Three different topographic reduction methods in geoid determination were investigated. The first method is the classical Helmert second method of condensation yielding the geoid, the second is the Residual Terrain Model (RTM) method yielding the quasigeoid and the third is the Rudzki inversion method. The different types of indirect effects (indirect effect on gravity and indirect effect on geoid) in Helmert's method were also investigated. All three methods use the remove-restore technique and the EGM96 geopotential model as the reference gravity field. A mountainous area, ranging from 32°S to 42°S in latitude and 72°W to 68°W in longitude, was chosen as test area. The area was selected due to its high topography, with a maximum height of 6795 meters and a mean height of 1188 meters, and due to the existence of GPS/leveling points in three different networks. The topography in the test area is-represented by a digital terrain model (DTM) with a grid spacing of 1 km x 1 km. Another test was carried out in a flat area with denser data coverage. The external accuracy of the three gravimetric geoids was evaluated by comparing them to undulations derived from GPS/leveling.En el siguiente trabajo se investigan tres métodos diferentes de reducciones gravimétricas para la determinación práctica del geoide gravimétrico: el clásico segundo método de condensación de Helmert, el modelo residual de terreno y el método de inversión de Rudzki. Los tres métodos utilizan la técnica remover-restaurar y el modelo de geopotencial EGM96. Fueron seleccionadas dos áreas, una en una zona montañosa de alta topografía con una altura máxima de 6795 metros y una altura promedio de 1188 metros y otra en una zona plana con cobertura más densa. La topografía está representada por un modelo digital de terreno con un espaciamiento de grillado de un kilómetro por un kilómetro. La evaluación externa del geoide gravimétrico se realiza comparándolo con ondulaciones obtenidas a partir de puntos GPS sobre nivelación.Asociación Argentina de Geofísicos y Geodesta

Font, Graciela

Sideris, Michael G.

Tocho, Claudia Noemí

English

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GEOACTA 28, 73 - 78,2003©2003 Asociación Argentina de Geofísicos y Geodestas.ISSN 0326-7237DIFFERENT TOPOGRAPHIC REDUCTION METHODS IN PRACTICAL GRAVIMETRIC GEOID DETERMINATIONClaudia TOCHO1, Michael G. SIDERIS2 y Graciela FONT1lFacultad de Ciencias Astronómicas y Geofísicas, Paseo del Bosque s/n, 1900 La Plata, Argentina, ctocho@fcaglp.unlp.edu.ar and graciela@fcaglp.unlp.edu.arDepartment of Geomatics Engineering, University of Calgary, 2500 University Drive N.W., Calgary, Alberta, Canada T2N1N4, sideris@ucalgary.caABSTRACTThree different topographic reduction methods in geoid determination were investigated. The first method is the classical Helmert second method of condensation yielding the geoid, the second is the Residual Terrain Model (RTM) method yielding the quasigeoid and the third is the Rudzki inversion method. The different types of indirect effects (indirect effect on gravity and indirect effect on geoid) in Helmert's method were also investigated. All three methods use the remove-restore technique and the EGM96 geopotential model as the reference gravity field. A mountainous area, ranging from 32°S to 42°S in latitude and 72°W to 68°W in longitude, was chosen as test area. The area was selected due to its high topography, with a maximum height of 6795 meters and a mean height of 1188 meters, and due to the existence of GPS/leveling points in three different networks. The topography in the test area is-represented by a digital terrain model (DTM) with a grid spacing of 1 km x 1 km. Another test was carried out in a flat area with denser data coverage. The external accuracy of the three gravimetric geoids was evaluated by comparing them to undulations derived from GPS/leveling.Keywords: Helmert - RTM - Rudzki - geoid - quasigeoid - direct terrain effects - indirect effects.RESUMENEn el siguiente trabajo se investigan tres métodos diferentes de reducciones gravimétricas para la determinación práctica del geoide gravimétrico: el clásico segundo método de condensación de Helmert, el modelo residual de terreno y el método de inversión de Rudzki. Los tres métodos