List of participants .......................................... xix
Part I Key lectures ............................................ 1
1 4D-VAR: four-dimensional variational assimilation
O. TALAGRAND ................................................. 3
1.1 Introduction ............................................ 5
1.2 Variational assimilation in the context of statistical
linear estimation ....................................... 5
1.3 Minimization methods. The adjoint approach ............. 12
1.4 Practical implementation ............................... 17
1.5 Further considerations on variational assimilation ..... 19
1.6 More on the adjoint method ............................. 23
1.7 Conclusions ............................................ 25
References .................................................. 26
2 Four-dimensional variational data assimilation
A.C. LORENC ................................................. 31
2.1 4D-Var: background and motivation ...................... 33
2.2 4D-Var: derivation ..................................... 47
2.3 4D-Var: advanced aspects ............................... 52
2.4 4D-Var. coupling with ensembles ........................ 63
References .................................................. 68
3 Introduction to the Kalman filter
C. SNYDER ................................................... 75
3.1 A Bayesian view of data assimilation ................... 77
3.2 The Kalman filter (I) .................................. 84
3.3 A closer look at the forecast and update steps ......... 89
3.4 The Kalman filter (II) ................................. 94
3.5 Assorted topics ........................................ 96
3.6 Nonlinearity and non-Gaussianity ...................... 103
3.7 Basics of the ensemble Kalman filter .................. 106
3.8 Assorted derivations and identities ................... 118
References ................................................. 119
4 Smoothers
E. COSME ................................................... 121
4.1 Introduction .......................................... 123
4.2 Smoothing algorithms in a Bayesian framework .......... 124
4.3 Linear Gaussian smoothers ............................. 127
4.4 Ensemble smoothers .................................... 131
4.5 Advantages, drawbacks, and high-dimensional
applications .......................................... 133
References ................................................. 135
5 Observation influence diagnostic of a data assimilation
system
C. CARDINALI ............................................... 137
5.1 Introduction .......................................... 139
5.2 Classical statistical definitions of influence matrix
and self-sensitivity ................................. 141
5.3 Observational influence and self-sensitivity
for a DA scheme ....................................... 143
5.4 Results ............................................... 148
5.5 Conclusions ........................................... 156
Acknowledgements ........................................... 158
Appendix 1: Influence matrix calculation in weighted
regression DA scheme ....................................... 158
Appendix 2: Approximate calculation of self-sensitivity in
a large variational analysis system ........................ 160
References ................................................. 162
6 Observation impact on the short-range forecast
C. CARDINALI ............................................... 165
6.1 Introduction .......................................... 167
6.2 Observational impact on the forecast .................. 168
6.3 Results ............................................... 173
6.4 Conclusion ............................................ 178
Acknowledgements ........................................... 179
References ................................................. 180
Part II Specialized lectures ................................. 183
7 Background error covariances: estimation and specification
L. BERRE ................................................... 185
7.1 Error equations and their simulation .................. 187
7.2 Innovation-based estimations .......................... 192
7.3 Diagnosis of background error covariances ............. 195
7.4 Modelling and filtering covariances ................... 199
7.5 Conclusions ........................................... 206
References ................................................. 206
8 Observation error specifications
G. DESROZIERS .............................................. 209
8.1 General framework ..................................... 211
8.2 Methods for estimating observation error statistics ... 212
8.3 Diagnosis of observation error variances .............. 219
8.4 Diagnosis of observation error correlations ........... 219
8.5 Observation error correlation specification in the
assimilation .......................................... 221
8.6 Conclusion ............................................ 226
References ................................................. 227
9 Brrors. A posteriori diagnostics
O. TALAGRAND ............................................... 229
9.1 Introduction .......................................... 231
9.2 Reminder on statistical linear estimation ............. 231
9.3 Objective evaluation of assimilation algorithms ....... 235
9.4 Estimation of the statistics of data errors ........... 237
9.5 Diagnostics of internal consistency ................... 238
9.6 Diagnostics of optimality of assimilation
algorithms ............................................ 250
9.7 Conclusions ........................................... 252
Acknowledgements ........................................... 253
References ................................................. 253
10 Error dynamics in ensemble Kalman-filter systems:
localization
P. HOUTEKAMER ............................................. 255
10.1 Motivation ............................................ 257
10.2 Estimation of scalars and matrices .................... 257
10.3 Assimilation of one observation ....................... 258
10.4 Experiments with the Lorenz III model ................. 260
10.5 Discussion ............................................ 264
References ................................................. 264
11 Short-range error statistics in an ensemble Kalman
filter
P. HOUTEKAMER .............................................. 267
11.1 Introduction .......................................... 269
11.2 Experimental environment .............................. 270
11.3 Horizontal correlations ............................... 271
11.4 vertical correlations ................................. 273
11.5 Temporal correlations ................................. 275
11.6 Stratospheric wind analysis ........................... 276
11.7 Discussion ............................................ 277
References ................................................. 278
12 Error dynamics in ensemble Kalman filter systems: system
error
P. HOUTEKAMER .............................................. 279
12.1 Introduction .......................................... 281
12.2 Monte Carlo methods ................................... 281
12.3 Review of model error ................................. 282
12.4 Review of data-assimilation error ..................... 284
12.5 Evidence of bias ...................................... 285
12.6 Evidence of horizontal error correlations ............. 286
12.7 Discussion ............................................ 287
References ................................................. 288
13 Particle filters for the geosciences
P.J. VAN LEEUWEN ........................................... 291
