Catalogus

In this paper, we make some important remarks about linear versus nonlinear systems, emphasizing the fact that a data assimilation problem may become extremely complicated for strongly nonlinear problems. Statistical moments of any order may develop from Gaussian initial conditions during nonlinear evolution, and important information may be discarded by calculating an estimate based on only the Gaussian part of the full probability distribution. We demonstrate that a Monte Carlo approach can provide information about the system under consideration. For example, an ensemble of states, which is a representative of the true probability density function, can be visualized in one, two or three dimensions. Also, one can find estimates for the degree of nonnormality of the ensemble, which may act as indicators of the validity of performing a data assimilation based on the Gaussian part of the full probability distribution.