utilizan la técnica remover-restaurar y el modelo de geopotencial EGM96. Fueron seleccionadas dos áreas, una en una zona montañosa de alta topografía con una altura máxima de 6795 metros y una altura promedio de 1188 metros y otra en una zona plana con cobertura más densa. La topografía está representada por un modelo digital de terreno con un espaciamiento de grillado de un kilómetro por un kilómetro. La evaluación externa del geoide gravimétrico se realiza comparándolo con ondulaciones obtenidas a partir de puntos GPS sobre nivelación.Palabras claves: Helmert - RTM-Rudzki - Geoide - Casi geoide- Efectos directos sobre el terreno - Efectos indirectosINTRODUCTIONThe use of Stokes's formula in gravimetric geoid determination requires that the gravity anomalies represent boundary values on the geoid. This means that the measured gravity (usually taken on the surface of the Earth) must be reduced to the geoid and there must be no masses outside the geoid (Heiskanen and Moritz, 1967).There are several gravity reductions used in physical geodesy: the Bouguer reduction, isostatic reductions, the Rudzki inversion method, the second method of Helmert's condensation, etc. The Residual Terrain Model (Forsberg, 1984) is another type of reduction, which takes into account the high frequencies of the topography. RTM yields the quasigeoid, thus the correction between quasigeoid and geoid was applied to compare the results from the three methods with GPS/leveling data.Theoretically, all reduction methods should lead us to the same geoid if the gravity reductions were applied consistently (Heiskanen and Moritz, 1967) evenRecibido: 4 de septiembre 2003 Aceptado: 2 de diciembre 2003 though each reduction treats the topography in a different way.DATA SETSTwo areas were selected as test areas. One is an area covering part of the Mendoza and Neuquen provinces. This test area was bounded by latitudes 32° to 42° S and longitudes 68° and 72° W and it was selected due to the presence of GPS/leveling data, sparse gravity coverage coming from different sources, and rough topography.The other test area is a flat area, with denser gravity data ranging from 34° to 38.5° S in latitude and 59° to 64.5° W in longitude.Surface gravity measurementsThe point gravity measurements, provided by different sources, were referenced to the International Standardisation Net 1971 (I.G.S.N.71). Most of the gravimetric data comes from the database of the Argentine Military Geographic Institute.73Claudia TOCHO, Michael G. SIDERIS y Graciela FONTA total of 1452 measured gravity points, with a mean data spacing of approximately 20 km are used in the mountainous area, and 4302 gravity points with a spacing of approximately 8 km are selected in the flat area. The distribution of the gravity points is shown in Figure 1 and Figure 3.Gravity anomaliesFree-air gravity anomalies are calculated using the parameters of the Geodetic Reference System 1980 (GRS80). The point free-air gravity anomalies were calculated using the following expression (Torge, 1989):where the height h is in m.Geopotential modelThe reference gravity field is computed from the EGM96 geopotential model (Lemoine et complete to degree and order 360.From the contribution of the EGM96 geopotential model, a reference gravity anomaly (Ag GM) and a reference geoidal undulation (NGM can be calculated. The gravity anomaly estimated at a position ((j>p,Xp) is expressed in spherical approximation (Heiskanen and Moritz, 1967) as:and the reference geoidal undulation as:(4)where R is the mean radius of the Earth, Cnm andSnm are the fully normalised sphericajj harmonic coefficients of the disturbing potential, Pnm are the fully normalized associated Legendre functions, and nmax denotes the maximum degree and order of expansion of the geopotential solution.Digital Elevation ModelThe global digital elevation model GTOPO30 with a horizontal grid spacing of 30 arc seconds (approximately 1 kilometre) is used to represent the topography in the test area. Detailed information on the characteristics of GTOPO30 including the data distribution format, the data sources, production methods, accuracy, and hints for users, is found in the GTOPO30 documentation file, (http://edcdaac.usgs. gov/gtopo30/gtopo30.html).The statistics