Data assimilation : making sense of observations

Cover Page\r......Page 1
F\rront Matter......Page 2
Contents......Page 6
Contributors......Page 10
Introduction......Page 14
Part I Theory......Page 16
2 Need for Information......Page 17
3 Sources of Information......Page 18
4 Characteristics of Information......Page 19
5 Objective Ways of Filling in Information Gaps......Page 20
6 Simple Examples of Data Assimilation......Page 21
7 Benefits of Combining Information......Page 25
References......Page 26
1 Introduction......Page 27
2.1 Basic Least-Squares Formulation for Perfect Models......Page 28
2.2 Properties of the Basic Least-Squares Formulation......Page 30
2.3 Best Linear Least-Squares Estimate......Page 31
2.4 Statistical Interpretation......Page 32
3 Sequential Data Assimilation Schemes......Page 34
3.1 Optimal Sequential Assimilation Scheme......Page 35
3.2 Practical Implementation......Page 37
3.3 Ensemble Filters and Sampling Methods......Page 38
4.1 4D-Var and the Adjoint Method......Page 39
4.2 Incremental Variational Methods......Page 41
4.3 Control Variable Transforms......Page 42
4.4 Model Reduction......Page 43
5.1 Least-Squares Formulation for Models with Errors......Page 44
5.2 Optimal Solution of the Assimilation Problem......Page 46
5.3 Systematic Model Error and State Augmentation......Page 47
5.4 Data Assimilation for Parameter Estimation......Page 49
References......Page 50
1 Introduction......Page 54
2 Variational Assimilation in the Context of Statistical Linear Estimation......Page 55
3.1 Gradient Methods for Minimization......Page 62
3.2 The Adjoint Method......Page 63
4 Practical Implementation......Page 67
4.1 The Incremental Approach......Page 68
4.2 First-Guess-At-the-Right-Time 3D-Var......Page 69
5 Further Considerations on Variational Assimilation......Page 70
6 More on the Adjoint Method......Page 74
7 Conclusion......Page 76
References......Page 77
1 Introduction......Page 81
2 Brief Review of Ensemble Kalman Filtering......Page 82
LETKF Algorithm......Page 86
3 Adaptation of 4D-Var Techniques into EnKF......Page 87
3.1 4D-LETKF and No-Cost Smoother......Page 88
3.2 Application of the No-Cost Smoother to the Acceleration of the Spin-Up......Page 89
3.3 \'\'Outer Loop\'\' and Dealing with Non-linear Ensemble Perturbations......Page 92
3.4 Adjoint Forecast Sensitivity to Observations Without Adjoint Model......Page 94
3.5 Use of a Lower Resolution Analysis......Page 96
3.6 Model and Observational Error......Page 98
4 Summary and Discussion......Page 99
References......Page 101
1 Introduction......Page 105
1.2 Importance of Background and Observation Error Statistics in Data Assimilation......Page 106
2.1 Estimation of Background and Observation Error Statistics from Innovations......Page 107
2.3 Estimation of Background Error Covariances with Monte Carlo Approaches......Page 109
2.4 Other Approaches for the Estimation of Background Error Covariances......Page 110
2.5 Estimation of Observation-Error Correlations......Page 111
3.1 Spectral Representation: Homogeneous and Isotropic Error Correlations......Page 113
3.2 Physical-Space Representation......Page 114
3.4 Theoretically-Based Correlation Modelling......Page 115
4.1 Estimated Error Variances......Page 117
4.2 Single Observation Experiments......Page 118
5 Summary......Page 121
References......Page 122
1 Introduction......Page 125
2.1 Bias Detection Using Innovations......Page 126
2.2 Bias Detection Using Analysis Increments......Page 127
3 Bias Analysis......Page 129
3.1 Variational Formulation......Page 130
3.2 Sequential Formulation......Page 132
4.1 Static Bias Correction Scheme......Page 134
4.3 Adaptive On-Line Bias Correction Scheme or Variational Correction Scheme......Page 136
5.1 Static Schemes......Page 137
5.2 Dynamical Schemes......Page 139
6 Conclusions......Page 144
References......Page 145
1 Introduction......Page 148
1.1 Applications......Page 149
1.2 Theory......Page 152
2.1 The Principle of Energetic Consistency......Page 153
2.2 Minimum Variance State Estimation......Page 156
2.3 Discretization......Page 161
