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.
1. Chapter 1: Introduction - p. iv 2. Chapter 2: Overview of Linear Algebra - p. 4 3. Chapter 3: Univariate Distribution Theory - p. 27 4. Abstract - p. 27 5. 3.1 Random Variables - p. 31 6. 3.2 Discrete Probability Theory - p. 32 7. 3.3 Continuous Probability Theory - p. 40 8. 3.4 Discrete Distribution Theory - p. 42 9. 3.5 Expectation and Variance of Discrete Random Variables - p. 46 10. 3.6 Moments and Moment-Generating Functions - p. 53 11. 3.7 Continuous Distribution Theory - p. 60 12. 3.8 Lognormal Distribution - p. 78 13. 3.9 Exponential Distribution - p. 86 14. 3.10 Gamma Distribution - p. 86 15. 3.11 Beta Distribution - p. 92 16. 3.12 Chi-Squared (χ2) Distribution - p. 97 17. 3.13 Rayleigh Distribution - p. 101 18. 3.14 Weibull Distribution - p. 107 19. 3.15 Gumbel Distribution - p. 111 20. 3.16 Summary of the Descriptive Statistics, Moment-Generating Functions, and Moments for the Univariate Distribution - p. 122 21. 3.17 Summary - p. 124 22. Chapter 4: Multivariate Distribution Theory - p. 124 23. Abstract - p. 124 24. 4.1 Descriptive Statistics for Multivariate Density Functions - p. 126 25. 4.2 Gaussian Distribution - p. 134 26. 4.3 Lognormal Distribution - p. 139 27. 4.4 Mixed Gaussian-Lognormal Distribution - p. 145 28. 4.5 Multivariate Mixed Gaussian-Lognormal Distribution - p. 156 29. 4.6 Gamma Distribution - p. 159 30. 4.7 Summary - p. 161 31. Chapter 5: Introduction to Calculus of Variation - p. 161 32. Abstract - p. 161 33. 5.1 Examples of Calculus of Variation Problems - p. 161 34. 5.2 Solving Calculus of Variation Problems - p. 167 35. 5.3 Functional With Higher-Order Derivatives - p. 180 36. 5.4 Three-Dimensional Problems - p. 182 37. 5.5 Functionals With Constraints - p. 185 38. 5.6 Functional With Extremals That Are Functions of Two or More Variables - p. 189 39. 5.7 Summary - p. 196 40. Chapter 6: Introduction to Control Theory - p. 196 41. Abstract - p. 196 42. 6.1 The Control Problem - p. 196 43. 6.2 The Uncontrolled Problem - p. 201 44. 6.3 The Controlled Problem - p. 212 45. 6.4 Observability - p. 220 46. 6.5 Duality - p. 222 47. 6.6 Stability - p. 224 48. 6.7 Feedback - p. 228 49. 6.8 Summary - p. 234 50. Chapter 7: Optimal Control Theory - p. 234 51. Abstract - p. 234 52. 7.1 Optimizing Scalar Control Problems - p. 238 53. 7.2 Multivariate Case - p. 241 54. 7.3 Autonomous (Time-Invariant) Problem - p. 243 55. 7.4 Extension to General Boundary Conditions - p. 246 56. 7.5 Free End Time Optimal Control Problems - p. 249 57. 7.6 Piecewise Smooth Calculus of Variation Problems - p. 254 58. 7.7 Maximization of Constrained Control Problems - p. 261 59. 7.8 Two Classical Optimal Control Problems - p. 267 60. 7.9 Summary - p. 272 61. Chapter 8: Numerical Solutions to Initial Value Problems - p. 272 62. Abstract - p. 272 63. 8.1 Local and Truncation Errors - p. 275 64. 8.2 Linear Multistep Methods - p. 277 65. 8.3 Stability - p. 280 66. 8.4 Convergence - p. 283 67. 8.5 Runge-Kutta Schemes - p. 290 68. 8.6 Numerical Solutions to Initial Value Partial Differential Equations - p. 296 69. 8.7 Wave Equation - p. 311 70. 8.8 Courant Friedrichs Lewy Condition - p. 314 71. 8.9 Summary - p. 315 72. Chapter 9: Numerical Solutions to Boundary Value Problems - p. 315 73. Abstract - p. 315 74. 9.1 Types of Differential Equations - p. 318 75. 9.2 Shooting Methods - p. 320 76. 