  During the last few years discontinuous Galerkin (DG) methods have received
increased interest from the geophysical community. In these methods the
solution in each grid cell is approximated as a linear combination of basis
functions. Ensemble data assimilation (EnDA) aims to approximate the true state
by combining model outputs with observations using error statistics estimated
from an ensemble of model runs. Ensemble data assimilation in geophysical
models faces several well-documented issues. In this work we exploit the
expansion of the solution in DG basis functions to address some of these
issues. Specifically, it is investigated whether a DA-DG combination 1)
mitigates the need for observation thinning, 2) reduces errors in the field's
gradients, 3) can be used to set up scale-dependent localisation. Numerical
experiments are carried out using stochastically generated ensembles of model
states, with different noise properties, and with Legendre polynomials as basis
functions. It is found that strong reduction in the analysis error is achieved
by using DA-DG and that the benefit increases with increasing DG order. This is
especially the case when small scales dominate the background error. The DA
improvement in the 1st derivative is on the other hand marginal. We think this
to be a counter-effect of the power of DG to fit the observations closely,
which can deteriorate the estimates of the derivatives. Applying optimal
localisation to the different polynomial orders, thus exploiting their
different spatial length, is beneficial: it results in a covariance matrix
closer to the true covariance than the matrix obtained using traditional
optimal localisation in state-space.
