Data Assimilation Fundamentals : : A Unified Formulation of the State and Parameter Estimation Problem.

• Superior document: Springer Textbooks in Earth Sciences, Geography and Environment Series
• Author: Evensen, Geir.
• TeilnehmendeR: Vossepoel, Femke C.; van Leeuwen, Peter Jan.
• Place / Publishing House: Cham : : Springer International Publishing AG,, 2022.; Ã2022.
• Year of Publication: 2022
• Edition: 1st ed.
• Language: English
• Series: Springer Textbooks in Earth Sciences, Geography and Environment Series
• Physical Description: 1 online resource (251 pages)

Table of Contents

Intro -- Preface -- Contents -- Symbols -- List of Approximations -- 1 Introduction -- 2 Problem Formulation -- 2.1 Bayesian Formulation -- 2.1.1 Assimilation Windows -- 2.1.2 Model with Uncertain Inputs -- 2.1.3 Model State -- 2.1.4 State Vector -- 2.1.5 Formulation Over Multiple Assimilation Windows -- 2.1.6 Measurements with Errors -- 2.1.7 Bayesian Inference -- 2.2 Recursive Bayesian Formulation -- 2.2.1 Markov Model -- 2.2.2 Independent Measurements -- 2.2.3 Recursive form of Bayes' -- 2.2.4 Marginal Bayes' for Filtering -- 2.3 Error Propagation -- 2.3.1 Fokker-Planck Equation -- 2.3.2 Covariance Evolution Equation -- 2.3.3 Ensemble Predictions -- 2.4 Various Problem Formulations -- 2.4.1 General Smoother Formulation -- 2.4.2 Filter Formulation -- 2.4.3 Recursive Smoother Formulation -- 2.4.4 A Smoother Formulation for Perfect Models -- 2.4.5 Parameter Estimation -- 2.4.6 Estimating Initial Conditions, Parameters, Controls, and Errors -- 2.5 Including the Predicted Measurements in Bayes Theorem -- 3 Maximum a Posteriori Solution -- 3.1 Maximum a Posteriori (MAP) Estimate -- 3.2 Gaussian Prior and Likelihood -- 3.3 Iterative Solutions -- 3.4 Gauss-Newton Iterations -- 3.5 Incremental Form of Gauss-Newton Iterations -- 4 Strong-Constraint 4DVar -- 4.1 Standard Strong-Constraint 4DVar Method -- 4.1.1 Data-Assimilation Problem -- 4.1.2 Lagrangian Formulation -- 4.1.3 Explaining the Measurement Operator -- 4.1.4 Euler-Lagrange Equations -- 4.2 Incremental Strong-Constraint 4DVar -- 4.2.1 Incremental Formulation -- 4.2.2 Lagrangian Formulation for the Inner Iterations -- 4.2.3 Euler-Lagrange Equations for the Inner Iterations -- 4.3 Preconditioning in Incremental SC-4DVar -- 4.4 Summary of SC-4DVar -- 5 Weak Constraint 4DVar -- 5.1 Forcing Formulation -- 5.2 State-Space Formulation -- 5.3 Incremental Form of the Generalized Inverse.

5.4 Minimizing the Cost Function for the Increment -- 5.5 Observation Space Formulation -- 5.5.1 Original Representer Method -- 5.5.2 Efficient Weak-Constraint Solution in Observation Space -- 6 Kalman Filters and 3DVar -- 6.1 Linear Update from Predicted Measurements -- 6.2 3DVar -- 6.3 Kalman Filter -- 6.4 Optimal Interpolation -- 6.5 Extended Kalman Filter -- 7 Randomized-Maximum-Likelihood Sampling -- 7.1 RML Sampling -- 7.2 Approximate EKF Sampling -- 7.3 Approximate Gauss-Newton Sampling -- 7.4 Least-Squares Best-Fit Model Sensitivity -- 8 Low-Rank Ensemble Methods -- 8.1 Ensemble Approximation -- 8.2 Definition of Ensemble Matrices -- 8.3 Cost Function in the Ensemble Subspace -- 8.4 Ensemble Subspace RML -- 8.5 Ensemble Kalman Filter (EnKF) Update -- 8.6 Ensemble DA with Multiple Updating (ESMDA) -- 8.7 Ensemble 4DVar with Consistent Error Statistics -- 8.8 Square-Root EnKF -- 8.9 Ensemble Subspace Inversion -- 8.10 A Note on the EnKF Analysis Equation -- 9 Fully Nonlinear Data Assimilation -- 9.1 Particle Approximation -- 9.2 Particle Filters -- 9.2.1 The Standard Particle Filter -- 9.2.2 Proposal Densities -- 9.2.3 The Optimal Proposal Density -- 9.2.4 Other Particle Filter Schemes -- 9.3 Particle-Flow Filters -- 9.3.1 Particle Flow Filters via Likelihood Factorization -- 9.3.2 Particle Flows via Distance Minimization -- 10 Localization and Inflation -- 10.1 Background -- 10.2 Various Forms of the EnKF Update -- 10.3 Impact of Sampling Errors in the EnKF Update -- 10.3.1 Spurious Correlations -- 10.3.2 Update Confined to Ensemble Subspace -- 10.3.3 Ensemble Representation of the Measurement Information -- 10.4 Localization in Ensemble Kalman Filters -- 10.4.1 Covariance Localization -- 10.4.2 Localization in Observation Space -- 10.4.3 Localization in Ensemble Space -- 10.4.4 Local Analysis -- 10.5 Adaptive Localization.

