Data Assimilation for the Geosciences

From Theory to Application

2nd Edition - November 16, 2022
Imprint: Elsevier
Author: Steven J. Fletcher
Language: English
Paperback ISBN:
9 7 8 - 0 - 3 2 3 - 9 1 7 2 0 - 9
eBook ISBN:
9 7 8 - 0 - 3 2 3 - 9 7 2 5 3 - 6

Data Assimilation for the Geosciences: From Theory to Application, Second Edition brings together all of the mathematical and statistical background knowledge needed to formul… Read more

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Data Assimilation for the Geosciences: From Theory to Application, Second Edition brings together all of the mathematical and statistical background knowledge needed to formulate data assimilation systems into one place. It includes practical exercises enabling readers to apply theory in both a theoretical formulation as well as teach them how to code the theory with toy problems to verify their understanding. It also demonstrates how data assimilation systems are implemented in larger scale fluid dynamical problems related to land surface, the atmosphere, ocean and other geophysical situations. The second edition of Data Assimilation for the Geosciences has been revised with up to date research that is going on in data assimilation, as well as how to apply the techniques. The new edition features an introduction of how machine learning and artificial intelligence are interfacing and aiding data assimilation. In addition to appealing to students and researchers across the geosciences, this now also appeals to new students and scientists in the field of data assimilation as it will now have even more information on the techniques, research, and applications, consolidated into one source.

Title of Book
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Table of Contents
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Chapter 1: Introduction
Chapter 2: Overview of Linear Algebra
2.1. Properties of Matrices
2.1.1. Matrix Multiplication
2.1.2. Transpose of a Matrix
2.1.3. Determinants of Matrices
2.1.4. Inversions of Matrices
2.1.5. Rank, Linear Independence and Dependence
2.1.6. Matrix Structures
2.2. Matrix and Vector Norms
2.2.1. Vector Norms
2.2.2. Matrix Norms
2.2.3. Conditioning of Matrices
2.2.4. Matrix Condition Number
2.3. Eigenvalues and Eigenvectors
2.4. Matrix Decompositions
2.4.1. Gaussian Elimination and the LU Decomposition
2.4.2. Cholesky Decomposition
2.4.3. The QR Decomposition
2.4.4. Diagonalization
2.4.5. Singular Value Decomposition
2.5. Sherman-Morrison-Woodbury Formula
2.6. Summary
Chapter 3: Univariate Distribution Theory
3.1. Random Variables
3.2. Discrete Probability Theory
3.2.1. Discrete Random Variables
3.3. Continuous Probability Theory
3.4. Discrete Distribution Theory
3.4.1. Binomial Distribution
3.4.2. Geometric Distribution
3.4.3. Poisson Distribution
3.4.4. Discrete Uniform Distribution
3.5. Expectation and Variance of Discrete Random Variables
3.5.1. Mean of the Binomial Distribution
3.5.2. Mean of the Geometric Distribution
3.5.3. Mean of the Poisson Distribution
3.5.4. Mean of the Discrete Uniform Distribution
3.5.5. Variance of a Discrete Probability Mass Function
3.5.6. Variance of the Binomial Distribution
