最优Tikhonov正则化矩阵及其在卫星导航定位模糊度解算中的应用

Optimal Tikhonov Regularization Matrix and Its Application in GNSS Ambiguity Resolution

首先,用贝叶斯(Bayes)统计理论的观点,把未知参数看作随机变量,引入未知参数的无信息先验分布函数,从数学上推导了均方误差最小意义下的正则化矩阵;然后,结合最优正则化矩阵和快速截断奇异值算法,提出了一种新的正则化方法;最后,探讨了新方法在全球卫星导航系统(Global Navigation Satellite System, GNSS)模糊度解算中的应用。通过一组GNSS模糊度解算实验,比较了最小二乘(least squares,LS)方法、L曲线岭估计和新方法的性能。结果表明,新方法解算成功率略高于L曲线岭估计,远高于LS方法;计算耗时略大于LS方法,远小于L曲线岭估计。

This contribution can be mainly divided into 3 aspects:(1) Based on Bayesian theory, unknown parameters are treated as random varies and their non-informative prior distribution function is introduced. Mathematical analysis is carried out to drive the optimal Tikhonov regularization matrix in the sense of minimizing the mean square error(MSE) of the solutions.(2) Combining the efficient truncated singular value decomposition(eTSVD), a new regularization method is proposed.(3) Glo-bal Navigation Satellite System(GNSS) ambiguity resolution application of the new method is discussed. Least squares(LS) estimation, ridge estimation of L curve and the new algorithm are compared by a group of GNSS ambiguity resolution experiments. The results show that the MSE of the new algorithm is slightly smaller than ridge estimation of L curve and much smaller than LS, however, the computational cost of the new algorithm is slightly more than LS but much less than ridge estimation of L curve.