9.3 Finite Difference Methods - p. 326 77. 9.4 Self-Adjoint Problems - p. 332 78. 9.5 Error Analysis - p. 334 79. 9.6 Partial Differential Equations - p. 339 80. 9.7 Self-Adjoint Problem in Two Dimensions - p. 349 81. 9.8 Periodic Boundary Conditions - p. 358 82. 9.9 Summary - p. 360 83. Chapter 10: Introduction to Semi-Lagrangian Advection Methods - p. 360 84. Abstract - p. 360 85. 10.1 History of Semi-Lagrangian Approaches - p. 362 86. 10.2 Derivation of Semi-Lagrangian Approach - p. 363 87. 10.3 Interpolation Polynomials - p. 367 88. 10.4 Stability of Semi-Lagrangian Schemes - p. 389 89. 10.5 Consistency Analysis of Semi-Lagrangian Schemes - p. 407 90. 10.6 Semi-Lagrangian Schemes for Non-Constant Advection Velocity - p. 409 91. 10.7 Semi-Lagrangian Scheme for Non-Zero Forcing - p. 411 92. 10.8 Example: 2D Quasi-Geostrophic Potential Vorticity (Eady Model) - p. 416 93. 10.9 Summary - p. 440 94. Chapter 11: Introduction to Finite Element Modeling - p. 441 95. Abstract - p. 441 96. 11.1 Solving the Boundary Value Problem - p. 441 97. 11.2 Weak Solutions of Differential Equation - p. 450 98. 11.3 Accuracy of the Finite Element Approach - p. 460 99. 11.4 Pin Tong - p. 466 100. 11.5 Finite Element Basis Functions - p. 469 101. 11.6 Coding Finite Element Approximations for Triangle Elements - p. 471 102. 11.7 Isoparametric Elements - p. 477 103. 11.8 Summary - p. 481 104. Chapter 12: Numerical Modeling on the Sphere - p. 482 105. Abstract - p. 482 106. 12.1 Vector Operators in Spherical Coordinates - p. 482 107. 12.2 Spherical Vector Derivative Operators - p. 484 108. 12.3 Finite Differencing on the Sphere - p. 486 109. 12.4 Introduction to Fourier Analysis - p. 502 110. 12.5 Spectral Modeling - p. 535 111. 12.6 Summary - p. 554 112. Chapter 13: Tangent Linear Modeling and Adjoints - p. 554 113. Abstract - p. 554 114. 13.1 Additive Tangent Linear and Adjoint Modeling Theory - p. 556 115. 13.2 Multiplicative Tangent Linear and Adjoint Modeling Theory - p. 563 116. 13.3 Examples of Adjoint Derivations - p. 565 117. 13.4 Perturbation Forecast Modeling - p. 588 118. 13.5 Adjoint Sensitivities - p. 591 119. 13.6 Singular Vectors - p. 592 120. 13.7 Summary - p. 598 121. Chapter 14: Observations - p. 598 122. Chapter 15: Non-variational Sequential Data Assimilation Methods - p. 626 123. Abstract - p. 626 124. 15.1 Direct Insertion - p. 628 125. 15.2 Nudging - p. 630 126. 15.3 Successive Correction - p. 632 127. 15.4 Linear and Nonlinear Least Squares - p. 639 128. 15.5 Regression - p. 653 129. 15.6 Optimal (Optimum) Interpolation/Statistical Interpolation/Analysis Correction - p. 659 130. 15.7 Summary - p. 671 131. Chapter 16: Variational Data Assimilation - p. 671 132. Abstract - p. 671 133. 16.1 Sasaki and the Strong and Weak Constraints - p. 675 134. 16.2 Three-Dimensional Data Assimilation - p. 677 135. 16.3 Four-Dimensional Data Assimilation - p. 681 136. 16.4 Incremental VAR - p. 685 137. 16.5 Weak Constraint—Model Error 4D VAR - p. 690 138. 16.6 Observational Errors - p. 696 139. 16.7 4D VAR as an Optimal Control Problem - p. 699 140. 16.8 Summary - p. 702 141. Chapter 17: Subcomponents of Variational Data Assimilation - p. 703 142. Abstract - p. 703 143. 17.1 Balance - p. 706 144. 17.2 Control Variable Transforms - p. 717 145. 17.3 Background Error Covariance Modeling - p. 734 146. 17.4 Preconditioning - p. 740 147. 17.5 Minimization Algorithms - p. 743 148. 