3.5.7. Variance of the Geometric Distribution
3.5.8. Variance of the Poisson Distribution
3.5.9. Variance of the Discrete Uniform Distribution
3.6. Moments and Moment-Generating Functions
3.6.1. Moment-Generating Functions for Probability Mass Functions
3.6.2. Binomial Distribution Moment-Generating Function
3.6.3. Geometric Distribution Moment-Generating Function
3.6.4. Poisson Moment-Generating Function
3.6.5. Discrete Uniform Distribution Moment-Generating Function
3.7. Continuous Distribution Theory
3.7.1. Gaussian (Normal) Distribution
3.7.2. Moments of the Gaussian Distribution
3.7.3. Moment-Generating Functions for Continuous Probability Density Functions
3.7.4. Median of the Gaussian Distribution
3.7.5. Mode of the Univariate Gaussian Distribution
3.8. Lognormal Distribution
3.8.1. Moments of the Lognormal Distribution
3.8.2. Geometric Behavior of the Lognormal
3.8.3. Median of the Univariate Lognormal Distribution
3.8.4. Mode of the Lognormal Distribution
3.9. Reverse Lognormal Distribution
3.9.1. Mean of the Reverse Lognormal Distribution
3.9.2. Variance of the Reverse Lognormal Distribution
3.9.3. Skewness of the Reverse Lognormal Distribution
3.9.4. Kurtosis of the Reverse Lognormal Distribution
3.9.5. Median of the Reverse Lognormal Distribution
3.9.6. Mode of the Reverse Lognormal Distribution
3.10. Exponential Distribution
3.11. Gamma Distribution
3.11.1. Moment-Generating Function for the Gamma Distribution
3.11.2. Skewness of the Gamma Distribution
3.11.3. Kurtosis of the Gamma Distribution
3.11.4. Median of the Gamma Distribution
3.11.5. Mode of the Gamma Distribution
3.11.6. Remarks About the Gamma Distribution and the Gaussian Distribution
3.11.7. Properties of Gamma-Distributed Random Variables
3.12. Inverse Gamma Distribution
3.12.1. Moments of the Inverse-Gamma Distribution
3.12.2. Skewness of the Inverse-Gamma Distribution
3.12.3. Kurtosis of the Inverse-Gamma Distribution
3.12.4. Mode of the Inverse-Gamma Distribution
3.13. Beta Distribution
3.13.1. Moments of the Beta Distribution
3.13.2. Median of the Beta Distribution
3.13.3. Mode of the Beta Distribution
3.14. Chi-Squared (χ2) Distribution
3.14.1. Moments of the Chi-Squared Distribution
3.14.2. Median of the Chi-Squared Distribution
3.14.3. Mode of the Chi-Squared Distribution
3.14.4. Relationships to Other Distributions
3.15. Rayleigh Distribution
3.15.1. Moment-Generating Function for the Rayleigh Distribution
3.15.2. Moments of the Rayleigh Distribution
3.15.3. Skewness of the Rayleigh Distribution
3.15.4. Kurtosis of the Rayleigh Distribution
3.15.5. Median of the Rayleigh Distribution
3.15.6. Mode of the Rayleigh Distribution
3.16. Weibull Distribution
3.16.1. Moments of the Weibull Distribution
3.16.2. Skewness and Kurtosis of the Weibull Distribution
3.16.3. Mode of the Weibull Distribution
3.17. Gumbel Distribution
3.17.1. Moments of the Gumbel Distribution
3.17.2. Differentiating Gamma Functions
3.17.3. Returning to the Moments of the Gumbel Distribution
3.17.4. Skewness of a Gumbel Distribution
3.17.5. Kurtosis of the Gumbel Distribution
3.17.6. Median of the Gumbel Distribution
3.17.7. Mode of the Gumbel Distribution
3.18. Summary of the Descriptive Statistics, Moment-Generating Functions, and Moments for the Univariate Distribution
