Abstract
In this study, a new data-adaptive network design methodology called k-SRBF is presented for the spherical radial basis functions (SRBFs) in regional gravity field modeling. In this methodology, the cluster centers (centroids) obtained by the k-means clustering algorithm are post-processed to construct a network of SRBFs by replacing the centroids with the SRBFs. The post-processing procedure is inspired by the heuristic method, Iterative Self-Organizing Data Analysis Technique (ISODATA), which splits clusters within the user-defined criteria to avoid over- and under-parameterization. These criteria are the minimum spherical distance between the centroids and the minimum number of samples for each cluster. The bandwidth (depth) of each SRBF is determined using the generalized cross-validation (GCV) technique in which only the observations within the radius of impact area (RIA) are used. The numerical tests are carried out with real and simulated data sets to investigate the effect of the user-defined criteria on the network design. Different bandwidth limits are also examined, and the appropriate lower and upper bandwidth limits are chosen based on the empirical signal covariance function and user-defined criteria. Also, additional tests are performed to verify the performance of the proposed methodology in combining different types of observations, such as terrestrial and airborne data available in Colorado. The results reveal that k-SRBF is an effective methodology to establish a data-adaptive network for SRBFs. Moreover, the proposed methodology improves the condition number of normal equation matrix so that the least-squares procedure can be applied without regularization considering the user-defined criteria and bandwidth limits.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Data availability
The data sets used in this study are open-source data. The data sets generated during the current study are available from the corresponding author on reasonable request.
References
Antunes C, Pail R, Catalão J (2003) Point mass method applied to the regional gravimetric determination of the geoid. Stud Geophys Geod 47(3):495–509. https://doi.org/10.1023/A:1024836032617
Ball GH, Hall DJ (1965) ISODATA, a novel method of data analysis and pattern classification. Technical Report NTIS No. AD 699616, Menlo Park, CA: Stanford Research Institute
Barthelmes F, Dietrich R (1991) Use of point masses on optimized positions for the approximation of the gravity field. Determination of the geoid. Springer, Berlin, pp 484–493
Barthelmes F (1986) Untersuchungen zur Approximation des äußeren Gravitationsfeldes der Erde durch Punktmassen mit optimierten Positionen. Dissertion, Veröffentlichungen des Zentralinstituts für Physik der Erde 92 Zentralinstitut für Physik der Erde. Potsdam: Akademie der Wissenschaften der DDR
Bentel K, Schmidt M, Gerlach C (2013a) Different radial basis functions and their applicability for regional gravity field representation on the sphere. GEM Int J Geomath 4:67–96. https://doi.org/10.1007/s13137-012-0046-1
Bentel K, Schmidt M, Rolstad Denby C (2013b) Artifacts in regional gravity representations with spherical radial basis functions. J Geod Sci. https://doi.org/10.2478/jogs-2013-0029
Bucha B, Sansò F (2021) Gravitational field modelling near irregularly shaped bodies using spherical harmonics: a case study for the asteroid (101955) Bennu. J Geod 95:56. https://doi.org/10.1007/s00190-021-01493-w
Bucha B, Janák J, Papčo J, Bezděk A (2016) High-resolution regional gravity field modelling in a mountainous area from terrestrial gravity data. Geophys J Int 207:949–966. https://doi.org/10.1093/gji/ggw311
Chambodut A, Panet I, Mandea M et al (2005) Wavelet frames: An alternative to spherical harmonic representation of potential fields. Geophys J Int 163:875–899. https://doi.org/10.1111/j.1365-246X.2005.02754.x
Damiani TM, Youngman MA (2011) GRAV-D General airborne gravity data user manual. Version 1:1–28
Dampney CNG (1969) The equivalent source technique. geophysics 34:39–53. https://doi.org/10.1190/1.1439996
Denker H (2013) Regional gravity field modeling: theory and practical results. Sciences of geodesy - II. Springer, Berlin, pp 185–291
