Redundant source-wavelet amplitude influence in wave-equation migration/demigration flow and its removal

In wave-equation migration and demigration, the cross-correlation imaging/forwarding step implicitly injects an additional copy of the source wavelet, so that the amplitude spectrum of the wavelet is applied redundantly (effectively imposing a wavelet-spectrum weighting, often akin to an amplitude-squared bias). This redundancy degrades structural fidelity and amplitude balance yet is frequently overlooked. We (i) formalize the mechanism by which cross-correlation duplicates the source-wavelet amplitude effect in both migration and demigration, and (ii) introduce a source-equalized operator that removes the redundancy by deconvolving (or dividing by) the wavelet amplitude spectrum in the imaging condition and its demigration counterpart, while leaving phase/kinematics intact. Using a band-limited Ricker wavelet on a two-layer model and on Marmousi, we show that, if unmanaged, the redundant wavelet spectrum broadens main lobes, introduces ringing, and suppresses vertical resolution in migrated images, and inflates spectrum mismatches between demigrated and observed data even when peak times agree. With our correction, images recover observed-data-consistent bandwidth and sharpened interfaces, and demigrated data also exhibit improved spectrum conformity and reduced amplitude misfit. The results clarify when source amplitudes matter, why cross-correlation makes them redundantly matter, and how a lightweight spectral correction restores physically meaningful amplitude behavior in wave-equation migration/demigration.