3.19. Summary
Chapter 4: Multivariate Distribution Theory
4.1. Descriptive Statistics for Multivariate Density Functions
4.1.1. Multivariate Moment-Generating Functions
4.1.2. Moments of Multivariate Distributions
4.1.3. Second-Order Moments: Variance and Covariance
4.1.4. Third-Order Moments: Skewness and Co-Skewness
4.1.5. Fourth-Order Moments: Kurtosis and Co-kurtosis
4.1.6. Mode of Multivariate Distribution
4.1.7. Median of Multivariate Distribution
4.2. Gaussian Distribution
4.2.1. Bivariate Gaussian Distribution
4.2.2. Medians of the Bivariate Gaussian Distribution
4.2.3. Mode of the Bivariate Lognormal
4.2.4. Multivariate Gaussian Distribution
4.3. Lognormal Distribution
4.3.1. Bivariate Lognormal Distribution
4.3.2. Moments of the Bivariate Lognormal Distribution
4.3.3. Median of the Bivariate Lognormal Distribution
4.3.4. Maximum Likelihood State of a Bivariate Lognormal Distribution
4.3.5. Multivariate Lognormal Distribution
4.4. Mixed Gaussian-Lognormal Distribution
4.4.1. Moments of the Bivariate Mixed Gaussian-Lognormal Distribution
4.4.2. Median of the Mixed Gaussian-Lognormal Distribution
4.4.3. Maximum Likelihood Estimate for the Mixed Gaussian and Lognormal Distribution
4.4.4. Diagrams of the Bivariate Gaussian-Lognormal Distribution
4.5. Multivariate Mixed Gaussian-Lognormal Distribution
4.5.1. Trivariate and Quadvariate Mixed Distribution
4.5.2. Mode of the Multivariate Mixed Distribution
4.6. Reverse Lognormal Distribution
4.6.1. Multivariate Reverse Lognormal Distribution
4.6.2. Combining With Gaussian Distribution
4.6.3. Combining With a Lognormal Distribution
4.6.4. Combining Multivariate Gaussian, Lognormal, and Reverse-Lognormal Distributions
4.7. Gamma Distribution
4.7.1. Bivariate Gamma Distribution
4.7.2. Multivariate Gamma Distribution
4.8. Summary
Chapter 5: Introduction to Calculus of Variation
5.1. Examples of Calculus of Variation Problems
5.1.1. Shortest/Minimum Distance
5.1.2. Brachistochrone Problem
5.1.3. Minimum Surface Area
5.1.4. Dido's Problem—Maximum Enclosed Area for a Given Perimeter Length
5.1.5. General Form of Calculus of Variation Problems
5.2. Solving Calculus of Variation Problems
5.2.1. Special Cases for Euler's Equations
5.2.2. Transversality Conditions
5.3. Functional With Higher-Order Derivatives
5.4. Three-Dimensional Problems
5.5. Functionals With Constraints
5.6. Functional With Extremals That Are Functions of Two or More Variables
5.6.1. Three-Dimensional Problems
5.7. Summary
Chapter 6: Introduction to Control Theory
6.1. The Control Problem
6.2. The Uncontrolled Problem
6.2.1. Fundamental Solutions
6.2.2. Properties of the State Transition Matrix
6.2.3. Time-Invariant Case
6.2.4. Properties of Exponential Matrices
6.2.5. Eigenvalues/Vectors Approach for Finding the State Transition Matrix
6.3. The Controlled Problem
6.3.1. Controllability
6.3.2. Equivalence
6.4. Observability
6.5. Duality
6.6. Stability
6.6.1. Algebraic Stability Conditions
6.7. Feedback
6.7.1. Observers and State Estimators
6.8. Summary
Chapter 7: Optimal Control Theory
7.1. Optimizing Scalar Control Problems
7.2. Multivariate Case
7.3. Autonomous (Time-Invariant) Problem
7.4. Extension to General Boundary Conditions
