Hybrid 4DVAR and nonlinear ensemble Kalman smoother methods

It is known that 4DVAR and the Kalman smoother are equivalent in the linear case with gaussian distributions, and that the Gauss-Newton method for 4DVAR as a nonlinear least squares problem requires solving linear least squares problems which are the same as in the Kalman smoother. While the least squares character of 4DVAR implies the gaussian assumption for the data and the model errors, 4DVAR is more suitable for nonlinear problems than the Kalman smoother because the observation and the model operators are nonlinear and thus the state distribution may not be gaussian.

We present a variant of 4DVAR with the linear solver by the ensemble Kalman smoother (EnKS) for the linearized (tangent) problem. However, a common form of the EnKS works with nonlinear operators directly and it does not require any tangent operators. We apply this EnKS to a secant linearization of 4DVAR, which gives a new combined method that implements 4DVAR efficiently in parallel, without requiring any tangent or adjoint operators, and thus can be applied to a model as a black box. We also propose another combination of EnKS and 4DVAR, a new EnKS method with nonlinear correction by 4DVAR and model reduction on the subspace generated by the ensemble. Suitability of the methods will be assessed on applications.