Application Research On Total Least Squares In The Coordinate System Transformation Under Different Models Influenced By Random Errors

Coordinate system transformation is widely used in geodesy, for example, the coordinates often needs to be converted into the national coordinate system or the local coordinate system through the GNSS positioning technology. The core problem is to calculate transformation parameters. It is often handled by the classical least squares(LS) method. However, in practical engineering measurement, coordinates of a coincidence point in the two coordinate systems are obtained by measuring or calculating, and contain errors inevitably, which leads to the coefficient matrix of the error equation contaminating errors. And LS is no longer the optimal unbiased estimation method. In order to make up for the theoretical defect of LS method without considering the error of coefficient matrix, the total least squares method(TLS) attempts to obtain more reliable estimates considering both the coefficient matrix error and the observation vector error.However, whether TLS method can get more effective solution of transformation parameters relative to LS?In different applications, some scholars at home and abroad conclude that the solution of TLS with smaller mean square error of unit weight is more close to the true values of the parameters than that of LS. However, in some applications such as data-fitting, parameter estimates of TLS have tiny discrepancy in numerical value and precision with LS. In addition, calculation methods of mean square error of unit weight about LS and TLS are different, so it is not reasonable to measure precision. Based on the above analysis, it is still urgent and necessary to research the efficiency and applicability of TLS and LS in coordinate system transformation.Firstly, the specific example of four-parameter coordinate system transformation model with 4 coincidence points, in different kinds of error model by random errors(EIV, EIOO,and EIVO), is taken to analyze the similarities and differences between LS and TLS with mean square of residual true error(MSRTE).Secondly, the effectiveness of TLS and LS is analyzed with simulation experiments under three kinds of error models by random errors which take five different kinds of four- and seven-parameter coordinate system transformation models with different numbers of observations(the number of coincidence points is 3, 6, 9, 12 and 15, respectively) and different distributions of error(mean square error of unit weight is 1.0 cm and 3.0 cm and 5.0 cm) as examples.Simulation experimental results show that under the EIV model, the EIVO model and the EIOO model, TLS is slightly inferior than LS in the four-parameter models, and TLS is not better than LS in the seven-parameter models. In short, TLS has no advantages compared to LS in the four- and seven-parameter coordinate system transformation, and LS is more effective than TLS.