Detection noise bias and variance in the power spectrum and bispectrum in optical interferometry

Context. Long-baseline optical interferometry uses the power spectrum and bispectrum constructs as fundamental observables. Noise arising in the detection of the fringe pattern results in both variance and biases in the power spectrum and bispectrum. Previous work on correcting the biases and estimating the variances for these quantities typically includes restrictive assumptions about the sampling of the interferogram and/or about the relative importance of Poisson and Gaussian noise sources. Until now it has been difficult to accurately compensate for systematic biases in data which violates these assumptions. Aims. We seek a formalism to allow the construction of bias-free estimators of the bispectrum and power spectrum, and to estimate their variances, under less restrictive conditions, which include both unevenly-sampled data and measurements affected by a combination of noise sources with Poisson and Gaussian statistics. Methods. We used a method based on the moments of the noise distributions to derive formulae for the biases introduced to the power spectrum and bispectrum when the complex fringe amplitude is derived from an arbitrary linear combination of a set of discrete interferogram measurements. Results. We have derived formulae for bias-free estimators of the power spectrum and bispectrum, which can be used with any linear estimator of the fringe complex amplitude. We have demonstrated the importance of bias-free estimators for the case of the detection of faint companions (for example exoplanets) using closure phase nulling. We have derived formulae for the variance of the power spectrum and have shown how the variance of the bispectrum can be calculated.