Downward continuation of gravity field

重力场向下延续

• 批准号: 8300-2013
• 负责人: Vanicek, Petr
• 金额: $ 1.84万
• 依托单位: University of New Brunswick
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2015
• 资助国家: 加拿大
• 起止时间: 2015-01-01 至 2016-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=568578
• 关键词: Downward; continuation; gravity; field

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.

重力是在地球表面观察到的，但对于大地水准面的计算，它需要在海平面上(在大地水准面上)：它必须通过地形质量“持续向下”到海平面上。在我们确定大地水准面的方法中，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)代表了“NT空间”中的情况。事实上，我们用大地水准面上的冷凝层代替了地形，并将这个新空间称为“赫尔默特空间”。因此，赫尔默特空间就是由冷凝层扩大的NT空间。在NT和赫尔默特空间中，向下的连续都是通过我们所能做到的质量尽可能少的空间的一部分进行的。 向下延续的过程与向上延续的过程相反。向上延拓是一种平滑操作，向下延拓是一种提升高频的操作。这从数字上来说就是麻烦：下行延续的过程必然是不稳定的。从数学的观点来看，无质量空间的向上延拓可以用经典的泊松积分来描述。反问题用Fredholm型第一类积分方程解的不稳定性来描述。然而，根据公认的定义，积分方程代表了一个适定的问题，它有唯一的、有限的但不稳定的解。 在我们的大地水准面估计中，我们需要在Helmert空间中计算向下延拓，并且我们使用Fredholm线性方程组的Jacobi迭代反演法。463)。即使这种方法给出了合理的大地水准面计算结果，但它缺乏物理意义。我们希望继续这项研究，增加一些物理上有意义的约束(对协调性的要求，在大地水准面上的二维拉普拉斯最小化等)。为解决方案增加一些物理意义。