High-resolution regional gravity field recovery from Poisson wavelets using heterogeneous observational techniques
© The Author(s) 2017
Received: 8 November 2016
Accepted: 13 February 2017
Published: 24 February 2017
KeywordsRegional gravity field recovery Poisson wavelets Tikhonov regularization Quasi-geoid/geoid GPS/leveling data
High-resolution regional gravity field recovery is of considerable importance not only for surveying and mapping, but also for research fields, such as oceanography (understanding ocean circulation and currents), geophysics (investigating the structure of seismic activities and the lithosphere), and geodynamics (Kuroishi 2009; Panet et al. 2011; Shih et al. 2015).
Typically, middle- and short-wavelength gravity field signals down to a few kilometers are extracted from high-resolution ground-based measurements, e.g., terrestrial and shipborne gravity data, which are only available in geographically limited regions (Wang et al. 2012; Odera and Fukuda 2014; Lieb et al. 2016). In contrast, long-wavelength signals from tens of kilometers or larger are often recovered using global geopotential models (GGMs) derived from satellite observations. Over the last 10 years, launches of the Gravity Field and Climate Experiment (GRACE) (Tapley et al. 2004) and Gravity Field and Steady-State Ocean Circulation Explorer (GOCE) (Rummel et al. 2002) missions have greatly contributed to improving the spatial resolution and accuracy of GGMs. Moreover, developing various observational techniques, e.g., GPS, airborne gravimetric measurements, and satellite altimetry missions, can further improve regional gravity fields (Hwang et al. 2006; Jiang and Wang 2016; Wu and Luo 2016). Combined, these data sets form a solid basis for modeling high-resolution and high-quality regional gravity fields. However, these data have heterogeneous spatial coverage and resolutions, various error characteristics, and different spectral contents, which make their use an open issue. Thus, the aim of this study is to adopt an approach that combines heterogeneous data and extracts different spectral contents from various observational techniques for regional gravity field recovery.
The Stokes/Molodensky integral makes it difficult to combine heterogeneous data, while the least squares collocation (LSC) is numerically inefficient in managing cases that involve a large number of point-wise data (Wittwer 2009). The new gravity field described in this study is parameterized using Poisson wavelets. Poisson wavelets are radially symmetric basis functions that have localizing properties in both the spatial and frequency domains, which have been used extensively in regional gravity field modeling and potential field analysis (Tenzer and Klees 2008; Hayn et al. 2012; Bentel et al. 2013).
We also investigate several aspects that affect the solution quality derived from Poisson wavelets. To begin with, as heterogeneous data have different spatial coverage and resolutions, the derived least squares system from Poisson wavelets is typically ill-conditioned, where regularization is mandatory for deriving reliable results (Wittwer 2009). One of the key points that affect the quality of regularization is the choice of regularization matrices (Chambodut et al. 2005). Unlike using diagonal regularization matrices in global gravity field modeling from spherical harmonics (Kusche and Klees 2002), the regularization matrices derived from various constraints in regional scale are no longer entirely diagonal. The choice of regularization matrices may affect the solution quality, which is investigated in this study.
Moreover, mainly due to commission errors in the GGMs and uncorrected systematic errors in the data, the computed gravimetric quasi-geoid/geoid usually deviates from local values observed from GPS/leveling data by a centimeter level or larger (Wu et al. 2016). Generally speaking, corrector surface (Featherstone 2000; Fotopoulos 2005; Nahavandchi and Soltanpour 2006) or more complicated algorithms, e.g., least squares collocation (Tscherning 1978) and boundary-value methodology (Klees and Prutkin 2008; Prutkin and Klees 2008), can be applied to reduce systematic errors and properly combine GPS/leveling data and gravimetric solutions. However, given the difficulty in choosing a proper corrector surface, as well as the associated algorithmic complexity for the least-squares collocation and boundary-value approach, we propose a direct methodology for removing the inconsistency between these two data sets. The GPS/leveling data are treated as an independent observation group and added to the functional model for the gravity field computation. Using this method, the quasi-geoid/geoid that fits the local leveling system is computed in one step, no systematic errors exist between the data, and post-processing procedures, e.g., corrector surface or LSC approaches, for calibrating systematic errors are not required.
