Preface -- Part 1. Data Assimilation -- Chapter 1. Methods for Assimilation of Observations: Application to Numerical Weather Prediction -- 1.1. Introduction -- 1.2. The linear and Gaussian case -- 1.2.1. Variational form -- 1.3. Optimal interpolation -- three-dimensional variational assimilation -- 1.4. Taking the dynamics of the flow into account -- 1.4.1. The Kalman Filter -- 1.4.2. Four-dimensional variational assimilation -- 1.4.3. Ensemble methods -- 1.4.4. Stability and instability -- 1.5. Particle filters -- 1.6. Artificial intelligence -- 1.7. Extensions and applications -- 1.8. References -- Chapter 2. Ensemble Data Assimilation in Volcanology -- 2.1. Volcano monitoring and eruption forecasting -- 2.2. Ensemble data assimilation -- 2.2.1. Volcanic data assimilation using the ensemble Kalman filter -- 2.2.2. The dynamic model -- 2.2.3. Data observations -- 2.3. Potentiality assessment of volcanic data assimilation for eruption forecasting based on synthetic simulations -- 2.3.1. EnKF formulation -- 2.3.2. Synthetic observations -- 2.3.3. Experiment setup