Note: Intro -- Preface -- Contents -- List of Contributors -- Chapter 1 A Survey of Observers for Nonlinear Dynamical Systems -- 1.1 Introduction -- 1.2 Observability -- 1.3 Asymptotic Observers -- 1.3.1 Luenberger Observer -- 1.3.2 Observers with Linear Error Dynamics -- 1.3.3 Observers Based on Lyapunov Functions -- 1.4 Optimal Filtering -- 1.4.1 Kalman Filters -- 1.4.2 H∞ Filter -- 1.4.3 Minimum Energy Estimation -- 1.5 Observer Construction for PDE Systems -- 1.5.1 Linear Case -- 1.5.2 Nonlinear Case -- 1.6 Conclusions -- References -- Chapter 2 Nudging Methods: A Critical Overview -- 2.1 Introduction -- 2.2 Early Empirical Method -- 2.3 Estimating Optimal Nudging Coefficient: Problems and Challenges -- 2.3.1 Estimation of g Using the Variational Approach -- 2.3.2 Estimation of G Using Kalman-Like Nudging Scheme -- 2.4 Observability and Observer-Based Nudging -- 2.4.1 Conditions for Observability -- 2.4.2 Observer-Based Nudging: Linear Dynamics -- 2.4.3 Observer Based Nudging: Nonlinear Dynamics -- 2.5 Back and Forth Nudging Scheme -- 2.5.1 Analysis of Forward Nudging -- 2.5.2 Analysis of Backward Nudging -- 2.5.3 Back and Forth Nudging Scheme -- 2.6 Discussion and Conclusions -- References -- Chapter 3 Markov Chain Monte Carlo Methods: Theory and Applications -- 3.1 Introduction and History -- 3.2 Theoretical Basis of MCMC in Bayesian Inference -- 3.3 Practical Issues -- 3.3.1 Choice and Tuning of Proposal Distribution -- 3.3.2 The Initial Sample -- 3.3.3 Single Versus Multiple Chains -- 3.3.4 Pseudo-convergence -- 3.3.5 Diagnosing Convergence -- 3.3.5.1 Time Series Plots -- 3.3.5.2 Running or Batch Moments -- 3.3.5.3 Multi-chain Convergence Diagnostics: The R-Statistic -- 3.3.6 Working with the Posterior Sample -- 3.4 Select Applications of MCMC in the Atmospheric Sciences. , 3.4.1 Retrieval of Atmospheric State Variables from Satellite Measurements -- 3.4.2 Model Parameter Estimation and Uncertainty Analysis -- 3.5 Concluding Remarks -- References -- Chapter 4 Observation Influence Diagnostic of a Data Assimilation System -- 4.1 Introduction -- 4.2 Classical Statistical Definition of Influence Matrix and Self-Sensitivity -- 4.3 Observational Influence and Self-Sensitivity for a DA Scheme -- 4.3.1 Linear Statistical Estimation in Numerical Weather Prediction -- 4.3.2 R Diagonal -- 4.3.3 Toy Model -- 4.4 Results -- 4.4.1 Trace diagnostic: Observation Influence and DFS -- 4.4.2 Geographical Map of OI -- 4.5 Conclusions -- References -- Chapter 5 A Question of Adequacy of Observations in Variational Data Assimilation -- 5.1 Introduction -- 5.2 Model Dynamics: Air/Sea Interaction -- 5.2.1 Background Physical Processes -- 5.2.2 Governing Equation -- 5.2.3 Sensitivities -- 5.2.4 Cost Function for Data Assimilation -- 5.3 Forward Sensitivity Method (FSM) Applied to Air/Sea Interaction Model -- 5.4 Numerical Experiments -- 5.4.1 Prelude -- 5.4.2 Forecast Errors -- 5.4.3 Experiment 1: Insufficiency of Observations -- 5.4.4 Experiment 2: Sufficiency of Observations -- 5.5 Discussion and Conclusions -- References -- Chapter 6 Quantifying Observation Impact for a Limited Area Atmospheric Forecast Model -- 6.1 Adjoint Sensitivities -- 6.1.1 Tangent Linear and Adjoint of the Forecast Model -- 6.1.2 Observation Sensitivity -- 6.1.3 Adjoint Observation Impact -- 6.1.4 Applications -- 6.2 COAMPS and NAVDAS -- 6.2.1 COAMPS -- 6.2.2 NAVDAS -- 6.2.3 COAMPS Adjoint -- 6.2.4 NAVDAS Adjoint -- 6.3 Observation Impacts for COAMPS/NAVDAS -- 6.3.1 