Stochastic empirical models have shown some success at reproducing statistics of the time evolution of geophysical flows. These stochastic empirical models assume a Markhov process and are built from measured or simulated data. Given a sufficient amount of data to develop the model, the discretized dynamics of the traditional model can be replaced by a matrix of computed values that serve as a propagator matrix. The simplest form of such a model is linear, ie. s_t=Ls + C where s is a vector of unknowns (such as velocities at various locations or the spectral coefficients of the velocity), s_t is its time tendency, L is a linear matrix operator relating the two, and C is the additive constant. The dynamics are contained in the matrix, L. Nonlinearities are parameterized by corrections to L as well as within the noise portion of C. This simple linear form is easily fit using standard analytical techniques to minimize the least square error between the model and time series data. Such linear empirical models often compete well with linearized dynamical models in reproducing the statistics of the modeled field. In addition, much can be learned about the flow dynamics.

The linear forms of the models reach their limit on highly nonlinear problems. Stochastic empirical models can also be formulated using nonlinear dynamics. For instance, a quadratic empirical model can have the form: s_t=Ns ^Ts + Ls + C. The nonlinear interactions now occur explicitly through the nonlinear third order tensor operator, N. Although one can still find a closed form solution for this nonlinear problem, it involves inverting a fourth order tensor. For problems larger than a very low number of dimensions, such an inversion is not trivial. Therefore, a genetic algorithm is instead used to find parameters which minimize the least square error between the model and the data for this nonlinear model.

The utility of nonlinear stochastic empirical models is demonstrated on a range of problems. For simple low dimensional problems, a quadratically nonlinear empirical model is successful at reproducing a chaotic attractor. For intermediate problems, we demonstrate that empirical models can be used to diagnose the important dynamics of a flow system and identify the essential terms to include in a forward dynamical model.