Even after elaborate conventional processing, land seismic data in complex areas exhibit weak and distorted prestack reflections with low coherency. Usually, the local stacking methods reveal clear reflections. However, the absolute level of amplitude spectra after such stacking experiences a strong decline across the entire frequency band, reaching - 10-25 dB. In addition, stacking leads to a significant and progressive loss of higher frequencies. We describe mathematical and intuitive physical models for multiplicative noise that could consistently explain these field observations at least semi-quantitatively. Multiplicative noise is represented by random timeshifts (residual statics) and random phase perturbations different for each frequency. Residual statics explains the progressive loss of higher frequencies. On the other hand, phase perturbations lead to a severe loss of coherency on prestack gathers and produce a strong downward bias or loss of broadband amplitudes after stacking. We show that both types of multiplicative noise can be physically generated by near-surface scattering layers with small-to-medium-scale geological heterogeneities. We speculate that such multiplicative distortions can be referred to as seismic speckle noise well established in optics and ultrasonics. Furthermore, we derive fundamental properties of how multiplicative noise transforms while stacking. The first essential finding reveals that stacking produces an unbiased estimate of the clean signal phase. The second finding gives the mathematical relationship between frequency-dependent loss of stacked amplitude and standard deviation of residual statics and phase perturbations. These findings serve as a theoretical justification for previously proposed methods of phase substitution and phase corrections and open the way to efficiently address multiplicative noise in seismic processing.
