Synthetical Study to Estimate the Variance Components of Traverse Nets

This paper gives the function model to estimate the viarance components of traverse nets,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 σ₀₁² and σ₀₂² respectively,and P _1j⁽ⁱ⁾ and P _2j⁽ⁱ⁾ stand for the weights of jth observed value of traverse angles and legs respectively in the calculation of ith time.In Eqs.(19) and (20), P ₁ and P ₂ stand for weights matrix of observed values of traverse angles and legs respectively, P ₂⁽⁰⁾= P ₂₀ stands for the initial value of P ₂,k_i=k can be calculated by Eq.(17). Because only the weights of traverse legs are changed in every circulate processes to estimate the variance components of traverse nets,a circulate calculating method to use the increments of weights 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, V ₁₀ and V ₂₀ stand for the corrections of traverse angles and legs calculated by initial weights matrix respectively,and the weights increments matrix Δ P comes from Eq.(21).the circulate calculating method to use the weights 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 in Helmert's formula.These calculations not only are complicated but also need a lot of store space for computer.So this paper also advances the characteristic roots method to estimate the viarance components of traverse nets.This method makes complicated calculations between matrices in Helmert's formula become into the simple calculations between characteristic roots.Besides,the calculation for characteristic roots is only needed once.In line with the same thought,Förstener's formula can be simplified too.The calculated method about the characteristic roots can be found from Eqs. (34)~(37).At the end of this paper,a calculated example is given.It is used to prove that the new method to estimate the variance components is effective.Advancing a new simplified formula and the 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 components estimate and the method to calculate the variance components of traverse nets using the weights increments are the main achievements in the paper.Supplying a practical method to make two real and symmetric matrices become into two diagonal matrices together may be another achievement of this paper.