13.1 Introduction .......................................... 293
13.2 A simple particle filter based on importance
sampling .............................................. 294
13.3 Reducing the variance in the weights .................. 298
13.4 The proposal density .................................. 300
13.5 Conclusions ........................................... 316
References ................................................. 318
14 Second-order methods for error propagation in variational
data assimilation
F.-X. LE DIMET, I. GEJADZE, and V. SHUTYAEV ................ 319
14.1 Introduction .......................................... 321
14.2 Variational methods ................................... 322
14.3 Second-order methods .................................. 325
14.4 Sensitivity with respect to sources ................... 329
14.5 Stochastic methods .................................... 334
14.6 Covariances of the optimal solution error ............. 335
14.7 Effective inverse Hessian (EIH) method ................ 338
14.8 Numerical examples .................................... 341
14.9 Conclusions ........................................... 346
Acknowledgements ........................................... 347
References ................................................. 347
15 Adjoints by automatic differentiation
L. HASCOËT ................................................. 349
15.1 Introduction .......................................... 351
15.2 Elements of AD ........................................ 351
15.3 Application of adjoint AD to data assimilation ........ 359
15.4 Improving the adjoint AD code ......................... 362
15.5 AD tools .............................................. 364
15.6 Conclusion ............................................ 366
References ................................................. 368
16 Assimilation of images
A. VIDARD, O. TITAUD ....................................... 371
16.1 Motivations ........................................... 373
16.2 Images: level(s) of interpretation .................... 375
16.3 Current use of images in data assimilation: pseudo
observation ........................................... 377
16.4 Direct assimilation of images ......................... 382
References ................................................. 391
17 Multigrid algorithms and local mesh refinement methods
in the context of variational data assimilation
L. DEBREU, E. NEVEU, E. SIMON, and F.-X. LE DIMET .......... 395
17.1 Structure of the variational data assimilation
problem ............................................... 397
17.2 Multigrid methods and application to variational
data assimilation ..................................... 400
17.3 Data assimilation and local mesh refinement ........... 405
17.4 Coupling the two approaches ........................... 409
17.5 Conclusions and perspectives .......................... 410
References ................................................. 411
18 Selected topics in multiscale data assimilation
M. BOCQUET, L. WU, F. CHEVALLIER, and M.R. KOOKHAN ......... 413
18.1 Introduction .......................................... 415
18.2 Bayesian multiscale analysis .......................... 415
18.3 Application to Bayesian control space design .......... 422
18.4 Empirical multiscale statistics ....................... 427
18.5 Conclusion ............................................ 430
References ................................................. 431
19 Data assimilation in meteorology
F. RABIER and M. FISHER .................................... 433
19.1 Transforming data ..................................... 435
19.2 Comparing data and models ............................. 439
19.3 Thinning the dataset .................................. 444
19.4 Filtering the analysis ................................ 447
19.5 Nonlinearities and non-Gaussian densities in
variational data assimilation ......................... 449
19.6 Parallel algorithms for 4D-Var ........................ 453
19.7 Conclusion ............................................ 456
References ................................................. 457
20 An introduction to inverse modelling and parameter
estimation for atmosphere and ocean sciences
M. BOCQUET ................................................. 461
20.1 Introduction .......................................... 463
20.2 Bayesian approach to inverse problems ................. 463
20.3 Alternative approaches ................................ 473
20.4 Estimation of second-order statistics ................. 479
20.5 Inverse modelling in atmospheric and ocean
sciences: a selection ................................. 489
20.6 Conclusion ............................................ 493
Acknowledgements ........................................... 493
References ................................................. 493
21 Greenhouse gas flux inversion
F. CHEVALLIER .............................................. 497
21.1 Introduction .......................................... 499
21.2 Observations .......................................... 499
21.3 Uncertainties ......................................... 500
21.4 Methods ............................................... 501
21.5 Conclusion ............................................ 502
References ................................................. 503
22 Data assimilation in atmospheric chemistry and air
quality
H. ELBERN, E. FRIESE, L. NIERADZIK, and J. SCHWINGER ....... 507
22.1 Introduction .......................................... 509
22.2 Advanced chemistry data assimilation .................. 513
22.3 A posteriori validation in atmospheric chemistry ...... 518
22.4 Tropospheric chemical data assimilation ............... 520
22.5 Aerosol data assimilation ............................. 523
References ................................................. 528
23 Combining models and data in large-scale oceanography:
examples from the consortium for Estimating the
Circulation and Climate of the Ocean (ECCO)
I. FUKUMORI ................................................ 535
23.1 Introduction .......................................... 537
23.2 Physical consistency .................................. 537
23.3 ECCO products ......................................... 539
23.4 Examples of ECCO applications ......................... 540
23.5 Practical considerations in employing advanced
estimation methods .................................... 544
23.6 Summary ............................................... 550
Acknowledgements ........................................... 551
References ................................................. 551
24 Data assimilation in coastal oceanography: IS4DVAR
in the Regional Ocean Modelling System (ROMS)
J. ZAVALA-GARAY, J. WILKIN, and J. LEVIN ................... 555
24.1 The Regional Ocean Modelling System and the IS4DVAR
data assimilation algorithm ........................... 557
24.2 ROMS IS4DVAR in a quasi-geostrophic domain:
the East Australia Current ............................ 560
24.3 ROMS IS4DVAR in a complex coastal domain:
the Middle Atlantic Bight ............................. 564
References ................................................. 573
25 Data assimilation in glaciology
B. BONAN, M. NODET, O. OZENDA, and C. RITZ ................. 577
25.1 Introduction .......................................... 579
25.2 Ice-sheet model ....................................... 579
25.3 Adjoint method and adjoint model ...................... 581
25.4 Numerical results for twin experiments ................ 582
References ................................................. 584
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