Implicit particle filtering for models with partial noise, and an application to geomagnetic data assimilation

TL;DR

This paper introduces an improved implicit particle filtering method tailored for models with partial noise, demonstrating its efficiency and accuracy through geomagnetic data assimilation.

Contribution

It develops a new approach combining gradient descent with random maps to apply implicit particle filtering to high-dimensional models with partial noise.

Findings

Efficiently handles high-dimensional models with partial noise.

Accurately assimilates geomagnetic data using the new filter.

Operates in a reduced subspace, improving computational feasibility.

Abstract

Implicit particle filtering is a sequential Monte Carlo method for data assim- ilation, designed to keep the number of particles manageable by focussing attention on regions of large probability. These regions are found by min- imizing, for each particle, a scalar function F of the state variables. Some previous implementations of the implicit filter rely on finding the Hessians of these functions. The calculation of the Hessians can be cumbersome if the state dimension is large or if the underlying physics are such that derivatives of F are difficult to calculate. This is the case in many geophysical applica- tions, in particular for models with partial noise, i.e. with a singular state covariance matrix. Examples of models with partial noise include stochastic partial differential equations driven by spatially smooth noise processes and models for which uncertain dynamic equations are supplemented by con- servation laws with zero uncertainty. We make the implicit particle filter applicable to such situations by combining gradient descent minimization with random maps and show that the filter is efficient, accurate and reliable because it operates in a subspace whose dimension is smaller than the state dimension. As an example, we assimilate data for a system of nonlinear partial differential equations that appears in models of geomagnetism.