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Use of Similarity Transformations to Improve GPS Heighting
4 Tests Carried out on Control Data
As previously mentioned the control GPS points are only located in the state of Johor, whereas the comparisons mentioned in Sections 2.1, 2.2 and 2.3 encompass the whole of the Peninsular Malaysia. So as to prevent any confusion the final comparisons will only include the 25 points for the state of Johor. Before any schemes were tested all the GPS points were screened to make sure they all fall within three standard deviations of the mean of differences between NEGM96 derived using EGM96 and Nh-H derived using h and H. This is a quick and dirty data snooping technique and since there were no outliers there was no statistical reasoning to exclude any points although when compared the residual can be large. Table 1 shows the statistics of the comparisons from the residuals concerned with the above discussion. Table 1: Statistics of the residuals for the comparison of the outlier test conducted between NEGM96 and (h-H) ie: [(h – H) – NEGM96], (All units are in cm).
The largest residual is 0.653m at TG05, located in the far north of the state, there are also other points indicating large residuals and if these points along with the respective residuals are plotted as in Figure 1, it is possible to identify visually which points could be excluded in order to improve the results of the corrector surfaces. As better data in the adjustment will give better-adjusted coefficients, however there is the risk of excluding perfectly good data, for this study we have included all data, except for some data excluded for the purpose of external assessment of the adjustment of the coefficients.  Figure 1: Location of GPS points (25) showing residual contour plot [(h – H) – NEGM96], contour interval 10 cm It is evident in Figure 1, that the further north the points progress the larger the residuals [(h – H) – NEGM96] become, this perhaps is indicative of some systematic bias in the Peninsular Malaysia vertical datum, or it could just show there are problems in the adjustment of the vertical datum, this however is not investigated here. 5 Results from Comparisons of Corrector Surfaces Two types of comparisons are made, (1) all 25 GPS data points are used in the adjustment and the resulting residuals, Eq (3) are investigated, (2) 3 GPS points are excluded from the adjustment, namely GP48, GP58 and GP51 and are compared with the adjusted coefficients, this provides an external comparison of how well the coefficients are able to account for any deformations and dissimilarities there are between the GPS, vertical datum and co-geoid models used. Adjustments will be made using both available co-geoid models (CG1 & CG2) for Peninsular Malaysia. 5.1 Adjustment Results of Correction Surfaces Comparisons are carried out using the following for original misclosures and adjusted residuals respectively in Eq (12) and Eq (13). 
In the following tables No CS refers to no correction surface is applied and the original misclosures statistics are listed using (Eq 12), where as for the correction surfaces Eq (13) is being employed and the statistics all represent those of the residuals from the adjustment. Table 2 shows the statistics for CG2 and the results of the adjustments. Table 2: Statistical summary using CG2 to derive the coefficients for the similarity transformation and polynomial fitting at 25 GPS points.
Table 3: Statistical summary using CG1 to derive the coefficients for the similarity transformation and polynomial fitting at 25 GPS points.
Evidently, from Table 2, it is clear that the corrector surface CS8 has managed to remove the bias term, as do all corrector surfaces, however CS8 has achieved the smallest standard deviation of the adjusted residuals. Using no corrector surface (No CS) shows how there is still a great amount of variability between the different components (ie: h, H and N) for Peninsular Malaysia. Table 3 shows the same results only this time for CG1. The results from CG1 clearly improve, although only slightly but it seems that CG1 is a better fit to the local GPS data than that of CG2, as is indicated by its reduced mean (-1.8 cm). |