(PDF) An automated height transformation using precise geoid models

.Scientific Research and Essays Vol. 6(6), pp. 1351-1363, 18 March, 2011 Available online at http://www.academicjournals.org/SRE DOI: 10.5897/SRE10.1119 ISSN 1992-2248 ©2011 Academic Journals Full Length Research Paper An automated height transformation using precise geoid models Bihter Erol Department of Geomatics Engineering, Civil Engineering Faculty, Istanbul Technical University, Maslak 34469, Istanbul, Turkey. E-mail:

. Tel: +90 212 285 3821. Fax: +90 212 285 6587. Accepted 4 March, 2011 In most of the countries, national height systems are referenced to the mean sea level (geoid) whereas using GPS technique in positioning provides ellipsoidal heights. However, these GPS ellipsoidal heights h can be transformed into the orthometric heights from the geoid H depending on a simple relation among the heights, and this transformation requires to know the geoid undulations N ( H = h − N ). The methods for deriving the geoid undulation at a point are various and practicality of the method is one of the major concerns as well as its precision. In this study, an experiment on calculating local geoids using an algorithm based on surface polynomials with weighted corrections was performed and a program, which computes the geoid undulation at a point in the local area using the model was developed, and an executable file of the program was inserted into the commercial GPS data processing software as “height transformation module”. With this ad hoc module, the software provides the orthometric heights to the user at the same time with the three dimensional geodetic coordinates of the points. Hence, an automated height transformation, without user intervention, simultaneously with GPS data post-processes is provided. The experiment data includes dense and well distributed 301 GPS/levelling benchmarks in the west of Turkey. All the decisions regarding the data preparation and modelling process using polynomials (such as detecting the blunders in the data, determining the optimal degree of the polynomial and testing significance of the polynomial coefficients etc.) are critical in practice and were dealt in the study as well. The orthometric heights of the test benchmarks were provided with 3.5 cm accuracy by the automated height transformation module. Key words: Height transformation, geoid modelling, orthometric height, GPS/levelling, surface polynomials, programming, Turkey. INTRODUCTION The use of heights, such as orthometric heights that are In order to counteract these drawbacks and because of connected to the Earth’s gravity field is important in many the wide and increasing use of global navigation satellite fields, not only in all Earth sciences but also in other systems (GNSS) in all kind of geodetic and surveying applications such as cartography, oceanography, civil applications, the modern technique called GNSS levelling engineering, hydraulics, high-precision surveys and last can be considered as an alternative for practical height but not least geographical information systems. determination. In GNSS levelling, the orthometric heights Traditionally, these heights are determined by combining H based on the geoid which is approximated by mean geometric levelling and gravity observations with sea level are determined by converting the ellipsoidal millimetre precision in smaller regions. This technique, heights h with respect to a reference ellipsoid by applying however, is very time consuming, expensive and makes the fundamental equation H = h − N, instead of levelling providing vertical control difficult, especially in areas and gravity observations (Heiskanen and Moritz, 1967; which are hard to access. Another disadvantage is the Hofmann-Wellenhof and Moritz, 2006). In the formulation, loss of precision over longer distances since each height N is the deviation between the two reference surfaces, system usually refers to a benchmark point close to the geoid and reference ellipsoid, along the ellipsoidal normal sea level which is connected to a tide gauge station and called geoid undulation (or height). Hence, representing the mean sea level. determining the geoid height with an accurate geoid 1352 Sci. Res. Essays model make GNSS levelling possible, that is a simple polynomials are obviously easy models to determine and and cost-effective way of obtaining the orthometric use, comparing to the other more complicated ones such heights (Gilliland, 1986; Schwarz et al., 1987; Zilkoski as method of finite elements, geostatistical kriging, least and Hothem, 1989; Gilliland and Jaksa, 1994; Ananga squares collocations, artificial neural networks etc. (Çepni and Sakurai, 1996; Collier and Croft, 1997; Featherstone and Deniz, 2005; Kavzaoğlu and Saka, 2005; Kutoğlu, et al., 1998; Lee and Mezera, 2000; Erol et al., 2008; 2007). Gucek and Bašić, 2009). Since