Long-window tandem variational data assimilation methods for chaotic climate models tested with the Lorenz 63 system

We apply 4D variational data assimilation to the Lorenz 63 model to introduce a new method for parameter estimation in chaotic climate models. The approach aims to optimise an Earth system model (ESM) for which no adjoint exists by utilising the adjoint of a different, potentially simpler ESM. This relies on the synchronisation of the model to observed data. Dynamical state and parameter estimation (DSPE) is used to stabilise the tangent linear system by reducing all positive Lyapunov exponents to negative values, thereby improving parameter estimation by enabling long assimilation windows. The method introduces a second layer of synchronisation between the two models, with and without an adjoint, to facilitate linearisation around the trajectory of the model for which no adjoint exists. This is achieved by synchronising two Lorenz 63 systems, one with and the other without an adjoint model. Results are presented for an idealised case of identical, perfect models and for a more realistic case in which they differ from one another. If employed in a high-resolution ESM for which a coarse-resolution adjoint exists, the method will save computational resources as only one forward run with the full high-resolution ESM per iteration is needed. It is demonstrated that there is negligible error and uncertainty change compared to the traditional optimisation of a full ESM with an adjoint. Stemming from this approach, it is shown that the synchronisation between two identical models can be used to filter noisy data in a dynamical way which reduces the parametric uncertainty of the optimised model by approximately one-third. Such a precision gain could prove to be valuable for seasonal, annual, and decadal predictions.