  Ensemble transform Kalman filtering (ETKF) data assimilation is often used to
combine available observations with numerical simulations to obtain
statistically accurate and reliable state representations in dynamical systems.
However, it is well known that the commonly used Gaussian distribution
assumption introduces biases for state variables that admit discontinuous
profiles, which are prevalent in nonlinear partial differential equations. This
investigation designs a new structurally informed non-Gaussian prior that
exploits statistical information from the simulated state variables. In
particular, we construct a new weighting matrix based on the second moment of
the gradient information of the state variable to replace the prior covariance
matrix used for model/data compromise in the ETKF data assimilation framework.
We further adapt our weighting matrix to include information in discontinuity
regions via a clustering technique. Our numerical experiments demonstrate that
this new approach yields more accurate estimates than those obtained using ETKF
on shallow water equations, even when ETKF is enhanced with inflation and
localization techniques.
