The majority of data assimilation (DA) methods in the geosciences are based on Gaussian assumptions. While such approximations facilitate efficient algorithms, they cause analysis biases and subsequent forecast degradations. Nonparametric, 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 fields of measure transport and generative artificial intelligence, this paper develops a new estimation-theoretic framework which can incorporate general invertible transformations in a principled way. Specifically, a conjugate transform filter (CTF) is derived and shown to extend the celebrated Kalman filter to a much broader class of non-Gaussian distributions. The new filter has several desirable properties, such as its ability to preserve statistical relationships in the prior state and converge to highly accurate observations. An ensemble approximation of the new filtering framework is also presented and validated through idealized examples. The numerical demonstrations feature bounded quantities with non-Gaussian distributions, which is a typical challenge in Earth system models. Results suggest that the greatest benefits from the new filtering framework occur when the observation errors are small relative to the forecast uncertainty and when state variables exhibit strong nonlinear dependencies.
