附有等式约束病态模型正则化解的单位权中误差无偏估计

利用平差参数间合理的等式约束虽能提升病态模型解的精度，但其本质仍是通过引入正则化参数来改善模型的病态性，由于改变了观测方程的结构，所得的估值残差及单位权中误差均有偏。针对这一不足，在病态模型正则化解的无偏单位权方差估计式基础上引入等式约束条件，根据约束正则化解的残差二次型期望公式，导出约束正则化解的无偏单位权中误差估计式，并用数值算例和病态测边网算例验证其正确性。结果表明，本文公式所估的单位权中误差精度优于传统公式所估结果。

Unbiased Estimation of Unit Weight Medium Error for Regularized Solution of Ill-Posed Problem with Equality Constraints

The accuracy of solution for ill-posed model can be significantly improved by introducing the reasonable equality constraint between the adjustment parameters. However, the essence is to improve the ill-posedness of model by introducing regularizated parameters, which results in the change of structure of the observation equation. Therefore, the solution of ill-posed model with equality constraints is biased and its residual is amplified accordingly, which results that the unit weight medium error estimated by traditional formula is biased. In this paper, the unbiased estimation of unit weight medium error of constrained regularizated solution is derived according to mathematic expectancy formula of residual matrix in quadratic form, which is based on the unbiased estimation of unit weight variance for regularizated solution. The correctness of the formula is verified by numerical example and ill-posed trilateration network example and results show that the accuracy of unit weight medium error estimated by formula derived by this paper is better than traditional formula.