Lateral Boundaries -- 6.3.2 Location -- 6.3.3 Model Resolution -- 6.3.4 Other Metrics -- 6.4 Summary and Considerations -- References. , Chapter 7 Skewness of the Prior Through Position Errors and Its Impact on Data Assimilation -- 7.1 Introduction -- 7.2 Understanding Data Assimilation Through Bayes' Rule -- 7.2.1 Bayes' Rule: The Posterior Distribution -- 7.2.2 Data Assimilation as a Problem in Nonlinear Regression -- 7.2.2.1 Linear Regression -- 7.2.2.2 Quadratic Nonlinear Regression -- 7.3 Distributions Arising from Phase Errors -- 7.4 DA in the Presence of Phase Errors -- 7.4.1 Idealized Cases -- 7.4.1.1 Estimating the Posterior Mean -- 7.4.1.2 Ensemble Update -- 7.4.2 Hurricane Katia (2011) -- 7.5 Summary and Conclusions -- References -- Chapter 8 Background Error Correlation Modeling with DiffusionOperators -- 8.1 Introduction -- 8.2 Diffusion Operator and Covariance Modeling -- 8.2.1 Correlation Functions and Normalization -- 8.2.2 The Gaussian Model and Its Binomial Approximations -- 8.2.3 The Inverse Polynomial Model -- 8.3 Diagonal Estimation -- 8.3.1 Stochastic Methods -- 8.3.2 Locally Homogeneous Approximations -- 8.3.3 Numerical Results -- 8.3.3.1 Experimental Setting in 2d -- 8.3.3.2 Statistical Methods -- 8.3.3.3 Asymptotic Expansion Method -- 8.3.4 Numerical Efficiency -- 8.3.5 LH Experiments in 3d Setting -- 8.4 Summary and Discussion -- Appendix 1 -- Appendix 2 -- Appendix 3 -- References -- Chapter 9 The Adjoint Sensitivity Guidance to Diagnosis and Tuning of Error Covariance Parameters -- 9.1 Introduction -- 9.2 The Analysis Equation -- 9.2.1 Adjoint-DAS Observation Impact Estimation -- 9.2.2 Suboptimal Observation Performance: A Scalar Example -- 9.3 Adjoint-DAS Sensitivity Analysis -- 9.3.1 Sensitivity to Observations and Background -- 9.3.2 Forecast R- and B-Sensitivity and Impact Estimation -- 9.3.2.1 Forecast R-Sensitivity and Impact Estimation -- 9.3.2.2 Forecast B-Sensitivity and Impact Estimation -- 9.3.3 Forecast Sensitivity to Error Covariance Parameters. , 9.3.3.1 Sensitivity to Multiplicative Error Covariance Parameters -- 9.3.3.2 Sensitivity to the Observation Error Correlation Specification -- 9.3.3.3 Sensitivity to the Background Error Correlation Specification -- 9.3.4 The Adjoint Sensitivity Guidance: A Proof-of-Concept -- 9.4 Results with the Adjoint NAVDAS-AR/NOGAPS -- 9.5 Summary and Research Perspectives -- References -- Chapter 10 Treating Nonlinearities in Data-Space Variational Assimilation -- 10.1 Introduction -- 10.2 Background -- 10.3 Experiments -- 10.4 Posterior Statistics -- 10.5 Discussion -- References -- Chapter 11 Linearized Physics for Data Assimilation at ECMWF -- 11.1 Introduction -- 11.2 The Need for Physics in Variational Data Assimilation -- 11.3 Implication of the Linearity Constraint -- 11.3.1 Simplification -- 11.3.2 Regularization -- 11.4 Methodology for the Development of Linearized Simplified Parameterizations -- 11.4.1 Simplified Non-linear Version -- 11.4.2 Linearization Techniques -- 11.4.3 Tangent-Linear Version -- 11.4.4 Adjoint Version -- 11.4.5 Singular Vectors -- 11.5 ECMWF's Linearized Physics Package -- 11.5.1 Description -- 11.5.1.1 Radiation -- 11.5.1.2 Vertical Diffusion -- 11.5.1.3 Subgrid Scale Orographic Effects -- 11.5.1.4 Non-orographic Gravity Wave Drag -- 11.5.1.5 Moist Convection -- 11.5.1.6 Large-Scale Condensation and Precipitation -- 11.5.2 A Few Remarks -- 11.5.3 Benefits of Regularization -- 11.6 Performance of the Linearized Physics -- 