The coefficient of coherence; its estimation and use in geophysical data processing

Abstract The coefficient of coherence between two stationary time series was introduced by Wiener in 1930. It is related to the signal-to-noise ratio, to the minimum prediction error, and has important invariance properties. As an estimate of this parameter, most geophysicists have used the so-called 'sample coherence.' An approximate distribution of the sample coherence for Gaussian data has been derived by N. R. Goodman. We have tested this distribution by means of Monte Carlo experiments for validity and robustness (insensitivity to the Gaussian assumption). It has passed the tests.The Goodman distribution provides a means of constructing estimates of the true coherence which are better than the widely used sample coherence. It can also be used to calculate confidence intervals. Finally, it forms a basis for choosing the lag window and data window necessary for best estimation of the true coherence.For good estimates of the true coherence, two precautions must be observed:1. The cross-spectrum and power spectra of the two time series must be smoothly varying over the width of the spectral window.2. The ratio of the length of the data window to the lag window must be large.For most seismic work the second requirement severely limits the spectral resolution. Examples show that large errors can result if this resolution is not sufficient to satisfy the first requirement.In many geophysical studies the parameter of interest is the signal-to-noise ratio. Because of its relation to the coherence, the Goodman distribution provides a basis for its estimation as well.