Bias-Eliminating Techniques in the Computation of Power Spectra for Characterizing Gravity Waves: Interleaved Methods and Error Analyses

Abstract

Observational data inherently contain noise which manifests as uncertainties in the measured parameters and creates positive biases or noise floors in second-order products like variances, fluxes, and spectra. Historical methods estimate and subsequently subtract noise floors, but struggle with accuracy. Gardner and Chu (2020, doi.org/10.1364/AO.400375) proposed an interleaved data processing method, which inherently eliminates biases from variances and fluxes, and suggested that the method could also eliminate noise floors of power spectra. We investigate the interleaved method for spectral analysis of atmospheric waves through theoretical studies, forward modeling, and demonstration with lidar data. Our work shows that calculating the cross-power spectral density (CPSD) from two interleaved subsamples does reduce the spectral noise floor significantly. However, only the Co-PSD (the real part of CPSD) eliminates the noise floor completely, while taking the absolute magnitude of CPSD adds a reduced noise floor back to the spectrum when the sample number is finite. This reduced noise floor can be further minimized through averaging over more observations, completely different from traditional spectrum calculations whose noise floor cannot be reduced by incorporating more samples. We demonstrate the first application of the interleaved method to spectral data, successfully eliminating the noise floor using the Co-PSD in a forward model and in lidar observations of the vertical wavenumber of gravity waves at McMurdo, Antarctica. This high accuracy is gained by sacrificing precision due to photon-count splitting, requiring additional observations to counter this effect. We provide quantitative assessment of accuracy and precision as well as application recommendations.