽  
Error Covariance ◽  
Analysis Error ◽  
The Cost

AbstractThe 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.

scholarly journals Quasi-static ensemble variational data assimilation: a theoretical and numerical study with the iterative ensemble Kalman smoother

2018 ◽  
Vol 25 (2) ◽  
pp. 315-334 ◽  
Author(s):  
Anthony Fillion ◽  
Marc Bocquet ◽  
Serge Gratton
Keyword(s):  
Data Assimilation ◽  
Cost Function ◽  
Global Minimum ◽  
Numerical Study ◽  
Variational Data Assimilation ◽  
Kalman Smoother ◽  
Local Extrema ◽  
Starting Point ◽  
The Cost ◽  
Temporal Extent

Abstract. The analysis in nonlinear variational data assimilation is the solution of a non-quadratic minimization. Thus, the analysis efficiency relies on its ability to locate a global minimum of the cost function. If this minimization uses a Gauss–Newton (GN) method, it is critical for the starting point to be in the attraction basin of a global minimum. Otherwise the method may converge to a local extremum, which degrades the analysis. With chaotic models, the number of local extrema often increases with the temporal extent of the data assimilation window, making the former condition harder to satisfy. This is unfortunate because the assimilation performance also increases with this temporal extent. However, a quasi-static (QS) minimization may overcome these local extrema. It accomplishes this by gradually injecting the observations in the cost function. This method was introduced by Pires et al. (1996) in a 4D-Var context. We generalize this approach to four-dimensional strong-constraint nonlinear ensemble variational (EnVar) methods, which are based on both a nonlinear variational analysis and the propagation of dynamical error statistics via an ensemble. This forces one to consider the cost function minimizations in the broader context of cycled data assimilation algorithms. We adapt this QS approach to the iterative ensemble Kalman smoother (IEnKS), an exemplar of nonlinear deterministic four-dimensional EnVar methods. Using low-order models, we quantify the positive impact of the QS approach on the IEnKS, especially for long data assimilation windows. We also examine the computational cost of QS implementations and suggest cheaper algorithms.

scholarly journals Sensitivity analysis with respect to observations in variational data assimilation for parameter estimation

2018 ◽  
Vol 25 (2) ◽  
pp. 429-439 ◽  
Author(s):  
Victor Shutyaev ◽  
Francois-Xavier Le Dimet ◽  
Eugene Parmuzin
Keyword(s):  
Data Assimilation ◽  
Response Function ◽  
Optimal Solution ◽  
Coupled System ◽  
Variational Data Assimilation ◽  
Adjoint Equations ◽  
Observation Data ◽  
Unknown Parameters ◽  
The Baltic ◽  
Nonstandard Problem

Abstract. The problem of variational data assimilation for a nonlinear evolution model is formulated as an optimal control problem to find unknown parameters of the model. The observation data, and hence the optimal solution, may contain uncertainties. A response function is considered as a functional of the optimal solution after assimilation. Based on the second-order adjoint techniques, the sensitivity of the response function to the observation data is studied. The gradient of the response function is related to the solution of a nonstandard problem involving the coupled system of direct and adjoint equations. The nonstandard problem is studied, based on the Hessian of the original cost function. An algorithm to compute the gradient of the response function with respect to observations is presented. A numerical example is given for the variational data assimilation problem related to sea surface temperature for the Baltic Sea thermodynamics model.

Sensitivity with respect to observations in variational data assimilation

2017 ◽  
Vol 32 (1) ◽  
Author(s):  
Victor Shutyaev ◽  
Francois-Xavier Le Dimet ◽  
Elena Shubina
Keyword(s):  
Data Assimilation ◽  
Response Function ◽  
Nonlinear Evolution ◽  
Function Analysis ◽  
Optimal Solution ◽  
Coupled System ◽  
Variational Data Assimilation ◽  
Adjoint Equations ◽  
Observation Data ◽  
Standard Problem

AbstractThe 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 observation data, and hence the optimal solution, may contain uncertainties. A response function is considered as a functional of the optimal solution after assimilation. Based on the second-order adjoint techniques, the sensitivity of the response function to the observation data 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, based on the Hessian of the original cost function. An algorithm to compute the gradient of the response function with respect to observations is developed and justified.

Cycling the Representer Algorithm for Variational Data Assimilation with the Lorenz Attractor

2007 ◽  
Vol 135 (2) ◽  
pp. 373-386 ◽  
Author(s):  
H. E. Ngodock ◽  
S. R. Smith ◽  
G. A. Jacobs
Keyword(s):  
Data Assimilation ◽  
Variational Data Assimilation ◽  
Lorenz Attractor ◽  
Strong Constraint ◽  
Strongly Nonlinear ◽  
Time Period ◽  
Representer Method ◽  
The Cost ◽  
Solution Accuracy ◽  
Perturbation Magnitude