7.4.1. Extension of Calculus of Variation Theory
7.4.2. Optimal Control Problems With General Boundary Conditions
7.5. Free End Time Optimal Control Problems
7.5.1. Extension of the Calculus of Variation Theory
7.5.2. Applying the Theory to Control Problems
7.6. Piecewise Smooth Calculus of Variation Problems
7.6.1. Extension of Calculus of Variation Techniques
7.6.2. Application to the Optimal Control Problem
7.7. Maximization of Constrained Control Problems
7.7.1. Constrained Control Problems
7.8. Two Classical Optimal Control Problems
7.9. Summary
Chapter 8: Numerical Solutions to Initial Value Problems
8.1. Local and Truncation Errors
8.2. Linear Multistep Methods
8.3. Stability
8.4. Convergence
8.4.1. Explicit and Implicit Numerical Scheme
8.4.2. Dahlquist Convergence Theorem Example
8.5. Runge-Kutta Schemes
8.5.1. Explicit Runge-Kutta Methods
8.5.2. Consistency and Stability of Explicit Runge-Kutta Methods
8.5.3. Derivation of the Fourth-Order Runge-Kutta Scheme
8.6. Numerical Solutions to Initial Value Partial Differential Equations
8.6.1. Heat Equation
8.6.2. Numerical Approach
8.6.3. Norms and the Maximum Principle
8.6.4. Implementing and Solving the Implicit Equation
8.6.5. θ-Methods
8.6.6. More Generous Stability Condition
8.7. Wave Equation
8.7.1. Forward-Time, Centered-Space
8.7.2. Explicit Upwind
8.7.3. Implicit Upwind
8.7.4. Box Scheme
8.7.5. Lax-Wendroff Scheme
8.8. Courant Friedrichs Lewy Condition
8.9. Summary
Chapter 9: Numerical Solutions to Boundary Value Problems
9.1. Types of Differential Equations
9.2. Shooting Methods
9.2.1. Nonlinear Problems
9.3. Finite Difference Methods
9.3.1. Truncation Error
9.3.2. Mixed Boundary Conditions
9.4. Self-Adjoint Problems
9.5. Error Analysis
9.5.1. Irreducibility
9.6. Partial Differential Equations
9.6.1. Truncation Error
9.6.2. General Natural Ordering
9.6.3. Error Bound on Numerical Solution
9.6.4. Mixed Boundary Conditions
9.7. Self-Adjoint Problem in Two Dimensions
9.7.1. Solution Methods for Linear Matrix Equations
9.7.2. Jacobi Method
9.7.3. Gauss-Seidel
9.7.4. Successive Over-Relaxation Method
9.8. Periodic Boundary Conditions
9.9. Summary
Chapter 10: Introduction to Semi-Lagrangian Advection Methods
10.1. History of Semi-Lagrangian Approaches
10.2. Derivation of Semi-Lagrangian Approach
10.3. Interpolation Polynomials
10.3.1. Lagrange Interpolation Polynomials
10.3.2. Newton Divided Difference Polynomials
10.3.3. Hermite Interpolating Polynomials
10.3.4. Cubic Spline Interpolation Polynomials
10.3.5. Shape-Conserving Semi-Lagrangian Advection
10.4. Stability of Semi-Lagrangian Schemes
10.4.1. Stability Analysis of the Linear Lagrange Interpolation
10.4.2. Stability Analysis of the Quadratic Lagrange Interpolation
10.4.3. Stability Analysis of the Cubic Lagrange Interpolation
10.4.4. Stability Analysis of the Cubic Hermite Semi-Lagrangian Interpolation Scheme
10.4.5. Stability Analysis of the Cubic Spline Semi-Lagrangian Interpolation Scheme
10.5. Consistency Analysis of Semi-Lagrangian Schemes
10.6. Semi-Lagrangian Schemes for Non-Constant Advection Velocity
10.7. Semi-Lagrangian Scheme for Non-Zero Forcing
10.8. Example: 2D Quasi-Geostrophic Potential Vorticity (Eady Model)