of the topographic data in the test areaTable 1. Statistics of the GTOPO30. Unit: [m]DEM Max Min Mean Standard DeviationGTOPO30 (rough area) 6795 0 1183.530 880.990GTOPO30 (flat area) 1617 0 136.109 115.393are given are Table 1.GPS/leveling dataA total of 166 GPS/leveling points in three differ­ent networks were used for comparison with the gravimetric geoid in the rough area, and a total of 119 points were used in the flat area. The distributions, of the GPS points can be seen in Figure 2 and Figure 4, respectively. There are no GPS/leveling points aboveFigure 1. Distribution of the gravity pointsin the flat area on elevation map (m)Figure 2. Distribution of GPS/leveling points in the flat area on elevation map (m)the elevation of 1885 m.COMPUTATIONAL FORMULAS FOR TERRAIN EFFECTSThe three methods compared in this investigation use the remove-restore technique for the determination of the gravimetric geoid/quasigeoid. Each method handles the topography in a different way. 74 GEOACTA 28, 73 - 78,2003This formula uses the second order free-air reduction, applies atmospheric correction (8gatrn ) and evaluates normal gravity with Somigliana's closed formula, using the parameters of the GRS80. The atmospheric correction is applied as follows (Torge, 1989):(2)(1)(3)Different topographie reduction methods in practical gravimetric geoid determinationThe geoid or quasigeoid is calculated from Stokes's formula with gravity anomalies as input. Before applying this formula, gravity anomalies must be reduced, in the remove step, by(5)where AgFA is the free-air gravity anomaly, AGM is the reference gravity anomaly computed from the geopotential model and AgT is the terrain effect, which depends on the reduction method used.The gravimetric geoid is obtained, in the restore step, by(6)(12)where NGM is the reference geoidal undulation implied by the geopotential model, Nind is the indirect effect on the geoid and depends on the reduction method used, and NAg Srepresents residual geoid computed with the residual gravity anomalies given in Eq. (5).Stokes's formula with the rigorous spherical kernel was evaluated by the one-dimensional Fast Fourier Transform algorithm (Haagmans et al., 1993). The indirect effect on the geoid is(7)AW is the change of the potential at the geoid due to the terrain reduction applied.The indirect effect on gravity, which reduces gravity anomalies from the geoid to the cogeoid is expressed by(8)The RTM method estimates the quasigeoid. The reduced gravity anomalies refer to the surface of the topography. The final quasigeoid is obtained by(9)In order to compare the results of this method with the other two reduction methods and to make comparisons with the GPS/leveling derived geoid, the quasigeoid is converted to geoid using the quasigeoid-geoid separation given in Heiskanen and Moritz (1967) as(10)where AgB is the Bouguer anomaly and y is the mean normal gravity (9.81m/seg2)Helmert's second method of condensationHelmert's second method of condensation con­denses the topographic masses on a layer on thé geoid. Before applying Stokes's formula, the indirect effect on gravity has been considered.The terrain effect on gravity gT is given by(H)<5Ag is the indirect effect on gravity (Sideris and She, 1995) and Cp is the classical terrain correction given by the following equation:where (x,y,z) is the topographical density at the running point, here is assumed constant at 2.67 gr/cm3, G is the gravitational constant and E denotes the integration area.The indirect effect on the geoid, up the second order is in planar approximation (Wichiencharoen, 1982)where r0 is the planar distance between computation point and data point.Rudzki inversion methodThe Rudzki inversion method is a reduction where the indirect effect is zero. Rudzki shifts all the topographic masses inside the geoid.