Robust Data Assimilation Using $L_1$ and Huber Norms

Abstract

Abstract. Data assimilation is the process to fuse information from priors, observations of nature, and numerical models, in order to obtain best estimates of the parameters or state of a physical system of interest. Presence of large errors in some observational data, e.g., data collected from a faulty instrument, negatively affect the quality of the overall assimilation results. This work develops a systematic framework for robust data assimilation. The new algorithms continue to produce good analyses in the presence of observation outliers. The approach is based on replacing the traditional L 2 norm formulation of data assimilation problems with formulations based on L 1 and Huber norms. Numerical experiments using the Lorenz-96 and the shallow water on the sphere models illustrate how the new algorithms outperform traditional data assimilation approaches in the presence of data outliers.1. Introduction. Dynamic data-driven application systems (DDDAS [3]) integrate computational simulations and physical measurements in symbiotic and dynamic feedback control systems. Within the DDDAS paradigm, data assimilation (DA) defines a class of inverse problems that fuses information from an imperfect computational model based on differential equations (which encapsulates our knowledge of the physical laws that govern the evolution of the real system), from noisy observations (sparse snapshots of reality), and from an uncertain prior (which encapsulates our current knowledge of reality). Data assimilation integrates these three sources of information and the associated uncertainties in a Bayesian framework to provide the posterior, i.e., the probability distribution conditioned on the uncertainties in the model and observations. Two approaches to data assimilation have gained widespread popularity: ensemblebased estimation and variational methods. The ensemble-based methods are rooted in statistical theory, whereas the variational approach is derived from optimal control theory. The variational approach formulates data assimilation as a nonlinear optimization problem constrained by a numerical model. The initial conditions (as well as boundary conditions, forcing, or model parameters) are adjusted to minimize the discrepancy between the model trajectory and a set of time-distributed observations. In real-time operational settings the data assimilation process is performed in cycles: observations within an assimilation window are used to obtain an optimal trajectory, which provides the initial condition for the next time window, and the process is repeated in the subsequent cycles.Large errors in some observations can adversely impact the overall solution to the data assimilation system, e.g., can lead to spurious features in the analysis [15]. Various factors contribute to uncertainties in observations. Faulty and malfunctioning

Robust calibration of numerical models based on relative regret

“…A first approach is to set α, and then to estimate the function k → Γ α (k) as defined in Eq. (17). For instance, when N samples from U are available, namely…”

“…Indeed, one definition of the robustness of an estimate is a measure of the sensibility of said estimate to outliers [16]. This leads to the introduction of robust norms in data assimilation [17]. In a Bayesian framework, robustness may refer to the sensitivity to a wrong specification of the priors [18].…”

Numerical linear algebra in data assimilation

“…Data assimilation using different regularization terms was considered in [30,148], efficient solution techniques for such approaches require ideas from numerical linear algebra.…”

A data assimilation framework that uses the Kullback-Leibler divergence

“…However, minimization involving a L 1 -norm for the regularization (or background) term has also been used and this is found to be particularly useful for tracking sharp fronts and discontinuities (e.g., [ 9 , 10 ]). Rao et al [ 11 ] found L 1 -norm data assimilation to be beneficial when dealing with outlier observations, but had the drawback that solutions lacked smoothness near the mean, this desirable property was retained by using the Huber-norm, a hybrid that utilizes L 1 in the presence of outliers and L 2 close to the mean. An alternative approach to data assimilation, explored by Feyeux et al [ 12 ], utilizes optimal transport theory and within this context the Wasserstein distance (minimizing kinetic energy) replaces the L 2 distance.…”