the ellipsoidal heights are Therefore this investigation on developing an nowadays mostly obtained from GPS technique and the automated height transformation utility integrated on GPS ellipsoidal heights of the benchmarks used in this study processing software employed surface polynomials with are provided by GPS, GNSS term is replaced by GPS in additive corrections approach. The data of numerical the sequel. evaluations consist of dense and well distributed 301 The methods for geoid modelling are various and each benchmarks of geodetic GPS/levelling network in the method has its merits and limitations. The studies on west of Turkey. In the content, detailed discussions on determining a precise geoid model for Turkey has been the critical decisions of both data preparation and started in 1970’s and various regional geoid models modelling phases of the investigation, which directly based on different methods have been computed since affect the quality of final model and thus the accuracy of then (Ayan, 1976; Ayhan, 1993; Ayhan and Kilicoglu, the transformed heights, are also included. At the end, a 1993; TNFGN, 2002; TNGC, 2003). In 2003, the General program, which applies the local geoid model to derive Command of Mapping released the most recent regional geoid undulations at the points, was developed in Visual gravimetric geoid model of Turkey, TG03. This is a hybrid C environment, and added to a commercial GPS data geoid model which was gravimetrically determined and processing software to automate the height transforma- fitted to the regional vertical datum at 197 homoge- tion without user intervention and to derive the point neously distributed GPS/levelling benchmarks throughout heights in regional vertical datum simultaneously with the the country, by the adjustable tension continuous computation of their three dimensional GPS coordinates. curvature surface gridding algorithm (TNGC, 2003). Hence the possible personal mistakes, which may cause However, the absolute accuracy of TG03 is reported as gross errors, are cancelled out from the transformation 10 cm in the central territories and 20 cm along the coast- results, and also the speed of transformation process is lines and boundaries of the country (TNGC, 2003) that considerably increased. In the results of numerical the decimetre level accuracy of transformed orthometric experiment, the comparisons at the test benchmarks heights stay rough in many applications (Erol and Çelik, provided a 3.5 cm accuracy of the transformed ortho- 2004; Erol et al., 2005a, b). metric heights using the automated height transformation Therefore, because of the accuracy concerns in GPS module. levelling, in Large Scale Maps and Spatial Data Production Regulation of Turkey (legalized in 2005) (LSMSDPR, 2005; Deniz and Çelik, 2008), the DATA AND METHODOLOGY determination and use of GPS/levelling surface type In relatively small areas, a geometrical method for deriving geoid models for height transformation is encouraged in GPS/levelling surface (so called local geoid) based on bivariate respectively small areas where the dense and precise polynomial equations in various orders is often used (Yilmaz, 2005; data is available (LSMSDPR, 2005). Local GPS/levelling Kavzaoğlu and Saka, 2005; Stopar et al., 2006; Kutoğlu, 2007; geoid models are often used in Turkey and provide a Deniz and Çelik, 2008; Erol et al., 2008). The solution can also be practical and fast solution to the height transformation performed using either weighted average interpolators or since the absence of precise regional geoid yet in the multivariate regression equations in numerous forms, which are not going to be mentioned here. The geometrical method provides a country (Ayan et al., 1996a; Ayan et al., 1996b; Ayan et practical and fast transformation of GPS heights where a regional al., 1999; Çelik et al., 2002; Ayan et al., 2001; Ayan et al., precise geoid model is not available. The data used in numerical 2006). tests on height transformation using geometrically derived local The accuracy of local GPS/levelling geoid models as geoid are 329 (and 28 of these benchmarks were removed as height transformation tool is restricted by many factors, blunder in data screening) GPS/levelling benchmarks of Đzmir geodetic reference system 2001 (IzGRS01) network (Ayan et al., such as the data accuracies, the density and distribution 2001). of reference benchmarks, modelling methods etc. There These are the common points of C1, C2 and C3 orders GPS are many researches, done on the effects of data quality, benchmarks of Turkish National Fundamental GPS Network the density and distribution of reference benchmarks in (TUTGA99A) and high order levelling benchmarks of Turkish local geoid modelling, so far and can be found in the National Vertical Control Network (TUDKA99). Thus, the Helmert literature (Featherstone et al., 