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Use of Similarity Transformations to Improve GPS Heighting
2 Evaluation of Geoid Models and GPS Data
DSMM kindly provided data for the southern region of Peninsular Malaysia, containing geodetic latitude, longitude, orthometric height and ellipsoidal height. The geoid heights were provided from independent sources namely (Kadir et al., 1999), (Vella et al 2003) and EGM96 geopotential model coefficients by (Lemoine et al, 1997). 2.1 South East Asian Co-Geoid This co-geoid is a high resolution and high precision model computed for the whole of South East Asia (Kadi et al, 1999). The EGM96 geopotential model truncated to degree and order 70 was combined with surface gravity data and a modified Stokes’s kernel was used in the computations. Comparisons of the gravimetric geoid with the GPS/Levelling derived geoid/ellipsoid separations at 140 points throughout Peninsular Malaysia show that the absolute agreement with respect to the GPS/Levelling datum is generally better than 40 cm RMS, which demonstrates an improvement over EGM96 and OSU91A geoid models for the same points. For ease of reference this co-geoid model will be referred to as CG1. 2.2 Peninsular Malaysia Co-Geoid Free-air gravity anomalies are usually found to be locally correlated with the elevation of observing points. The correlation between the observed height of the free-air anomaly and its corresponding value is derived using least squares. It is shown that a minimum height of 400 m yields the best correlation (R=0.886) between the free-air anomaly and the height and therefore this height of 400 m is used as a minimum height to interpolate data from the GTOPO30 global Digital Elevation Model (DEM). From the DEM, all heights with a height equal to or greater than 400m, are used to derive anomaly-height correlated values. When these values are combined with the original gravity data set and gridded, the gridding is better controlled, as results over Peninsular Malaysia show. Using the 1D FFT and stokes integral with no modification the co-geoid is computed, (Vella et al., in press). The resulting free-air co-geoid model is compared to 143 GPS points. The model attained an RMS of 46.1 cm when compared at the GPS points showing an improvement over EGM96. For ease of reference this co-geoid model will be referred to as CG2. 2.3 EGM96 Comparisons The NASA Goddard Space Flight Center, The National Imagery and Mapping Agency (NIMA) and the Ohio State University (OSU) have collaborated to produce EGM96, an improved degree 360 spherical harmonic model representing the earth’s gravitational potential (Lemoine et al, 1997). The comparisons between the EGM96 model and the difference between the orthometric height and ellipsoidal height show that the RMS is 51cm at 143 GPS points. 3 Overview of Modelling Considerations for the CS Presented herein are the basic models describing mathematically the different schemes used in the adjustment and transformation results. There are other models and methods which account for local deformations in the geoid and other error sources, that can be used for corrector surfaces as demonstrated by (Zhong, 1997), (Featherstone, 1998) and (Zhiheng and Duquenne., 1996). 3.1 Similarity Transformations The basic model used is of a modified form of Eq. (1) as follows: 
where hi, Hi, and Ni are as previously described, x is an n x 1 vector of unknown parameters, ai is an n x 1 vector of known coefficients, and vi is the residual random noise term, (see e.g. Kotsakis and Sideris 1999). As stated in the introduction test are conducted on three similarity transformation schemes and four polynomial schemes, all schemes are solved using parametric least squares techniques according to Eq. (3) but each with differing observation Eq.s, for obvious reasons. It has been widely held that the four parameter model is best suited to this type of modelling, this however is not necessarily the case as the results will show, however it is important to state clearly all the models and there mathematical representation. The four parameter (Eq. 4) model from now on called CS4, is an approximate similarity transformation model describing the geoid undulation transformation; this scheme is adequately discussed in Heiskanen and Moritz (1967, Sect. 5-9): 
The five parameter scheme (CS5), a rigorous similarity transformation model is also discussed in Heiskanen and Moritz and is described as follows: 
The eight parameter scheme (CS8), which is a rigorous non-rigid similarity transformation model, is described as follows: 
where the quantity Wi is given by the relationship 
and the quantities f, a and e in the above formulas correspond to the flattening, the semi-major axis and the first eccentricity, respectively, of the reference ellipsoid (either the ellipsoid used for the GPS heights, or the ellipsoid used for the gravimetric geoid model). 3.2 Polynomial Models The first degree polynomial (CS3) is a three parameter scheme described as follows: 
The second degree polynomial (CS6) is a six parameter scheme described as follows: 
The third degree polynomial (CS10) is a ten parameter scheme described as follows: 
Fourth degree polynomial (CS15) is a fifteen parameter scheme described as follows: 
In the above formulas jo and lo denote the average latitude and longitude, respectively, of the test area The schemes describing CS4,CS5,CS8,CS3,CS6,CS10 and CS15 are applied to all points in the network, in this case 25 points and are operated on by least squares in order to minimise the residuals, thus giving the best estimate of the different coefficients involved. The adjustment carried out of the coefficients is in no way optimal as pointed out by (Kotsakis and Sideris, 1999), there is no weighting of the residuals done, thus leaving some questionable doubt as to which term is contributing to the residual, whether it is the ellipsoidal height, the orthometric height or the geoid/ellipsoid separation, it may even be a combination of some or all of these terms. So although some of the schemes may be rigorous, the solution is in no terms a rigorous solution, in this case. |