Scaled Unscented Transformation of Nonlinear Error Propagation: Accuracy, Sensitivity, and Applications

The scaled unscented transformation (SUT) with scaled symmetric sampling strategy is introduced into Geomatics and expanded. Unlike the Taylor series expansion, this method treats nonlinear error propagation without a derivative calculation. The formula of variance estimated by the SUT is expanded to two types of second-order terms. Three new theorems are proposed to describe the accuracy of the variance estimated by the SUT under different conditions. The comparison of the SUT and the unscented transformation (UT) with symmetric sampling strategy is discussed. According to the ratio of disturbance defined in this paper, the SUT is found to be insensitive to different matrix decompositions. The effects of an inaccurate mean of a random variable on the mean and variance estimated by the SUT are expressed by second-order accurate formulas. The accuracy of variance estimated by the SUT is found to change according to the bias of change of the parameters. Based on theoretical analyses, a modified SUT algorithm and a systematized SUT algorithm are proposed to strengthen its practicability. Five examples are used to support the proposed theories and show the applicability of the SUT in statistics calculation and bias correction for nonlinear function of Geomatics.