Data assimilation into nonlinear stochastic models

With very few exceptions, data assimilation methods which have been used or proposed for use with ocean models have been based on some assumption of l inearity or near-linearity. The great majority of these schemes have at the ir root some least-squares assumption. While one can always perform least-s quares analysis on any problem, direct application of least squares may not yield satisfactory results in cases in which the underlying distributions are significantly non-Gaussian. In many cases in which the behavior of the system is governed by intrinsically nonlinear dynamics, distributions of so lutions which are initially Gaussian will not remain so as the system evolv es. The presence of noise is an additional and inevitable complicating fact or. Besides the imperfections in our models which result From physical or c omputational simplifying assumptions, there is uncertainty in forcing field s such as wind stress and heat flux which will remain with us for the fores eeable future. The real world is a noisy place, and the effects of noise up on highly nonlinear systems can be complex. We therefore consider the probl em of data assimilation into systems modeled as nonlinear stochastic differ ential equations. When the models are described in this way, the general as similation problem becomes that of estimating the probability density funct ion of the system conditioned on the observations. The quantity we choose a s the solution to the problem can be a mean, a median, a mode, or some othe r statistic. In the fully general formulation, no assumptions about moments or near-linearity are required. We present a series of simulation experime nts in which we demonstrate assimilation of data into simple nonlinear mode ls in which least-squares methods such as the (Extended) Kalman filter or t he weak-constraint variational methods will not perform well. We illustrate the basic method with three examples: a simple one-dimensional nonlinear s tochastic differential equation, the well known three-dimensional Lorenz mo del and a nonlinear quasigeostrophic channel model. Comparisons to the exte nded Kalman filter and an extension to the extended Kalman filter are prese nted.