An Ensemble Smoother with Error Estimates

A smoother introduced earlier by van Leeuwen and Evensen is applied to a problem in which real observations are used in an area with strongly nonlinear dynamics. The derivation is new, but it resembles an earlier derivation by van Leeuwen and Evensen. Again a Bayesian view is taken in which the prior probability density of the model and the probability density of the observations are combined to form a posterior density. The mean and the covariance of this density give the variance-minimizing model evolution and its errors. The assumption is made that the prior probability density is a Gaussian, leading to a linear update equation. Critical evaluation shows when the assumption is justified. This also sheds light on why Kalman filters, in which the same approximation is made, work for nonlinear models. By reference to the derivation, the impact of model and observational biases on the equations is discussed, and it is shown that Bayes’s formulation can still be used. A practical advantage of the ensemble smoother is that no adjoint equations have to be integrated and that error estimates are easily obtained. The present application shows that for process studies a smoother will give superior results compared to a filter, not only owing to the smooth transitions at observation points, but also because the origin of features can be followed back in time. Also its preference over a strong-constraint method is highlighted. Furthermore, it is argued that the proposed smoother is more efficient than gradient descent methods or than the representer method when error estimates are taken into account.

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