Conservation of Mass and Preservation of Positivity with Ensemble-Type Kalman Filter Algorithms

Abstract

This paper considers the incorporation of constraints to enforce physically based conservation laws in the ensemble Kalman filter. In particular, constraints are used to ensure that the ensemble members and the ensemble mean conserve mass and remain nonnegative through measurement updates. In certain situations filtering algorithms such as the ensemble Kalman filter (EnKF) and ensemble transform Kalman filter (ETKF) yield updated ensembles that conserve mass but are negative, even though the actual states must be nonnegative. In such situations if negative values are set to zero, or a log transform is introduced, the total mass will not be conserved. In this study, mass and positivity are both preserved by formulating the filter update as a set of quadratic programming problems that incorporate nonnegativity constraints. Simple numerical experiments indicate that this approach can have a significant positive impact on the posterior ensemble distribution, giving results that are more physically plausible both for individual ensemble members and for the ensemble mean. In two examples, an update that includes a nonnegativity constraint is able to properly describe the transport of a sharp feature (e.g., a triangle or cone). A number of implementation questions still need to be addressed, particularly the need to develop a computationally efficient quadratic programming update for large ensemble.

Constrained probabilistic collocation method for uncertainty quantification of geophysical models

“…We test these collocation methods in several cases of multiphase flow and solute transport in porous media with different spatial variabilities and dispersivities and also compare them with the Monte Carlo (MC) method. Although the idea of considering constraints is only shown for uncertainty quantification, it can be extended to data assimilation as well, where the oscillation problem also occurs (e.g., [14,27]). For example, the proposed approach can be directly applied to the probabilistic collocation-based Kalman filter [37].…”

HErZ: The German Hans-Ertel Centre for Weather Research

“…Data assimilation on this scale demands the assimilation of temporally and spatially high-resolution observations while restoring balance and physical properties and accounting for the inherent analysis and forecast uncertainty through ensemble methods or particle filters. The research group investigates conservation in data assimilation (Janjić et al 2014) and the use of toy models for testing (Würsch and Craig 2014). Geostationary satellite and weather radar observations are assimilated (Kostka et al 2014;Lange and Craig 2014) and their contribution to forecast quality is analyzed (Sommer and Weissmann 2014) including the representation of uncertainty in ensemble systems (Harnisch and Keil 2015;Kühnlein et al 2014).…”

A data assimilation methodology for reconstructing turbulent flows around aircraft

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