Duquenne H (2007) A data set to test geoid computation methods. In: Proceedings of the 1st International Symposium of the International Gravity Field Service (IGFS). Harita Dergisi, General Command of Mapping, Istanbul, Turkey, pp 61–65
Eicker A (2008) Gravity Field Refinement by Radial Basis Functions from In-situ Satellite Data. Ph.D. Thesis. University of Bonn
Fecher T, Pail R, Gruber T (2017) GOCO05c: a new combined gravity field model based on full normal equations and regionally varying weighting. Surv Geophys 38:571–590. https://doi.org/10.1007/s10712-016-9406-y
Foroughi I, Tenzer R (2014) Assessment of the direct inversion scheme for the quasigeoid modeling based on applying the Levenberg-Marquardt algorithm. Appl Geomatics 6:171–180. https://doi.org/10.1007/s12518-014-0131-2
Foroughi I, Safari A, Novák P, Santos M (2018) Application of radial basis functions for height datum unification. Geosciences 8:369. https://doi.org/10.3390/geosciences8100369
Forsberg R (1984) A study of terrain reductions, density anomalies and geophysical inversion methods in gravity field modelling. Report no. 355, Department of Geodetic Science and Surveying, The Ohio State University, Columbus, USA
Förstner W (1979) Ein verfahren zur schätzung von varianz-und kovarianzkomponenten. Allg Vermess 86:446–453
Freeden W, Schreiner M (2009) Spherical functions of mathematical geosciences. Springer, Berlin
Freeden W, Gervens T, Schreiner M (1998) Constructive approximation on the sphere: with applications to geomathematics. Clarendon
Golub GH, Heath M, Wahba G (1979) Generalized cross-validation as a method for choosing a good ridge parameter. Technometrics 21:215. https://doi.org/10.2307/1268518
Goyal R, Ågren J, Featherstone WE et al (2021) Empirical comparison between stochastic and deterministic modifiers over the French Auvergne geoid computation test-bed. Surv Rev. https://doi.org/10.1080/00396265.2021.1871821
GRAV-D Science Team (2017) Gravity for the redefinition of the American Vertical Datum (GRAV-D) project, airborne gravity data; Block MS05. https://geodesy.noaa.gov/GRAV-D/data_ms05.shtml. Accessed 20 Dec 2021
Grigoriadis VN, Vergos GS, Barzaghi R et al (2021) Collocation and FFT-based geoid estimation within the Colorado 1 cm geoid experiment. J Geod 95:52. https://doi.org/10.1007/s00190-021-01507-7
Heikkinen M (1981) Solving the Shape of the Earth by Using Digital Density Models. Reports of the Finnish Geodetic Institute, 81, Helsinki, Finland
Heiskanen WA, Moritz H (1967) Physical geodesy. W.H. Freeman and Co., San Francisco
Hipkin R, Haines K, Beggan C et al (2004) The geoid EDIN2000 and mean sea surface topography around the British Isles. Geophys J Int 157:565–577. https://doi.org/10.1111/j.1365-246X.2004.01989.x
Hirt C, Kuhn M, Claessens S et al (2014) Study of the Earth's short-scale gravity field using the ERTM2160 gravity model. Comput Geosci 73:71–80. https://doi.org/10.1016/j.cageo.2014.09.001
Ihde J, Schirmer U, Stefani F, Töppe F (1998) Geoid modelling with point masses. In: Proceedings of the Second Continental Workshop on the Geoid in Europe, Budapest, March. pp 199–204
Işık MS, Erol B, Erol S, Sakil FF (2021) High-resolution geoid modeling using least squares modification of Stokes and Hotine formulas in Colorado. J Geod 95:49. https://doi.org/10.1007/s00190-021-01501-z
Karslioglu MO (2005) An interactive program for GPS-based dynamic orbit determination of small satellites. Comput Geosci 31:309–317. https://doi.org/10.1016/j.cageo.2004.10.010
Klees R, Wittwer T (2007a) A data-adaptive design of a spherical basis function network for gravity field modelling. Int Assoc Geod Symp 130:322–328. https://doi.org/10.1007/978-3-540-49350-1_48
Klees R, Wittwer T (2007b) Local gravity field modelling with multi-pole wavelets. Dynamic planet. Springer, Berlin, pp 303–308
Klees R, Tenzer R, Prutkin I, Wittwer T (2008) A data-driven approach to local gravity field modelling using spherical radial basis functions. J Geod 82:457–471. https://doi.org/10.1007/s00190-007-0196-3
Klees R, Slobbe DC, Farahani HH (2018) How to deal with the high condition number of the noise covariance matrix of gravity field functionals synthesised from a satellite-only global gravity field model? J Geod. https://doi.org/10.1007/s00190-018-1136-0
Koch K-R (1990) Bayesian inference with geodetic applications. Springer, Berlin
