A new technique for calculating potential anomalies on a sphere due to finite amplitude relief has been developed. We show that by raising the topography to the nth power and expanding this field into spherical harmonics, potential anomalies due to topography on spherical density interfaces can be computed to arbitrary precision. Using a filter for downward continuing the Bouguer anomaly, we have computed a variety of crustal thickness maps for the Moon, assuming both a homogeneous as well as a dual-layered crust. The crustal thickness maps for the homogeneous model give plausible results, but this model is not consistent with the seismic data, petrologic evidence, and geoid to topography ratios, all of which suggest some form of crustal stratification. Several dual-layered models were investigated, and it was found that only models with both upper and lower crustal thickness variations could satisfy the gravity and topography data. These models predict that the entire upper crust has been excavated beneath the major nearside multiring basins. Additionally, significant amounts of lower crustal material was excavated from these basins, especially beneath Crisium. This model also predicts that mantle material should not have been excavated during the South-Pole Aitken basin forming event, and that lower crustal material should be exposed at the surface in this basin. ¿ 1998 American Geophysical Union