Searching for analogues, how long must we wait?

Abstract

A three-way relationship is derived between the size of a library (M years) of historical atmospheric data, the distance between an arbitrarily picked state of the atmosphere and its nearest neighbor (or analogue), and the size of the spatial domain, as measured by the number of spatial degrees of freedom (N). It is found that it would take a library of order 10³⁰ years to find 2 observed flows that match to within current observational error over a large area such as the Northern Hemisphere. Obviously, with only 10—100 years of data, the probability of finding natural analogous is very small, unless one is satisfied with analogy over small areas or in just 2 of 3 degrees of freedom as represented, for instance, by 2 or 3 leading empirical orthogonal modes. We further propose the notion that analogues can be constructed by combining a number of observed flow patterns. We have found at least one application where linearly constructed analogues are conclusively better at specifying US surface weather from concurrent 700 mb geopotential height than natural analogues are.