11.6.1 TL Approximation -- 11.6.2 Adjoint Sensitivities -- 11.6.3 Data Assimilation -- 11.7 Conclusions and Prospects -- References -- Chapter 12 Recent Applications in Representer-Based Variational Data Assimilation -- 12.1 Introduction -- 12.2 Solver Improvements -- 12.3 Diagnosis of Error Variances -- 12.3.1 Notation and Background Materials -- 12.3.2 Validation of Error Variances by Posterior Diagnosis. , 12.3.3 Practical Implementation and Application to NAVDAS-AR -- 12.4 Summary -- References -- Chapter 13 Variational Data Assimilation for the Global Ocean -- 13.1 Introduction -- 13.2 Method -- 13.3 Error Covariances -- 13.3.1 Horizontal Correlations -- 13.3.2 Vertical Correlations -- 13.3.3 Multivariate Correlations -- 13.3.4 Background Error Variances -- 13.3.5 Observation Error Variances -- 13.4 Ocean Observations -- 13.4.1 Surface Observations -- 13.4.2 Profile Observations -- 13.4.3 Altimeter Sea Surface Height -- 13.5 NCODA System -- 13.5.1 Analysis Error Covariance -- 13.5.2 Adjoint -- 13.5.3 Ensemble Transformation -- 13.5.4 Residual Vector -- 13.5.5 Internal Data Checks -- 13.6 Global HYCOM -- 13.7 Future Capabilities -- 13.7.1 HYCOM GOFS -- 13.7.2 Satellite SST Radiance Assimilation -- 13.7.3 SSH Velocity Assimilation -- 13.7.4 Hybrid Ensemble Four Dimensional Data Assimilation -- 13.8 Summary -- References -- Chapter 14 A 4D-Var Analysis System for the California Current: A Prototype for an Operational Regional Ocean Data Assimilation System -- 14.1 Introduction -- 14.2 ROMS 4D-Var -- 14.2.1 Primal Versus Dual Formulation -- 14.2.2 Inner- and Outer-Loops -- 14.2.3 Conjugate Gradient Descent and Preconditioning -- 14.2.4 Covariance Models and Balance Operators -- 14.2.5 Background Quality Control Checks -- 14.3 Configuration of ROMS CCS and 4D-Var -- 14.4 CCS Historical Analyses -- 14.4.1 WCRA13 -- 14.4.2 WCRA31 -- 14.4.3 WCRA Observations -- 14.4.4 WCRA 4D-Var Configuration -- 14.4.5 Background Quality Control of Observations for WCRA -- 14.4.6 Preliminary Results -- 14.5 The CCS Near Real-Time Analysis and Forecast System -- 14.6 Summary -- References -- Chapter 15 A Weak Constraint 4D-Var Assimilation System for the Navy Coastal Ocean Model Using the Representer Method -- 15.1 Introduction -- 15.2 The Model -- 15.3 The 4D-Var System. , 15.3.1 Linearization.

Note: Intro -- Preface -- Contents -- Contributors -- Data Assimilation for Chaotic Dynamics -- 1 Introduction -- 2 Chaos in Atmospheric and Oceanic Flows -- 2.1 Measuring Sensitivity to Initial Conditions -- 3 Data Assimilation in Chaotic Systems-How the Dynamics Impacts the Way We Assimilate Data -- 3.1 Linear Dynamics: The Effect of Chaos on the Kalman Filter and Smoother -- 3.2 Nonlinear Dynamics: The Effect of Chaos on the Ensemble Kalman Filter -- 4 Data Assimilation for Chaotic Systems-How Chaos Becomes an Opportunity -- 4.1 Targeting Observations Using the Unstable Subspace -- 4.2 Assimilation in the Unstable Subspace -- 5 Forward Looking -- 5.1 AUS in a Non-Gaussian Filter? -- 5.2 Data Assimilation and Random Attractors -- 6 Summary and Conclusion -- References -- Multifidelity Data Assimilation for Physical Systems -- 1 Introduction -- 1.1 Notation -- 1.2 The Data Assimilation Problem -- 1.3 Multifidelity Models -- 2 Control Variates -- 2.1 Ensemble Control Variates -- 3 Multifidelity Filtering -- 3.1 Multifidelity Kalman Filter -- 3.2 Multifidelity Ensemble Kalman Filter -- 3.3 Other `Multi-x' Data Assimilation Algorithms -- 4 Multifidelity Observations -- 5 Numerical Experiments -- 6 Discussion -- References -- Filtering with One-Step-Ahead Smoothing for Efficient Data Assimilation -- 1 Introduction -- 2 Problem Formulation -- 3 One-Step-Ahead Smoothing (OSAS) Formulation of Bayesian Filtering -- 3.1 The Generic Algorithm -- 3.2 State-Space Transform -- 4 OSAS-Like Filtering for Small-Dimensional Systems -- 4.1 The OSAS-Based Kalman Filter (KF-OSAS) -- 4.2 The OSAS-Based Particle Filter (PF-OSAS) -- 5 OSAS-Like Filtering for Large-Dimensional Systems -- 5.1 The OSAS-Based Ensemble Kalman Filter (EnKF-OSAS) -- 5.2 State-Parameters Estimation with OSAS-Based Ensemble Filtering -- 6 Summary -- References. , Sparsity-Based Kalman Filters for Data Assimilation -- 1 Introduction -- 2 The Sparsity of Error Covariance -- 3 Sparse-UKF -- 3.1 Sparse Matrix Algebra -- 3.2 UKF -- 3.3 Sparse-UKF -- 3.4 Lorenz-96 Model -- 4 Progressive-EKF -- 4.1 Basic Ideas -- 4.2 Progressive-EKF -- 4.3 Examples -- 5 Conclusions -- References -- Perturbations by the Ensemble Transform -- 1 Introduction -- 2 Ensemble Perturbations and Ensemble Transform -- 3 Perturbations in LETKF in NWP Models -- 3.1 Cases of SPEEDY-LETKF -- 3.2 Case of NHM-LETKF -- 4 Perturbations by Ensemble Transform in a Cloud Resolving Model -- 4.1 2 km NHM-LETKF -- 4.2 Cycle Experiments and Verification -- 5 Summary and Concluding Remarks -- References -- Stochastic Representations for Model Uncertainty in the Ensemble Data Assimilation System -- 1 Introduction -- 2 Methodology -- 2.1 Local Ensemble Transform Kalman Filter (LETKF) -- 2.2 Numerical Weather Prediction (NWP) Model -- 2.3 Stochastic Perturbation Hybrid Tendencies (SPHT) Scheme -- 3 Experimental Designs -- 4 Results -- 5 Summary -- References -- Second-Order Methods in Variational Data Assimilation -- 1 Introduction -- 2 Variational Data Assimilation -- 3 Computing the Hessian -- 4 Parameter Estimation -- 5 Sensitivity Analysis -- 6 Sensitivity with Respect to Observations -- 7 Application for a Sea Thermodynamics Model -- 8 Conclusions -- References -- Statistical Parameter Estimation for Observation Error Modelling: Application to Meteor Radars -- 1 Introduction -- 1.1 Definitions, Sources, and Characteristics -- 1.2 Error Correlation -- 1.3 Operational Treatment -- 1.4 Outlook -- 2 Diagnosing Observation Error Including Error of Representation -- 2.1 Innovation Based Estimation Methods -- 2.2 Ensemble Methods -- 2.3 Representation Error -- 2.4 Sensitivity Diagnostics -- 2.5 Other Methods -- 2.6 Current Operational Practice. , 2.7 Inter-Channel Radiance Assimilation -- 2.8 Spatial Correlations -- 3 Practical Application: Application to Meteor Radar Assimilation -- 3.1 Meteor Radar Observations -- 3.2 Assimilation System -- 3.3 A First Look at Error Estimates -- 3.4 Differences By Station -- 3.5 Experiments with Inflated Ensemble Variance and Inflated Observation Error Variance -- 3.6 Observation Impact -- 3.7 Root-Mean-Squared Error (RMSE) -- 3.8 Temporal Correlation -- 4 Discussion and Conclusions -- References -- Observability Gramian and Its Role in the Placement of Observations in Dynamic Data Assimilation -- 1 Introduction -- 2 Notations and Statement of Problem -- 2.1 Model -- 2.2 Observations -- 2.3 Innovation/Forecast Error -- 2.4 Cost Functional -- 2.5 Statement of Problem -- 3 Dynamics of Evolution of Forward Sensitivities -- 4 Relation Between Adjoint Sensitivity and Initial Control Error: Linear Case -- 5 Relation Between Adjoint Sensitivity and Initial