Assessment of the Improved Covariance in Local Geoid Modeling Using Least Squares Collocation-Case study: Tehran Province

Document Type : Research


1 M.Sc. Graduated, Department of Surveying and Geomatics Engineering, Faculty of Engineering, University of Tehran, Iran

2 Professor, Department of Surveying and Geomatics Engineering, Faculty of Engineering, University of Tehran, Iran


The idea of using Improved covariance (I_COV) through Least Squares Collocation (LSC) was first introduced and assessed on gravity anomalies (Ramouz et al, 2020) and geoid heights (Heydarizadeh et al, 2020) modeling over four regions with different data distribution and topography patterns in Iran. The results of these two researches showed that using I_COV could enhance gravity field modeling, specifically the medium to short wave-lengths of the signal which are embedded in the local and near-surface masses and surface density anomalies. For instance, implementing I_COV on a region with rough topography is more effective than classic covariance, in comparison with regions with smooth topography.
The gravity and GNSS/Leveling networks of Iran suffer from the lack of sufficient and well distributed observations. Moreover, existence of Alborz and Zagros mountain chains and the rough topography in the North, South and West of Iran, make regional gravity modeling that cover the whole country a difficult task. On the other hand, thanks to the development in satellite gravity technology and observations that have improved the accuracy of long-wavelength modeling of the Earth gravity field. So, quality processing and densifying terrestrial observations, incorporating high resolution Digital Elevation Models (DEM)s and improving geodetic boundary value problems are the available solutions to extract the medium to short-wavelength of the gravity signal to improve the gravity modeling. In this way, investigation of the effect of area size selection of the terrestrial observations, data density and distribution and topography roughness is classified in the spatial localization of the gravity field modeling.
The goal of this research is to analysis the contribution of the observations’ area size, density and distribution parameters on the accuracy of the local geoid height modeling and assess the possibility of model enhancement through execution I_COV procedure via LSC algorithm. As input, EIGEN-6C4 Global Gravity Model (GGM) up to degree/order 360, terrestrial gravimetric observations inside and around Tehran Province (measured by National Cartographic Center of Iran) and SRTM-1arc-min DEM are used via Remote-Compute-Restore technique.
To determine the analytical covariance function in order to applying LSC, first, an empirical covariance is computed from the terrestrial observations. Then, the Tscherning-Rapp 1974 (TR1974) covariance function is fitted to the empirical one and its three parameters are estimated to calculate the auto and cross-covariance of the LSC modeling formula. After LSC, the systematic parts of the signal i.e. global and topographic effects are restored. To implement the I_COV idea in gravity field localization, the value selection of TR1974 parameters are entered in iterative process to enhance the covariance model and improve the accuracy of the local model.
Assessment of the computed local model with the 141 GNSS/Leveling control points (measured by NCC) illustrates that STD of the model is about 8.9 cm inside the case study. Furthermore, if the comparison is limited to 40 control points inside Tehran City, STD of the model will be about 6.1 cm. To draw a comparative picture, the accuracy of this local model is 49% and 51% higher than EGM 2008 model (which has been the most accurate GGM in the region so far) over the same control points.