10.8.1. Numerical Approximations for the Eady Model
10.8.2. Numerical Approximations to the Advection Equation
10.8.3. Numerical Approximation to the Laplace Equation in the Interior
10.8.4. Buoyancy Advection on the Boundaries: b0′=0, b1′=αsin⁡(KΔx)
10.8.5. Conditioning
10.8.6. QGPV ≠0
10.9. Summary
Chapter 11: Introduction to Finite Element Modeling
11.1. Solving the Boundary Value Problem
11.2. Weak Solutions of Differential Equation
11.2.1. Heat Development Due to Hydration of Concrete
11.2.2. Torsion of a Bar of Equilateral Triangle Cross Section
11.3. Accuracy of the Finite Element Approach
11.4. Pin Tong
11.5. Finite Element Basis Functions
11.5.1. One Dimension
11.5.2. Two Dimensions
11.6. Coding Finite Element Approximations for Triangle Elements
11.6.1. Square Elements
11.7. Isoparametric Elements
11.8. Summary
Chapter 12: Numerical Modeling on the Sphere
12.1. Vector Operators in Spherical Coordinates
12.1.1. Spherical Unit Vectors
12.2. Spherical Vector Derivative Operators
12.3. Finite Differencing on the Sphere
12.3.1. Map Projections
12.3.2. Grid-Point Representations of the Sphere
12.3.3. Different Grid Configuration
12.3.4. Vertical Staggering Grids
12.4. Introduction to Fourier Analysis
12.4.1. Fourier Series
12.4.2. Fourier Transforms
12.4.3. Laplace Transforms
12.5. Spectral Modeling
12.5.1. Sturm-Liouville Theory
12.5.2. Legendre Differential Equation
12.5.3. Legendre Polynomials
12.5.4. Spherical Harmonics
12.5.5. Legendre Transforms
12.5.6. Spectral Methods on the Sphere
12.6. Summary
Chapter 13: Tangent Linear Modeling and Adjoints
13.1. Additive Tangent Linear and Adjoint Modeling Theory
13.1.1. Derivation of the Linearized Model
13.1.2. Adjoints
13.1.3. Differentiating the Code to Derive the Adjoint
13.1.4. Test of the Tangent Linear and Adjoints Models
13.2. Multiplicative Tangent Linear and Adjoint Modeling Theory
13.3. Examples of Adjoint Derivations
13.3.1. Lorenz 63 Model
13.3.2. Eady Model
13.3.3. Tangent Linear Approximations to Semi-Lagrangian Schemes
13.3.4. Adjoint of Spectral Transforms
13.4. Perturbation Forecast Modeling
13.4.1. Example With a 1D Shallow Water Equations Model
13.5. Adjoint Sensitivities
13.6. Singular Vectors
13.6.1. Observational Impact
13.7. Summary
Chapter 14: Observations
14.1. Conventional Observations
14.1.1. Radiosondes
14.1.2. Microwave Radiometer
14.1.3. Infrared Sky Imager
14.1.4. Micropulse Lidar
14.1.5. Photometer
14.1.6. SNOTEL
14.1.7. SCAN
14.1.8. Airborne Observations
14.1.9. Ocean
14.1.10. Radar
14.2. Remote Sensing
14.2.1. Radiative Transfer Modeling
14.2.2. Satellite Characteristics
14.2.3. Infrared
14.2.4. Microwave
14.2.5. Visible
14.2.6. Lidar
14.2.7. Global Positioning System
14.3. Quality Control
14.3.1. Variational Quality Control
14.3.2. Variational Bias Correction
14.4. Summary
Chapter 15: Non-Variational Sequential Data Assimilation Methods
15.1. Direct Insertion
15.2. Nudging
15.3. Successive Correction
15.3.1. Bergthórsson and Döös [32]
15.3.2. Cressman [79]
15.3.3. Barnes [24]
15.4. Linear and Nonlinear Least Squares
15.4.1. Univariate Linear Least Squares
15.4.2. Multidimensional Least Squares
15.4.3. Nonlinear Least Squares Theory
15.5. Regression
