Abstract
<jats:p>Conventional theories of seismic interferometry commonly assume that source fluctuations are uncorrelated in both time and space. However, actual seismic noise sources and excitations in scattering media may possess finite temporal and spatial correlations. In this study, we generalize the source correlation function to include finite correlations in both time and space and analytically investigate their effects on the observed cross-correlation waveforms. We show that finite temporal correlation produces frequency-dependent phase shifts and amplitude modifications. In particular, when the source correlation has a characteristic time scale, a parallel time shift (PTS), in which the positive-and negative-lag waveforms shift in the same temporal direction, becomes pronounced at low frequencies. By contrast, finite spatial correlation does not directly generate an additional phase delay but primarily attenuates and modifies the amplitude structure of the cross-correlation waveforms. These results provide a theoretical basis for interpreting observed cross-correlation waveforms in terms of finite source correlations and improve our understanding of the reliability and physical interpretation of seismic interferometric measurements.</jats:p>