Predicting stochastic systems by noise sampling, and application to the El Niño-Southern Oscillation

Abstract

Interannual and interdecadal prediction are major challenges of climate dynamics. In this article we develop a prediction method for climate processes that exhibit low-frequency variability (LFV). The method constructs a nonlinear stochastic model from past observations and estimates a path of the "weather" noise that drives this model over previous finite-time windows. The method has two steps: (i) select noise samples-or "snippets"-from the past noise, which have forced the system during short-time intervals that resemble the LFV phase just preceding the currently observed state; and (ii) use these snippets to drive the system from the current state into the future. The method is placed in the framework of pathwise linear-response theory and is then applied to an El Niño-Southern Oscillation (ENSO) model derived by the empirical model reduction (EMR) methodology; this nonlinear model has 40 coupled, slow, and fast variables. The domain of validity of this forecasting procedure depends on the nature of the system's pathwise response; it is shown numerically that the ENSO model's response is linear on interannual time scales. As a result, the method's skill at a 6-to 16-month lead is highly competitive when compared with currently used dynamic and statistic prediction methods for the Niño-3 index and the global sea surface temperature field. E NSO forecasting has a decade-long history and relies mainly on two classes of models: dynamical and statistical (1, 2). Still, a further distinction has to be made within the latter class: Some of the statistical models do not make any use of dynamical information, like Lorenz's method of analogues (3) and its followers (4-6), while others do use a dynamical model-previously fitted to the observations from the past-to drive the statistics in the future (2,7,8). Empirical stochastic models belong to this hybrid category, and linear versions of such models have been used in ENSO forecasting for two decades; see ref. 9 for a survey. More recently, Kravtsov et al. (10) have extended this approach to nonlinear models by developing an EMR methodology that can include quadratic nonlinearities as well as state-dependent noise that parameterizes small-scale effects, without assuming a priori scale separation (11).The purpose of this paper is to show that, under suitable circumstances, a better understanding of the role of the fast processes, weather or noise, can help predict the slow ones-namely, the climate. To achieve this purpose, we proceed in two steps: (i) develop a special prediction methodology, called past noise forecasting (PNF), using EMR models; and (ii) provide a theoretical framework for applying the PNF method-or any other forecasting method based on perturbations of the noise-to other empirical stochastic models. Of late, probabilistic forecasts in weather and climate prediction have become fairly widespread: They are grounded in an estimation of the probability density function (PDF: 12-14).We take here a distinct, pathwise approach instead, and will show that th...

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