Data Assimilation for the Geosciences - 1st Edition - ISBN: 9780128044445, 9780128044841

Data Assimilation for the Geosciences

1st Edition

From Theory to Application

Authors: Steven Fletcher
eBook ISBN: 9780128044841
Paperback ISBN: 9780128044445
Imprint: Elsevier
Published Date: 15th March 2017
Page Count: 976
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Description

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

Readership

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

Table of Contents

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 (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

Details

No. of pages:
976
Language:
English
Copyright:
© Elsevier 2017
Published:
Imprint:
Elsevier
eBook ISBN:
9780128044841
Paperback ISBN:
9780128044445

About the Author

Steven Fletcher

Dr. Fletcher obtained his Bachelor’s degree with honors in Mathematics and Statistics from the University of Reading in 1998. He obtained his Master’s degree in Numerical Solutions to Differential Equations from the University of Reading in 1999. He obtained his Ph.D. in Mathematics/Data Assimilation in 2004, again from the University of Reading.

Since 2004, Dr. Fletcher has been a postdoctoral fellow, Research Scientist II, and is currently a Research Scientist III at the Cooperative Institute for Research in the Atmosphere (CIRA) at Colorado State University, where he has worked extensively on extending the Gaussian bases for variational and PSAS based data assimilation methods to allow for the minimization of lognormally distributed errors. He has also derived the theory to allow for Gaussian and lognormally distributed errors to be minimized simultaneously through deriving a mixed Gaussian-lognormal multivariate distribution. This mixed distribution has been applied for 3D and 4DVAR theory, for both full field and incremental formulations. Recently, Dr. Fletcher was able to derive a representer based formulation of 4DVAR for the mixed distribution, which is the basis of NAVDAS-AR, the US Navy’s numerical weather prediction system. The mixed distribution has also been applied in a retrieval system at CIRA for the retrieval of temperature and mixing ratio values given microwave based brightness temperatures, and it was shown that the mixed approach would obtain better fits to the observations compared to a logarithmic transform and a Gaussian fits all approach.

Dr. Fletcher is an active member in the Nonlinear Geophysics focus group of the American Geophysical Union, where he fills the role of the focus group’s representative on the program committee for the Annual Fall meeting, which is the largest meeting of geoscientists in the world.

Affiliations and Expertise

Cooperative Institute for Research in the Atmosphere, Colorado State University, USA