The rocks near a fault plane are commonly damaged by multiple earthquake ruptures which form damaged zones Damage zones are important structures controlling various properties of a fault however this

The rocks near a fault plane are often damaged by multiple earthquake ruptures, resulting in damaged zones that are important structures controlling various properties of a fault. However, this fine-scale structure (with widths of tens to hundreds of meters) is difficult to resolve with ground surface seismic observations. To address this issue, we propose using earthquakes that occur at depth within a fault-damaged zone as virtual sources (VS) and using ground surface observations to extract Green's functions (GFs) between VS pairs (VSGF). This theory is similar to ambient noise tomography, and the recovered VSGF is closely related to the structures between event pairs. In this study, we develop a complete theory about how to extract VSGFs using surface stations deployed across a fault zone.

Firstly, in a half-space, we use stationary point and Fresnel zone theory to determine the upper and lower limits of the GF frequency band, which are controlled by the spacing and aperture of a given seismic array. Secondly, for VS in a fault zone, we demonstrate that the VSGF can be recovered by linear seismic arrays deployed across the fault. The VSGF is equivalent to waveforms emitted simultaneously by an array of mirrored sources of one event and received by the other. The half-space theory is directly used to determine the associated frequency band in the damage zone situation. Finally, we analyze different combinations of VS pair geometry and conclude that a larger VS distance (event pair distance much larger than the damaged zone width) is more practical to recover damaged zone structures at feasible frequency bands. In this situation, VSGFs are trapped waves that can be represented by the interference of mirrored sources. The trapped wave theory is similar to that of surface waves, which requires the available frequency band to cover dispersion curves of multiple modes, allowing for the recovery of the damaged zone structure.