The rest of the paper is organized as follows: the main principle of regional gravity modeling from Poisson wavelets is first introduced, and heterogeneous gravity-related observations are linked to the functional model parameterized by Poisson wavelets. The weights for different observation groups are determined through the variance component estimation (VCE) approach, and unknown coefficients are estimated through the least squares adjustment. Further, the Tikhonov regularization method is introduced to deal with the ill-conditioned system, where a rapid synthesis method for computing zero- and first-order regularization matrices is provided. In the following section, Hong Kong is selected as the study area and heterogeneous data are introduced. The numerical results are also shown in this part, where the new height reference surface, HKGEOID-2016, is determined. In addition, HKGEOID-2016 is compared with existing models, e.g., HKGEOID-2000 (Luo et al. 2005) and recently published GGMs, such as EIGEN-6C4 (European Improved Gravity Model of the Earth by New Techniques 6C4) and EIGEN-6C3STAT (Förste et al. 2012, 2014), for cross-validation. The last section contains the main summary and conclusions.
Functional model and parameters estimation
Choice of regularization matrix
The regularization matrix is the key element controlling the quality of the regularized solutions, which describes the signal energy decreasing from large to small scale (Chambodut et al. 2005). For selecting the optimal regularization matrix, the zero- and first-order regularization matrices are investigated.
Zero-order Tikhonov regularization
First-order Tikhonov regularization
The zero-order Tikhonov regularization derived above is suitable for functions restricted to a sphere σ U belonging to L 2(σ U). However, this function space is quite large and includes many unsmooth functions. To introduce a scalar product suitable for smoother functions, a target function is selected as the first-order derivative of the Poisson wavelets, and the inner products of the target functions are used to derive the entries of the first-order regularization matrix.
Results and discussion
Study area and data
Global geopotential model and digital terrain model
Statistics of the difference between altimetry and marine gravity data (Units: mGal)
Statistics of the residual gravity observations (Units: mGal)
Terrestrial Δg − Δg EGM2008
Terrestrial Δg − Δg EGM2008 − Δg RTM
Shipborne Δg − Δg EGM2008
Shipborne Δg − Δg EGM2008 − Δg RTM
Satellite altimetry Δg − Δg EGM2008
Satellite altimetry Δg − Δg EGM2008 − Δg RTM
Determination the regularization matrix and regularization parameter
Evaluation of different geoids with various regularization parameters when zero-order regularization is used (Units: cm)
Evaluation of different geoids with various regularization parameters when first-order regularization is used (Units: cm)
Optimal network design for Poisson wavelets
A new height reference surface over Hong Kong
Statistics of the residuals of gravity data (Units: mGal)
Statistics of the residuals of geoidal heights at GPS/leveling points (Units: cm)
External accuracy of HKGEOID-2016 (Units: cm)
Statistical differences between GPS/leveling data and geoidal heights based on various geoids (Units: cm)
Statistical differences between GPS/leveling data and GGM-derived geoidal heights (Units: cm)
Poisson wavelets are used for modeling the regional gravity field using data from various observational techniques. The method combines data with different spatial coverage, various noise levels, and spectral contents. As a case study, terrestrial, shipborne, and satellite altimetry gravity data, as well as GPS/leveling measurements, are incorporated in the Poisson wavelet model for regional gravity field recovery over Hong Kong.
