1. An Overview
2. Data Mining, Data assimilation and prediction
3. A classification of forecast errors
4. Finite Dimensional Vector Space
5. Matrices
6. Matrices Continued
7. Multi-variate Calculus
8. Optimization in Finite Dimensional Vector spaces
9. Deterministic, Static, linear Inverse (well-posed) Problems
10. Deterministic, Static, Linear Inverse (Ill-posed) Problems
11. A Geometric View - Projections
12. Deterministic, Static, nonlinear Inverse Problems
13. On-line Least Squares
14. Examples of static inverse problems
15. Interlude and a Way Forward
16. Matrix Decomposition Algorithms
17. Matrix Decomposition Algorithms Continued
18. Minimization algorithms
19. Minimization algorithms Continued
20. Inverse problems in deterministic
21. Inverse problems in deterministic Continued
22. Forward sensitivity method
23. Relation between FSM and 4DVAR
24. Statistical Estimation
25. Statistical Least Squares
26. Maximum Likelihood Method
27. Bayesian Estimation
28. From Gauss to Kalman-Linear Minimum Variance Estimation
29. Initialization Classical Method
30. Optimal interpolations
31. A Bayesian Formation-3D-VAR methods
32. Linear Stochastic Dynamics - Kalman Filter
33. Linear Stochastic Dynamics - Kalman Filter Continued
34. Linear Stochastic Dynamics - Kalman Filter Continued.
35. Covariance Square Root Filter
36. Nonlinear Filtering
37. Ensemble Reduced Rank Filter
38. Basic nudging methods
39. Deterministic predictability
40. Predictability: A stochastic view and summary

Our aim is to provide a broad based background on the mathematical principles and tools from linear algebra, multivariate calculus and finite dimensional optimization theory, estimation theory, non-linear dynamics and chaos that constitute the basis for dynamic data assimilation as we know today. Our aim is to present the ideas at the level of a first year graduate/final year undergraduate student aspiring to enter this exciting area.