基于虚拟观测的病态问题解法

在大地测量数据处理领域中,处理病态问题的主要方法有:岭估计方法、奇异值分解法(SVD)、Tik-honov正则化方法等,但是这些方法大多数是强调数学上的意义,没有充分联系大地测量的实际情况,因此不利于在测绘领域病态问题本质的理解和研究。为使病态问题的求解具有实际的物理意义,提出了基于虚拟观测的岭估计方法。该方法将先验约束条件作为一类互相独立的虚拟观测值,从而把病态问题转化为测量平差问题,然后运用Helmert方差估计法来确定岭参数。该方法还可以得到的参数之间的权矩阵,用它来代替虚拟观测值的权矩阵,重新对参数进行计算,则实现了该方法向广义岭估计的推广。实际算例分析的结果表明该方法不仅计算简单而且能保证结果精确。

A virtual observational approach to ill-conditioned problem

To resolve the ill conditioned problem exists in geodesy data processing,there are many methods such as ridge estimation,singular value decomposition,Tihonov regularization.Most of these methods pay attention to the mathematic theory while lack the true physical meaning.These lead to difficulties in understanding and study of the ill-conditioned problem.In order to set real meaning to the method for ill-conditioned problem,a new method of ridge estimation based on virtual observation is developed in this paper.The apriori information was taken as one class of virtual independent observations,and then the ill-conditioned problem was translated to a problem of adjustment.So the ridge parameter can be resolved using Helmert variance components estimation which is a very common method in adjustment.And the weight matrix between the parameters can be derived from this method.The new method also can be used in general ridge estimation when the weight matrix of the virtual observations was replaced by the weight matrix between the parameters.The result of the examples show the new method is much simpler and more effective than the traditional methods.