Koch KR, Kusche J (2002) Regularization of geopotential determination from satellite data by variance components. J Geod 76:259–268. https://doi.org/10.1007/s00190-002-0245-x
Kusche J (2003) A Monte-Carlo technique for weight estimation in satellite geodesy. J Geod 76:641–652. https://doi.org/10.1007/s00190-002-0302-5
Kusche J, Ilk KH, Rudolph S, Thalhammer M (1998) Application of spherical wavelets for regional gravity field recovery — a comparative study. Geodesy on the move. Springer, Berlin, pp 213–218
Lehmann R (1993) The method of free-positioned point masses - geoid studies on the Gulf of Bothnia. Bull Géodésique 67:31–40. https://doi.org/10.1007/BF00807295
Lieb V, Schmidt M, Dettmering D, Börger K (2016) Combination of various observation techniques for regional modeling of the gravity field. J Geophys Res Solid Earth 121:3825–3845. https://doi.org/10.1002/2015JB012586
Lieb V (2017) Enhanced regional gravity field modeling from the combination of real data via MRR. PhD thesis, Technische Universität München
Lin M, Denker H, Müller J (2019) A comparison of fixed- and free-positioned point mass methods for regional gravity field modeling. J Geodyn 125:32–47. https://doi.org/10.1016/j.jog.2019.01.001
Lin M (2015) Regional gravity field recovery using the point mass method. Institut für Erdmessung, Leibniz Universität Hannover Hannover, Germany, PhD Thesis, Nr. 319
Liu Q, Schmidt M, Sánchez L, Willberg M (2020) Regional gravity field refinement for (quasi-) geoid determination based on spherical radial basis functions in Colorado. J Geod 94:99. https://doi.org/10.1007/s00190-020-01431-2
Luzum B, Petit G (2012) The IERS Conventions (2010): reference systems and new models. Proc Int Astron Union 10:227–228. https://doi.org/10.1017/S1743921314005535
MacQueen J (1967) Some methods for classification and analysis of multivariate observations. In: Proceedings of the fifth Berkeley symposium on mathematical statistics and probability. Oakland, CA, USA, pp 281–297
Mahbuby H, Amerian Y, Nikoofard A, Eshagh M (2021) Application of the nonlinear optimisation in regional gravity field modelling using spherical radial base functions. Stud Geophys Geod. https://doi.org/10.1007/s11200-020-1077-y
Marchenko AN (1998) Parameterization of the Earth’s gravity field: point and line singularities. Lviv Astronomical and Geodetical Society, Lviv
Marchenko AN, Barthelmes F, Meyer U, Schwintzer P (2001) Regional geoid de- termination: an application to airborne gravity data in the Skagerrak. GFZ Potsdam, Germany, Scientific Technical Report No. 01/07
Moritz H (1980) Advanced Physical Geodesy. Herbert Wichmann Verlag, Karlsruhe, Germany, 500 p
NGS (2012) Technical details for Geoid12/12A/12B. https://www.ngs.noaa.gov/GEOID/GEOID12B/GEOID12B_TD.shtml. Accessed 15 Dec 2021
Pail R, Fecher T, Barnes D et al (2018) Short note: the experimental geopotential model XGM2016. J Geod 92:443–451. https://doi.org/10.1007/s00190-017-1070-6
Panet I, Chambodut A, Diament M et al (2006) New insights on intraplate volcanism in French Polynesia from wavelet analysis of GRACE, CHAMP, and sea surface data. J Geophys Res Solid Earth. https://doi.org/10.1029/2005JB004141
Rexer M, Hirt C, Claessens S, Tenzer R (2016) Layer-Based modelling of the earth’s gravitational potential up to 10-km scale in spherical harmonics in spherical and ellipsoidal approximation. Surv Geophys 37:1035–1074. https://doi.org/10.1007/s10712-016-9382-2
Rousseeuw PJ (1987) Silhouettes: a graphical aid to the interpretation and validation of cluster analysis. J Comput Appl Math 20:53–65. https://doi.org/10.1016/0377-0427(87)90125-7
Sánchez L, Sideris MG (2017) Vertical datum unification for the International Height Reference System (IHRS). Geophys J Int 209(2):570–586. https://doi.org/10.1093/gji/ggx025
Sánchez L, Ågren J, Huang J et al (2021) Strategy for the realisation of the International Height Reference System (IHRS). J Geod 95:33. https://doi.org/10.1007/s00190-021-01481-0
Sansò F, Sideris MG (2017) Geodetic boundary value problem: the Equivalence between Molodensky’s and Helmert’s Solutions. Springer, Cham
Schmidt M, Fengler M, Mayer-Gürr T et al (2007) Regional gravity modeling in terms of spherical base functions. J Geod 81:17–38. https://doi.org/10.1007/s00190-006-0101-5
