Understanding the nature of the crust has long been a goal for seismologists when imaging the Earth. This is particularly true in volcanic regions where imaging melt storage and migration can have important implications for the size and nature of an eruption. Receiver functions and the H-κ stacking (Hκ) technique are often used to constrain crustal thickness (H) and the ratio of P to S wave velocities (κ). In this paper, I show that it is essential to consider anisotropy when performing Hκ. I show that in a medium with horizontally transverse isotropy a strong variation in κ with back azimuth is present, which characterizes the anisotropic medium. In a vertically transverse isotropic medium, no variation in κ with back azimuth is observed, but κ is increased across all back azimuths. Thus, estimates of κ are more difficult to relate to composition than previously thought. I extend these models to melt-induced anisotropy and show that similar patterns are observed, but with more significant variations and increases in κ. Based on these observations, I develop a new anisotropic H-κ stacking technique which inverts Hκ data for melt fraction, aspect ratio, and orientation of melt inclusions. I apply this to data for the Afar Depression and show that melt is stored in interconnected stacked sills in the lower crust, which likely supply the recent volcanic eruptions and dike intrusions. This new technique can be applied to any anisotropic medium where it can provide constraints on the average crustal anisotropy.