Tailoring data assimilation to discontinuous Galerkin models

TL;DR

This paper explores how discontinuous Galerkin methods can enhance ensemble data assimilation in geophysical models by reducing errors and enabling scale-dependent localization, especially at higher polynomial orders.

Contribution

It demonstrates that integrating DA with DG basis functions significantly reduces analysis errors and improves localization, particularly for small-scale features.

Findings

DA-DG reduces analysis errors, especially at higher DG orders.

Using DG basis functions improves scale-dependent localization.

Optimal localization with polynomial order-specific covariance matrices enhances data assimilation.

Abstract

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.