Given a virtually smooth quasi-projective scheme M, and a morphism from M to a nonsingular quasi-projective variety B, we show it is possible to find an affine bundle M′/M that admits a perfect obstruction theory relative to B. We study the resulting virtual cycles on the fibers of M′/B and relate them to the image of the virtual cycle [M]vir under refined Gysin homomorphisms. Our main application is when M is a moduli space of stable codimension 1 sheaves on a nonsingular projective surface or Fano threefold.