2.4.1 General Formulation......Page 164
2.4.2 Ensemble Behaviour Between Observation Times......Page 166
2.4.3 Ensemble Behaviour at Observation Times......Page 167
2.4.4 Ensemble Behaviour for Dissipative Models......Page 169
3.1 Problem Setting......Page 174
3.2 Scalar and Hilbert Space-Valued Random Variables......Page 175
3.3 The Principle of Energetic Consistency in Hilbert Space......Page 178
3.4 A Natural Restriction on S......Page 179
4.1 Ordinary Differential Equations......Page 181
4.2.1 The Deterministic Initial-Value Problem......Page 184
4.2.2 The Solution Operator......Page 188
4.2.3 The Stochastic Initial-Value Problem......Page 189
5 The Shallow-Water Equations......Page 192
6 Concluding Remarks......Page 195
1a H-Valued Random Variables......Page 197
1b Second-Order H-Valued Random Variables......Page 199
1c Properties of Second-Order H-Valued Random Variables......Page 200
1d Construction of Second-Order H-Valued Random Variables......Page 204
2a Construction of the Hilbert Spaces......Page 209
2b The Case......Page 212
3a Measure Spaces......Page 217
3b Integration......Page 218
3c Probability......Page 221
3d Hilbert Space......Page 223
References......Page 226
1 Introduction......Page 228
2 Reminder on Statistical Linear Estimation......Page 229
3 Objective Evaluation of Assimilation Algorithms......Page 233
4 Estimation of the Statistics of Data Errors......Page 235
5 Diagnostics of Internal Consistency......Page 236
6 Diagnostics of Optimality of Assimilation Algorithms......Page 248
7 Conclusions......Page 249
References......Page 250
1 Introduction......Page 252
2.2 Static Initialization......Page 254
2.4 Variational Initialization......Page 255
3.1 The Laplace Tidal Equations......Page 256
3.2 Vorticity and Divergence......Page 257
3.3 Rossby-Haurwitz Modes......Page 258
3.4 Gravity Wave Modes......Page 259
4 Normal Mode Initialization......Page 260
5 Digital Filter Initialization......Page 261
5.1 Design of Non-recursive Filters......Page 262
5.2 Application of a Non-recursive Digital Filter to Initialization......Page 264
5.3 Initialization Example......Page 265
6 Constraints in 4D-Var......Page 267
7 Conclusion......Page 269
References......Page 270
Part II Observations......Page 272
2 In Situ Observations......Page 273
2.1 Surface and Marine Observations......Page 274
2.2 Radiosondes......Page 275
2.3 Aircraft Observations......Page 277
2.4 Targeted Observing......Page 278
3 Remote Sensing Observations......Page 279
3.1 Passive Technologies......Page 281
3.1.1 Atmospheric Sounding Channels from Passive Instruments......Page 282
3.1.2 Surface Sensing Channels from Passive Instruments......Page 283
3.2.1 Surface Instruments......Page 284
3.3 Limb Technologies......Page 285
3.3.2 GPS Technologies......Page 286
4.1 Development of the In Situ Component of the GOS......Page 287
4.2 Development of the Space Component of the GOS......Page 288
5 Concluding Remarks......Page 289
References......Page 290
2 Assimilation of Radiance Observations......Page 292
2.1 Constraints on the Inversion of Radiance Data......Page 293
2.2 Non-linear Dependence on the Background Humidity Field......Page 294
2.3 Temperature/Humidity Partitioning of Radiance Increments......Page 295
3 Assimilation of Hourly Surface Pressure Measurements......Page 297
3.1 Synoptic Analysis of Rapidly Developing Storm Using Surface-Pressure Data from a Single Station......Page 298
3.2 Background Errors in Observable Quantities......Page 299
4.1 Probability Density Function of Observation Error......Page 300
4.2 Variational Quality Control......Page 301
5 Impact of Observations on the Quality of Numerical Forecasts......Page 303
6 Final Remarks......Page 305
References......Page 306
2 Observations......Page 309
3.1 General Considerations......Page 310
3.2 Research Satellites......Page 312
3.3 Benefits of Research Satellites......Page 319
3.4 Research Satellites and the Global Climate Observing System......Page 320
3.5 Capacity Report for Satellite Missions......Page 321
3.5.1 Capabilities......Page 322
3.5.2 Limitations......Page 323
