Preserving Structure in Variational Data Assimilation of Hyperbolic Conservation Laws via Regularization
Hyperbolic conservation laws pose a challenging setting for variational data assimilation, since the Gaussian assumption imposes a smoothness that smears jump discontinuities. A sparsity promoting regularization can help to mitigate this problem by incorporating a structural prior into the variational objective. Care must be taken when choosing this prior because it serves to counterbalance both a structurally misspecified background-error covariance, which introduces spurious oscillations over smooth regions, as well as spatially sparse observations, which leave too little data to constrain the state. Moreover, state variables of hyperbolic conservation laws typically do not have piecewise constant structure, which is an assumption made when using standard sparsity-promoting operators, such as total variation. Indeed the underlying {\em variability} of hyperbolic conservation law solutions are neither known in advance nor fixed. Hence using higher order total variation is also not suitable. Here we introduce a new regularization term for data assimilation, built on the sparsity promoting residual transform that neither commits to a fixed smoothness order nor requires the underlying variability to be known a priori. We solve the resulting nonconvex objective within a three-dimensional variational framework using generalized sparse Bayesian learning. This approach replaces a global \(\ell _{1}\) penalty with a hierarchical conditional Gaussian prior on the residual transform, where a learned hyper-prior yields location-specific weights attached to each spatial point. Numerical experiments across scalar, shallow-water, and Euler test problems show that this regularization term resolves discontinuities more accurately than total variation, and that its advantage grows as the background-error covariance becomes misspecified and the observations become sparser.