2001; Fotopoulos et al., orthometric heights of the benchmarks are in TUDKA99 Datum and their absolute accuracy is 2.5 cm. GPS coordinates of the 2001; Fotopoulos et al., 2003; Fotopoulos, 2003; Yilmaz, benchmarks refer to the ITRF96 datum. The accuracies of the GPS- 2005; Erol et al., 2006; Erol, 2008). However, this study derived coordinates are 1.5 cm and 2.3 cm in the horizontal and concentrated mainly on the role of practicality and vertical directions (Ayan et al., 2001). The network area covers 50 applicability of a local geoid model in the fast solution of × 45 km2 and the density of the benchmarks is around 1 height transformation problem. In this manner surface benchmark per 8 km2 with relatively homogeneous distribution Erol 1353 Figure 1. GPS/levelling benchmarks, Đzmir, Turkey. (Figure 1). The topographic heights in the region range from the bigger than three times of standard deviations of the residuals (3σ sea level to 1500 m. test, with 99.7% confidence interval), as having blunder (Sen and Srivastava, 1990; Draper and Smith, 1998; Fotopoulos, 2003; Surfer, 2009). Data screening In preparing the reference data for geoid modelling, first of all Modelling local GPS/levelling geoid with surface screening the data and detecting the blunders possibly contained polynomials and additive corrections by the data is very essential step and affects the accuracy of the final model very much. In many of the blunder detection In modelling local GPS/levelling geoid with geometric approaches, cross validating the reference data is a very practical approach, a geoid reference benchmarks network having method which localizes the blundered data quickly. In the cross validation procedure, the relative quality of the observations at let’s coverage of entire area is constituted. The geoid say given n observation locations (the reference data set) are reference benchmarks are generally selected from the assessed by comparing the observed value with the interpolated common points of C1, C2, C3 order GPS benchmarks st nd value from the surrounding observations (residual = interpolated and the 1 and the 2 order levelling network points, and value − observed value). must have homogeneous distribution at the characteristic In the algorithm, the errors are calculated by removing the first locations of topography. The density of the reference observation from the data set, and using the remaining data and the specified algorithm to interpolate a value at the first observation benchmarks is suggested to be 6 benchmarks per 20 2 2 location. Then, the first observation is put back into the data set and km , and an additional bench-mark for each 15 km en- the second observation is removed from the data set. Using the largement of the model area, by LSMSDPR (2005). With remaining data (including the first observation), and the specified existing n reference benchmarks having GPS ellipsoidal algorithm, a value is interpolated at the second observation and levelling heights (and hence with known geoid location. Using the known observation value at this location, the interpolation error is computed as before. heights: NGPS/lev. = hGPS − Hlevelling ) in a local area, the The second observation is put back into the data set and the general equation of polynomial interpolation to estimate process is continued in this fashion for the third, fourth, fifth GPS/levelling geoid heights at unknown points in the observations, etc., all the way through up to and including observa- area can be given as: tion n. This process generates n residuals, which may provide a measure to assess whether the observation at a location has L M blunder or not. Thus, cross validation process can be summarized N ( u, v ) = ∑ ∑a mn u mvn (1) in four steps: i) selecting an interpolation method (such as inverse m= 0 n=0 distance weighting, triangulation with linear interpolation, Kriging, polynomial etc.), along with all of the defining parameters, ii) for where u and v represent the position coordinates, amn each observation location, interpolating the value using the neighbouring data, but not the observation itself, iii) computing the symbolize the polynomial coefficients, and L is the resulting errors (residuals), iv) assessing the quality of the degree of the polynomial. The position coordinates can observation using a pre-defined statistical criteria for the residual at be constituted in various ways, and in this study they are each data point, e.g. deciding the observation, having the residual obtained from the ellipsoidal geographical coordinates as 1354 Sci. Res. Essays follows: additive corrections ( dN P ) calculated with weighted averages of geoid undulation residuals at the u = k (ϕ − ϕo ) , v = k ( λ − λo ) (2) neighbouring geoid reference benchmarks can improve the precision of the geoid undulation at the calculation where ϕo and λo are the arithmetic averages of