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Data Assimilation for the Geosciences,

Edition 1 From Theory to Application

Editors: By Steven J. Fletcher

Publication Date: 15 Mar 2017

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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.

Key Features

Includes practical exercises, enabling readers to apply concepts in a theoretical formulation
Offers explanations for how to code certain parts of the theory
Presents a step-by-step guide on how, and why, data assimilation works and can be used

About the author

By Steven J. Fletcher, Research Scientist III, Cooperative Institute for Research in the Atmosphere (CIRA), Colorado State University – Fort Collins, Colorado, USA

Chapter 1: Introduction

Abstract

Chapter 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 (¿²) 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

ISBN: 9780128044445

Page Count: 976

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All geoscientists especially geophysicists, atmospheric scientists and mathematicians who are learning about data assimilation