Data Assimilation for the Geosciences

Data Assimilation for the Geosciences: From Theory to Application brings together all of the mathematical,statistical, and probability background knowledge needed to formulate data assimilation systems in one place. It includes practical exercises for understanding theoretical formulation and presents some aspects of coding the theory with a toy problem.The book also demonstrates how data assimilation systems are implemented in larger scale fluid dynamical problems related to the atmosphere, oceans, as well as the land surface and other geophysical situations. It offers a comprehensive presentation of the subject, from basic principles to advanced methods, such as Particle Filters and Markov-Chain Monte-Carlo methods. Additionally, Data Assimilation for the Geosciences: From Theory to Application covers the applications of data assimilation techniques in various disciplines of the geosciences, making the book useful to students, teachers, and research scientists.

Chapter 1: Introduction AbstractChapter 2: Overview of Linear Algebra Abstract 2.1 Properties of Matrices 2.2 Matrix and Vector Norms 2.3 Eigenvalues and Eigenvectors 2.4 Matrix Decompositions 2.5 Sherman-Morrison-Woodbury Formula 2.6 Summary Chapter 3: Univariate Distribution Theory Abstract 3.1 Random Variables 3.2 Discrete Probability Theory 3.3 Continuous Probability Theory 3.4 Discrete Distribution Theory 3.5 Expectation and Variance of Discrete Random Variables 3.6 Moments and Moment-Generating Functions 3.7 Continuous Distribution Theory 3.8 Lognormal Distribution 3.9 Exponential Distribution 3.10 Gamma Distribution 3.11 Beta Distribution 3.12 Chi-Squared (χ2) Distribution 3.13 Rayleigh Distribution 3.14 Weibull Distribution 3.15 Gumbel Distribution 3.16 Summary of the Descriptive Statistics, Moment-Generating Functions, and Moments for the Univariate Distribution 3.17 Summary Chapter 4: Multivariate Distribution Theory Abstract 4.1 Descriptive Statistics for Multivariate Density Functions 4.2 Gaussian Distribution 4.3 Lognormal Distribution 4.4 Mixed Gaussian-Lognormal Distribution 4.5 Multivariate Mixed Gaussian-Lognormal Distribution 4.6 Gamma Distribution 4.7 Summary Chapter 5: Introduction to Calculus of Variation Abstract 5.1 Examples of Calculus of Variation Problems 5.2 Solving Calculus of Variation Problems 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 5.7 Summary Chapter 6: Introduction to Control Theory Abstract 6.1 The Control Problem 6.2 The Uncontrolled Problem 6.3 The Controlled Problem 6.4 Observability 6.5 Duality 6.6 Stability 6.7 Feedback 6.8 Summary Chapter 7: Optimal Control Theory Abstract 7.1 Optimizing Scalar Control Problems 7.2 Multivariate Case 7.3 Autonomous (Time-Invariant) Problem 7.4 Extension to General Boundary Conditions 7.5 Free End Time Optimal Control Problems 7.6 Piecewise Smooth Calculus of Variation Problems 7.7 Maximization of Constrained Control Problems 7.8 Two Classical Optimal Control Problems 7.9 Summary Chapter 8: Numerical Solutions to Initial Value Problems Abstract 8.1 Local and Truncation Errors 8.2 Linear Multistep Methods 8.3 Stability 8.4 Convergence 8.5 Runge-Kutta Schemes 8.6 Numerical Solutions to Initial Value Partial Differential Equations 8.7 Wave Equation 8.8 Courant Friedrichs Lewy Condition 8.9 Summary Chapter 9: Numerical Solutions to Boundary Value Problems Abstract 9.1 Types of Differential Equations 9.2 Shooting Methods 9.3 Finite Difference Methods 9.4 Self-Adjoint Problems 9.5 Error Analysis 9.6 Partial Differential Equations 9.7 Self-Adjoint Problem in Two Dimensions 9.8 Periodic Boundary Conditions 9.9 Summary Chapter 10: Introduction to Semi-Lagrangian Advection Methods Abstract 10.1 History of Semi-Lagrangian Approaches 10.2 Derivation of Semi-Lagrangian Approach 10.3 Interpolation Polynomials 