The reduction of sample‐bias in polarization estimators for multichannel geophysical data with anisotropic noise

Summary. Estimates of the bivariate coherence, γ 2 , and the degree of polarization, β 2 , are often used to determine the amount of correlation between different time series. Both of these measures can be used to determine the presence of polarized waves in geophysical data, or to determine whether the multichannel data can be accurately represented by a reduced number of channels. The parameter β 2 is rotationally invariant, suggesting that it is a more suitable measure for analysing vector waves. If noise in the data is spatially anisotropic, then both γ 2 and β 2 can assume large values, even when no signal is present, and consequently the signal may be difficult to detect. This difficulty can be reduced by changing the metric in the instrument space. Similarly, since γ 2 and β 2 are estimated from a finite number of samples, or finite degrees of freedom v , the estimators can have large biases. Approximate corrections for these sample biases are given by expanding the expectation of the estimators in power series in v ‐1 .