**1. Introduction**

In this chapter, we will talk about the modelling and simulation using both observed data and numerical models, that is, the observations will be incorporated into numerical models for optimal modelling and simulation. In statistics, this is called state-space estimation. In the earth science, it is called data assimilation. For example, a strict definition of data assimilation in atmospheric and oceanic sciences is the process to estimate the state of a dynamic system such as atmospheric and oceanic flow by combining the observational and model forecast data [1].

In general, assimilation methods can be classified into two categories: variational and sequential. This chapter is a tutorial on the sequential data assimilation methods such as ensemble Kalman filter (EnKF) and its variants. A brief introduction of the particle filter (PF) is also provided in this chapter.

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This tutorial places emphasis on the rationale behind each method, including: (i) the principle for deriving the algorithm; (ii) the basic assumptions of each method; (iii) the connection and relation between different methods (e.g. extended Kalman filter (EKF) and EnKF, EnKF and sigma-point Kalman filters (SPKF), etc.); and (iv)the advantage and deficiency of each method.

This chapter has been written and organized through teaching for under-/graduatestudents in earth science courses. It can also be a good reference to researchers in the field of modelling and data assimilation.
