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Progress Towards A Centimeter Level Orthometric Heights Using a Single GPS Receiver 4. GPS-leveling fitting of the geoid The computed gravimetric geoid needs to be fitted to local GPS-leveling data for operational GPS height use, to eliminate datum shift, residual long-wavelength gravity errors, and as well possible systematic errors in the levelling. The basic principle used here is to model the gravimetric and GPS geoid difference by a smooth function consisting of a trend function f (a polynomial) and a residual e', to be modelled by least/squares collocation 
4.1. GPS-levelling outlier rejection It was apparent that a number of outliers were present in the GPS leveling data. In the first step a linear regression was performed on the difference (trend function as a third order polynomial in N and E, no collocation), and the residuals used to reject outliers. The outlier detection was done for mainland Dubai and Hatta, separately. Table 6 shows the number of apparent outliers, based on the cut-off criterion.
For the final geoid fitting it was chosen to use a cut-off of 20 cm for Dubai and 24 cm for Hatta, in order not to loose data in regions with a poor gravimetric geoid (especially at the outer limits of the area). Fig. 3 shows the covariance function for the plane-fit detrended GPS geoid data. It is apparent that the residual errors are close to white noise.  Fig. 3. Covariance function of GPS geoid residuals after planar fit (DM data). 4.2. Final geoid computation For the final geoid computation, the residual e' is modelled by least-squares collocation, using a 2nd order Markov covariance function ("geogrid") C(s) = Co (1+ks)e-ks where k is a constant, determined by correlation length, and s the distance. The factor Co is determined from the data, whereas k and the apriori noise on the GPS leveling may be determined by the user. These factors effectively act like a smoothing parameter, making sure the final geoid on average fits the GPS data, but is not affected by individual, random errors. After a number of tests, a correlation length (x1/2) of 25 km and GPS geoid noise (sigma) of 0.035 m seemed to give good results. The polynomial trend function used was a constant, to prevent the geoid to diverge too much outside the area of GPS coverage. Table 7 shows the fit of the GPS leveling data to the tailored geoid. It should be stressed that the 3.5 cm is not the final accuracy of the geoid - the geoid will affectively average a large number of GPS data, and we believe that the collocation error estimate of 2 cm r.m.s. (1-sigma) is a realistic estimate for the error of the geoid in the Dubai main area? In Hatta the geoid is difficult to compute due to lack of gravity data in the neighbouring regions (Oman and other UAE emirates), and due to the apparent larger noise in the GPS leveling. The geoid might only be accurate to 5-10 cm. here. Fig. 4. GPS corrector signal to the gravimetric geoid. It is seen that the major corrections - except for a bias - is occurring along the edges (and at Hatta), in accordance with expectations.
 Fig. 4. Dubai geoid computed from gravity, GPS and leveling. Contour interval 20 cm. 5. Conclusions The outcome of this investigation is a detailed centimetres gravimetric geoid for Dubai Emirate. The geoid is recovered using1km x 1 km gravity data and 20m X 20m Digital Terrain Elevation Model as well as GPS and leveling data combined with EGM96 (360,360) spherical harmonics potential coefficient set. To compute the gravimetric geoid in Dubai Emirate, the modified Stokes's kernel is being used instead of the original ellipsoidal Stokes's kernel, to reduce truncation errors as the former tapers off more rapidly than the latter (the influence of distant gravity anomalies on local geoid heights is reduced). The reduction is proportional to the degree L of the satellite model being used to recover the long wavelength component of the geoid. To recover the long wavelength contribution of the gravimetric geoid, the EGM96 (360,360) model coefficient set complete to degree and order 360, has been chosen out of the variety of potential field models published so far, as probably the most recent solution available at the time of our computations. This implies that the smallest gravity field features represented in EGM96 (360,360) model have spatial extent of 0.50 spherical distance or 55 km. The short wavelength component of the geoid is recovered by means of mean gravity anomalies after subtracting the corresponding EGM96 (360,360) model values from gravity anomalies. The total gravimetric geoid was obtained from the summation of the long wavelength, geoid component and the short wavelength geoid components. This solution was referred to WGS-84 datum. The comparison of GPS/levelling geoid heights with the corresponding gravimetric values showed a reasonable agreement with RMS of 3-4 cm rms. Finally, to meet the 1cm geoid, which has been the goal of geodesists and geophysicists, the effect of the atmosphere, the topography and the ellipticity of the reference surface on the gravity as well as the indirect effect on the computed geoid has been taken into account. To verify our gravimetric solution, an independent geoid was derived from Global Positioning System (GPS) network established throughout the Emirate with its stations located at known benchmark heights.The derived Geoid model is precise enough to replace conventional levelling, particularly for third order levelling in large areas. In the main land Dubai the accuracy could be of the order of 1-3 cm on average. In Hatta region the accuracy ranges between of 5-10 cm. Still more information and data is necessary to improve the model in this part of the Emirate. We believe this Geoid model could meet the requirement of many potential users who would intend to convert GPS heights into their corresponding Mean Sea Level heights References
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