15.5.1. Linear Regression Involving Two or More Variables
15.5.2. Nonlinear Regression
15.6. Optimal (Optimum) Interpolation/Statistical Interpolation/Analysis Correction
15.6.1. Derivation of the Optimum Interpolation From Alaka and Elvander [3]
15.6.2. Matrix Version of Optimum Interpolation
15.6.3. Implementation of OI
15.6.4. Analysis Correction (AC)
15.7. Summary
Chapter 16: Variational Data Assimilation
16.1. Sasaki and the Strong and Weak Constraints
16.2. Three-Dimensional Data Assimilation
16.2.1. Gaussian Framework
16.3. Four-Dimensional Data Assimilation
16.4. Incremental VAR
16.4.1. Incremental Spatial VAR, 1D, 2D, and 3D VAR
16.4.2. Incremental Temporal 4D VAR
16.4.3. Inner and Outer Loops
16.4.4. Nonlinearities and Outer Loops
16.4.5. First Guess at Appropriate Time
16.5. Weak Constraint—Model Error 4D VAR
16.5.1. Model-Bias Control Variable
16.5.2. Modeling the Model Error Covariance Matrix
16.5.3. Model Error Forcing Control Variable
16.5.4. Model State Control Variable
16.5.5. Time Lag Model Error Modeling
16.6. Observational Errors
16.6.1. Correlated Measurement Errors
16.7. Forecast Sensitivity Observation Impact (FSOI)
16.8. Saddle Point 4D VAR
16.9. Rapid Update Cycling (RUC)
16.10. Regularization
16.10.1. Optimal Transport
16.10.2. Lp-Norm Regularization
16.11. 4D VAR as an Optimal Control Problem
16.12. Summary
Chapter 17: Subcomponents of Variational Data Assimilation
17.1. Balance
17.1.1. Linear and Nonlinear Balances
17.1.2. Linear and Nonlinear Normal Mode Initialization
17.2. Control Variable Transforms
17.2.1. Kinematic Approach
17.2.2. Spectral-Based CVT
17.2.3. Wavelet
17.2.4. Nonlinear Balance on the Sphere
17.2.5. Ellipticity Conditions for Continuous PDEs
17.2.6. Higher-Order Balance Conditions
17.2.7. Geostrophic Coordinates
17.2.8. Linearization
17.3. Background Error Covariance Modeling
17.3.1. Error Modeling Functions
17.3.2. Determining Variances and Decorrelation Lengths
17.4. Preconditioning
17.4.1. Time-Parallel Preconditioning
17.5. Minimization Algorithms
17.5.1. Newton-Raphson
17.5.2. Quasi-Newton Methods
17.5.3. Steepest Descent
17.5.4. Conjugate Gradient
17.5.5. Lanczos Methods
17.6. Performance Metrics
17.6.1. Scorecard
17.7. Summary
Chapter 18: Observation Space Variational Data Assimilation Methods
18.1. Derivation of Observation Space-Based 3D VAR
18.2. 4D VAR in Observation Space
18.2.1. Solution to the Coupled Linear Euler-Lagrange System
18.2.2. Representer Solution to a Coupled Linearized Euler-Lagrange System
18.3. Duality of the VAR and PSAS Systems
18.4. Summary
Chapter 19: Kalman Filter and Smoother
19.1. Derivation of the Kalman Filter
19.2. Kalman Filter Derivation From a Statistical Approach
19.3. Extended Kalman Filter
19.4. Square Root Kalman Filter
19.5. Smoother
19.5.1. Forward Step: Kalman Filter
19.5.2. Backward Step: Reverse-Time Information Filter
19.5.3. Smoothing
19.6. Properties and Equivalencies of the Kalman Filter and Smoother
19.7. Summary
Chapter 20: Ensemble-Based Data Assimilation
20.1. Stochastic Dynamical Modeling
20.2. Ensemble Kalman Filter
20.2.1. Perturbed Observations-Based EnKF
20.3. Ensemble Square Root Filters
20.3.1. Localization and Inflation
20.4. Ensemble and Local Ensemble Transform Kalman Filter
