Downward continuation of gravity field

重力是在地球表面观测到的，但对于大地水准面的计算，它需要在海平面（在大地水准面上）：它必须通过地形质量“继续向下”到海平面。在我们的大地水准面确定方法中，从实际地形中减去地形质量的最佳近似值，我们认为结果（没有地形的地球）代表“NT空间”中的情况。实际上，我们用大地水准面上的凝结层来代替地形，并把这个新的空间称为“赫尔默特空间”。因此，赫尔默特的空间是由凝结层增加的NT空间。在NT空间和赫尔默特空间中，向下延拓是通过一部分尽可能没有质量的空间进行的，向下延拓过程是向上延拓过程的逆过程。向上延续是一种平滑操作，向下延续是一种提升高频的操作。从数字的角度来看，这意味着麻烦：向下延续的过程必然是不稳定的。从数学的观点出发，用经典的Poisson积分来描述无质量空间中的向上延拓。反问题由Fredholm的第一类积分方程描述，已知其解是不稳定的。然而，根据公认的定义，积分方程代表一个适定问题，具有唯一的，有限的，但不稳定的解决方案。在我们的大地水准面估计中，我们需要在赫尔默特空间中计算向下延拓，我们使用Fredholm线性方程组的Jacobi迭代反演（参考文献463）。这种方法虽然给出了合理的大地水准面计算结果，但缺乏物理意义。我们希望继续这项研究，增加一些物理上有意义的约束（要求调和性，最小化的二维拉普拉斯在大地水准面上，等等）。给解决方案增加一些物理意义。

Gravity is observed on the earth's surface but for the geoid computation it is needed at the sea level (on the geoid): it has to be "continued downward" onto the sea level through topographical masses. In our approach to geoid determination, the best approximation of topographical masses is subtracted from actual topography and we consider the result (Earth without topography) to represent the situation in the "NT space". In fact, we replace topography by a condensation layer on the geoid and call this new space the "Helmert space". Helmert's space is thus the NT space augmented by the condensation layer. In both the NT and Helmert's space, the downward continu-ation is carried out through a part of space that is as void of mass as we can possibly make it. The downward continuation process is the inverse of upward continuation. While upward continuation is a smoothing operation, downward continuation is an operation that boosts up high frequencies. This spells trouble from the numerical point of view: the process of downward continuation is bound to be unstable. From the mathematical point of view, the upward continuation in a space void of mass is described by the classical Poisson integral. The inverse problem is described by Fredholm's integral equation of first kind, the solution of which is known to be unstable. Yet, the integral equation represents a well-posed problem according to the accepted definition that has a unique, finite but unstable solution. In our geoid estimation we needed the downward continuation evaluated in Helmert's space and we use the Jacobi iterative inversion of the system of Fredholm's linear equations (ref. 463). Even though this approach gives reasonable results for the geoid computation, it is devoid of physical meaning. We wish to carry on with this research, adding some physically meaningful constraints (requirement of harmonicity, minimization of 2D Laplacean on the geoid, etc.) to add some physical sense to the solution.