Sjöberg LE (2005) A discussion on the approximations made in the practical implementation of the remove–compute–restore technique in regional geoid modelling. J Geod 78:645–653. https://doi.org/10.1007/s00190-004-0430-1
Slobbe C, Klees RH. Farahani H, et al (2019) The impact of noise in a GRACE/GOCE global gravity model on a local quasi-geoid. J Geophys Res Solid Earth. https://doi.org/10.1029/2018jb016470
Tenzer R, Klees R (2008) The choice of the spherical radial basis functions in local gravity field modeling. Stud Geophys Geod 52:287–304. https://doi.org/10.1007/s11200-008-0022-2
Thorndike RL (1953) Who belongs in the family? Psychometrika 18:267–276. https://doi.org/10.1007/BF02289263
Torge W (1989) Gravimetry. Walter de Gruyter, Berlin
Ulug R, Karslıoglu MO (2022) SRBF_Soft: a Python-based open-source software for regional gravity field modeling using spherical radial basis functions based on the data-adaptive network design methodology. Earth Sci Inform 15:1341–1353. https://doi.org/10.1007/s12145-022-00790-y
van Westrum D, Ahlgren K, Hirt C, Guillaume S (2021) A Geoid Slope Validation Survey (2017) in the rugged terrain of Colorado, USA. J Geod 95:1–19. https://doi.org/10.1007/s00190-020-01463-8
Vermeer M (1982) The use of mass point models for describing the Finnish gravity field. In: Proceedings 9th meeting of the Nordic Geodetic Commission, Gävle, Sweden. pp 13–17
Wang YM, Sánchez L, Ågren J et al (2021) Colorado geoid computation experiment: overview and summary. J Geod 95:127. https://doi.org/10.1007/s00190-021-01567-9
Wessel P, Luis JF, Uieda L et al (2019) The generic mapping tools version 6. Geochem Geophys Geosyst 20:5556–5564. https://doi.org/10.1029/2019GC008515
Willberg M, Zingerle P, Pail R (2020) Integration of airborne gravimetry data filtering into residual least-squares collocation: example from the 1 cm geoid experiment. J Geod 94:1–17. https://doi.org/10.1007/s00190-020-01396-2
Wittwer T (2009) Regional gravity field modelling with radial basis functions. Ph.D. Thesis. Delft University of Technology
Wu Y, Luo Z, Chen W, Chen Y (2017a) High-resolution regional gravity field recovery from Poisson wavelets using heterogeneous observational techniques. Earth Planets Space 69(1):1–15. https://doi.org/10.1186/s40623-017-0618-2
Wu Y, Zhou H, Zhong B, Luo Z (2017b) Regional gravity field recovery using the GOCE gravity gradient tensor and heterogeneous gravimetry and altimetry data. J Geophys Res Solid Earth 122:6928–6952. https://doi.org/10.1002/2017JB014196
Wu Y, Abulaitijiang A, Featherstone WE et al (2019) Coastal gravity field refinement by combining airborne and ground-based data. J Geod 93:2569–2584. https://doi.org/10.1007/s00190-019-01320-3
Xu R, Wunsch DC (2008) Clustering. Wiley, Hoboken
Yu H, Chang G, Zhang S, Qian N (2020) Sparsifying spherical radial basis functions based regional gravity models. J Spat Sci 00:1–16. https://doi.org/10.1080/14498596.2020.1760952
Zingerle P, Pail R, Gruber T, Oikonomidou X (2020) The combined global gravity field model XGM2019e. J Geod 94:66. https://doi.org/10.1007/s00190-020-01398-0
Acknowledgements
The authors would like to thank the National Geodetic Survey for providing terrestrial, airborne and GPS\leveling data sets in the Colorado area and Prof. H. Duquenne for the availability of terrestrial and GPS\leveling data sets in the Auvergne area. The map figures were generated using the Generic Mapping Tools (GMT) (Wessel et al. 2019). The authors also would like to thank the editors and anonymous reviewers for their comments which improved the quality of the manuscript.
Funding
This research received no external funding.
Author information
Authors and Affiliations
Contributions
RU designed and analyzed the algorithm, performed the calculations, compiled the figures and wrote the original manuscript. MOK supervised the entire work including the design and analysis of the algorithm. All authors read and approved the final manuscript.
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflicts of interest.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Ulug, R., Karslıoglu, M.O. A new data-adaptive network design methodology based on the k-means clustering and modified ISODATA algorithm for regional gravity field modeling via spherical radial basis functions. J Geod 96, 91 (2022). https://doi.org/10.1007/s00190-022-01681-2
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00190-022-01681-2