The effective number of spatial degrees of freedom of a time-varying field

The authors systematically investigate two easily computed measures of the effective number of spatial degrees of freedom (ESDOF), or number of indepe ndently varying spatial patterns, of a time-varying field of data. The firs t measure is based on matching the mean and variance of the time series of the spatially integrated squared anomaly of the held to a chi-squared distr ibution. The second measure, which is equivalent to the first for a long ti me sample of normally distributed field values, is based on the partitionin g of variance between the EOFs. Although these measures were proposed almos t 30 years ago, this paper aims to provide a comprehensive discussion of th em that may help promote their more widespread use. The authors summarize the theoretical basis of the two measures and conside rations when estimating them with a limited time sample or from nonnormally distributed data. It is shown 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 the two ES DOF measures. Also described is a method involving ESDOF for deciding wheth er two time-varying fields are significantly correlated to each other. A discussion of the parallels between ESDOF and the effective sample size o f an autocorrelated time series is given, and the authors review how an app ropriate measure of effective sample size can be computed for assessing the significance of correlations between two rime series.