Effective Number of Degrees of Freedom of a Spatial Field

Martin Widmann, Atmospheric Sciences, University of Washington, Seattle

email: widmann@atmos.washington.edu

We systematically investigate two easily computed measures of the effective number of spatial degrees of freedom (ESDOF), or number of independently varying spatial patterns, of a time varying field of data. The first measure is based on matching the mean and variance of the time series of the spatially integrated squared anomaly of the field to a chi-squared distribution. The second measure, which is equivalent to the first for a long time sample of normally distributed field values, is based on the partitioning of variance between the EOFs. Although these measures were proposed almost 30 years ago, our paper aims to provide a comprehensive discussion of them that may help promote their more widespread use.

We summarize the theoretical basis of the two measures and considerations when estimating them with a limited time sample or from non-normally distributed data. We show that standard statistical significance tests for the difference or correlation between two realizations of a field (e.g. a forecast and an observation) are approximately valid if the number of degrees of freedom is chosen using an appropriate combination of our two ESDOF measures.