The Tikhonov regularization is introduced to manage the ill-conditioned least squares system; in particular, the performances of various regularization matrices are investigated. The numerical results show solutions with first-order regularization provide better results, where the accuracy of the local solution increases by 0.2 cm compared to that obtained from zero-order regularization. These results also indicate that first-order regularization may be more preferable in regional gravity field recovery using Poisson wavelets. Moreover, a direct approach is proposed to properly combine the gravimetric quasi-geoid/geoid and GPS/leveling data; a subset of the GPS/leveling data is treated as an independent observation group to formulate the new functional model, and the quasi-geoid/geoid that fits the local leveling system can be modeled in a single step. The results show the SD is approximately 0.3 cm for the residuals on the first two GPS/leveling groups, regarded as observations. The zero mean value indicates the gravimetric model and GPS/leveling data can be properly combined through this direct approach. In addition, an external validation with 61 independent GPS/leveling points shows that the accuracy of the new geoid, HKGEOID-2016, is approximately 1.1 cm. Compared with the original solution, HKGEOID-2000, the SD of the differences between observed and computed geoidal heights at all GPS/leveling points is reduced from 2.4 to 0.6 cm when HKGEOID-2016 is incorporated. This is a significant improvement. In addition, the performances of three recently published GGMs, EGM2008, EIGEN-6C3STAT and EIGEN-6C4, are investigated in Hong Kong. The corresponding results show the accuracies of these GGMs are all below 4 cm. The deviation from the local geoid at the decimeter level also indicates the GGM alone cannot recover a high-quality gravity field in this region, and local refinement is necessary.
Several issues should be carefully considered to make further improvements to the local geoid. Usually, the geoid is poorly modeled in the coastal areas due to unfavorable data coverage (Hipkin et al. 2004; Filmer and Featherstone 2012). In the study region, the distribution of shipborne data is quite sparse, and the quality of the altimetry data degrades in the vicinity of coastal areas. Therefore, the densification of shipborne data would further improve the local geoid in future work. In addition, the developments of appropriate altimeter waveform retracking approaches may also contribute to improving the geoid over coastal areas (Hwang et al. 2006; Andersen and Knudsen 2009). Errors in the GGM inevitably propagate into regional solutions because we usually consider the GGM as error-free data implemented in the remove-restore framework. The magnitude of these commission errors in the GGMs reaches the centimeter scale or larger (Pavlis et al. 2012), which should be carefully considered when computing the quasi-geoid/geoid at centimeter accuracy. These commission errors are actually calibrated with GPS/leveling data using the direct approach proposed in this study. However, the errors in the GGM could also be quantified if the full error variance–covariance matrix of the spherical coefficients is known, and a more realistic error variance–covariance matrix of the data could be estimated through error propagation. In this manner, the weights of different observation groups may be more properly determined, and the accuracy of the quasi-geoid/geoid, especially for the purely gravimetric one, may be further improved.
Global Positioning System
global geopotential model
Gravity Field and Climate Experiment
Gravity Field and Steady-State Ocean Circulation Explorer
least squares collocation
variance component estimation
European Improved Gravity Model of the Earth by New Techniques
residual terrain model
Technical University of Denmark
degree and order
digital terrain model
Shuttle Radar Topography Mission
General Bathymetric Chart of the Oceans
mean elevation surface
YW and ZL initiated the study, designed the numerical experiments, and wrote the manuscript. WC and YC provided the data and supplied beneficial suggestions. YW finalized the manuscript. All authors read and approved the final manuscript.
The authors would like to give our sincerest thanks to the two anonymous reviewers for their beneficial suggestions and comments, which helped us improve the manuscript. We would also like to thank the anonymous editor and the editorial office for their work on language editing. We acknowledge the assistance of Prof. Roland Klees and Dr. Cornelis Slobbe from Delft University of Technology for kindly providing the original software. This research was primarily supported by the National Natural Science Foundation of China (41374023), China Postdoctoral Science Foundation (No. 2016M602301), Key Laboratory of Geospace Environment and Geodesy, Ministry of Education, Wuhan University (15-02-08), the State Scholarship Fund from Chinese Scholarship Council (201306270014) and project K-ZS0J, ‘Development of a Hong Kong Positioning Infrastructure based on GPS, Beidou, and Ground based Augmentation System.’ Generic Mapping Tools (GMT) was used to generate the figures.