Control Error: Non Linear Case -- 6 Air-Sea Interaction Example -- 7 Conclusions -- References -- Placement of Observations for Variational Data Assimilation: Application to Burgers' Equation and Seiche Phenomenon -- 1 Introduction -- 2 Burgers' Equation -- 3 Data Assimilation Experiment with Burgers' Equation -- 4 Seiche Dynamics -- 4.1 Data Assimilation for Seiche -- 5 Conclusions -- References -- Analysis, Lateral Boundary, and Observation Impacts in a Limited Area Model -- 1 Forecast Sensivity to Observation Impact -- 2 COAMPS and NAVDAS -- 2.1 COAMPS Atmospheric Model -- 2.2 NAVDAS -- 2.3 COAMPS FSOI and Lateral Boundary Impacts -- 2.4 Forecast Domain -- 3 Forecast Error Reduction -- 4 Model Space Impacts -- 5 Observation Impacts -- 5.1 Radiosonde Verification -- 6 Summary -- References -- Assimilation of In-Situ Observations -- 1 Introduction -- 2 Radiosonde Observations -- 3 Surface Observations. , 4 Aircraft-Based Observations -- 5 Summary and Discussion -- Appendix 1: Definitions of Acronyms -- Appendix 2: Station Metadata Considerations -- References -- GNSS-RO Sounding in the Troposphere and Stratosphere -- 1 Fundamentals of the Radio Occultation Measurement -- 2 Typical Use of GNSS-RO in NWP -- 2.1 GNSS-RO Processing -- 3 Assimilation Methods and Error Statistic Assumptions -- 3.1 Forward Operators: H(x) -- 3.2 Error Statistic Assumptions -- 4 GNSS-RO Impact in NWP Systems -- 5 Future Directions for the Observation and Methods -- References -- Impact of Assimilating the Special Radiosonde Observations on COAMPS Arctic Forecasts During the Year of Polar Prediction -- 1 Introduction -- 2 Synoptic Features -- 3 Experimental Design -- 4 Discussion of Results -- 5 Summary and Conclusion -- References -- Images Assimilation: An Ocean Perspective -- 1 Introduction -- 2 Images Source and Processing: The Ocean Example -- 3 Methods for Image Assimilation and Their Limitations -- 3.1 Indirect Assimilation of Image -- 3.2 Direct Assimilation of Images -- 4 The Cost Function -- 5 Conclusion -- References -- Sensitivity Analysis in Ocean Acoustic Propagation -- 1 Introduction -- 2 The Model -- 3 Sensitivity Analysis -- 4 Numerical Experiments -- 5 Discussion and Summary -- Appendix: Equation of Sound Speed with Its Tangent Linear and Adjoint -- References -- Difficulty with Sea Surface Height Assimilation When Relying on an Unrepresentative Climatology -- 1 Introduction -- 2 Numerical Model: NCOM-4DVAR -- 2.1 Forward Model -- 2.2 Data Assimilation Configuration -- 3 Observations: SSHA and Glider Temperature and Salinity -- 3.1 SSHA Observations -- 3.2 Glider Data -- 3.3 Co-location of Assimilated Data -- 4 Experiment Setup and Results -- 4.1 Fit to Assimilated Data -- 4.2 Fit to Unassimilated Data. , 4.3 Direct Comparison of Experiment Results Against Glider Jade Temperature and Salinity -- 4.4 Comparison Against Recovered SSH -- 4.5 Comparison Against MODAS Profiles -- 5 Discussion -- 6 Summary -- References -- Theoretical and Practical Aspects of Strongly Coupled Aerosol-Atmosphere Data Assimilation -- 1 Introduction -- 1.1 Background on Coupled Data Assimilation System -- 1.2 Theoretical Description of Coupled Data Assimilation System -- 1.3 Single Observation Experiment -- 2 Current Status on Aerosol-Atmosphere Coupled Data Assimilation -- 2.1 Operational Centers and Research Community -- 2.2 Global Versus Regional Applications -- 3 Aerosol Observation and Forward Operator -- 3.1 Retrievals Versus Direct Measurements -- 3.2 