Data Assimilation for the Geosciences From Theory to Application 2nd Edition by Steven J Fletcher ISBN 9780323917209 0323917208

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ISBN 10: 0323917208
ISBN 13: 9780323917209
Author: Steven J Fletcher

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.

Data Assimilation for the Geosciences From Theory to Application 2nd Edition Table of contents:

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

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Tags: Steven J Fletcher, Data Assimilation, Geosciences, Application