  In a recent methodological paper, we showed how to learn chaotic dynamics
along with the state trajectory from sequentially acquired observations, using
local ensemble Kalman filters. Here, we more systematically investigate the
possibility to use a local ensemble Kalman filter with either covariance
localisation or local domains, in order to retrieve the state and a mix of key
global and local parameters. Global parameters are meant to represent the
surrogate dynamical core, for instance through a neural network, which is
reminiscent of data-driven machine learning of dynamics, while the local
parameters typically stand for the forcings of the model. Aiming at joint state
and parameter estimation, a family of algorithms for covariance and local
domain localisation is proposed. In particular, we show how to rigorously
update global parameters using a local domain ensemble Kalman filter (EnKF)
such as the local ensemble transform Kalman filter (LETKF), an inherently local
method. The approach is tested with success on the 40-variable Lorenz model
using several of the local EnKF flavors. A two-dimensional illustration based
on a multi-layer Lorenz model is finally provided. It uses radiance-like
non-local observations. It features both local domains and covariance
localisation in order to learn the chaotic dynamics and the local forcings.
This paper more generally addresses the key question of online estimation of
both global and local model parameters.