10.6 Localization in Time -- 10.7 Inflation -- 10.8 Localization in Particle Filters -- 10.9 Summary -- 11 Methods' Summary -- 11.1 Discussion of Methods -- 11.2 So Which Method to Use? -- blackPart II Examples and Applications-1pt -- 12 A Kalman Filter with the Roessler Model -- 12.1 Roessler Model System -- 12.2 Kalman Filter with the Roessler System -- 12.3 Extended Kalman Filter with the Roessler System -- 13 Linear EnKF Update -- 13.1 EnKF Update Example -- 13.2 Solution Methods -- 13.3 Example 1 (Large Ensemble Size) -- 13.4 Example 2 (Ensemble Size of 100) -- 13.5 Example 3 (Augmenting the Measurement Perturbations) -- 13.6 Example 4 (Large Number of Measurements) -- 14 EnKF for an Advection Equation -- 14.1 Experiment Description -- 14.2 Assimilation Experiment -- 15 EnKF with the Lorenz Equations -- 15.1 The Lorenz'63 Model -- 15.2 Ensemble Smoother Solution -- 15.3 Ensemble Kalman Filter Solution -- 15.4 Ensemble Kalman Smoother Solution -- 16 3Dvar and SC-4DVar for the Lorenz 63 Model -- 16.1 Data Assimilation Set up -- 16.2 Comparing 3DVar and SC-4DVar -- 16.3 Sensitivity to Observation Density in SC-4DVar -- 16.4 3DVar and SC-4DVar with Partial Observations -- 16.5 Sensitivity to the Length of Assimilation Window -- 16.6 SC-4DVar with Multiple Assimilation Windows -- 16.7 A Comparison with Ensemble Methods -- 17 Representer Method with an Ekman-Flow Model -- 17.1 Ekman-Flow Model -- 17.2 Example Experiment -- 17.3 Assimilation of Real Measurements -- 18 Comparison of Methods on a Scalar Model -- 18.1 Scalar Model and Inverse Problem -- 18.2 Discussion of Data-Assimilation Examples -- 18.3 Summary -- 19 Particle Filter for Seismic-Cycle Estimation -- 19.1 Particle Filter for State and Parameter Estimation -- 19.2 Seismic Cycle Model -- 19.3 Data-Assimilation Experiments -- 19.4 Case A: State Estimation.

19.5 Case B: State Estimation with Increased Model Error -- 19.6 Case C: State- and Parameter Estimation -- 19.7 Summary -- 20 Particle Flow for a Quasi-Geostrophic Model -- 20.1 Introduction -- 20.2 Application to the QG Model -- 20.3 Data-Assimilation Experiment -- 20.4 Results -- 21 EnRML for History Matching Petroleum Models -- 21.1 Reservoir Modeling -- 21.2 History Matching Reservoir Models -- 21.3 Example -- 22 ESMDA with a SARS-COV-2 Pandemic Model -- 22.1 An Extended SEIR Model -- 22.2 Example -- 22.3 Sensitivity to Ensemble Size -- 22.4 Sensitivity to MDA Steps -- 22.5 Summary -- 23 Final Summary -- 23.1 Classification of the Nonlinearity -- 23.1.1 Linear to Weakly-Nonlinear Systems with Gaussian Priors -- 23.1.2 Weakly Nonlinear Systems with Gaussian Priors -- 23.1.3 Strongly Nonlinear Systems -- 23.2 Purpose of the Data Assimilation -- 23.2.1 Hindcasts and Re-analyses -- 23.2.2 Prediction Systems -- 23.2.3 Uncertainty Quantification and Risk Assessment -- 23.2.4 Model Improvement and Parameter Estimation -- 23.2.5 Scenario Forecasts and Optimal Controls -- 23.3 How to Reduce Computational Costs -- 23.4 What Will the Future Hold? -- References -- Author Index -- Author Index -- Index -- Index.