Reconstruction of the autocorrelation function from segmented data and its application to the Earth’s seismic hum

Summary We develop a new method for estimating the autocorrelation function (ACF) from segmented data with the assumption of stochastic stationarity. The ACF of a signal is represented as the summation of the cross terms of sub-segments of arbitrary length. To successfully remove undesired transients in the data, this method introduces a correction for the amplitude bias associated with the removal of sub-segments, based on the comparison between the expected stationary signal and the measured signal. The method reconstructs and accesses later lag times, provides finer frequency resolution, obtains a better signal-to-noise ratio, which enables the extraction of detailed temporal or spectral structures from noisy data sets. As an application, we successfully retrieved a spectrum of the Earth’s seismic hum on the vertical component with fine frequency resolution and compared it to synthetic autocorrelation for spatially isotropic and homogeneous excitation by random shear traction at the ocean bottom and random pressure at the Earth’s surface. Although both models can explain the observed fundamental spheroidal modes, shear traction is better at explaining the observed overtones above 3 mHz. From 2 to 3 mHz, the pressure source also contributes to the excitation of the overtones, and the shear traction becomes dominant again below 2 mHz. This new method is anticipated to be effective in extracting valuable information from rare records within the context of extraterrestrial seismology.