Concrete ensemble Kalman filters with rigorous catastrophic filter divergence.
The ensemble Kalman filter and ensemble square root filters are data assimilation methods used to combine high-dimensional, nonlinear dynamical models with observed data. Ensemble methods are indispensable tools in science and engineering and have enjoyed great success in geophysical sciences, because they allow for computationally cheap low-ensemble-state approximation for extremely high-dimensional turbulent forecast models. From a theoretical perspective, the dynamical properties of these methods are poorly understood. One of the central mysteries is the numerical phenomenon known as catastrophic filter divergence, whereby ensemble-state estimates explode to machine infinity, despite the true state remaining in a bounded region. In this article we provide a breakthrough insight into the phenomenon, by introducing a simple and natural forecast model that transparently exhibits catastrophic filter divergence under all ensemble methods and a large set of initializations. For this model, catastrophic filter divergence is not an artifact of numerical instability, but rather a true dynamical property of the filter. The divergence is not only validated numerically but also proven rigorously. The model cleanly illustrates mechanisms that give rise to catastrophic divergence and confirms intuitive accounts of the phenomena given in past literature.
Courant Institute of Mathematical Sciences, New York University, New York, NY 10012;
David
D
Kelly
Courant Institute of Mathematical Sciences, New York University, New York, NY 10012; Center for Atmosphere Ocean Science, New York University, New York, NY 10012 jonjon@cims.nyu.edu tong@cims.nyu.edu.
Andrew J
AJ
Majda
Courant Institute of Mathematical Sciences, New York University, New York, NY 10012; Center for Atmosphere Ocean Science, New York University, New York, NY 10012 jonjon@cims.nyu.edu tong@cims.nyu.edu.
Xin T
http://orcid.org/0000-0002-3409-0919
XT
Tong
data assimilation; ensemble Kalman filter; filter divergence