17.6 Performance Metrics - p. 748 149. 17.7 Summary - p. 750 150. Chapter 18: Observation Space Variational Data Assimilation Methods - p. 751 151. Chapter 19: Kalman Filter and Smoother - p. 763 152. Chapter 20: Ensemble-Based Data Assimilation - p. 782 153. Abstract - p. 782 154. 20.1 Stochastic Dynamical Modeling - p. 784 155. 20.2 Ensemble Kalman Filter - p. 785 156. 20.3 Ensemble Square Root Filters - p. 793 157. 20.4 Ensemble and Local Ensemble Transform Kalman Filter - p. 797 158. 20.5 Maximum Likelihood Ensemble Filter - p. 803 159. 20.6 Hybrid Ensemble and Variational Data Assimilation Methods - p. 809 160. 20.7 NDEnVAR - p. 815 161. 20.8 Ensemble Kalman Smoother - p. 818 162. 20.9 Ensemble Sensitivity - p. 819 163. 20.10 Summary - p. 821 164. Chapter 21: Non-Gaussian Variational Data Assimilation - p. 821 165. Abstract - p. 821 166. 21.1 Error Definitions - p. 824 167. 21.2 Full Field Lognormal 3D VAR - p. 826 168. 21.3 Logarithmic Transforms - p. 829 169. 21.4 Mixed Gaussian-Lognormal 3D VAR - p. 831 170. 21.5 Lognormal Calculus of Variation-Based 4D VAR - p. 835 171. 21.6 Bayesian-Based 4D VAR - p. 840 172. 21.7 Bayesian Networks Formulation of Weak Constraint/Model Error 4D VAR - p. 845 173. 21.8 Results of the Lorenz 1963 Model for 4D VAR - p. 848 174. 21.9 Incremental Lognormal and Mixed 3D and 4D VAR - p. 853 175. 21.10 Regions of Optimality for Lognormal Descriptive Statistics - p. 861 176. 21.11 Summary - p. 866 177. Chapter 22: Markov Chain Monte Carlo and Particle Filter Methods - p. 868 178. Chapter 23: Applications of Data Assimilation in the Geosciences - p. 885 179. Chapter 24: Solutions to Select Exercise - p. 916 180. Bibliography - p. 922 181. Index - p. 939
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.
1. Chapter 1: Introduction - p. iv 2. Chapter 2: Overview of Linear Algebra - p. 4 3. Chapter 3: Univariate Distribution Theory - p. 27 4. Abstract - p. 27 5. 3.1 Random Variables - p. 31 6. 3.2 Discrete Probability Theory - p. 32 7. 3.3 Continuous Probability Theory - p. 40 8. 3.4 Discrete Distribution Theory - p. 42 9. 3.5 Expectation and Variance of Discrete Random Variables - p. 46 10. 3.6 Moments and Moment-Generating Functions - p. 53 11. 3.7 Continuous Distribution Theory - p. 60 12. 3.8 Lognormal Distribution - p. 78 13. 3.9 Exponential Distribution - p. 86 14. 3.10 Gamma Distribution - p. 86 15. 3.11 Beta Distribution - p. 92 16. 3.12 Chi-Squared (χ2) Distribution - p. 97 17. 3.13 Rayleigh Distribution - p. 101 18. 3.14 Weibull Distribution - p. 107 19. 3.15 Gumbel Distribution - p. 111 20. 3.16 Summary of the Descriptive Statistics, Moment-Generating Functions, and Moments for the Univariate Distribution - p. 122 21. 3.17 Summary - p. 124 22. Chapter 4: Multivariate Distribution Theory - p. 124 23. Abstract - p. 124 24. 4.1 Descriptive Statistics for Multivariate Density Functions - p. 126 25. 4.2 Gaussian Distribution - p. 134 26. 4.3 Lognormal Distribution - p. 139 27. 4.4 Mixed Gaussian-Lognormal Distribution - p. 145 28. 4.5 Multivariate Mixed Gaussian-Lognormal Distribution - p. 156 29. 4.6 Gamma Distribution - p. 159 30. 4.7 Summary - p. 161 31. Chapter 5: Introduction to Calculus of Variation - p. 161 32. Abstract - p. 161 33. 5.1 Examples of Calculus of Variation Problems - p. 161 34. 5.2 Solving Calculus of Variation Problems - p. 167 35. 5.3 Functional With Higher-Order Derivatives - p. 180 36. 