A knowledge of the background velocity model is crucial to achieve the accurate reservoir description now expected from imaging and inversion. Conventional methods for reconstructing the background velocity model, like migration-velocity methods, often assume an isotropic subsurface and can yield inaccurate reservoir descriptions when the subsurface contains anisotropic rock formations. Here, we generalize the migration-velocity concept by permitting the background velocity to be transversely isotropic (TI) with respect to the vertical axis or othorhombic. The scheme consists of scanning different anisotropic velocity models using a phase-shift migration and of picking anisotropic parameters based on amplitude variations of migrated results (focusing analysis). As the anisotropic background velocity model is generally described by several elastic coefficients, it is important to adopt an efficient scanning procedure. We have chosen to work with common azimuthal sections. For a given common azimuthal section, we sequentially scan two parameters: normal move-out velocity and the anisotropic parameter known as anellipticity. These two scans allow us to reconstruct an azimuthally isotropic velocity model. The procedure is then repeated for different common azimuthal sections; each common azimuthal section leads to a new azimuthally isotropic velocity model if the medium is azimuthally anisotropic. The number of common azimuthal sections, and therefore the number of azimuthally isotropic velocity models, needed to reconstruct an azimuthally anisotropic velocity model is dependent on the type of symmetries. For example, only three common azimuthal sections are needed for an orthorhombic medium. As the contribution of isotropic component of most rock formations is generally more important than the anisotropic one, we have proposed to base the picking of values of anellipticity on the subtraction of the result of isotropic migration from that of TI migration instead of using directly the TI migrated results.