  Characteristic scale is a notion that pervades the geophysical sciences, but
it has no widely accepted precise definition. The wavelet transform decomposes
a time series into coefficients that are associated with different scales. The
variance of these coefficients can be used to decompose the variance of the
time series across different scales. A practical definition for characteristic
scale can be formulated in terms of peaks in plots of the wavelet variance
versus scale. This paper presents basic theory for characteristic scales based
upon the discrete wavelet transform, proposes a natural estimator for these
scales and provides a large sample theory for this estimator that permits the
construction of confidence intervals for a true unknown characteristic scale.
Computer experiments are presented that demonstrate the efficacy of the large
sample theory for finite sample sizes. Examples of characteristic scale
estimation are given for global temperature records, coherent structures in
river flows, the Madden-Julian oscillation in an atmospheric time series and
transects of one type of Arctic sea ice.