the point: latitudes and longitudes of the data set, k = 100/ρo is scaling and unit adjustment factor. The polynomial V1 V V  coefficients are determined according to Least Squares S 2 + 22 ... S P− 2 ∑  S 2   Adjustment (LSA) method. The summation of observation dN P = P −1 = (5) (Ni) and its correction (residual, Vi) at each reference 1 1  1  benchmark is described with the function of unknowns as + S P2 −1 S P2 − 2 ... ∑  S 2  follows: where Vi is the geoid undulation residual at the Ni + Vi = a00 + a10u + a11v corresponding reference benchmark, SP-i is the distance th + a20u 2 + a21uv + a22v 2 between the calculation point P and the i reference benchmark in km, and hence dNP is the additive + a30u 3 + a31u 2 v + a32uv 2 + a33v3 (3a) correction to the geoid undulation at point P. At the case study, the S distance was limited with 3 km, and the only + a40u 4 + a41u 3v + a42u 2 v 2 + a43uv 3 + a44 v 4 reference benchmarks, staying in the circle having the ... computation point in the center and the radius of 3 km, were considered in calculation of the additive corrections. N +V = A X (3b) The distance limit can be decided considering the topo- graphic character of the area. With practicality concerns of height transformation using the local geoid model, the  N1  V1  1 u1 v1 ...  a00  residuals at the geoid reference benchmarks were      ... a  derived and included in an input file of the developed  N 2  V2  1 u2 v2   10  program codes. .  + .  =  . . . ...  .  (3c)         .  .   . . . ...  .  Testing the polynomials for an optimal geoid model  N i  Vi  1 ui vi ...   amn      One of the main difficulties of using polynomial interpo- lation is determining the optimal form of the model having and hence the unknown polynomial coefficients (amn as an appropriate degree and significant coefficients, as it the elements of X vector in Equation 4a) with their defines the accuracy of the approximation. Whilst the use covariance information QXX (in Equation 4b) are simply of a low-degree polynomial usually results in an insuffi- determined according to LSA principles: cient or rough approximation of the surface, the use of a higher-degree function may produce an over fitted −1 X = ( AT A) AT l (4a) surface. After determining the model with its coefficients using LSA (Equation 3), it is tested with statistical tests to assess its performance and to select the best model for −1 the data, and these tests are quite arbitrary. In this study, QXX = ( AT A ) (4b) the test procedure as suggested by Fotopoulos (2003) was adopted. Figure 3 summarizes the evaluation of the where A is called coefficients matrix and ℓ is the polynomials in various orders, determined in the result of observations vector of which elements are the geoid LSA, for determining an optimal model of geoid data. heights (Ni). The degree of the polynomial is one of the In the evaluation of the polynomials, at first, testing the critical parameters which should be decided in local geoid statistical significance of the polynomial coefficients is modelling using surface polynomials. Figure 2 shows critical, since the insignificant parameters may bias samples of surface plots with varying polynomial degrees others in the model. With the purpose of significance test from the first to the fourth degree (Fotopoulos, 2003). of the model parameters, F-test with the null hypothesis Decision procedure of an optimum degree of the H0 : X = 0 and the alternative hypothesis H1 : X ≠ 0 was surface polynomial will be discussed with details applied (Draper and Smith, 1998; Koch, 1999; subsequently. Fotopoulos, 2003). The F-statistic is used to verify the After calculating geoid undulation at a new point using null hypothesis and computed as a function of the derived surface polynomial (in Equation 3a), the observations (Dermanis and Rossikopoulos, 1991): Erol 1355 Figure 2. (a) First, (b) second, (c) third and (d) fourth degree surface polynomial fit samples. Determining the model with LSA: ℓ+V = N+V = AX Testing parameter significance: Empirical tests: Comparing the models using empirical test results: • at the reference benchmarks: F ≤ Ft α,r or F > Ftα,r • at the test points: deciding the optimal V = Nmodel − NGPS/lev. polynomial model Figure 3. Performance assessment of the polynomial model. X iT QX−1i X i X i where X i is the estimate of the parameter vector, QX i X i F= 2 (6) 2 t σˆ is the cofactor matrix of X i , σˆ is a-posteriori variance 1356 Sci. Res. Essays Table 1. The statistics of geoid heights and residuals before and precision of the model. Therefore, the tests with the after removing blunders in the data. residuals at the independent test benchmarks result more objective and realistic measures on the accuracy or Reference data (meter) prediction capability of the model. Only the handicap with Min Max Mean σ selecting homogeneously distributed test points among Before (329 BM*) 37.10 38.81 38.07 0.18 the data may distort the densification and homogeneity of After (301 BM) 37.59 38.72 38.06 0.16 geoid reference benchmarks, used for determination of the model. In this case, cross validation can be used to Geoid height residuals (centimeter) derive a realistic criterion for the performance of the Before (329 BM) -69.5 109.1 0.1 11.6 polynomials, and the estimated geoid height values in the After (301 BM) -8.9 10.4 0.0 3.0 results of iterative procedure of cross validation at each benchmark can be compared with the known geoid *BM: Benchmark heights. In the empirical tests of the polynomials with cross-validation, considering the point sets, consist of few benchmarks, instead of single point at each iteration step is recommended to reduce the correlation of the results. Table 2. The statistics of geoid height residuals at the cross- validated benchmarks for the polynomials in varying degrees (centimeter). CASE STUDY: HEIGHT TRANSFORMATION WITH Min Max Mean σ GPS/LEVELLING GEOID MODEL First degree -43.3 33.2 0.0 13.5 The determination and use of a GPS/levelling geoid Second degree -28.4 18.9 0.0 8.4 model for height transformation in the local area in the Third degree -24.2 14.1 0.0 5.3 west of Turkey is explained using the reference GPS/ Fourth degree -15.2 18.5 0.0 4.6 levelling network of the case study. Data preparation for Fifth degree -10.6 9.1 0.0 3.5 the determination of the geoid model constitutes the Sixth degree -19.5 12.1 0.0 3.3 blunder detection and format arrangements of the input data files, including the matrices for LSA calculation of the polynomial coefficients. In the data screening and factor, t is the number of parameters tested. The null blunder detection stage, 28 of the 329 reference α α benchmarks were detected as blunders and removed hypothesis is accepted if F ≤ Ft , r , where Ft ,r is obtained from the data. from the standard statistical tables for a confidence level Figure 4 visualizes the GPS/levelling surface before α and degrees of freedom r, which means the and after removing the blunders from the reference data corresponding parameters of the test are insignificant and with cross-validation, and absurd changes like erections deleted from the model. If the contrary is true and and collapses stem from the blunders in the reference F > Ftα,r is fulfilled, then the tested parameters remain in data were disappeared after removing the blunders and hence the geoid surface became smoother. Table 1 the model. F-test can be applied to the parameters of the shows the data statistics before and after blunder model with stepwise algorithm based on backward detection, where the third order polynomial was used and elimination and forward selection procedures. The test the geoid height residuals of the original data and the starts with the lowest degree form of the model (forward polynomial model at the cross validated benchmarks are selection), and the parameters to be tested are selected also included in the table, for the data sets, consisting one by one or a few at a time and examined for 329 and 301 benchmarks, respectively. significance. The backward elimination is embedded in After preparing the reference geoid data, the regression the stepwise algorithm. After removing insignificant equations in the form of bivariate orthogonal polynomials parameters in the result of the statistical test and in varying degrees up to six were calculated using LSA determining the final form of the polynomial model, its method and tested with the procedure summarized in performance is tested empirically, considering the Figure 3. The basic statistics of the geoid height residuals residuals at the benchmarks of the network at the cross-validated benchmarks for each polynomial ( V = N model − N GPS / lev . ). Among the determined are shown in Table 2, that provide a comparison among polynomials in various degrees of expansions, the model the polynomial models for geoid data. Another having the smallest residuals can be used as the most comparison among the polynomials was done 2 appropriate model for the geoid data. considering the coefficients of determinations, R , which However, it should be noticed that the empirical tests of provides a measure on the goodness of the parametric 2 the models using the residuals at the reference model to fit the data (0 ≤ R ≤ 1). The equation to derive benchmarks introduce optimistic measures on the the coefficient of determination is as follows: Erol 1357 Figure 4. GPS/levelling surface before and after removing the blunders in the data. j 2 and in the equation, j is the number of observations, ˆl i is ∑(l ˆ i −l i ) the computed value of the geoid height using the model R2 = 1− i =1 (7) j ( ˆl = N ) , and l is the mean value of the observations, ∑(l 2 − l) i model i l = NGPS /lev. (Sen and Srivastava, 1990). i =1 1358 Sci. Res. Essays 1.00 0.95 