Polynomial Chaos Quantification of the Growth of Uncertainty Investigated with a Lorenz Model

Abstract

A time-dependent physical model whose initial condition is only approximately known can predict the evolving physical state to only within certain error bounds. In the prediction of weather, as well as its ocean counterpart, quantifying this uncertainty or the predictability is of critical importance because such quantitative knowledge is needed to provide limits on the forecast accuracy. Monte Carlo simulation, the accepted standard for uncertainty determination, is impractical to apply to the atmospheric and ocean models, particularly in an operational setting, because of these models’ high degrees of freedom and computational demands. Instead, methods developed in the literature have relied on a limited ensemble of simulations, selected from initial errors that are likely to have grown the most at the forecast time. In this paper, the authors present an alternative approach, the polynomial chaos method, to the quantification of the growth of uncertainty. The method seeks to express the initial errors in functional form in terms of stochastic basis expansions and solve for the uncertainty growth directly from the equations of motion. It is shown via a Lorenz model that the resulting solution can provide all the error statistics as in Monte Carlo simulation but requires much less computation. Moreover, it is shown that the functional form of the solution facilitates the uncertainty analysis. This is discussed in detail for the tangent linear case of interest to ensemble forecasting. The relevance of the uncertainty covariance result to data assimilation is also noted.