The authors declare that they have no competing interests.
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- Andersen OB, Knudsen P (2009) The DNSC08 mean sea surface and mean dynamic topography. J Geophys Res 114(C11):327–343. doi:10.1029/2008JC005179 View ArticleGoogle Scholar
- Andersen OB, Knudsen P, Stenseng L (2013) The DTU13 global mean sea surface from 20 years of satellite altimetry. In: OSTST Meeting, Boulder, Colo
- Bentel K, Schmidt M, Gerlach C (2013) Different radial basis functions and their applicability for regional gravity field representation on the sphere. Int J Geomath 4(1):67–96. doi:10.1007/s13137-012-0046-1 View ArticleGoogle Scholar
- Chambodut A, Panet I, Mandea M, Diament M, Holschneider M, Jamet O (2005) Wavelet frames: an alternative to spherical harmonic representation of potential fields. Geophys J Int 163(3):875–899. doi:10.1111/j.1365-246X.2005.02754.x View ArticleGoogle Scholar
- Chen Y, Luo Z (2004) A hybrid method to determine a local geoid model-Case study. Earth Planets Space 56(4):419–427. doi:10.1186/BF03352495 View ArticleGoogle Scholar
- Eshagh M, Zoghi S (2016) Local error calibration of EGM08 geoid using GNSS/levelling data. J Appl Geophys 130(5):209–217. doi:10.1016/j.jappgeo.2016.05.002 View ArticleGoogle Scholar
- Evans RB (1990) Hong Kong gravity observations in July 1990 with BGS Lacoste and Romberg meter No. 97 and international connections to IGSN 71. Report, British and Geology Survey, Hong Kong, China
- Featherstone WE (2000) Refinement of a gravimetric geoid using GNSS and levelling data. J Surv Eng 126(2):27–56. doi:10.1061/(ASCE)0733-9453(2000)126:2(27) View ArticleGoogle Scholar
- Filmer MS, Featherstone WE (2012) A re-evaluation of the offset in the Australian Height Datum between mainland Australia and Tasmania. Mar Geod 35(1):107–119. doi:10.1080/01490419.2011.634961 View ArticleGoogle Scholar
- 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, Columbus, Ohio, USA
- Forsberg R, Tscherning CC (1981) The use of height data in gravity field approximation by collocation. J Geophys Res 86(B9):7843–7854View ArticleGoogle Scholar
- Förste C, Bruinsma SL, Flechtner F, Marty JC, Lemoine JM, Dahle C, Abrikosov O, Neumayer KH, Biancale R, Barthelmes F, Balmino G (2012) A preliminary update of the Direct approach GOCE Processing and a new release of EIGEN-6C. AGU General Assembly, San FranciscoGoogle Scholar
- Förste C, Bruinsma SL, Abrikosov O, Lemoine JM, Schaller T, Götze HJ, Ebbing J, Marty JC, Flechtner F, Balmino G, 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. The 5th GOCE User Workshop, Paris, France
- Fotopoulos G (2005) Calibration of geoid error models via a combined adjustment of ellipsoidal, orthometric and gravimetric geoid height data. J Geod 79(1):111–123. doi:10.1007/s00190-005-0449-y View ArticleGoogle Scholar
- Hansen PC, Jensen TK, Rodriguez G (2007) An adaptive pruning algorithm for the discrete L-curve criterion. J Comput Appl Math 198(2):483–492. doi:10.1016/j.cam.2005.09.026 View ArticleGoogle Scholar
- Hayn M, Panet I, Diament M, Holschneider M, Mandea M, Davaille A (2012) Wavelet-based directional analysis of the gravity field: evidence for large-scale undulations. Geophys J Int 189(3):1430–1456. doi:10.1111/j.1365-246X.2012.05455.x View ArticleGoogle Scholar
- Heck B, Seitz K (2007) A comparison of the tesseroid, prism and point-mass approaches for mass reductions in gravity field modelling. J Geod 81(2):121–136. doi:10.1007/s00190-006-0094-0 View ArticleGoogle Scholar