0.92 0.94 0.89 0.90 0.80 0.72 0.70 0.60 R2 0.50 0.40 0.30 0.26 0.20 0.10 Model 0.00 performance 1 2 3 4 5 6 Degree of polynomial Figure 5. The coefficients of determinations (R2) versus polynomial degrees. Figure 6. Local GPS/levelling geoid (5th degree polynomial). Figure 5 shows the computed coefficients of the sixth order polynomial models are minor in terms of determinations using each polynomial. standard deviation of geoid height residuals and coeffi- In the comparisons among the polynomial models con- cient of determination. Hence, the fifth order polynomial sidering the validation results against the observed data, having 18 significant parameters is decided as the and coefficients for determinations, it is seen that the fifth optimal model for the local geoid in the territory. The and the sixth order polynomials provided a better fit at the coefficients of the determined polynomial model are validation benchmarks, with higher coefficient of deter- provided in Table 3. The local geoid surface using the mination. However, the differences between the fifth and calculated polynomial is shown in Figure 6. Erol 1359 Table 3. Coefficients of the fifth degree polynomial. th The 5 degree polynomial coefficients a00 37.971 a20 3.824 a30 42.292 a40 -0.482 a50 -175.544 a10 -0.600* a21 9.063 a31 -7.854 a41 41.343 a51 416.621 a11 -0.175* a22 2.059* a32 -6.265 a42 4.689 a52 150.027 a33 8.102 a43 -67.811 a53 -151.087 a44 -4.096 a54 -9.877 a55 -25.992 * The coefficients are assigned insignificant and ignored in the model. th The 5 degree polynomial model N i = a00 + a10 u + a11v + a 20 u 2 + a 21uv + a 22 v 2 + a30 u 3 + a31u 2 v + a32 uv 2 + a33 v 3 + a40 u 4 + a 41u 3 v + a 42 u 2 v 2 + a 43uv 3 + a 44 v 4 + a50 u 5 + a51u 4 v + a52 u 3 v 2 + a53 u 2 v 3 + a54 uv 4 + a55 v 5 u = k (ϕ − ϕo ) , v = k ( λ − λo ) , k = 100 ρ o , ϕo = 38o.4143 , λo = 27 o.1063 Program development for practical implementation of using a geoid model for transforming the ellipsoidal height transformation using local GPS/levelling geoid heights from the GPS data processing to the orthometric heights in regional datum, and in order that the user can With the purpose of practical implementation of height either use one of the valid geoid models in the data base transformation using local GPS/leveling geoid model, a of the software, which are generally global geoid models program was developed in Visual C by Geodesy division having accuracy in metre (Erol et al., 2009)), or a regional at Istanbul Technical University (Ayan et al., 2001). geoid model, developed as external program and added Hence, in addition to accelerate transformation of the to the software. Hence, the orthometric heights of the heights, it was also aimed to reduce the calculation benchmarks are derived at the same time with their mistakes by the users of the model and the differences in geodetic coordinates from the GPS data. the results stem from the numerical ignoring. The first In this study, the developed program based on the version of the program accepts manual input of the polynomial type local GPS/levelling geoid model was geodetic latitude, longitude and ellipsoidal height of the added to the employed GPS data processing software, computation point in ITRF96 datum via keyboard. When Leica SKI Pro. Figure 8a describes the procedure, which the program runs, an initial screen provides the program is followed in introducing a new geoid model to the related information (Figure 7a), and gives the instructions software as an executable program file. In the procedure, to input the required data. In the process, at first, it is the path of the executable file of the geoid program detected whether the computation point is in the cover should be introduced following the “Coordinate System area of the local geoid model. If it is not in the model Management” option of the Tools pull-down menu, and area, the process is stopped and a warning message the geoid model related information, such as the name appears on the screen, because extrapolating the geoid and reference ellipsoid of the new geoid model, must be heights is not allowed by the program. When the entered via the “New Geoid Model” window. In this study, computation is done, the geoid height of the computation we use GRS 1980 ellipsoid for the geoid model, because point derived from the polynomial model and refined this model will then be attached to Turkish national geoid height with additive corrections, separately, are datum, which is ITRF96 with its reference ellipsoid GRS shown in addition to residuals at contributed reference 1980. After adding the new geoid model file to the GPS benchmarks in computation of the additive correction, on processing software, it is ready to be attached and run in the screen (Figure 7b). a project to compute the geoid heights and the A modified version of the program accepts the data of a orthometric