| Preface | p. ix |
| Partial list of symbols | p. xi |
| Introduction | p. 1 |
| Atmospheric characteristics | p. 2 |
| The ultimate problem in meteorology | p. 6 |
| The observing system | p. 8 |
| Subjective analysis | p. 17 |
| Objective analysis: first attempts | p. 19 |
| The data assimilation cycle | p. 24 |
| Spatial analysis | p. 29 |
| Function Fitting | p. 32 |
| Local polynomial fitting | p. 32 |
| Least squares estimation | p. 34 |
| The Gram matrix | p. 41 |
| Underfitting, overfitting, and other problems | p. 45 |
| The a posteriori analysis weights | p. 49 |
| Multivariate function fitting | p. 52 |
| Constrained minimization: penalty functions | p. 58 |
| Exercises | p. 62 |
| The Method of Successive Corrections | p. 64 |
| Basic formulation for a single correction | p. 65 |
| The iteration cycle | p. 69 |
| Spectral response: infinite continuous networks | p. 71 |
| Spectral response: finite discrete networks | p. 78 |
| Convergence properties | p. 86 |
| Barnes' algorithm | p. 90 |
| Iteration to the optimal solution | p. 93 |
| Exercises | p. 96 |
| Statistical Interpolation: Univariate | p. 98 |
| Minimum variance estimation | p. 99 |
| The statistical interpolation algorithm | p. 101 |
| Background error covariances and correlations | p. 107 |
| A continuous analogue | p. 118 |
| Filtering and interpolation properties | p. 125 |
| One, two, and multiple observation problems | p. 131 |
| The vertical question | p. 136 |
| Correlated observation error | p. 137 |
| Misspecification of the background and observation error covariances | p. 143 |
| Exercises | p. 149 |
| Statistical Interpolation: Multivariate | p. 150 |
| The multivariate algorithm | p. 150 |
| Statistics of a homogeneous two-dimensional windfield | p. 155 |
| The geopotential/wind covariances | p. 164 |
| Multivariate filtering properties | p. 169 |
| Multivariate versus univariate analysis | p. 175 |
| A generalized algorithm | p. 180 |
| Exercises | p. 184 |
| The Initialization Problem | p. 186 |
| Characterization of the synoptic and planetary scale motion in the atmosphere | p. 186 |
| Filtered models and initialized primitive equation models | p. 188 |
| The benefits of initialization | p. 190 |
| A linearized shallow water model | p. 194 |
| Initialization of the linearized shallow water model | p. 200 |
| Geostrophic adjustment | p. 202 |
| A brief history of initialization | p. 208 |
| Exercises | p. 210 |
| Quasi-Geostrophic Constraints | p. 212 |
| The Hinkelmann-Phillips model | p. 212 |
| Initialization of the Hinkelmann-Phillips model | p. 215 |
| Scale analysis of the primitive equations | p. 219 |
| Quasi-geostrophic initialization of the primitive equations | p. 223 |
| The linear balance equation | p. 227 |
| The nonlinear balance equation | p. 230 |
| The quasi-geostrophic omega equation | p. 236 |
| The quasi-geostrophic potential vorticity equation | p. 238 |
| The limitations of quasi-geostrophic initialization | p. 240 |
| Exercises | p. 241 |
| Variational Procedures | p. 242 |
| The stationary value of a function | p. 243 |
| The stationary value of a definite integral | p. 246 |
| Application to atmospheric analysis problems | p. 251 |
| Weak constraint formulations | p. 255 |
| Inequality constraints | p. 257 |
| Four-dimensional variational analysis: Thompson's scheme | p. 258 |
| Exercises | p. 261 |
| Normal Mode Initialization: Theory | p. 263 |
| The linearized baroclinic primitive equations | p. 263 |
| The vertical structure equation | p. 267 |
| The horizontal structure equation | p. 270 |
| Horizontal structure functions (Hough modes) | p. 274 |
| Normal mode form of the model equations | p. 281 |
| The Machenhauer balance condition | p. 284 |
| The Baer-Tribbia scheme | p. 288 |
| Exercises | p. 290 |
| Normal Mode Initialization: Applications | p. 292 |
| Some results from normal mode initialization experiments | p. 292 |
| Separation of slow and fast time scales | p. 297 |
| The slow manifold | p. 301 |
| Normal mode and quasi-geostrophic initialization | p. 306 |
| Variational normal mode initialization | p. 308 |
| Initialization for limited area models | p. 314 |
| Convergence properties and diabatic initialization | p. 316 |
| Exercises | p. 321 |
| Dynamic Initialization | p. 322 |
| Damping time integration procedures | p. 322 |
| Application to simple models | p. 326 |
| Experiments with sophisticated models | p. 329 |
| Exercises | p. 332 |
| Continuous Data Assimilation | p. 333 |
| The basic philosophy | p. 333 |
| Damping time integration schemes | p. 338 |
| Identical twin configurations: convergence of solutions | p. 339 |
| The effect of nonlinearities | p. 345 |
| The effect of observation error | p. 348 |
| Real data experiments: the rejection problem | p. 353 |
| The geostrophic wind correction | p. 355 |
| Dynamic relaxation | p. 357 |
| Data assimilation on the slow manifold | p. 360 |
| Exercises | p. 362 |
| Future Directions | p. 363 |
| Descent methods | p. 364 |
| Four-dimensional variational analysis | p. 369 |
| The Kalman-Bucy filter | p. 376 |
| Initialization by Laplace transform | p. 384 |
| The bounded derivative initialization method | p. 390 |
| Initialization of the hydrological cycle | p. 391 |
| Mesoscale data assimilation | p. 394 |
| Data assimilation in the oceans | p. 399 |
| New challenges | p. 402 |
| Exercises | p. 404 |
| Appendices | p. 407 |
| The normal modes of the spherical horizontal structure equations | p. 407 |
| The effect of temporally correlated observation error on data assimilation | p. 410 |
| Four-dimensional variational analysis with a nonlinear model | p. 413 |
| Some statistical concepts | p. 416 |
| Classical interpolation | p. 421 |
| An iterative approach to statistical interpolation | p. 425 |
| Bessel functions and Hankel transforms | p. 428 |
| Aliasing, network design, and Observation System Simulation Experiments (OSSE) | p. 432 |
| References | p. 439 |
| Index | p. 455 |
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