20.4.1. ETKF
20.4.2. LETKF
20.5. Maximum Likelihood Ensemble Filter
20.5.1. Forecast Step
20.5.2. Analysis Step
20.5.3. Lyapunov and Bred Vectors
20.5.4. Hybrid Lyapunov-Bred Vectors
20.5.5. MLEF, Information Theory, and Entropy Reduction
20.6. Hybrid Ensemble and Variational Data Assimilation Methods
20.6.1. α Control Variables
20.6.2. Hybrid Ensemble Transform PSAS
20.6.3. Ensembles of 4D VARs (EDA)
20.7. NDEnVAR
20.8. Scale Dependent Background Error Covariance Localization
20.9. Ensemble Kalman Smoother
20.10. Ensemble Sensitivity
20.11. Ensemble Forecast Sensitivity to Observations (EFSO)
20.12. Local Ensemble Tangent Linear Model (LETLM)
20.13. Summary
Chapter 21: Non-Gaussian Based Data Assimilation
21.1. Error Definitions
21.2. Full Field Lognormal 3D VAR
21.2.1. Lognormal Observational Error
21.2.2. Lognormal Background Errors
21.3. Logarithmic Transforms
21.4. Mixed Gaussian-Lognormal 3D VAR
21.4.1. Experiments With the Lorenz 1963 Model
21.5. Lognormal Calculus of Variation-Based 4D VAR
21.5.1. Near Weighted Least Squares Functional Formulation for Non-Gaussian 4D VAR
21.5.2. Functional Form of a Modal Approach for Non-Gaussian Distribution-Based 4D VAR
21.6. Bayesian-Based 4D VAR
21.6.1. Bayesian Networks
21.6.2. Equivalence of the Weighted Least Squares and Probability Models for Multivariate Gaussian Errors
21.6.3. Equivalence of the Lognormal Functional Approach
21.6.4. Mixed Distribution Equivalency to Weighted Least Squares Approach
21.7. Bayesian Networks Formulation of Weak Constraint/Model Error 4D VAR
21.8. Results of the Lorenz 1963 Model for 4D VAR
21.9. Incremental Lognormal and Mixed 3D and 4D VAR
21.9.1. Multiplicative Incremental 3D VAR
21.9.2. Multiplicative Incremental 4D VAR
21.9.3. Mixed Additive and Multiplicative Incremental VAR
21.9.4. Analysis Mean of a Lognormal Data Assimilation System Not Equal to Zero
21.9.5. Comparison of a Mixed Incremental System With Gaussian-Only Scheme
21.10. Reverse Lognormal Variational Data Assimilation
21.10.1. 3D and 4D Mixed Gaussian-Reverse Lognormal Cost Functions
21.10.2. 3D and 4D Mixed Lognormal-Reverse Lognormal Cost Functions
21.10.3. 3D and 4D Mixed Gaussian-Lognormal-Reverse-Lognormal Cost Functions
21.11. Lognormal and Mixed Gaussian-Lognormal Kalman Filters
21.11.1. Attempted Derivation at a Lognormal Based Kalman Filter
21.11.2. Lognormal Kalman Filter - Median Based Approach
21.11.3. Mixed Gaussian-Lognormal Kalman Filter (MXKF)
21.12. Gaussian Anamorphosis
21.13. Gamma-Inverse-Gamma-Gaussian (GIGG) Filter
21.14. Regions of Optimality for Lognormal Descriptive Statistics
21.15. Summary
Chapter 22: Markov Chain Monte Carlo, Particle Filters, Particle Smoothers, and Sigma Point Filters
22.1. Markov Chain Monte Carlo Methods
22.1.1. MC Methods for Inverse Problems
22.1.2. Sample Methods
22.1.3. Application of MCMC in the Geosciences
22.2. Particle Filters
22.2.1. Resampling
22.2.2. Proposal Densities
22.2.3. Optimal Proposal Density
22.2.4. Implicit Particle Filter
22.2.5. Transportation Particle Filters
22.2.6. Tempering of the Likelihood
22.2.7. Particle Flow Filters
22.3. Local Particle Filter
22.4. Particle Smoother
22.5. Sigma Point Kalman Filters (SPKF)
22.5.1. Sigma-Point Unscented KF (SP-UKF)
