用于重力向下延拓的离散化方法(英文)

DISCRETIZATION SCHEMES USED IN DOWNWARD CONTINUATION OF GRAVITY

泊松积分通常用于把重力值从地球表面转化到大地水准面 (即称之为重力向下延拓 )的过程中。由于这是一个反问题 ,一些数字技术比如将积分离散化为一个线性方程组是必需的。目前 ,已经提出了两种离散化方案(单点和双重平均 )。虽然这两种方案在数学上都是可解的 ,但用它们处理相同的输入地面重力值时 ,却得出不同的大地水准面上的重力值。这种差异的产生是由于对泊松核函数的不同离散化方法造成的 ,而且这一问题尚未引起足够的关注。实际上 ,数学上的可解性并不能保证得到正确的解。问题在于此方程组是否构成得很好 ,或者说 ,离散化是否合理。因而本文研究泊松积分的离散方法。为此目的 ,本文提出一个单点平均的方案来对泊松积分进行估值。单点平均方案基本上与双点平均方案是相同的 ,但它在计量上更为简单 ,因为其计算工作大为减少。比较单点和双点平均方案后表明 ,对于一个有限的地表网格范围情况 ,单点离散方案会带来严重的理论问题 ,因为会相当大地低估大地水准面上的重力值甚至在极端情况时会给出不正确的结果。我们得到 。

The Poisson integral is usually used to transfer gravity from the topographic surface to the geoid in a so called downward continuation of gravity. Since it is an inverse problem, numerical techniques, such as discretization of the integral into a system of linear equations, are necessary. Two (point and double mean) discretization schemes of the Poisson integral have been proposed to date. Although the two schemes are mathematically solvable, they produce different gravity on the geoid for the same input data (gravity on topography). This discrepancy arises because of different discretization techniques of the Poisson kernel; still, this problem has not received adequate attention. Actually, the mathematical solvability does not ensure a correct solution. The question is whether the system is well structured, or, whether the discretization is reasonable. Methods to discretize the Poisson integral are investigated in this study. For this purpose, a new single mean scheme is presented to evaluate numerically the Poisson integral. The single mean scheme is basically the same as the double one, but it is numerically simpler since it greatly reduces numerical effort. A comparison between the point and mean schemes shows that, for a limit topographical grid size, the point discretization scheme results in a serious theoretical problem since it greatly underestimates gravity on the geoid, and even gives incorrect results for extreme cases. A careful construction of the coefficient matrix for the discrete system is much more important than using point gravity as input.