However, solutions from these models do not necessarily agree with observations. There are many reasons why this is the case, including the fact that data cannot be collected at all points in space and time, so we do not have accurate initial and boundary conditions. Even if we could observe our world at all points in space and time, these data would contain measurement errors, and there may not be a direct relationship between the data and variables in the models.
On the other hand, when it comes to model solutions, even if we have good numerical solutions, the underlying models are typically not accurate due to incomplete physics or unknown parameters. Data assimilation is often used to address these issues. I will describe a weak-constraint, 4 dimensional variational data assimilation system. This system can be viewed from control theory or a statistical perspective, and is a global method that is made efficient with representer solutions which search a data subspace to find optimal solutions. This general theory can be used for parameter estimation, data interpolation, and model error approximation. I will show some results from it applied to Lagrangian ocean dynamics.