Abstract Realistic dynamic systems are often strongly nonlinear, particularly those for the ocean and atmosphere. Applying variational data assimilation to these systems requires a tangent linearization of the nonlinear dynamics about a background state for the cost function minimization. The tangent linearization may be accurate for limited time scales. Here it is proposed that linearized assimilation systems may be accurate if the assimilation time period is less than the tangent linear accuracy time limit. In this paper, the cycling representer method is used to test this assumption with the Lorenz attractor. The outer loops usually required to accommodate the linear assimilation for a nonlinear problem may be dropped beyond the early cycles once the solution (and forecast used as the background in the tangent linearization) is sufficiently accurate. The combination of cycling the representer method and limiting the number of outer loops significantly lowers the cost of the overall assimilation problem. In addition, this study shows that weak constraint assimilation corrects tangent linear model inaccuracies and allows extension of the limited assimilation period. Hence, the weak constraint outperforms the strong constraint method. Assimilated solution accuracy at the first cycle end is computed as a function of the initial condition error, model parameter perturbation magnitude, and outer loops. Results indicate that at least five outer loops are needed to achieve solution accuracy in the first cycle for the selected error range. In addition, this study clearly shows that one outer loop in the first cycle does not preclude accuracy convergence in future cycles.

scholarly journals Multiplicative and Additive Incremental Variational Data Assimilation for Mixed Lognormal–Gaussian Errors

2014 ◽  
Vol 142 (7) ◽  
pp. 2521-2544 ◽  
Author(s):  
Steven J. Fletcher ◽  
Andrew S. Jones
Keyword(s):  
Data Assimilation ◽  
Lognormal Distribution ◽  
Weather Prediction ◽  
Variational Data Assimilation ◽  
Incremental Formulation ◽  
Incremental Scheme ◽  
Observational Errors ◽  
Mixed Approach ◽  
The Cost ◽  
Gaussian Space

Abstract An advance that made Gaussian-based three- and four-dimensional variational data assimilation (3D- and 4DVAR, respectively) operationally viable for numerical weather prediction was the introduction of the incremental formulation. This reduces the computational costs of the variational methods by searching for a small increment to a background state whose evolution is approximately linear. In this paper, incremental formulations for 3D- and 4DVAR with lognormal and mixed lognormal–Gaussian-distributed background and observation errors are presented. As the lognormal distribution has geometric properties, a geometric version for the tangent linear model (TLM) is proven that enables the linearization of the observational component of the cost functions with respect to a geometric increment. This is combined with the additive TLM for the mixed distribution–based cost function. Results using the mixed incremental scheme with the Lorenz’63 model are presented for different observational error variances, observation set sizes, and assimilation window lengths. It is shown that for sparse accurate observations the scheme has a relative error of ±0.5% for an assimilation window of 100 time steps. This improves to ±0.3% with more frequent observations. The distributions of the analysis errors are presented that appear to approximate a lognormal distribution with a mode at 1, which, given that the background and observational errors are unbiased in Gaussian space, shows that the scheme is approximating a mode and not a median. The mixed approach is also compared against a Gaussian-only incremental scheme where it is shown that as the z-component observational errors become more lognormal, the mixed approach appears to be more accurate than the Gaussian approach.

scholarly journals Behavior of the iterative ensemble-based variational method in nonlinear problems

2020 ◽  
Author(s):  
Shin'ya Nakano
Keyword(s):  
Data Assimilation ◽  
Variational Method ◽  
Objective Function ◽  
Iterative Algorithm ◽  
Linear Approximation ◽  
Local Maximum ◽  
Nonlinear Problems ◽  
Variational Data Assimilation ◽  
Monotonic Convergence ◽  
System Nonlinearity

Abstract. The behavior of the iterative ensemble-based data assimilation algorithm is discussed. The ensemble-based method for variational data assimilation problems, referred to as the 4-dimensional ensemble variational method (4DEnVar), is a useful tool for data assimilation problems. Although the 4DEnVar is derived based on a linear approximation, highly uncertain problems, where system nonlinearity is significant, are solved by applying this method iteratively. However, it is not necessarily trivial how the algorithm works in highly uncertain problems where nonlinearity is not negligible. In the present study, an ensemble-based iterative algorithm is reformulated to allow us to analyze its behavior in nonlinear problems. The conditions for monotonic convergence to a local maximum of the objective function are discussed in nonlinear context. The findings as the results of the present study were also experimentally supported.

scholarly journals Sensitivity of functionals of variational data assimilation problems

2019 ◽  
Vol 486 (4) ◽  
pp. 421-425
Author(s):  
V. P. Shutyaev ◽  
F.-X. Le Dimet
Keyword(s):  
Data Assimilation ◽  
Observational Data ◽  
Response Function ◽  
Optimal Solution ◽  
Evolutionary Model ◽  
Variational Data Assimilation ◽  
Adjoint Equations ◽  
Unknown Parameters ◽  
Initial State ◽  
Nonstandard Problem

The problem of variational data assimilation for a nonlinear evolutionary model is formulated as an optimal control problem to find simultaneously unknown parameters and the initial state of the model. The response function is considered as a functional of the optimal solution found as a result of assimilation. The sensitivity of the functional to observational data is studied. The gradient of the functional with respect to observations is associated with the solution of a nonstandard problem involving a system of direct and adjoint equations. On the basis of the Hessian of the original cost function, the solvability of the nonstandard problem is studied. An algorithm for calculating the gradient of the response function with respect to observational data is formulated and justified.

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