AOD Observation Operator -- 3.3 AOD Error and Bias Estimation -- 4 Challenges -- 4.1 Choice of Control Variables -- 4.2 Background Error Covariance -- 4.3 Non-Gaussianity and Non-Linearity -- 4.4 Insufficient Data for Independent Verification -- 5 Experiments and Results -- 5.1 Case Study -- 5.2 Overview of the RAMS-MLEF System -- 5.3 Application of the RAMS-MLEF System -- 5.4 Synthetic Geostationary Satellite Imagery. -- 5.5 Model Response to Adjustments from Data Assimilation. -- 6 Summary and Future Directions -- References -- Improving Near-Surface Weather Forecasts with Strongly Coupled Land-Atmosphere Data Assimilation -- 1 Introduction -- 2 The Relationship Between Soil Moisture and Near-Surface Atmospheric Conditions -- 3 Strongly Coupled Versus Weakly Coupled Land-Atmosphere Data Assimilation -- 3.1 Characteristics of Background Error Covariance of Soil Moisture and Atmospheric States in Strongly Coupled Land-Atmosphere Data Assimilation -- 3.2 Soil Moisture Data Assimilation: Weakly Versus Strongly Coupled Data Assimilation. , 4 Enhanced Near-Surface Weather Forecasts Using Strongly Coupled Land-Atmosphere Data Assimilation.

Note: Intro -- Preface -- In Memory of Yoshi -- References -- Reminiscences on Dr. Yoshi Sasaki -- References -- Photos with Prof. Sasaki and JMA's Condolences to His Wife -- Yoshi's NRL Monterey Connection -- Reference -- Yoshi, My Mentor -- Contents -- Contributors -- Variational Data Assimilation: Optimization and Optimal Control -- 1 Introduction -- 1.1 Historical Perspective -- 1.2 Variational Methods in Meteorology: A Perspective -- 1.3 Variational Methods in Meteorology: The Optimization Theory View Point -- 2 Ingredients of a Variational Method -- 2.1 Definition of a Variational Method -- 3 Variational Analysis -- 4 Optimal Control Techniques -- 4.1 General Results -- 4.2 Control of the Initial Condition -- 4.3 Control of the Boundary -- 5 Weak Constraints in Variational Data Assimilation -- 5.1 Three Basic Methods in Constrained Optimization -- 5.2 Direct Control of the Error in VDA -- 5.3 Weak Constraint: Control of Systematic Error -- 5.4 Example: Saint-Venant's Equations -- 6 Second Order Methods -- 6.1 Sensitivity analysis -- 6.2 Sensitivity in the Presence of Data -- 7 Sensitivity with Respect to Sources -- 7.1 Identification of the Fields -- 7.2 Formulation of the Sensitivity Problem -- 8 Incremental Methods -- 8.1 Description of the Method -- 9 Developments in Variational Data Assimilation in Last 2 Decades -- 9.1 Estimation of Background and Observation Error Covariances -- 9.2 Observation Error Covariance -- 10 Hybrid Data Assimilation -- 11 Numerical Experiments -- 11.1 Burgers Model -- 11.2 Shallow Water Equations Model -- 12 Outlook of Modern Data Assimilation Topics -- 12.1 Data Assimilation Applied to Other Fields -- 12.2 Further Applications of Variational Data Assimilation -- References -- 2 Data Assimilation for Coupled Modeling Systems -- Abstract -- 1 Introduction -- 2 Motivation -- 3 Challenges -- 3.1 Control Variable. , 3.2 Forecast Error Covariance -- 3.3 High-Dimensional State Vector -- 3.4 Non-gaussian Errors -- 3.5 Spatiotemporal Scales -- 4 Two-Component Coupled System Data Assimilation -- 5 Structure of Coupled Forecast Error Covariance -- 6 Summary and Future -- Acknowledgements -- References -- 3 Representer-Based Variational Data Assimilation Systems: A Review -- Abstract -- 1 Introduction -- 2 Systems -- 2.1 IOM -- 2.1.1 Implementation -- 2.1.2 Applications -- 