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The Resource Forecast Error Correction using Dynamic Data Assimilation

Forecast Error Correction using Dynamic Data Assimilation

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Forecast Error Correction using Dynamic Data Assimilation
Title
Forecast Error Correction using Dynamic Data Assimilation
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Language
eng
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MiAaPQ
Literary form
non fiction
Nature of contents
dictionaries
Series statement
Springer Atmospheric Sciences
Forecast Error Correction using Dynamic Data Assimilation
Label
Forecast Error Correction using Dynamic Data Assimilation
Link
http://libproxy.rpi.edu/login?url=https://ebookcentral.proquest.com/lib/rpi/detail.action?docID=4722672
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Carrier category
online resource
Carrier category code
cr
Carrier MARC source
rdacarrier
Color
multicolored
Content category
text
Content type code
txt
Content type MARC source
rdacontent
Contents
  • Preface -- Acknowledgments -- Contents -- Part I Introduction to Forward Sensitivity Method -- 1 Introduction -- 1.1 Predictability Limits -- 1.2 Sources of Error that Limit Predictability -- 1.3 Data Assimilation in Service to Prediction -- 1.4 Overview and Goals -- 1.5 A Classification of Forecast Dynamics and Errors -- 1.5.1 Forecast Dynamics -- 1.5.2 Forecast Errors -- 1.6 Organization of the Monograph -- 1.7 Exercises -- 1.7.1 Demonstrations and Problems -- 2 Forward Sensitivity Method: Scalar Case -- 2.1 Evolution of First-Order Sensitivities -- 2.2 First-Order Sensitivities Used in FSM -- 2.2.1 First-Order Analysis -- Case 1: Single Observation -- Case 2: Multiple Observations -- 2.3 Evolution of Second-Order Sensitivities -- 2.3.1 Evolution of {u2202}2 x(t) / {u2202}x2(0) -- 2.3.2 Evolution of {u2202}2 x(t) / {u2202}Ü2 -- 2.3.3 Evolution of {u2202}2 x(t) / {u2202}Ü {u2202}x(0) -- 2.4 Data Assimilation Using FSM: A Second-Order Method -- 2.5 FSM: Discrete Time Formulation -- 2.5.1 Discrete Evolution of First-Order Forward Sensitivities -- 2.5.2 Discrete Evolution of Second-Order Forward Sensitivity -- 2.6 Sensitivity to Initial Conditions and Lyapunov Index -- 2.6.1 Continuous Time Model -- 2.6.2 Discrete Time Model -- 2.7 Exercises -- 2.7.1 Demonstrations -- 2.8 Notes and References -- 3 On the Relation Between Adjoint and Forward Sensitivity -- 3.1 On the Structure of Adjoint Sensitivity -- 3.1.1 Adjoint Method -- 3.1.2 Computing x(0) J in (3.1.20) -- 3.1.3 Computation of ÜJ in (3.1.22) -- 3.1.4 4D-VAR method -- 3.2 On the Relation Between FSM and 4D-VAR -- 3.3 Investigation of the Impact of Observations Using 4D-VAR and FSM -- 3.3.1 Experiments -- 3.4 Exercises -- 3.4.1 Demonstrations -- IM.2 Nondimensional Form of the Solution in (IM.21) -- IM.3 True Control -- IM.4 Generate Current Observations -- IM.5 Forecast from Erroneous Control
  • IM.6 Evolution of Forward Sensitivities -- IM.7 Data Assimilation -- Generation of Observations -- Forecast Error Correction Using FSM -- Data Assimilation Using FSM -- 3.5 Notes and References -- 4 Forward Sensitivity Method: General Case -- 4.1 Dynamics of Evolution of First Order Forward Sensitivities -- 4.1.1 Dynamics of Evolution of u1(k) -- 4.1.2 Dynamics of Evolution of v1(k) -- 4.1.3 Propagation of Perturbation and Forward Sensitivity -- 4.2 On the Relation Between Adjoint and Forward Sensitivities -- 4.3 Data Assimilation Using FSM -- 4.3.1 Case 1 Single Observation -- 4.3.2 Case 2: Multiple Observations -- 4.4 Exercises -- 4.4.1 Demonstrations -- 4.5 Notes and References -- 5 Forecast Error Correction Using Optimal Tracking -- 5.1 Pontryagin's Minimum Principle (PMP) in Discrete Time -- 5.1.1 Condition 1: Model Dynamics -- 5.1.2 Condition 2: Co-State or Adjoint Dynamics -- 5.1.3 Condition 3: Stationarity Condition -- 5.1.4 Condition 4: Boundary Conditions -- 5.1.4.1 A Framework for Optimal Tracking -- 5.2 Connection to 4D-VAR -- 5.2.1 Condition 1: Model Dynamics -- 5.2.2 Condition 2: Co-State or Adjoint Dynamics -- 5.2.3 Condition 3: Boundary Conditions -- 5.3 Optimal Tracking: Linear Case -- 5.3.1 Structure of the Optimal Control -- 5.3.2 The TPBVP -- 5.3.3 Affine Relation Between nk and xk -- 5.3.4 Conversion of TPBVP to Two Initial Value Problems -- 5.3.5 Structure of the Off-Line Optimal Control -- 5.3.6 Optimal Trajectory -- 5.3.7 Identification of Model Error -- 5.4 Computation of Model Errors -- 5.4.1 Gradient of Ü( S, x) -- 5.4.2 Gradient of Ý( S, x, y) -- 5.4.3 Gradient of Q( S) -- 5.5 An Application: Linear Advection Equation Model and Its Solution -- 5.5.1 The Low-Order Model (LOM) -- 5.5.1.1 Conservation of Energy -- 5.5.2 Solution of LOM(n) in (5.5.8) -- 5.5.3 Identification of Model Error -- 5.6 Exercises
  • 5.6.1 Demonstrations -- 5.7 Notes and References -- Appendix -- Solution of the LOM in (5.5.8) -- Eigenstructure of the Matrix A -- Jordan Canonical Form for A -- Solution of (5.5.8) -- Part II Applications of Forward Sensitivity Method -- 6 The Gulf of Mexico Problem: Return Flow Analysis -- 6.1 Observations in Support of Mixed Layer Modeling -- 6.2 The Five Variable Model -- 6.3 Evolution of Forward Sensitivities -- 6.3.1 Sensitivity with Respect to the Parameters -- 6.3.2 Sensitivity with Respect to Initial Conditions -- 6.3.3 Sensitivity at the End of the Simulation -- 6.4 Data Assimilation Using Forward Sensitivity Method: Numerical Experiments -- 6.4.1 Observations at Multiple Time -- 6.5 Problems -- 6.6 Notes and References -- Appendix -- Model Jacobians -- Elements of the Jacobian Dx(M) -- Elements of the Jacobian DÜ -- 7 Lagrangian Tracer Dynamics -- 7.1 Shallow Water Model and Tracer Dynamics -- 7.1.1 Low-Order Model (LOM) -- 7.1.2 Solution of LOM -- 7.1.3 Tracer Dynamics -- 7.2 Analysis of Equilibria of Tracer Dynamics -- 7.2.1 Case 1: Equilibria in Geostrophic Mode -- 7.2.1.1 Type 1 Equilibria -- 7.2.1.2 Type 2 Equilibria -- 7.2.2 Case 2: Equilibria in Inertia-Gravity Mode -- 7.2.2.1 Type a: Equilibria in the Upper Half Plane -- 7.2.2.2 Type b: Equilibria in the Lower Half Plane -- 7.2.3 Case 3: Equilibria in General Case -- 7.2.3.1 Type A: Equilibria in the First Quadrant -- 7.2.3.2 Type B: Equilibria in the Fourth Quadrant -- 7.2.3.3 Type C: Equilibria in the Second Quadrant -- 7.2.3.4 Type D: Equilibria in the Fourth Quadrant -- 7.2.4 Conditions for the Sign Definiteness of Ei(t) -- 7.3 Analysis of Bifurcation -- 7.3.1 Case 1 -- 7.3.2 Case 2 -- 7.4 Dynamics of Evolution of Forward Sensitivities -- 7.5 Sensitivity Plots -- 7.5.1 Sensitivity to Initial Conditions -- 7.5.2 Sensitivity to Parameters -- 7.6 Data Assimilation Using FSM
  • 7.6.1 Methodology -- 7.6.2 Data Assimilation Experiment -- 7.7 Notes and References -- Appendix -- Bounds on u1(t) in (7.1.13) -- Definition and Properties of Standard Hyperbola -- Epilogue -- A Basic Notation -- A.1 Basic Notation -- A.2 Inner Product -- A.3 Gradient, Jacobian, Hessian -- A.4 Taylor Expansion -- A.5 First Variations -- A.6 Conditions for Minima -- References -- Index
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9783319399973
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Media MARC source
rdamedia
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