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 linearity or near-linearity. The great majority of these schemes have at their root some least-squares assumption. While one can always perform least-squares 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 solutions which are initially Gaussian will not remain so as the system evolves. The presence of noise is an additional and inevitable complicating factor. Besides the imperfections in our models which result From physical or computational simplifying assumptions, there is uncertainty in forcing fields such as wind stress and heat flux which will remain with us for the foreseeable future. The real world is a noisy place, and the effects of noise upon highly nonlinear systems can be complex. We therefore consider the problem of data assimilation into systems modeled as nonlinear stochastic differential equations. When the models are described in this way, the general assimilation problem becomes that of estimating the probability density function of the system conditioned on the observations. The quantity we choose as the solution to the problem can be a mean, a median, a mode, or some other statistic. In the fully general formulation, no assumptions about moments or near-linearity are required. We present a series of simulation experiments in which we demonstrate assimilation of data into simple nonlinear models in which least-squares methods such as the (Extended) Kalman filter or the weak-constraint variational methods will not perform well. We illustrate the basic method with three examples: a simple one-dimensional nonlinear stochastic differential equation, the well known three-dimensional Lorenz model and a nonlinear quasigeostrophic channel model. Comparisons to the extended Kalman filter and an extension to the extended Kalman filter are presented.