Spatial and Temporal Averaging Windows and Their Impact on Forecasting: Exactly Solvable Examples

Abstract

Abstract In making weather and climate predictions, the goal is often not to predict the instantaneous, local value of temperature, wind speed, or rainfall; instead, the goal is often to predict these quantities after averaging in time and/or space-for example, over one day or one week. What is the impact of spatial and/or temporal averaging on forecasting skill?Here this question is investigated using simple stochastic models that can be solved exactly analytically. While the models are idealized, their exact solutions allow clear results that are not affected by errors from numerical simulations or from random sampling. As a model of time series of oscillatory weather fluctuations, the complex Ornstein-Uhlenbeck process is used. To furthermore investigate spatial averaging, the stochastic heat equation is used as an idealized spatiotemporal model for moisture and rainfall. Space averaging and time averaging are shown to have distinctly different impacts on prediction skill. Spatial averaging leads to improved forecast skill, in line with some forms of basic intuition. Time averaging, on the other hand, is more subtle: it may either increase or decrease forecast skill. The subtle effects of time averaging are seen to arise from the relative definitions of the time averaging window and the lead time. These results should help in understanding and comparing forecasts with different temporal and spatial averaging windows.