Preface
1 Introduction through historical perspective

1.1 From Gauss to Kolmogorov

1.2 Approaching the meteorological system

1.3 Numerical Weather Prediction models

1.4 What, Where, When

2 Representation of the physical system

2.1 The observational system and errors

2.2 Variational approach: 3-D VAR and 4-D VAR

2.3 Assimilation as an inverse problem

3 Sequential interpolation

3.1 An effective introduction of a Kalman Filter

3.2 More Kalman Filters

4 Advanced data assimilation methods
4.1 Recursive Bayesian Estimation
4.2 Ensemble Kalman Filter
4.3 Issues due to small ensembles
4.4 Methods to reduce problems of undersampling

5 Applications
5.1 Lorenz model
5.2 Biology and Medicine
5.3 Mars data assimilation: the General Circulation Model
5.4 Earthquake forecast

A Appendix
A.1 Hadamard product
A.2 Differential calculus
A.3 The method of characteristics
A.4 Calculus of variations
A.5 The solution for the simplified equation of oceanographic

This book endeavours to give a concise contribution to understanding the data assimilation and related methodologies. The mathematical concepts and related algorithms are fully presented, especially for those facing this theme for the first time. 

The first chapter gives a wide overview of the data assimilation steps starting from Gauss' first methods to the most recent as those developed under the Monte Carlo methods.  The second chapter treats the representation of the physical  system as an ontological basis of the problem. The third chapter deals with the classical Kalman filter, while the fourth chapter deals with the advanced methods based on  recursive Bayesian Estimation. A special chapter, the fifth, deals with the possible applications, from the first Lorenz model, passing trough the biology and medicine up to planetary assimilation, mainly on Mars.

This book serves both teachers and college students, and other interested parties providing the algorithms and formulas to manage the data assimilation everywhere a dynamic system is present.