Determination of Shape and Mass Density

The ill-posed problem of determining the shape of a body immersed in a ""at earth" and the mass density within this body from gravity anomaly measurements on the surface of the earth is investigated. Under some a priori assumptions on the structure of the body's boundary a general theorem on uniqueness of the solution is established and some further theorems are derived as immediate consequences. As often is the case with inverse and ill-posed problems, this question of uniqueness is particulary important whereas because of unavoidable measurement errors, the question of existence for given date, is less so. Let V be a body of mass density 1 , lying below the surface of the earth represented by a half-space, and let the density of the medium surrounding V be 2. Call the relative density, i.e. = 1 ? 2. Several problems have been posed concerning V and , The foregoing problems are fundamental problems in Applied Geophysics and it is known that there is no uniqueness for the problems posed in such generality. Let us look at Problem 1 and clarify the meaning of the phrase "Knowing ". This means that a function is given on all of R 3 while the density function is deened only on the unknown