Theory and Practice of Phase-Aware Ensemble Forecasting

The timing of events represents a source of uncertainty in ensemble forecasting that can produce misleading ensemble statistics. In this paper, a general theory is presented to overcome drawbacks of traditional ensemble forecasting statistics that perform poorly in the presence of timing disagreements among ensemble members. It was shown, in particular, that ensemble forecasts containing substantial uncertainty in timing can produce non-trivial higher-order statistical moments, rendering the ensemble mean inappropriate as a best available estimate of the future state of the forecast parameter in question. A set of theoretical experiments showed that the existence of large timing differences among ensemble members can produce negative ensemble skewness even when the ensemble members are sinusoids whose amplitudes are drawn from a normal distribution: Consistently, the ensemble mean will tend to fall on the left tail of the normal distribution representing the originally sampled amplitudes, rather than at the mean (or, equally, the median). To remedy the left-tail placement problem of the ensemble mean, a new generally applicable ensemble statistic - the phase-aware ensemble mean - is proposed that is more robust against ensemble skewness resulting from timing spread. The computation of the phase-aware mean involves the transformation of all ensemble members to wavelet space and the subsequent inverse wavelet transformation of the product of the ensemble mean wavelet phase and modulus back to the time domain. A quasi-bootstrap and percentile-based procedure was also introduced to extend a small number of numerical-model-based ensemble predictions through the addition of statistically-drafted ensemble members. The new methods were applied to storm surge reforecasts for Hurricane Irene and Sandy at 8 stations located around the New York City metropolitan area. The phase-aware ensemble mean was found to perform better at detecting the magnitude of events compared to the traditional ensemble mean, consistent with the results from theoretical experiments. The ensemble mean, moreover, was found to be consistently located on the left tail of distributions representing future peak storm surge outcomes. The quasi-bootstrap and percentile-based procedures were found to be capable of transforming an initially under-spread ensemble of 21 numerical-model-based members to a better-dispersed ensemble. The results suggest that the method may be used to alternatively produce a more reliable ensemble, decreasing the need for additional computationally expensive numerical model runs.