- Heiskanen WA, Moritz H (1967) Physical geodesy. WH Freeman and Co., San FranciscoGoogle Scholar
- Hipkin RG, Haines K, Beggan C, Bingley R, Hernandez F, Holt J, Baker T (2004) The geoid EDIN2000 and mean sea surface topography around the British Isles. Geophys J Int 157(2):565–577. doi:10.1111/j.1365-246X.2004.01989.x View ArticleGoogle Scholar
- Holschneider M, Iglewska-Nowak I (2004) Poisson wavelets on the sphere. J Fourier Anal Appl 13(4):405–419. doi:10.1007/s00041-006-6909-9 View ArticleGoogle Scholar
- Hwang C, Guo J, Deng X, Hsu H, Liu Y (2006) Coastal gravity anomalies from retracked geosat/GM altimetry: improvement, limitation and the role of airborne gravity data. J Geod 80(4):204–216. doi:10.1007/s00190-006-0052-x View ArticleGoogle Scholar
- Jiang T, Wang YM (2016) On the spectral combination of satellite gravity model, terrestrial and airborne gravity data for local gravimetric geoid computation. J Geod. doi:10.1007/s00190-016-0932-7 Google Scholar
- Klees R, Prutkin I (2008) The combination of GNSS-levelling data and gravimetric (quasi-) geoid heights in the presence of noise. J Geod 84(12):731–749. doi:10.1007/s00190-010-0406-2 View ArticleGoogle Scholar
- 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(8):457–471. doi:10.1007/s00190-007-0196-3 View ArticleGoogle Scholar
- Koch KR, Kusche J (2002) Regularization of geopotential determination from satellite data by variance components. J Geod 76(5):259–268. doi:10.1007/s00190-002-0245-x View ArticleGoogle Scholar
- Kuroishi Y (2009) Improved geoid model determination for Japan from GRACE and a regional gravity field model. Earth Planets Space 61(7):807–813. doi:10.1186/BF03353191 View ArticleGoogle Scholar
- Kusche J (2003) A Monte-Carlo technique for weight estimation in satellite geodesy. J Geod 76(11):641–652. doi:10.1007/s00190-002-0302-5 View ArticleGoogle Scholar
- Kusche J, Klees R (2002) Regularization of gravity field estimation from satellite gravity gradients. J Geod 76(6):359–368. doi:10.1007/s00190-002-0257-6 View ArticleGoogle Scholar
- 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(5):3825–3845. doi:10.1002/2015JB012586 View ArticleGoogle Scholar
- Luo Z, Ning J, Chen Y, Yang Z (2005) High precision geoid models HKGEOID-2000 for Hong Kong and SZGEOID-2000 for Shenzhen, China. Mar Geod 28(2):191–200. doi:10.1080/01490410590953758 View ArticleGoogle Scholar
- Nahavandchi N, Soltanpour A (2006) Improved determination of heights using a conversion surface by combining gravimetric quasi-geoid/geoid and GNSS-levelling height differences. Stud Geophys Geod 50(2):165–180. doi:10.1007/s11200-006-0010-3 View ArticleGoogle Scholar
- Odera PA, Fukuda Y (2014) Improvement of the geoid model over Japan using integral formulae and combination of GGMs. Earth Planets Space 66(1):1–7. doi:10.1186/1880-5981-66-22 View ArticleGoogle Scholar
- Omang OCD, Forsberg R (2000) How to handle topography in practical geoid determination: three examples. J Geod 74(6):458–466. doi:10.1007/s001900000107 View ArticleGoogle Scholar
- Panet I, Kuroishi Y, Holschneider M (2011) Wavelet modelling of the gravity field by domain decomposition methods: an example over Japan. Geophys J Int 184(1):203–219. doi:10.1111/j.1365-246X.2010.04840.x View ArticleGoogle Scholar