4 Data Assimilation of Research Satellites......Page 324
References......Page 327
2.1 General Details......Page 330
2.2 Influence of Rotation......Page 333
3.1 The Thermally-Driven Circulation in the Tropics......Page 337
3.2 Angular Momentum Balance......Page 338
3.3 Rossby Waves and Mid Latitude Systems......Page 339
3.4 The Extra-Tropical Meridional Circulation......Page 342
3.5 Other Tropical Circulations......Page 343
4.1 Introduction to the Middle Atmosphere......Page 345
4.2 Winter and Summer Stratosphere......Page 346
4.3 Humidity......Page 349
4.4 Ozone......Page 351
5 Conclusions......Page 352
References......Page 353
Part III Meteorology and Atmospheric Dynamics......Page 355
2 Definition and Description of the Model......Page 356
3 The Role of the Model in Data Assimilation......Page 360
4 Component Structure of an Atmospheric Model......Page 365
5 Consideration of the Observation-Model Interface......Page 372
6 Physical Consistency and Data Assimilation......Page 375
Example 1: Observational Correction to the Thermodynamic Equation......Page 377
Example 2: Horizontal Divergence and the Vertical Wind......Page 379
7 Summary......Page 382
References......Page 383
1 Introduction......Page 385
2.1 Operational Observing System......Page 386
2.2 Quality Control......Page 387
3.1 Introduction......Page 389
3.2 Variational Methods......Page 390
3.3 Assimilation of Satellite Soundings......Page 393
3.4 Ensemble Assimilation Methods......Page 394
4.1 Development of Numerical Models......Page 396
4.2 Model Configurations......Page 397
5.1 Benefits of Ensemble Forecasts......Page 398
5.2 Initial Condition Perturbations......Page 399
5.3 Accounting for Model Errors......Page 400
6.1 Weather Forecasts......Page 402
6.2 Site-Specific Information......Page 403
6.3 Probabilistic Forecasts......Page 404
6.4 Warnings of High-Impact Weather......Page 405
6.5 Improving the Prediction of High-Impact Weather......Page 407
References......Page 408
Part IV Atmospheric Chemistry......Page 411
1 Importance of Chemistry......Page 412
2 Atmospheric Processes Affecting the Composition......Page 413
2.1 Elementary Chemical Processes......Page 414
2.2 Stratospheric Chemistry......Page 416
2.3 Tropospheric Chemistry......Page 417
2.4 Surface Emissions......Page 420
2.5 Transport of Chemicals from Sources in the PBL and Convection......Page 423
2.6 Circulation and Transport......Page 424
2.6.1 Tropospheric Circulation and Mixing......Page 425
2.6.2 Stratospheric Circulation and Mixing......Page 427
2.6.3 Transport and Chemistry Across the Tropopause......Page 429
References......Page 432
1 Introduction......Page 434
2 Linear Formalism for Error Evolution in Box Chemical Models......Page 435
3 Variance Evolution and Applications to Measurement Information Content......Page 440
4 Error Representation in 3-D Chemistry-Transport Models......Page 445
5 Discussion......Page 448
References......Page 450
1 Introduction......Page 452
2.1 Introduction......Page 457
2.2 Assimilation of Humidity......Page 458
2.3 Assimilation of Ozone......Page 459
3 Chemical Model Approaches......Page 465
4 Evaluation of Models, Observations and Analyses......Page 470
5.1 Tropospheric Pollution......Page 475
5.2 Analyses of Constituents......Page 476
5.3 Stratospheric Ozone Monitoring......Page 481
5.4 Ozone Forecasting......Page 483
6 Future Directions......Page 484
References......Page 486
1.1 General Remarks......Page 494
1.2 Features of Tropospheric Chemical Data Assimilation......Page 495
1.3 Observations......Page 497
2.1 Tropospheric Gas Phase Data Assimilation......Page 498
2.2 Tropospheric Aerosol Data Assimilation......Page 501
3.1 Kalman Filter Equations......Page 502
3.2 Ensemble Kalman Filter......Page 503
3.3 Reduced Rank Square Root Kalman Filter......Page 504
3.4 4D Variational Data Assimilation......Page 505
4 Examples......Page 507
4.1 Nested Application of 4D-Var......Page 508
4.2 Emission Rate Estimates......Page 509
4.3 Tropospheric Satellite Data Assimilation......Page 511
4.4 Aerosol Assimilation......Page 512
5 Outlook......Page 513
