关于引力反问题
On the Gravitational Inverse Problem
摘要：
我们讨论了一些数学方面的问题，反演引力场数据，以提取潜在的质量分布。虽然从给定的质量分布计算重力场的正演问题在数学上是简单的，但是这个正演映射的逆映射有一些有趣的特征，使得逆映射成为一个困难的问题。特别地，正向映射具有无限维核，这使得反演基本上是不唯一的。我们完全表征的内核的两个引力前向映射，一个映射质量密度的牛顿标量势，和其他映射质量密度的重力梯度张量，这是最常见的量在现场观测。此外，我们提出了一些唯一的反演约束条件下的结果，并评论的作用的内核的前向映射和非唯一性发挥离散化方法的连续反演问题。
英文摘要：
We discuss some mathematical aspects of the problem of inverting gravitational field data to extract the underlying mass distribution. While the forward problem of computing the gravity field from a given mass distribution is mathematically straightforward, the inverse of this forward map has some interesting features that make inversion a difficult problem. In particular, the forward map has an infinite-dimensional kernel which makes the inversion fundamentally non-unique. We characterize completely the kernels of two gravitational forward maps, one mapping mass density to the Newtonian scalar potential, and the other mapping mass density to the gravity gradient tensor, which is the quantity most commonly measured in field observations. In addition, we present some results on unique inversion under constrained conditions, and comment on the roles the kernel of the forward map and non-uniqueness play in discretized approaches to the continuum inverse problem.
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