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.