  The majority of data assimilation (DA) methods in the geosciences are based
on Gaussian assumptions. While these assumptions facilitate efficient
algorithms, they cause analysis biases and subsequent forecast degradations.
Non-parametric, particle-based DA algorithms have superior accuracy, but their
application to high-dimensional models still poses operational challenges.
Drawing inspiration from recent advances in the field of generative artificial
intelligence (AI), this article introduces a new nonlinear estimation theory
which attempts to bridge the existing gap in DA methodology. Specifically, a
Conjugate Transform Filter (CTF) is derived and shown to generalize the
celebrated Kalman filter to arbitrarily non-Gaussian distributions. The new
filter has several desirable properties, such as its ability to preserve
statistical relationships in the prior state and convergence to highly accurate
observations. An ensemble approximation of the new theory (ECTF) is also
presented and validated using idealized statistical experiments that feature
bounded quantities with non-Gaussian distributions, a prevalent challenge in
Earth system models. Results from these experiments indicate that the greatest
benefits from ECTF occur when observation errors are small relative to the
forecast uncertainty and when state variables exhibit strong nonlinear
dependencies. Ultimately, the new filtering theory offers exciting avenues for
improving conventional DA algorithms through their principled integration with
AI techniques.