- Pavlis NK, Factor JK, Holmes SA (2007) Terrain-related gravimetric quantities computed for the next EGM. Proceedings of the 1st international symposium of the international gravity field service, Istanbul, pp. 318–323
- Pavlis NK, Holmes SA, Kenyon SC, Factor JF (2008) An Earth gravitational model to degree 2,160: EGM2008. Presented at the 2008 General Assembly of the European Geosciences Union, Vienna, April 13–18
- Pavlis NK, Holmes SA, Kenyon SC, Factor JF (2012) The development and evaluation of Earth Gravitational Model (EGM2008). J Geophys Res 117:B04406. doi:10.1029/2011JB008916 View ArticleGoogle Scholar
- Prutkin I, Klees R (2008) On the non-uniqueness of local quasi-geoids computed from terrestrial gravity anomalies. J Geod 82(3):147–156. doi:10.1007/s00190-007-0161-1 View ArticleGoogle Scholar
- Rummel R, Balmino G, Johannessen J, Visser P, Woodworth P (2002) Dedicated gravity field missions-Principle and aims. J Geodyn 33(1):3–20. doi:10.1016/S0264-3707(01)00050-3 View ArticleGoogle Scholar
- Shih HC, Hwang C, Barriot JP, Mouyen M, Corréia P, Lequeux D, Sichoix L (2015) High-resolution gravity and geoid models in Tahiti obtained from new airborne and land gravity observations: data fusion by spectral combination. Earth Planets Space 67(1):1–16. doi:10.1186/s40623-015-0297-9 View ArticleGoogle Scholar
- Slobbe DC (2013) Roadmap to a mutually consistent set of offshore vertical reference frames. Dissertation, Delft University of Technology, Delft
- Tapley BD, Bettadpur S, Watkins M, Reigber C (2004) The gravity recovery and climate experiment: mission overview and early results. Geophys Res Lett 31:L09607. doi:10.1029/2004GL019920 View ArticleGoogle Scholar
- Tenzer R, Klees R (2008) The choice of the spherical radial basis functions in local gravity field modeling. Stud Geophys Geod 52(3):287–304. doi:10.1007/s11200-008-0022-2 View ArticleGoogle Scholar
- Tenzer R, Klees R, Wittwer T (2012) Local gravity field modelling in rugged terrain using spherical radial basis functions: case study for the Canadian rocky mountains. In: Kenyon S (eds) Geodesy for Planet Earth, International Association of Geodesy Symposia 136, Springer, Berlin, pp 401–409
- Tscherning CC (1978) Introduction to functional analysis with a view to its application in approximation theory. In: Moritz H, Sünkel H (eds) Approximation methods in geodesy. Karlsruhe, GermanyGoogle Scholar
- Wang Y, Saleh J, Li X, Roman DR (2012) The US Gravimetric Geoid of 2009 (USGG2009): model development and evaluation. J Geod 86(3):165–180. doi:10.1007/s00190-011-0506-7 View ArticleGoogle Scholar
- Wittwer T (2009) Regional gravity field modelling with radial basis functions, Dissertation, Delft University of Technology, Delft
- Wu Y, Luo Z (2016) The approach of regional geoid refinement based on combining multi-satellite altimetry observations and heterogeneous gravity data sets. Chin J Geophys (in Chinese) 59(5):1596–1607. doi:10.6038/cjg20160505 Google Scholar
- Wu Y, Luo Z, Zhou B (2016) Regional gravity modelling based on heterogeneous data sets by using Poisson wavelets radial basis functions. Chin J Geophys (in Chinese) 59(3):852–864. doi:10.6038/cjg20160308 Google Scholar
- Xu P (1992) The value of minimum norm estimation of geopotential fields. Geophys J Int 111(1):170–178. doi:10.1111/j.1365-246X.1992.tb00563.x View ArticleGoogle Scholar
- Xu P (1998) Truncated SVD methods for discrete linear ill-posed problems. Geophys J Int 135(2):505–514. doi:10.1046/j.1365-246X.1998.00652.x View ArticleGoogle Scholar