5.4 Three-Dimensional Problems - p. 182 37. 5.5 Functionals With Constraints - p. 185 38. 5.6 Functional With Extremals That Are Functions of Two or More Variables - p. 189 39. 5.7 Summary - p. 196 40. Chapter 6: Introduction to Control Theory - p. 196 41. Abstract - p. 196 42. 6.1 The Control Problem - p. 196 43. 6.2 The Uncontrolled Problem - p. 201 44. 6.3 The Controlled Problem - p. 212 45. 6.4 Observability - p. 220 46. 6.5 Duality - p. 222 47. 6.6 Stability - p. 224 48. 6.7 Feedback - p. 228 49. 6.8 Summary - p. 234 50. Chapter 7: Optimal Control Theory - p. 234 51. Abstract - p. 234 52. 7.1 Optimizing Scalar Control Problems - p. 238 53. 7.2 Multivariate Case - p. 241 54. 7.3 Autonomous (Time-Invariant) Problem - p. 243 55. 7.4 Extension to General Boundary Conditions - p. 246 56. 7.5 Free End Time Optimal Control Problems - p. 249 57. 7.6 Piecewise Smooth Calculus of Variation Problems - p. 254 58. 7.7 Maximization of Constrained Control Problems - p. 261 59. 7.8 Two Classical Optimal Control Problems - p. 267 60. 7.9 Summary - p. 272 61. Chapter 8: Numerical Solutions to Initial Value Problems - p. 272 62. Abstract - p. 272 63. 8.1 Local and Truncation Errors - p. 275 64. 8.2 Linear Multistep Methods - p. 277 65. 8.3 Stability - p. 280 66. 8.4 Convergence - p. 283 67. 8.5 Runge-Kutta Schemes - p. 290 68. 8.6 Numerical Solutions to Initial Value Partial Differential Equations - p. 296 69. 8.7 Wave Equation - p. 311 70. 8.8 Courant Friedrichs Lewy Condition - p. 314 71. 8.9 Summary - p. 315 72. Chapter 9: Numerical Solutions to Boundary Value Problems - p. 315 73. Abstract - p. 315 74. 9.1 Types of Differential Equations - p. 318 75. 9.2 Shooting Methods - p. 320 76. 9.3 Finite Difference Methods - p. 326 77. 9.4 Self-Adjoint Problems - p. 332 78. 9.5 Error Analysis - p. 334 79. 9.6 Partial Differential Equations - p. 339 80. 9.7 Self-Adjoint Problem in Two Dimensions - p. 349 81. 9.8 Periodic Boundary Conditions - p. 358 82. 9.9 Summary - p. 360 83. Chapter 10: Introduction to Semi-Lagrangian Advection Methods - p. 360 84. Abstract - p. 360 85. 10.1 History of Semi-Lagrangian Approaches - p. 362 86. 10.2 Derivation of Semi-Lagrangian Approach - p. 363 87. 10.3 Interpolation Polynomials - p. 367 88. 10.4 Stability of Semi-Lagrangian Schemes - p. 389 89. 10.5 Consistency Analysis of Semi-Lagrangian Schemes - p. 407 90. 10.6 Semi-Lagrangian Schemes for Non-Constant Advection Velocity - p. 409 91. 10.7 Semi-Lagrangian Scheme for Non-Zero Forcing - p. 411 92. 10.8 Example: 2D Quasi-Geostrophic Potential Vorticity (Eady Model) - p. 416 93. 10.9 Summary - p. 440 94. Chapter 11: Introduction to Finite Element Modeling - p. 441 95. Abstract - p. 441 96. 11.1 Solving the Boundary Value Problem - p. 441 97. 11.2 Weak Solutions of Differential Equation - p. 450 98. 11.3 Accuracy of the Finite Element Approach - p. 460 99. 11.4 Pin Tong - p. 466 100. 11.5 Finite Element Basis Functions - p. 469 101. 11.6 Coding Finite Element Approximations for Triangle Elements - p. 471 102. 11.7 Isoparametric Elements - p. 477 103. 11.8 Summary - p. 481 104. Chapter 12: Numerical Modeling on the Sphere - p. 482 105. Abstract - p. 482 106. 12.1 Vector Operators in Spherical Coordinates - p. 482 107. 12.2 Spherical Vector Derivative Operators - p. 484 108. 12.3 Finite Differencing on the Sphere - p. 486 109. 12.4 Introduction to Fourier Analysis - p. 502 110. 12.5 Spectral Modeling - p. 535 111. 