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This study presents innovative approximate iterative methods for data assimilation in atmospheric modeling, specifically using the Gauss-Newton iteration. It explores two common approximations and the consequences of truncating inner minimization and approximating linear models. By addressing nonlinear least squares problems, we find the best estimate of the atmosphere’s true state, consistent with temporal observations and system dynamics. Our findings suggest that the truncated Gauss-Newton method enhances overall convergence while maintaining solution accuracy, even with approximate linear models.
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Approximate iterative methods for data assimilation Amos Lawless1, Serge Gratton2 and Nancy Nichols1 1Department of Mathematics, University of Reading 2CERFACS, France
Outline • Data assimilation and the Gauss-Newton iteration • Two common approximations • Truncation of inner minimization • Approximation of linear model • Conclusions
Data Assimilation Aim: Find the best estimate (analysis) of the true state of the atmosphere, consistent with both observations distributed in time and system dynamics.
Nonlinear least squares problem subject to - Background state - Observations - Observation operator - Background error covariance matrix - Observation error covariance matrix
Incremental 4D-Var Set (usually equal to background) Fork = 0, …, K Solve inner loop minimization problem with ; Update
General nonlinear least squares 4D-Var cost function can be written in this form with
Gauss-Newton iteration The Gauss-Newton iteration is
1D Shallow Water Model Nonlinear continuous equations with We discretize using a semi-implicit semi-Lagrangian scheme.
Assimilation experiments • Observations are generated from a model run with the ‘true’ initial state • First guess estimate is truth with phase error. • No background term included. • Inner problem solved using minimization by CONMIN algorithm, with stopping criterion on relative change in objective function. • Assimilation window is 100 time steps.
Truncated Gauss-Newton Linear quadratic problem is not solved exactly on each iteration. The residual error is given by rk. Solve such that on each iteration
Convergence of Truncated Gauss-Newton Theorem (GLN): (i) implies G-N converges. (ii) implies TG-N converges.
Convergence of Truncated Gauss-Newton (2) Theorem (GLN): Conditions of D&S hold, then implies TG-N converges. The rate of convergence can also be established. Proof: Extension of D&S
Perturbed Gauss-Newton The linear model is approximated. Equivalent to replacing by ExactG-N solves Perturbed G-N solves
Convergence of Perturbed Gauss-Newton Theorem (GLN): implies convergence of PG-N to x* . Theorem (GLN): Distance between fixed points depends on distance between pseudo-inverses(JTJ)-1JT and (JTJ)-1JT calculated at x* and on residual f(x*). ~ ~ ~ ~
Conclusions • Incremental 4D-Var without approximations is equivalent to a Gauss-Newton iteration. • In operational implementation we usually approximate the solution procedure. • Truncation of inner minimization may improve overall convergence. • Good solution obtained even with approximate linear model (PFM). • Theoretical results obtained by reference to Gauss-Newton method.
Convergence - Case 1 12 G-N iterations