s 375 The Isostatic Green’s Function and the Inverse Potential Problem L. M. Dorman and B. T. R. Lewis Although the Earth‘s gravity field is simply and directly related to the Earth‘s density, the usefulness of gravity for examination of the density variations within the Earth is limited by the inherent non-uniqueness of solutions to the inverse potential problem. When we are studying isostasy, however, we have the additional constraint that the density variations we are interested in must be systematically related to the variations in elevation of the Earth‘s surface. This additional constraint is sufficient to allow us to extract the isostatic Green’s function from observational data (gravity and surface elevation). The Green’s function thus computed can be used to calculate isostatic anomalies which are not based on any specific model. It is, however, of more use to us than that. By remembering that the Green’s function relates the forcing function of our problem (the topography) to the observable (the gravity anomaly). We can see that the Green’s function must represent the gravity anomaly associated with a concentrated load on the Earth’s surface. If we have knowledge of, or make assumptions about, the mechanical properties of the Earth‘s crust, we can invert this Green’s function and find the density changes as a function of depth which are associated with this concentrated load. For the assumption of local compensation (compensation occurring directly beneath the load), this inversion requires us to invert an integral transform similar to the Laplace transform where the function to be inverted is derived from observations and hence contains errors. Using gravity and elevation from the continental U.S., we have extracted the Green’s function and, using the elegant techniques of Backus and Gilbert, inverted it. These calculations show that compensation occurs at at least two depths. Near the base of the crust, there is over compensation by about 20 per cent. This extra 20 per cent is cancelled out deeper in the mantle, at about 450 km depth. Comparison of the density differences between the Earth beneath low-lying regions and that beneath elevated regions with the differences in seismic P and S wave velocities between these regions gives support for a two-depth compensation mechanism. L. M. Dorman: B. T. R. Lewis: NOAA Atlantic Oceanographic University of Washington, and Meteorological Laboratory, Miami, Florida 33 149, USA Seattle, Washington 98105, USA