  We propose a straightforward and effective approach for quantifying the band
inversion induced by spin-orbit coupling in band insulators. In this approach
we define a quantity as a function of wavevector in the Brillouin zone
measuring the spillage between the occupied states of a system with and without
SOC. Plots of the spillage throughout the BZ provide a ready indication of the
number and location of band inversions driven by SOC. To illustrate the method,
we apply this approach to the 2D Kane-Mele model, a 2D Bi bilayer with applied
Zeeman field, and to first-principles calculations of some 3D materials
including both trivial and \Z2\ topological insulators. We argue that the
distribution of spillage in the BZ is closely related to the topological
indices in these systems. Our approach provides a fresh perspective for
understanding topological character in band theory, and should be helpful in
searching for new materials with non-trivial band topology.