Main Subjects

صفری، ع.، راموز، ص. و جمعه‌گی، ع.، 1393، بهبود مدل‌سازی محلی میدان گرانی به‌روش تبدیل هم‌جایی از راه چگالی پوسته، مدل‌های ژئوپتانسیل جهانی و تلفیق مشاهدات ژئودتیک منطقه تحقیقاتی: پارس ساحلی، م. فیزیک زمین و فضا، 40(4)،. 83-98.
صفری، ع.، 1395، ژئودزی فیزیکی، دانشگاه تهران، چاپ سوم، شابک 9640362976-978.
Barzaghi, R., Borghi, A. and Sona, G., 2001, New Covariance Models for Local Applications of Collocation, in: IV Hotine-Marussi Symposium on Mathematical Geodesy, edited by: Benciolini, B., IAG Symposia, Springer, Berlin, Heidelberg, 122, 91–101, https://
Darbeheshti, N. and Featherstone, W. E., 2009, Non-stationary covariance function modelling in 2D least-squares collocation, J. Geod., 83, 495–508, s00190-008-0267-0.
Featherstone, W., 1998, Do We Need a Gravimetric Geoid or a Model of the Australian Height Datum to Transform GPS Heights in Australia?, The Australian Surveyor 43 (4). Inpress.
Foroughi, I., Afrasteh, Y., Ramouz, S. and Safari, A., 2017, Local evaluation of Earth Gravitational Models, case study: Iran, Geodesy Cartogr. 43, 1–13, 20296991.2017.1299839.
Forsberg, R., 1984, A Study of Terrain Reductions, Density Anomalies and Geophysical Inversion Methods in Gravity Field Modelling, Report 355, Department of Geodetic Science and Surveying, The Ohio State University, Columbus.
Förste, C., Bruinsma, S., Abrikosov, O., Lemoine, J. M., Marty, J. C., Flechtner, F., Balmino, G., Barthelmes, F., and Biancale, R., 2014, EIGEN-6C4 The latest combined global gravity field model including GOCE data up to degree and order 2190 of GFZ Potsdam and GRGS Toulouse, GFZ Data Services,
Hatam C. Y., 2010, Etablissement des nouveaux reseaux multi-observations geodesiques et gravimetriques et determination du geoide en Iran. PhD Thesis. Geophysics, University Montpellier 2, Montpellier, France (in French).
Heydarizadeh Shali H., Ramouz S., Safari A. and Barzaghi R., 2020, Assessment of Tscherning-Rapp covariance in Earth gravity modeling using gravity gradient and GPS/leveling observations, European Geosciences Union General Assembly, Vienna, Austria, doi:10.5194/egusphere-egu2020-1059.
Heiskanen W.A. and Moritz H., 1967, Physical Geodesy. W.H. Freeman, San Francisco, CA.
Hirt C., 2011, Assessment of EGM2008 over Germany using accurate quasigeoid heights from vertical deflections, GCG05 and GPS/levelling. Zeitschrift für Geodäsie, Geoinformation und Landmanagement, 136(3), 138149.
Ince, E. S., Barthelmes, F., Reißland, S., Elger, K., Förste, C., Flechtner, F. and Schuh, H., 2019, ICGEM – 15 years of successful collection and distribution of global gravitational models, associated services and future plans.-Earth System Science Data, 11, pp. 647-674,DOI:
Keller, W., 2002, A Wavelet Solution to 1D Non-Stationary Collocation with Extension to the 2D Case, in: Gravity, Geoid and Geodynamics 2000, edited by: Sideris, M. G., IAG Symposia, Springer, Berlin, Heidelberg, 123, 79–84,
Kiamehr, R., 2006, Precise Gravimetric Geoid Model for Iran Based on GRACE and SRTM Data and the Least-Squares Modification of Stokes’ Formula: with Some Geodynamic Interpretations. PhD Thesis. Royal Institute of Technology, Stockholm, Sweden.
Kotsakis, C., 2007, Least-squares collocation with covariance-matching constraints, J. Geod., 81, 661–677,
Moritz, H., 1980, Advanced Physical Geodesy, Herbert Wichmann Verlag, Karlsruhe.
Nahavandchi, H. and Soltanpour, A., 2005, Improved determination of heights using a conversion surface by combining gravimetric quasi/geoid and GPS-levelling height differences. Stud. Geophys. Geod., 50, 165180.
NASA, 2013, NASA Shuttle Radar Topography Mission Global 1 arc second, Data set, NASA LP DAAC,
Pavlis, N. K., Holmes, S. A., Kenyon, S. C. and Factor, J. K., 2012, The development and evaluation of the Earth Gravitational Model 2008 (EGM2008). J. Geophys. Res.-Solid Earth, 117, B04406, DOI:10.1029/2011JB 008916.
Ramouz, S., Afrasteh, Y., Reguzzoni, M., Safari, A. and Saadat, A., 2019, IRG2018: A regional geoid model in Iran using Least Squares Collocation, Studia Geophysica et Geodaetica, 63, 191–214,
Ramouz, S., Reguzzoni M., Afrasteh, Y, Safari, A., 2020, Assessment of Local Covariance Estimation Through Least Squares Collocation Over Iran, Adv. Geosci., 50, 65-75, https://doi:10.5194/adgeo-50-65-2020.
Saadat, A., Safari, A. and Needell, D., 2018, IRG2016: RBF-based regional geoid model of Iran, Studia Geophysica et Geodaetica, 62, 380–407,
Safari, A., Ardalan A. A. and Grafarend, E. W., 2005, A new ellipsoidal gravimetric, satellite altimetry and astronomic boundary value problem, a case study: The geoid of Iran. J. Geodyn., 39, 545568.
Sansò, F. and Sideris, M. G., 2013, Geoid Determination: Theory and Methods, Springer-Verlag, Berlin Heidelberg,
Tscherning, C. C., 1999, Construction of anisotropic covariance functions using sums of Riesz-representers, J. Geod., 73, 332–336,
Tscherning, C. C., 2015, Least-squares collocation. In: Grafarend E. (Ed.), Encyclopedia of Geodesy. Springer, Cham, Switzerland, DOI: 10.1007/978-3-319-02370-0_51-1.
Tscherning, C. C., Forsberg, R. and Knudsen, P., 1992, The GRAVSOFT package for geoid determination. Proceedings of the 1st Continental Workshop on the Geoid in Europe, Prague. Research Institute of Geodesy, Topography and Cartography, Prague, Czech Republic.
Tscherning, C. C. and Rapp, R., 1974, Closed Covariance Expressions for Gravity Anomalies, Geoid Undulations, and Deflections of the Vertical Implied by Anomaly Degree Variance Models, Report 208, Department of Geodetic Science, The Ohio State University, Columbus.
Yildiz, H., Forsberg, R., Ågren, J., Tscherning, C. C. and Sjöberg, L. E., 2012, Comparison of Remote Compute Restore and Least Squares Modification Stokes' Formula Techniques to Quasi-Geoid Determination over Auvergne Test Area, J. Geod. Sci., 2, 53–64,