## Dynamic Data Assimilation: A Least Squares Approach, Volume 13Dynamic data assimilation is the assessment, combination and synthesis of observational data, scientific laws and mathematical models to determine the state of a complex physical system, for instance as a preliminary step in making predictions about the system's behaviour. The topic has assumed increasing importance in fields such as numerical weather prediction where conscientious efforts are being made to extend the term of reliable weather forecasts beyond the few days that are presently feasible. This book is designed to be a basic one-stop reference for graduate students and researchers. It is based on graduate courses taught over a decade to mathematicians, scientists, and engineers, and its modular structure accommodates the various audience requirements. Thus Part I is a broad introduction to the history, development and philosophy of data assimilation, illustrated by examples; Part II considers the classical, static approaches, both linear and nonlinear; and Part III describes computational techniques. Parts IV to VII are concerned with how statistical and dynamic ideas can be incorporated into the classical framework. Key themes covered here include estimation theory, stochastic and dynamic models, and sequential filtering. The final part addresses the predictability of dynamical systems. Chapters end with a section that provides pointers to the literature, and a set of exercises with instructive hints. |

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### Contents

Brief history of data assimilation | 81 |

method of normal equations | 99 |

projection and invariance | 121 |

Nonlinear least squares estimation | 133 |

Recursive least squares estimation | 141 |

Matrix methods | 149 |

steepest descent method | 169 |

Conjugate directiongradient methods | 190 |

the straight line problem | 365 |

linear dynamics | 382 |

nonlinear dynamics | 401 |

Secondorder adjoint method | 422 |

a statistical and a recursive view | 445 |

Kalman ﬁlter | 463 |

part II | 485 |

Nonlinear ﬁltering | 509 |

Newton and quasiNewton methods | 209 |

Principles of statistical estimation | 227 |

Statistical least squares estimation | 240 |

Maximum likelihood method | 254 |

Bayesian estimation method | 261 |

sequential linear minimum | 271 |

concepts and formulation | 285 |

Classical algorithms for data assimilation | 300 |

a Bayesian formulation | 322 |

Spatial digital ﬁlters | 340 |

Reducedrank ﬁlters | 534 |

a stochastic view | 563 |

a deterministic view | 581 |

Epilogue | 628 |

Preface page | xiii |

Appendix A Finitedimensional vector space | xviii |

Acknowledgements | xxi |

636 | |

642 | |

### Common terms and phrases

adjoint method algorithm analysis Appendix approximation assumed called Chapter components compute condition number conjugate gradient conjugate gradient method constraints convergence covariance matrix data assimilation data assimilation problem define deﬁned denote density derivation descent direction described deterministic diagonal diagonal matrix dynamical system eigenvalues ensemble equation error example Exercise f(xk ﬁeld variable Figure ﬁlter ﬁnd ﬁrst ﬁrst-order follows forecast function given grid point Hence Hessian initial condition inverse iterative Kalman ﬁlter known least squares estimate linear least squares linear system M(xk minimization Newton’s noise nonlinear obtain optimal orthogonal matrix parameter perturbation positive deﬁnite matrix prediction properties quadratic quadratic form quasi-Newton methods random Recall recursive refer relation Rn×n scalar Section sequence solution solving square root Step Substituting symmetric and positive symmetric matrix Tikhonov regularization trajectory unbiased vector veriﬁed Verify xk+1 zero