Exact nonlinear state estimation

TL;DR

This paper introduces a new nonlinear estimation theory, the Conjugate Transform Filter (CTF), which generalizes the Kalman filter to non-Gaussian distributions, improving data assimilation accuracy in geosciences.

Contribution

It develops the CTF and its ensemble approximation ECTF, bridging the gap between traditional Gaussian-based methods and non-parametric AI-inspired approaches.

Findings

ECTF performs well with small observation errors and strong nonlinear dependencies.

ECTF preserves statistical relationships better than Gaussian-based filters.

Experimental results show ECTF's superior accuracy in non-Gaussian, nonlinear scenarios.

Abstract

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.