  Long duration noisy-looking waveforms such as those obtained in randomly
multiply scattering and reverberant media are complex; they resist direct
interpretation. Nevertheless, such waveforms are sensitive to small changes in
the source of the waves or in the medium in which they propagate. Monitoring
such waveforms, whether obtained directly or obtained indirectly by noise
correlation, is emerging as a technique for detecting changes in media.
Interpretation of changes is in principle problematic; it is not always clear
whether a change is due to sources or to the medium. Of particular interest is
the detection of small changes in propagation speeds. An expression is derived
here for the apparent, but illusory, waveform dilation due to a change of
source. The expression permits changes in waveforms due to changes in wavespeed
to be distinguished with high precision from changes due to other reasons. The
theory is successfully compared with analysis of a laboratory ultrasonic data
set and a seismic data set from Parkfield California.