heights of the benchmarks having three list of computation points in a specified input file and dimensional geodetic coordinates, calculated. Figure 8b provides their orthometric heights in an output file, and shows running the geoid model program for deriving the hence deriving the orthometric heights of a point group in orthometric heights, following the “Compute Geoid a project is possible. Recently, many commercial GPS Separations” option in the tools menu. After a message data processing software in the market provide an option, appeared on the screen, saying that the computation is 1360 Sci. Res. Essays (a) (b) Figure 7. User interface of the height transformation program based on Local Geoid model during the computations. (a) Introductory screen of the program for height transformation with local geoid model. (b) Input data and the results of the computation on the screen. successfully completed, the geoid undulations and the transformation via developed program working in a GPS orthometric heights are included in the “Point Lists” view processing software was reviewed. Then, the article is in addition to the geodetic coordinates of the followed by a numerical case study, which was done benchmarks. using 301 high-order reference GPS/levelling benchmarks in the west of Turkey, and consists of data Conclusions preparation, decision of an optimal polynomial model for the geoid data and model performance tests, as well. The In this article, determining local geoid model with precise determined polynomial model with additive corrections GPS/levelling data and it’s use for automated height and the software program can be applied for deriving the Erol 1361 (a) (b) Figure 8. Automated height transformation using Leica SKI Pro GPS data processing software and added local geoid model program: (a) Adding a new geoid model to the Leica SKI Pro GPS data processing software. (b) Computing the point orthometric heights in the active project, using the geoid model program. orthometric heights with 3.5 cm accuracy in the described fast and precise determination of the heights in regional local area, and presented methodology is suggested for vertical datum. In addition, the following points on precise 1362 Sci. Res. Essays modelling and use of geoid are also emphasized in the Özlüdemir MT, Erol S, Erol B, Acar M, Mercan H, Tekdal E (2006). Đstanbul GPS Triangulation Network (IGNA) 2005-2006: Re- results of the study: measurement and Data analysis. Scientific and Technical Report, Istanbul Technical University, Faculty of Civil Engineering, Volume I 1. Multivariate polynomial equations provides practical and II, Đstanbul, Turkey. solution in modelling the local geoid with precise, Ayan T, Deniz R, Çelik RN, Denli HH, Özlüdemir MT, Erol S, Erol homogeneously distributed and dense GPS/levelling (Özöner) B, Akyılmaz O, Güney C (2001). Đzmir Geodetic Reference System 2001 (ĐzJRS-2001) Project. Scientific and Technical Report, data, in the areas where a precise regional geoid model Istanbul Technical University, Faculty of Civil Engineering, Đstanbul, is not available, and hence precise transformation of GPS Turkey, p. 117. ellipsoidal heights to regional vertical datum. Ayan T, Aksoy A, Deniz R, Arslan E, Çelik RN, Özşamlı C, Denli H, Erol S, Erol (Özöner) B (1999). Đstanbul GPS Triangulation Network 2. However modelling procedure consists of critical (ĐGNA) Project. Scientific and Technical Report, No. 1997/3882, decisions, such as detecting and removing the blunders Istanbul Technical University, Faculty of Civil Engineering, Đstanbul, in the reference data properly, determining optimal form Turkey, p.119. of the polynomial and clarifying the performance of the Ayan T, Çelik RN, Alanko MG (1996a). Izmir Metropolitan Triangulation Network Densification Using GPS Technique Project. 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Wolf Geodesy Symposium, Chamber of derived simultaneously with their geodetic coordinates Turkish Surveying Engineers held in Istanbul, Turkey, pp. 409-435. from the GPS data processes, without user intervention. Collier P, Croft M (1997). Heights from GPS in an engineering Hence the transformation process is accelerated, the environment. Survey Rev., 264: 76-86, 265: 11-18. personal mistakes, possible during the calculations using Çelik RN, Ayan T, Erol B (2002). Geodetic Infrastructure Project of the parametric model, are reduced and uniform results Marmara Earthquake Region Land Information System (MERLIS). . Scientific and Technical Report, No. ITU 2002/06/209, Istanbul are provided. Technical University, Faculty of Civil Engineering, Đstanbul, Turkey. 4. Beside the precision and practicality advantages of Çepni MS, Deniz R (2005). Examination of continuity on geodetic local GPS/levelling geoids, the handicap in their use is transformations. 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