Dynamic Data Assimilation

Overview

Introduction:Data Mining, Data Assimilation, Inverse problems and Prediction - Static vs. dynamic and deterministic vs. stochastic problems- formulation & classification;Mathematical tools:Finite dimensional vector space basic concepts - Overview of properties and operations on matrices - Special classes of matrices, Eigen decomposition, and matrix square root - Gradient, Jacobian, Hessian, Quadratic forms and their properties;Static, deterministic models: least Squares method formulation and properties - Linear least squares (LLS) - over determined case, weighted and unweighted formulation, orthogonal and oblique projections - LLS- Underdetermined case -Lagrangian multiplier - Nonlinear least squares problem (NLS) - formulation - Approximation first and second-order methods for solving NLS - Examples of LLS and NLS - satellite retrieval

Matrix methods solving LLS:Normal equations symmetric positive definite (SPD) systems multiplicative matrix decomposition - Cholesky decomposition- matrix square root - Gramm-Schmidt orthogonalization process - QR decomposition - Singular value decomposition (SVD) - Solution of retrieval problems;Direct minimization methods for solving LLS:LLS as a quadratic minimization problem - Gradient method, its properties - Convergence and speed of convergence of gradient method - Conjugate gradient and Quasi-Newton methods - Practice problems and programming exercises;Deterministic, dynamic models:adjoint method - Dynamic models, role of observations, and least squares objective function, estimation of initial condition (IC) and parameters, adjoint sensitivity - A straight line problem a warm up - Linear model, first-order adjoint dynamics and computation of the gradient of the least squares objective function - Nonlinear model and first-order adjoint dynamics - Illustrative examples and practice problems, programming exercises

Deterministic, Dynamic models:Other methods - Forward sensitivity method for estimation of IC and parameters, forward sensitivity dynamics - Example of Carbon dynamics,Relation between adjoint and forward sensitivity, Predictability, Lyapunov index - Method of nudging and overview of nudging methods;Static, stochastic models:Bayesian framework - Bayesian method linear, Gaussian case - Linear minimum variance estimation (LMVE) and prelude to Kalman filter - Model space vs. observation space formulation -Duality between Bayesian and LMVE;Dynamic, Stochastic models:Kalman filtering - Derivation of the Kalman filter equations - Derivation of Nonlinear filter - Computational requirements - Ensemble Kalman filtering;Dynamic stochastic models:Other methods - Unscented Kalman filtering - Particle filtering - An overview and assessment of methods