12.6 Summary - p. 554 112. Chapter 13: Tangent Linear Modeling and Adjoints - p. 554 113. Abstract - p. 554 114. 13.1 Additive Tangent Linear and Adjoint Modeling Theory - p. 556 115. 13.2 Multiplicative Tangent Linear and Adjoint Modeling Theory - p. 563 116. 13.3 Examples of Adjoint Derivations - p. 565 117. 13.4 Perturbation Forecast Modeling - p. 588 118. 13.5 Adjoint Sensitivities - p. 591 119. 13.6 Singular Vectors - p. 592 120. 13.7 Summary - p. 598 121. Chapter 14: Observations - p. 598 122. Chapter 15: Non-variational Sequential Data Assimilation Methods - p. 626 123. Abstract - p. 626 124. 15.1 Direct Insertion - p. 628 125. 15.2 Nudging - p. 630 126. 15.3 Successive Correction - p. 632 127. 15.4 Linear and Nonlinear Least Squares - p. 639 128. 15.5 Regression - p. 653 129. 15.6 Optimal (Optimum) Interpolation/Statistical Interpolation/Analysis Correction - p. 659 130. 15.7 Summary - p. 671 131. Chapter 16: Variational Data Assimilation - p. 671 132. Abstract - p. 671 133. 16.1 Sasaki and the Strong and Weak Constraints - p. 675 134. 16.2 Three-Dimensional Data Assimilation - p. 677 135. 16.3 Four-Dimensional Data Assimilation - p. 681 136. 16.4 Incremental VAR - p. 685 137. 16.5 Weak Constraint—Model Error 4D VAR - p. 690 138. 16.6 Observational Errors - p. 696 139. 16.7 4D VAR as an Optimal Control Problem - p. 699 140. 16.8 Summary - p. 702 141. Chapter 17: Subcomponents of Variational Data Assimilation - p. 703 142. Abstract - p. 703 143. 17.1 Balance - p. 706 144. 17.2 Control Variable Transforms - p. 717 145. 17.3 Background Error Covariance Modeling - p. 734 146. 17.4 Preconditioning - p. 740 147. 17.5 Minimization Algorithms - p. 743 148. 17.6 Performance Metrics - p. 748 149. 17.7 Summary - p. 750 150. Chapter 18: Observation Space Variational Data Assimilation Methods - p. 751 151. Chapter 19: Kalman Filter and Smoother - p. 763 152. Chapter 20: Ensemble-Based Data Assimilation - p. 782 153. Abstract - p. 782 154. 20.1 Stochastic Dynamical Modeling - p. 784 155. 20.2 Ensemble Kalman Filter - p. 785 156. 20.3 Ensemble Square Root Filters - p. 793 157. 20.4 Ensemble and Local Ensemble Transform Kalman Filter - p. 797 158. 20.5 Maximum Likelihood Ensemble Filter - p. 803 159. 20.6 Hybrid Ensemble and Variational Data Assimilation Methods - p. 809 160. 20.7 NDEnVAR - p. 815 161. 20.8 Ensemble Kalman Smoother - p. 818 162. 20.9 Ensemble Sensitivity - p. 819 163. 20.10 Summary - p. 821 164. Chapter 21: Non-Gaussian Variational Data Assimilation - p. 821 165. Abstract - p. 821 166. 21.1 Error Definitions - p. 824 167. 21.2 Full Field Lognormal 3D VAR - p. 826 168. 21.3 Logarithmic Transforms - p. 829 169. 21.4 Mixed Gaussian-Lognormal 3D VAR - p. 831 170. 21.5 Lognormal Calculus of Variation-Based 4D VAR - p. 835 171. 21.6 Bayesian-Based 4D VAR - p. 840 172. 21.7 Bayesian Networks Formulation of Weak Constraint/Model Error 4D VAR - p. 845 173. 21.8 Results of the Lorenz 1963 Model for 4D VAR - p. 848 174. 21.9 Incremental Lognormal and Mixed 3D and 4D VAR - p. 853 175. 21.10 Regions of Optimality for Lognormal Descriptive Statistics - p. 861 176. 21.11 Summary - p. 866 177. Chapter 22: Markov Chain Monte Carlo and Particle Filter Methods - p. 868 178. Chapter 23: Applications of Data Assimilation in the Geosciences - p. 885 179. Chapter 24: Solutions to Select Exercise - p. 916 180. Bibliography - p. 922 181. Index - p. 939