22.5.2. Sigma Point Central Difference KF (SP-CDKF)
22.6. Summary
Chapter 23: Lagrangian Data Assimilation
23.1. Extended Kalman Filter Approach
23.2. Variational Lagrangian Data Assimilation
23.2.1. Converting Lagrangian Data to Eulerian to Assimilate
23.2.2. Direct Assimilation of Lagrangian Observations
23.2.3. Direct Lagrangian Trajectory Variational Assimilation
23.3. Lagrangian Ensemble Kalman Filter
23.4. Localized Ensemble Transform Kalman Filter Lagrangian Data Assimilation (LETKF-LaDA)
23.5. Hybrid Particle Filters and Ensemble Kalman Filters Lagrangian Data Assimilation
23.6. Summary
Chapter 24: Artificial Intelligence and Data Assimilation
24.1. Helpful Definitions
24.2. Introduction to Machine Learning Algorithms
24.2.1. Linear Regression
24.2.2. Logistic Regression
24.2.3. Support Vector Machine
24.2.4. Classification and Regression Trees (CART)
24.2.5. K-Nearest Neighbors
24.2.6. Random Forests
24.3. Introduction to Deep Learning
24.3.1. Neural Networks (NN)
24.3.2. Restricted Boltzmann Machine (RBM)
24.3.3. Training Algorithms
24.4. Applications of Artificial Intelligence With Data Assimilation
24.4.1. Detection of Non-Gaussian Signals
24.4.2. Deep Data Assimilation
24.4.3. Latent Space Data Assimilation by Using Deep Learning
24.4.4. Deep Learning for Fast Radiative Transfer
24.4.5. Using ML to Correct Model Error
24.4.6. k-Nearest Neighbor for Data Driven Data Assimilation (DD-DA)
24.4.7. Other Applications
24.5. Summary
Chapter 25: Applications of Data Assimilation in the Geosciences
25.1. Atmospheric Science
25.1.1. Operational Numerical Weather Prediction Centers
25.1.2. Limited Area Synoptic Scale Data Assimilation
25.1.3. Mesoscale Data Assimilation
25.1.4. Cloud Resolving Data Assimilation
25.1.5. Retrievals
25.1.6. Atmospheric Chemistry and Aerosols Assimilation
25.2. Joint Effort for Data Assimilation Integration (JEDI)
25.2.1. OOPS Abstract Interfaces
25.2.2. Observations Space Interfaces
25.2.3. Error Covariances
25.2.4. UFO, IODA, and SABER
25.3. Observing-System Experiments (OSE)
25.4. Observing System Simulation Experiments (OSSE)
25.5. Oceans
25.5.1. Global Ocean Data Assimilation
25.5.2. Regional Ocean Data Assimilation
25.5.3. Sea Ice Data Assimilation
25.6. Hydrological Applications
25.7. Coupled Data Assimilation
25.7.1. Coupled Atmosphere-Ocean Data Assimilation
25.7.2. Coupled Land and Atmosphere Data Assimilation
25.7.3. Coupled Atmosphere-Land-Ocean-Sea Ice Data Assimilation
25.8. Reanalysis
25.9. Ionospheric Data Assimilation
25.10. Renewable Energy Data Application
25.11. Earthquakes
25.11.1. Optimal Interpolation
25.11.2. Greens Function Data Assimilation
25.12. Oil and Natural Gas
25.13. Biogeoscience Application of Data Assimilation
25.14. Other Applications of Data Assimilation
25.15. Summary
Chapter 26: Solutions to Select Exercise
Chapter 2
Chapter 3
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Bibliography
Index

Life Sciences

Physical Sciences & Engineering

Social Sciences & Humanities

Health

Data Assimilation for the Geosciences

From Theory to Application

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Steven J. Fletcher

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