Rocks near a fault plane are commonly damaged by multiple earthquake ruptures, forming damage zones. The damage zone is an important structure controlling various properties of a fault, yet its fine scale (tens to hundreds of meters) structure is difficult to resolve with surface seismic observations. We propose to use earthquakes that occur at depth within a fault zone as virtual seismometers (VSs) and use surface observations to extract Green's function (GFs) between VS pairs (VSGFs). This method resembles that of ambient noise tomography and the retrieved VSGFs are related to the structures between event pairs. In this study, we develop the theory about how to extract VSGFs using surface stations deployed across a fault zone. First, we use a half-space model and Fresnel zone analysis to determine the upper and lower limits of the GF frequency band, which is controlled by the station spacing and aperture of a given seismic array. Then, for VS in a fault zone, we demonstrate that the VSGF can be retrieved by linear seismic arrays deployed across the fault, and that the VSGF is equivalent to waves emitted simultaneously from an array of mirror sources of one event and received by the other. Secondly, the half-space result is directly adopted to determine the corresponding frequency band in the damage zone situation. Thirdly, we analyse different combinations of VS pair geometry and conclude that a relatively larger VS distance (much larger than the damage zone width) is more effective to recover damage zone structures for the available frequency bands. In this situation, VSGFs are trapped waves, that is represented by the interference of mirror sources. In such a case, the trapped waves are equivalent to surface waves, which have dispersion features to extract damage zone structures. Finally, we adopt the VSGF method to the Ridgecrest earthquake aftershock monitoring array and use a profile of aftershocks to extract six pairs of VSGFs. The spatial variation of VSGFs may reflect the depth-dependent variation of damaged zone. Our analysis shows a promising direction to use VSGFs to extract spatial variations of fault damaged zones. [ABSTRACT FROM AUTHOR]

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