2.2 NAVDAS-AR -- 2.2.1 Implementation -- 2.2.2 Applications -- 2.3 NCOM 4D-Var -- 2.3.1 Implementation -- 2.3.2 Applications -- 2.4 ROMS 4D-Var -- 2.4.1 Implementation -- 2.4.2 Applications -- 3 Summary -- Acknowledgements -- References -- Adjoint-Free 4D Variational Data Assimilation into Regional Models -- 1 Introduction -- 2 Variational Data Assimilation -- 2.1 Adjoint Methods -- 2.2 Adjoint-Free Methods -- 3 a4dVar and 4dVar Assimilation of Real Data in the Adriatic Sea -- 3.1 Model and Data -- 3.2 Assimilation Parameters -- 3.3 Comparison with 4dVar -- 4 a4dVar Analysis of Simulated Wave Data in the Chukchi Sea -- 4.1 The WAM Model and Simulated Data -- 4.2 Comparison with Sequential Method -- 5 Summary and Discussion -- References -- 5 Convergence of a Class of Weak Solutions to the Strong Solution of a Linear Constrained Quadratic Minimization Problem: A Direct Proof Using Matrix Identities -- Abstract -- 1 Introduction -- 2 Proof of Convergence in (1.12) -- 3 Applications -- Acknowledgements -- Appendix A -- References -- Information Quantification for Data Assimilation -- 1 Introduction -- 2 Review of Observability -- 2.1 Comparison to Observation Sensitivity and Observation Impact -- 3 Partial Observability -- 4 Observability Optimization -- 4.1 Sensor Placement -- 4.2 Data Thinning -- 5 Final Remarks -- References. , 7 Quantification of Forecast Uncertainty and Data Assimilation Using Wiener's Polynomial Chaos Expansion -- Abstract -- 1 Introduction -- 2 PC Framework for Forecast Analysis -- 3 Examples -- 3.1 Experiment 3.1 A Linear Problem -- 3.2 Experiment 3.2 Nonlinear Problem -- 4 Data Assimilation Using PC and Ensemble Method -- 5 Conclusions -- Acknowledgements -- Appendix A -- Appendix B -- Hermite Polynomial Chaos -- Appendix C -- Appendix D -- References -- 8 The Treatment, Estimation, and Issues with Representation Error Modelling -- Abstract -- 1 Introduction -- 1.1 Definition of a "True State" -- 1.2 Modifications of the Kalman Filter Equations -- 1.3 Estimating Observation Error Covariance Matrices in an Operational Setting -- 2 Representation Error in Data Assimilation -- 3 Estimation of Representation Error -- 3.1 Desroziers' Method -- 3.2 HL Method -- 4 Issues with Incorrectly Specified Prior Error Variances -- 5 Summary and Conclusions -- References -- 9 Soil Moisture Data Assimilation -- Abstract -- 1 Background -- 2 Land Data Assimilation Systems for Soil Moisture Estimation -- 3 Data Assimilation Skill Evaluation -- 4 Root-Zone Soil Moisture Estimation -- 5 Uncertainty Characterization in the Precipitation Forcing -- 6 Multi-scale Soil Moisture Data Assimilation -- 7 Summary -- References -- 10 Surface Data Assimilation and Near-Surface Weather Prediction over Complex Terrain -- Abstract -- 1 Introduction -- 2 Characteristics of Errors in Near-Surface Atmospheric Conditions -- 3 Surface Data Assimilation: 3DVAR Versus EnKF -- 4 Results from the MATERHORN Field Program -- 4.1 Evaluation of Real-Time WRF Forecasts During MATERHORN -- 4.1.1 General Statistics and Diurnal Variations -- 4.1.2 Playa Versus Sagebrush -- 4.2 Surface Data Assimilation with EnKF -- 5 Concluding Remarks -- Acknowledgements -- References. , Recent Developments in Bottom Topography Mapping Using Inverse Methods -- 1 Introduction -- 2 Bottom Topography Estimation in Various Domains -- 2.1 Rivers -- 2.2 Estuaries -- 2.3 Nearshore/Surf Zone -- 3 Discussion -- 4 Summary -- References -- 12 The Impact of Doppler Wind Lidar Measurements