References......Page 514
Part V Wider Applications......Page 517
1 Introduction to the Ocean Circulation......Page 518
2 Ocean Modelling Methods......Page 520
3 Observational Ocean Data......Page 523
4 Ocean Data Assimilation: Applications and Current Issues......Page 529
5.1 General Considerations......Page 532
5.2 Physical Relationships Between Variables......Page 534
6 In Situ Temperature and Salinity Assimilation......Page 541
References......Page 544
1 Introduction......Page 549
2 Background: Land Surface Observations......Page 550
3 Background: Land Surface Modelling......Page 552
4 History of Land Surface Data Assimilation......Page 553
4.1 Early Land Surface State Estimation Studies......Page 555
4.2 Data Assimilation Beyond State Estimation......Page 556
5 General Concept of Land Surface Data Assimilation......Page 557
5.1 Direct Observer Assimilation......Page 558
5.2 Dynamic Observer Assimilation......Page 559
5.3 Features of Data Assimilation......Page 560
5.4 Quality Control for Data Assimilation......Page 561
5.5 Validation Using Data Assimilation......Page 562
6.1 Land Surface System......Page 563
6.2 Direct Observer Data Assimilation......Page 565
6.3 Dynamic Observer Assimilation Methods......Page 571
7 Assimilation of Land Surface Observations......Page 572
7.1.2 Statistical Correction, Nudging, Optimal Interpolation......Page 573
7.1.3 Kalman Filter......Page 574
7.1.4 3D/4D-Var......Page 578
7.2 Soil Temperature Observations......Page 579
7.4 Land Surface Flux Observations......Page 580
7.7 Snow Water Equivalent/Snow Cover Observations......Page 581
7.8 Ground Water Storage Observations......Page 582
8.1 Case Study 1: Soil Moisture Assimilation......Page 583
8.2 Case Study 2: Streamflow Assimilation......Page 584
8.3 Case Study 3: Snow Assimilation......Page 586
8.4 Case Study 4: Skin Temperature Assimilation......Page 588
9 Summary......Page 589
References......Page 590
1 Introduction......Page 598
2 Background......Page 599
3.1 Elementary Processes......Page 600
3.2 Transport and Solar Effects......Page 603
4 Modelling the Ionosphere......Page 605
5 GPS Data......Page 608
6 Ionospheric Data Assimilation......Page 609
7 Impact of Ionosphere on Telecommunications, Scintillations......Page 611
8 Application to Single-Frequency GPS Positioning......Page 613
9 Future Directions......Page 616
References......Page 617
Part VI The Longer View......Page 619
1 Introduction......Page 620
2 Special Aspects of the Reanalysis Problem......Page 622
3 Lessons from Applications......Page 630
4 Summary......Page 639
References......Page 640
1 Definition and Motivation of OSSEs......Page 644
2 Historical Summary of OSSEs......Page 647
3.1 Characteristics of the Nature Run......Page 650
3.2 Evaluation and Potential Adjustment of the Nature Run......Page 651
3.3 Requirements for a Future Nature Run......Page 653
4 Assignment of Realistic Observation Errors......Page 654
5.1 Basic Guidelines......Page 657
5.2 Specific Issues Related to Different Observational Types......Page 659
5.2.1 Simulation of Conventional Observations......Page 660
5.2.2 Simulation of Radiance Data......Page 661
5.2.3 Simulation of Doppler Wind Lidar (DWL) Data......Page 662
6.1 Initial Conditions......Page 663
6.2 Spin-Up Period......Page 664
7.1 Data Denial (or Adding) Experiments (DDEs)......Page 665
7.2 Adjoint--Based Techniques......Page 666
8 Calibration of OSSEs......Page 667
9.2 Calibration Performed for NCEP OSSE......Page 668
9.3 Evaluation of DWL Impact Using the NCEP OSSE......Page 670
10 Summary and Concluding Remarks for OSSEs......Page 671
References......Page 673
1 Introduction......Page 677
2 Motivation for the Assimilation of Extra-Terrestrial Data......Page 678
3 Data Assimilation for the Atmosphere of Mars......Page 679
3.1 Data Assimilation Schemes for Mars......Page 682
3.2 Results from Martian Data Assimilation......Page 684
4 Future Prospects for Other Planets......Page 690
5 Implications for Terrestrial Data Assimilation......Page 691
References......Page 692
List of Acronyms......Page 696
Index......Page 700