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This book addresses the problem of inferring the state of ocean circulation, understanding it dynamically, and forecasting it through a quantitative combination of theory and observation. It focuses on so-called inverse methods and related methods of statistical inference. The author considers both time-independent and time-dependent problems, including Gauss-Markov estimation, sequential estimators and adjoint/Pontryagin principle methods. This book is intended for use as a graduate level text for students of oceanography and related fields. It will also be of interest to working physical oceanographers.