on High-Impact Weather Forecasting: Regional OSSE and Data Assimilation Studies -- Abstract -- 1 Introduction -- 2 Overview of the Impact of Ground-Based and Airborne DWL Wind Profiles on High-Impact Weather Forecasts -- 2.1 Ground-Based DWL Wind Profiles -- 2.2 Airborne DWL Wind Profiles -- 3 The Impact of Satellite-Based Wind Profiles on Hurricane Forecasts: Results from OSSEs with the WRF ARW Model -- 3.1 Brief Overview of the Regional OSSE Concept and Early Studies -- 3.2 The Impact of Resolution and Errors in DWL Wind Measurements on the Numerical Prediction of a Tropical Cyclone -- 4 Recent OSSE Results with the HWRF Model: 3-D Wind Profiles Versus Ocean-Surface Winds -- 5 Summary and Concluding Remarks -- Acknowledgements -- References -- 13 A Three-Dimensional Variational Radar Data Assimilation Scheme Developed for Convective Scale NWP -- Abstract -- 1 Introduction -- 2 Description of Data Assimilation Method -- 3 Several Examples of Idealized and Real Data Case Studies -- 4 Summary and Future Work -- Acknowledgements -- References -- 14 Data Assimilation Experiments of Refractivity Observed by JMA Operational Radar -- Abstract -- 1 Introduction -- 2 An Estimation Method of Temporal Variations of Refractivity -- 3 Temporal Variations of Refractivity on August 4th 2008 -- 4 Data Assimilation Methods and Impact of Refractivity -- 5 Summary -- Acknowledgements -- References -- 15 Assessment of Radiative Effect of Hydrometeors in Rapid Radiative Transfer Model in Support of Satellite Cloud and Precipitation Microwave Data Assimilation -- Abstract. , 1 Introduction -- 2 The Process of the Radiative Effect of Hydrometeors in the RRTM -- 3 Data and Methods -- 3.1 Satellite Observation -- 3.2 Case Description -- 3.3 Numerical Forecast Model -- 3.4 Inputs to the RRTM -- 4 Analysis of the Simulation of Cloudy and Rainy Satellite Microwave Observations -- 4.1 Influence of Hydrometeors on the Satellite Microwave Simulated Brightness Temperature -- 4.2 Deviation and Root-Mean-Squared Error -- 4.3 Contribution Ratio -- 5 Sensitivity of the Satellite Simulation to Hydrometeors' Properties -- 5.1 Sensitivity to Water Content -- 5.2 Sensitivity to Particle Size -- 5.3 Sensitivity to the Vertical Distribution of Hydrometeors -- 6 Inter-Comparison Between RTTOV and CRTM -- 6.1 Simulated Satellite Brightness Temperature -- 6.2 Deviation and RMSE -- 6.3 Response Function of Hydrometeors -- 7 Summary and Discussion -- Acknowledgements -- References -- Toward New Applications of the Adjoint Sensitivity Tools in Data Assimilation -- 1 Introduction -- 2 A Posteriori Observation Error Covariance Diagnosis -- 3 Forecast Sensitivity to Observation Error Covariance (FSR) -- 3.1 Estimation of the Forecast Impact of the Model widetildeR -- 3.2 Proof-of-Concept with Lorenz Model -- 4 Results with NAVDAS-AR/NAVGEM -- 4.1 Estimates of the Observational Error Standard Deviation -- 4.2 Estimates of the Inter-Channel Error Correlations -- 4.3 The FSR Guidance and Forecast Error Impact Estimates -- 5 Summary and Research Perspectives -- References -- 17 GPS PWV Assimilation with the JMA Nonhydrostatic 4DVAR and Cloud Resolving Ensemble Forecast for the 2008 August Tokyo Metropolitan Area Local Heavy Rainfalls -- Abstract -- 1 Introduction -- 2 Tokyo Metropolitan Area Local Heavy Rainfalls on 5 August 2008 -- 3 Forecast from Operational Mesoscale Analysis of JMA -- 3.1 Downscale Forecast -- 3.2 Ensemble Forecast. , 4 Analysis and Assimilation of GPS PWV.