We present a new approach to enhancing weak prestack reflection signals without sacrificing higher frequencies. As a first step, we employ known multidimensional local stacking to obtain an approximate model of the signal. Guided by phase spectra from this model, we can detect very weak signals and make them visible and coherent by repairing corrupted phase of original data. Both presented approaches phase substitution and phase sign corrections show good performance on complex synthetic and field data suffering from severe near-surface scattering where conventional processing methods are rendered ineffective. The methods are mathematically formulated as a special case of time-frequency masking (common in speech processing) combined with the signal model from local stacking. This powerful combination opens the avenue for a completely new family of approaches for multi-channel seismic processing that can address seismic processing of land data with nodes and single sensors in the desert environment.