10.4 Stability of Semi-Lagrangian Schemes 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) 10.9 Summary Chapter 11: Introduction to Finite Element Modeling Abstract 11.1 Solving the Boundary Value Problem 11.2 Weak Solutions of Differential Equation 11.3 Accuracy of the Finite Element Approach 11.4 Pin Tong 11.5 Finite Element Basis Functions 11.6 Coding Finite Element Approximations for Triangle Elements 11.7 Isoparametric Elements 11.8 Summary Chapter 12: Numerical Modeling on the Sphere Abstract 12.1 Vector Operators in Spherical Coordinates 12.2 Spherical Vector Derivative Operators 12.3 Finite Differencing on the Sphere 12.4 Introduction to Fourier Analysis 12.5 Spectral Modeling 12.6 Summary Chapter 13: Tangent Linear Modeling and Adjoints Abstract 13.1 Additive Tangent Linear and Adjoint Modeling Theory 13.2 Multiplicative Tangent Linear and Adjoint Modeling Theory 13.3 Examples of Adjoint Derivations 13.4 Perturbation Forecast Modeling 13.5 Adjoint Sensitivities 13.6 Singular Vectors 13.7 Summary Chapter 14: Observations Abstract 14.1 Conventional Observations 14.2 Remote Sensing 14.3 Quality Control 14.4 Summary Chapter 15: Non-variational Sequential Data Assimilation Methods Abstract 15.1 Direct Insertion 15.2 Nudging 15.3 Successive Correction 15.4 Linear and Nonlinear Least Squares 15.5 Regression 15.6 Optimal (Optimum) Interpolation/Statistical Interpolation/Analysis Correction 15.7 Summary Chapter 16: Variational Data Assimilation Abstract 16.1 Sasaki and the Strong and Weak Constraints 16.2 Three-Dimensional Data Assimilation 16.3 Four-Dimensional Data Assimilation 16.4 Incremental VAR 16.5 Weak Constraint—Model Error 4D VAR 16.6 Observational Errors 16.7 4D VAR as an Optimal Control Problem 16.8 Summary Chapter 17: Subcomponents of Variational Data Assimilation Abstract 17.1 Balance 17.2 Control Variable Transforms 17.3 Background Error Covariance Modeling 17.4 Preconditioning 17.5 Minimization Algorithms 17.6 Performance Metrics 17.7 Summary Chapter 18: Observation Space Variational Data Assimilation Methods Abstract 18.1 Derivation of Observation Space-Based 3D VAR 18.2 4D VAR in Observation Space 18.3 Duality of the VAR and PSAS Systems 18.4 Summary Chapter 19: Kalman Filter and Smoother Abstract 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 19.6 Properties and Equivalencies of the Kalman Filter and Smoother 19.7 Summary Chapter 20: Ensemble-Based Data Assimilation Abstract 20.1 Stochastic Dynamical Modeling 20.2 Ensemble Kalman Filter 20.3 Ensemble Square Root Filters 20.4 Ensemble and Local Ensemble Transform Kalman Filter 20.5 Maximum Likelihood Ensemble Filter 20.6 Hybrid Ensemble and Variational Data Assimilation Methods 20.7 NDEnVAR 20.8 Ensemble Kalman Smoother 20.9 Ensemble Sensitivity 20.10 Summary Chapter 21: Non-Gaussian Variational Data Assimilation Abstract 21.1 Error Definitions 21.2 Full Field Lognormal 3D VAR 21.3 Logarithmic Transforms 21.4 Mixed Gaussian-Lognormal 3D VAR 21.5 Lognormal Calculus of Variation-Based 4D VAR 21.6 Bayesian-Based 4D VAR 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 21.10 Regions of Optimality for Lognormal Descriptive Statistics 21.11 Summary Chapter 22: Markov Chain Monte Carlo and Particle Filter Methods Abstract 22.1 Markov Chain Monte Carlo Methods 22.2 Particle Filters 22.3 Summary Chapter 23: Applications of Data Assimilation in the Geosciences Abstract 23.1 Atmospheric Science 23.2 Oceans 23.3 Hydrological Applications 23.4 Coupled Data Assimilation 23.5 Reanalysis 23.6 Ionospheric Data Assimilation 23.7 Renewable Energy Data Application 23.8 Oil and Natural Gas 23.9 Biogeoscience Application of Data Assimilation 23.10 Other Applications of Data Assimilation 23.11 Summary Chapter 24: Solutions to Select Exercise Chapter 2 Chapter 3 Chapter 5 Chapter 6 Chapter 7 Chapter 8 Chapter 9