A data-driven framework for the stochastic reconstruction of small-scale features with application to climate data sets

Abstract

Turbulent fluid flows in atmospheric and oceanic sciences are characterized by strongly transient features with spatial inhomogeneity, spanning a wide range of spatial and temporal scales. While large-scale dynamics are often well approximated by closure schemes there is still a need to efficiently represent the corresponding small-scale features, when it comes to the risk analysis for extreme events. We introduce a data-driven framework for the stochastic reconstruction of the small spatial scales in terms of the large ones. The framework employs a spherical wavelet decomposition to partition field quantities, obtained from reanalysis data into non-overlapping spectral components. Using these time-series we formulate, for each spatial location, a machine-learning scheme that naturally ‘splits’ the small-scales into a predictable part, which can be effectively parametrized in terms of the large-scales time-series, and a stochastic residual, which cannot be uniquely determined using the large-scale information. The later is represented using a conditionally Gaussian process, a choice that allows us to overcome the need for a vast amount of training data, which for climate applications, is naturally limited to a single realization for each spatial location. Using a second round of machine-learning we parametrize, for each location, the covariance of the stochastic component in terms of the large scales. We employ the machine-learned statistics to parsimoniously reconstruct random realizations of the small scales. We demonstrate the approach on reanalysis data involving vorticity over Western Europe and we show that the reconstructed random samples for the small scales result in excellent agreement to the spatial spectrum, single-point probability density functions, and temporal spectral content.