On the Mathematical Theory of Ensemble (Linear-Gaussian) Kalman-Bucy Filtering

The purpose of this review is to present a comprehensive overview of the theory of ensemble Kalman-Bucy filtering for linear-Gaussian signal models. We present a system of equations that describe the flow of individual particles and the flow of the sample covariance and the sample mean in continuous-time ensemble filtering. We consider these equations and their characteristics in a number of popular ensemble Kalman filtering variants. Given these equations, we study their asymptotic convergence to the optimal Bayesian filter. We also study in detail some non-asymptotic time-uniform fluctuation, stability, and contraction results on the sample covariance and sample mean (or sample error track). We focus on testable signal/observation model conditions, and we accommodate fully unstable (latent) signal models. We discuss the relevance and importance of these results in characterising the filter's behaviour, e.g. it's signal tracking performance, and we contrast these results with those in classical studies of stability in Kalman-Bucy filtering. We provide intuition for how these results extend to nonlinear signal models and comment on their consequence on some typical filter behaviours seen in practice, e.g. catastrophic divergence

Thermophysical modelling and parameter estimation of small solar system bodies via data assimilation

Deriving thermophysical properties such as thermal inertia from thermal infrared observations provides useful insights into the structure of the surface material on planetary bodies. The estimation of these properties is usually done by fitting temperature variations calculated by thermophysical models to infrared observations. For multiple free model parameters, traditional methods such as Least-Squares fitting or Markov-Chain Monte-Carlo methods become computationally too expensive. Consequently, the simultaneous estimation of several thermophysical parameters together with their corresponding uncertainties and correlations is often not computationally feasible and the analysis is usually reduced to fitting one or two parameters. Data assimilation methods have been shown to be robust while sufficiently accurate and computationally affordable even for a large number of parameters. This paper will introduce a standard sequential data assimilation method, the Ensemble Square Root Filter, to thermophysical modelling of asteroid surfaces. This method is used to re-analyse infrared observations of the MARA instrument, which measured the diurnal temperature variation of a single boulder on the surface of near-Earth asteroid (162173) Ryugu. The thermal inertia is estimated to be $295 \pm 18$ $\mathrm{J\,m^{-2}\,K^{-1}\,s^{-1/2}}$, while all five free parameters of the initial analysis are varied and estimated simultaneously. Based on this thermal inertia estimate the thermal conductivity of the boulder is estimated to be between 0.07 and 0.12 $\mathrm{W\,m^{-1}\,K^{-1}}$ and the porosity to be between 0.30 and 0.52. For the first time in thermophysical parameter derivation, correlations and uncertainties of all free model parameters are incorporated in the estimation procedure which is more than 5000 times more efficient than a comparable parameter sweep

Data assimilation in the solar wind: challenges and first results

Data Assimilation (DA) is used extensively in numerical weather prediction (NWP) to improve forecast skill. Indeed, improvements in forecast skill in NWP models over the past 30 years have directly coincided with improvements in DA schemes. At present, due to data availability and technical challenges, DA is underused in space weather applications, particularly for solar wind prediction. This paper investigates the potential of advanced DA methods currently used in operational NWP centres to improve solar wind prediction. To develop the technical capability, as well as quantify the potential benefit, twin experiments are conducted to assess the performance of the Local Ensemble Transform Kalman Filter (LETKF) in the solar wind model ENLIL. Boundary conditions are provided by the Wang-Sheeley-Arge coronal model and synthetic observations of density, temperature and momentum generated every 4.5hr at 0.6AU. While in-situ spacecraft observations are unlikely to be routinely available at 0.6AU, these techniques can be applied to remote sensing of the solar wind, such as with Heliospheric Imagers or Interplanetary Scintillation. The LETKF can be seen to improve the state at the observation location and advect that improvement towards the Earth, leading to an improvement in forecast skill in near Earth space for both the observed and unobserved variables. However, sharp gradients caused by the analysis of a single observation in space resulted in artificial wave-like structures being advected towards Earth. This paper is the first attempt to apply DA to solar wind prediction, and provides the first in-depth analysis of the challenges and potential solutions

Ensemble Kalman Methods: A Mean Field Perspective

This paper provides a unifying mean field based framework for the derivation and analysis of ensemble Kalman methods. Both state estimation and parameter estimation problems are considered, and formulations in both discrete and continuous time are employed. For state estimation problems both the control and filtering approaches are studied; analogously, for parameter estimation (inverse) problems the optimization and Bayesian perspectives are both studied. The approach taken unifies a wide-ranging literature in the field, provides a framework for analysis of ensemble Kalman methods, and suggests open problems

Uncertainty Quantification

Uncertainty quantification (UQ) is concerned with including and characterising uncertainties in mathematical models. Major steps comprise proper description of system uncertainties, analysis and efficient quantification of uncertainties in predictions and design problems, and statistical inference on uncertain parameters starting from available measurements. Research in UQ addresses fundamental mathematical and statistical challenges, but has also wide applicability in areas such as engineering, environmental, physical and biological applications. This workshop focussed on mathematical challenges at the interface of applied mathematics, probability and statistics, numerical analysis, scientific computing and application domains. The workshop served to bring together experts from those disciplines in order to enhance their interaction, to exchange ideas and to develop new, powerful methods for UQ

DATeS: a highly extensible data assimilation testing suite v1.0

A flexible and highly extensible data assimilation testing suite, named DATeS, is described in this paper. DATeS aims to offer a unified testing environment that allows researchers to compare different data assimilation methodologies and understand their performance in various settings. The core of DATeS is implemented in Python and takes advantage of its object-oriented capabilities. The main components of the package (the numerical models, the data assimilation algorithms, the linear algebra solvers, and the time discretization routines) are independent of each other, which offers great flexibility to configure data assimilation applications. DATeS can interface easily with large third-party numerical models written in Fortran or in C, and with a plethora of external solvers.</p