Sdliner, W., Tsvankin, I. and Filpo Ferreira da Silva, E., 2010. Multi-azimuth prestack time migration for anisotropic, weakly heterogeneous media. Journal of Seismic Exploration, 19: 187-206. Conventional prestack time-migration velocity analysis is designed to estimate diffraction time functions in a fixed azimuthal direction from narrow-azimuth reflection data. Therefore, it can build accurate 3D migration operators only if the subsurface is isotropic (or azimuthally isotropic) and laterally homogeneous. Here, we extend time-migration methodology to multi-azimuth or wide-azimuth data from azimuthally anisotropic, weakly heterogeneous media. We derive the azimuthally varying diffraction time function from the most general form of Hamilton’s principal equation and apply a Taylor series expansion to the traveltime in the vicinity of the image ray. This approach helps relate the Taylor series coefficients to the corresponding multi-azimuth imaging parameters. The second-order coefficients, which define the 'migration-velocity ellipse', are obtained from time-migration velocity analysis in at least three distinct azimuthal directions. Our multi-azimuth prestack time migration (MAPSTM) solves the mismatch problem that occurs in conventional processing when the same depth point creates different time images in different azimuths. The algorithm is tested on synthetic data from a horizontally layered, azimuthally anisotropic model and an isotropic medium with a dipping interface.