导线网方差分量估计的综合研究

提出方差分量估计的多余观测分量平均匹配公式;总结出导线网方差分量估计定权的基本规律,并给出其权增量循环算法和特征根法以及将两个实对称矩阵同时对角化的实用方法。

Synthetical Study toEstimate the Variance Components of Traverse Nets

This paper gives the function model to estimate the viarance components of traversenets, narrates the basic processes of viarance components estimate simply, and enumerates the formulae to estimate the viarance components in common use. Supposing every measured values have the same redundant observation components, a new simplified formula to estimate viarance components is advanced. In Tab. 1, the changing law of weights of observed values is depicted when calculating the traverse nets by variance components estimate, d01 and d02 stand for 021 and σ022 respectively, and P )and P ) stand for the weights of jth observed value of traverse angles and legsrespectively in the calculation ofith time. In Eqs. (19) and (20), P1 and P2 stand for weightsmatrix of observed values of traverse angles and legs respectively, p(0)2 ＝ P20 stands for the initialvalue ofP2, ki ＝ k can be calculated by Eq.(17).　　Because only the weights of traverse legs are changed in every circulate processes to estimatethe variance components of traverse nets, a circulate calculating method to use the increments ofweights is summed up,and the related formulae are Eqs. (22) to (25) ,as well as the whole circulate calculating processes are listed in Tab. 2. In the formulae or Tab. 2, V10 and V20 stand for thecorrections of traverse angles and legs calculated by initial weights matrix respectively, and theweights increments matrix Ap comes from Eq. (21). the circulate calculating method to use theweights increments has an obvious advantage, which is that the normal equations do not need constituted in every circulate processes.　　There are many continued multiplication and inversion and finding trace of matrices inHelmert's formula. These calculations not only are complicated but also need a lot of store spacefor computer. So this paper also advances the characteristic roots method to estimate the viarancecomponents of traverse nets. This method makes complicated calculations between matrices inHelmert' s formula become into the simple calculations between characteristic roots.Besides, thecalculation for characteristic roots is only needed once. In line with the same thought, Forstener' sformula can be simplified too. The calculated method about the characteristic roots can be foundfrom Eqs. (34)～(37).　　At the end of this paper, a calculated example is given. It is used to prove that the newmethod to estimate the variance components is effective. Advancing a new simplified formula andthe characteristic roots method for variance components estimate, summing up the basic law to calculate the weights of the observed values for traverse nets in the processes of variance componentsestimate and the method to calculate the variance components of traverse nets using the weightsincrements are the main achievements in the paper. Supplying a practical method to make two realand symmetric matrices become into two diagonal matrices together may be another achievementof this paper.