  The Ensemble Kalman filter and Ensemble square root filters are data
assimilation methods used to combine high dimensional nonlinear models with
observed data. These methods have proved to be indispensable tools in science
and engineering as they allow computationally cheap, low dimensional ensemble
state approximation for extremely high dimensional turbulent forecast models.
From a theoretical perspective, these methods are poorly understood, with the
exception of a recently established but still incomplete nonlinear stability
theory. Moreover, recent numerical and theoretical studies of catastrophic
filter divergence have indicated that stability is a genuine mathematical
concern and can not be taken for granted in implementation. In this article we
propose a simple modification of ensemble based methods which resolves these
stability issues entirely. The method involves a new type of adaptive
covariance inflation, which comes with minimal additional cost. We develop a
complete nonlinear stability theory for the adaptive method, yielding Lyapunov
functions and geometric ergodicity under weak assumptions. We present numerical
evidence which suggests the adaptive methods have improved accuracy over
standard methods and completely eliminate catastrophic filter divergence. This
enhanced stability allows for the use of extremely cheap, unstable forecast
integrators, which would otherwise lead to widespread filter malfunction.