(15)where AT is the attraction of the topographic masses, Ac is the attraction of the inverted masses, with the density of the topographic masses being equal to the density of the inverted masses and the thickness of the inverted masses is equal to the height of the topography(16)GEOACTA 28, 73 - 78,2003 75(13)Claudia TOCHO, Michael G. SIDERIS y Graciela FONT(17)RTM methodElevation Model (DEM) using the Tc2DFTPL program (Li, 1993). The statistics of terrain corrections in gravity stations can be seen in Table 2. For the rough area, we used up to the third order term of a mass prism model and for the flat area we used up to second order term of a mass line model.The RTM reduction method takes into account the high frequencies of the topography. The effect of the topography above a long wavelength topographic surface is first removed and later restored.The RTM gravity terrain effect, AgT is given by the approximate expression (Forsberg, 1984)Table 2. Statistics of terrain corrections cp in gravity stations. Unit: [mGal} in both test areasThe indirect effect on gravity due to Helmert's second method of condensation was considered before applying Stokes's formula. The statistics of this effect, together with the indirect effect on the geoid in the rough area, can be seen in Table 3. In the mountainous area, the computation of the indirect effect on the geoid should be done up to at least second order term. Figure 5 and Figure 6 show the indirect effect on the geoid and on gravity for Helmert's method. The maximum indirect effects are correlated with the topography. In the flat area, the topographic indirect effect on the geoid does not add any significant contribution to the gravimetric geoid undulations. The computation was done only consider the first term in Eq. (14) and the indirect effect on gravity was neglected.The TC program was used to compute the RTM effects (Forsberg, 1984) and it was modified toFigure 4. Distribution of GPS/leveling points in the rough area on elevation map Color scale Unit:[m]Figure 3. Distribution of gravity pointsin the rough area on elevation map Color scale Unit:[m]76 GEOACTA 28, 73 - 78,2003where h is the topographic height given by a digital terrain model, href is the height of a smooth mean reference surface and cp is the classical terrain correction.The RTM geoid effect is expressed in linear approximation as:where r0 is the planar distanceNUMERICAL EXAMPLESTerrain corrections (cp) were calculated by FFT for each gravity point from the GTOPO30 DigitalMax Min Mean Standard DeviationcP (rough area) 35.39 0.07 2.85 4.12cP (flat area) 1.51 0.00 0.02 0.04(18)(19)Different topographie reduction methods in practical gravimetric geoid determinationFigure 5. Indirect effect on gravity due to the Helmert's condens ation method in the rough area Unit: [mGal]Table 3. Statistics of indirect effects due to the Helmert's condensation methodMax Min Mean Standard DeviationOn gravity [mGal] (rough area) 0.20 0.00 0.02 0.03On geoid [m] (rough area) 0.000 -1.162 0.110 0.225On gravity [mGal] (flat area) 0.00 0.00 0.00 0.00On geoid [m] (flat area) -0.001 -0.004 -0.002 -0.001compute the Rudzki inversion method (Bajracharya et al., 2001). The statistics of the gravity anomalies calculated with the three topographic gravity reductions are presented in Table 4. RTM anomalies are the smoothest gravity anomalies with a standard deviation of 29.5 mGal. Faye anomalies have the maximum standard deviation compared to the other reduction methods. The removal of the reference field Figure 6. Indirect effect on the geoid due to the Helmert's condensation method in the rough area Unit: [m](EGM96) does not improve the statistics of RTM reduced gravity anomalies but it does improve the statistics for Helmert and Rudzki reduced gravity anomalies.The systematic datum differences between the gravimetric geoid and the GPS/leveling geoid and the long wavelengths errors of the geoid were removed by a four-parameter transformation model (Heiskanen and Moritz, 1967). The statistics of the absolute differences between the GPS/leveling-derived geoid and the gravimetric geoids computed using different methods of handling the topography are summarised in Table 5. The numbers in parentheses refer to the results after the least-squares fitting of the four-parameter transformation model has been applied Table 4. Statistics of the gravity anomalies calculated with the three topographic reductions. Unit [mGal]Faye anomalies (rough area)Rudzki anomalies (rough area)RTM anomalies (rough area) Faye anomalies (flat area) Rudzki anomalies (flat area) RTM anomalies (flat area)Max Min Mean StandardDeviation173.59 -124.76 7.05 40.31192.02 -192.21 -17.20 37.53127.09 -122.73 11.36 38.30140.65 -131.77 -12.89 28.77155.40 -71.57 31.28 29.53100.97 -175.03 7.03 35.8596.84 -28.18 10.01 11.8694.99 -41.19 1.82 7.5496.91 -27.72 10.01 11.8695.07 -40.73 1.81 7.5497.43 -33.34 10.13 11.6195.58 -46.36 1.94 7.83GEOACTA 28, 73 - 78,2003 77Claudia TOCHO, Michael G. SIDERIS y Graciela FONTto the original differences. Before applying the four-parameter transformation model, two GPS on benchmark points with large gross errors in either the GPS or the leveling data were removed.In the mountainous area, gravimetric geoid computed with Rudzki inversion topographic reduction shows the smallest differences from GPS/leveling before fit and RTM geoid shows the highest differences compared to Helmert and Rudzki methods. These large differences could be due to discrepancies between the model elevation and the actual elevation at the station.CONCLUSIONSThree different gravity reduction methods have been presented. They treat the topography in a very different way. Helmert's second method of condensation and the RTM method are the most used reduction techniques for the determination of a gravimetric geoid. In this paper, we applied the Rudzki inversion method as well, which is not very often used, even though it has the advantage of no indirect effects.In the mountainous area, the gravimetric geoid computed with the Rudzki inversion method gave better results compared with the GPS/leveling-derived geoid before and after fit and was the only method that improved the gravimetric geoid considerably compared to the EGM96 results. The gravimetric data needs to be improved in the area of the Andes in order to see further improvements in the geoid. In the flat area, the three reduction methods gave identical results as expected. In the future it is planned to test two other gravity reduction methods, namely the Airy- Heiskanen and Pratt-Hayford topographic-isostatic reductions.Acknowledgements: We wish to thank all the organi­zations and people who provided the data for this work: Instituto Geográfico Militar for the gravity data set and Raúl Perdomo, Daniel del Cogliano, Luis Lenzano and Daniel Querejeta for the GPS/leveling data.REFERENCESBajracharya, S., Kotsakis C., Sideris M. G., 2001. Geoid Determination Using Different Gravity Reduction Techniques. Presented in I AG meeting, Budapest.Forsberg, R., 1984. A study of terrain reductions, density anomalies and geophysical inversion methods in gravity field modeling. Report No. 355, Department of Geodetic Science and Surveying, The Ohio State University, Colombus, Ohio, USA.GTOPO30, 1996. (http://edcdaac.usgs.gov/gtopo30/ gtopo30.html).Haagmans, R., de Min, E., van Gelderen, M., 1993. Fast evaluation of convolution integrals on the sphere using ID FFT and a comparison with existing methods for Stokes' integral. Manuscripta Geodaetica, 18: 227-241.Heiskanen, W.A. and Moritz, H., 1967. Physical Geodesy. Freeman and Company, San Francisco.Lemoine, F. G., Kenyon, S. C., Factim, J. K., Trimmer, R. G., Pavlis, N. K„ Chinn, D. S., Cox, C. M., Klosko, S. M., Luthcke, S. B., Torrence, M. H., Wang, Y. M., Williamson, R. G., Pavlis, E.C.,  Rapp, H. and Olson, T. R., 1998. The development of the joint NASA GSFC and the National Imagery and Mapping Agency (NIMA) Geopotential Model EGM 96. Pub. Goddard Space Flight Center.Li, Y., 1993. HFTGVBP Software package for the solution of GVBP by means of fast Hartley/Fourier Transform. TOPOGEOP Software packages to evaluate the TOPOgraphic effects on GEOdetic /GEOPhysical Observation. Department of Geomatics Engineering, The University of Calgary.Sideris, M.G., She, B.B., 1995. A new high-resolution geoid for Canada and part of U.S. by the 1D- FFT method. Bull Geod 69, 92-108.Torge, W., 1989. Gravimetry, Walter de Gruyter, Berlin.Wichiencharoen, C., 1982. The indirect effects on the computation of geoid undulations. OSU Report. No 336, Department of Geodetic Science and Suveying, The Ohio State University, Columbus, Ohio, USA.78 GEOACTA 28, 73 - 78,2003

Different topographic reduction methods in practical gravimetric geoid determination

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Different topographic reduction methods in practical gravimetric geoid determination

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