变密度体重力勘探反演的多解性问题

本文利用经典位论中格林层的概念,证明了: 1.在一个物体内可以有多种密度分布,使其总质量为零,在物体外任一点的位为零; 2.当总质量不变时,在一个固定形状及体积的物体中,可以有多种密度分布,并且处处密度为正,使物体外任一点的位相同; 3.当总质量不变时,在不同形状及不同体积的物体内,可以有多种密度分布,并且处处密度为正,使物体外任一点的位相同。 由此得出结论:总质量不变,若物体的密度不均匀（而且处处为正）,根据重力异常作反演,不能唯一地求出物体的形状;当物体的形状已知时,也不能唯一地求出其密度分布。

ON THE NONUNIQUE SOLUTIONS IN THE INVERSE PROBLEM OF GRAVIMETRY FOR THE BODIES WITH IMHOMOGENEOUS DENSITY

Based on the idea of Green's shell and the relationship between the surface density and the volume density, we have proved the following theorems:1. There are more than one way in which a given mass can be distributed imho-mogeneously in a given body with the total mass equal to zero and gives a zero potential field outside the body.2. In a given body with given mass, its density can assume more than one kind of distribution but still gives the same potential field outside the body.3. If the regions occupied by different bodies have a common point P as the pole of the Green's functions relative to the different surfaces enclosing the different regions, a given mass can be distributed imhomogeneously in the bodies in more than one way such that their potential fields outside the bodies are the same.Prom what stated above, we get the following conclusion:If a given mass distributes imhomogeneously in a given body, it is impossible to get unique solutions of the shape and density of the body, it is also impossible to get a unique solution of the density even if the shape of the body is given.