Abstract
The spherical cap harmonics (SCH) method can be used in regional geoid modeling. The core of this approach is the computation of its associated Legendre functions (ALF) with non-integer degree. However, it is unlikely to obtain a large number of zero-root values for the non-integer ALF. To overcome this problem, a new approach called virtual spherical harmonics (VSH) is proposed in this paper to transform the cap range into the whole sphere so that unlimited numbers of zero-root values can be obtained. The new approach was tested using four cap ranges with the radii of
Funding source: National Natural Science Foundation of China
Award Identifier / Grant number: 41304020
Award Identifier / Grant number: 41404026
Award Identifier / Grant number: 41464001
Funding source: China Scholarship Council
Award Identifier / Grant number: 201500880003
Funding statement: This research is supported by the National Natural Science Foundation of China (41304020, 41404026, 41464001) and the China Scholarship Council (201500880003).
Acknowledgment
The authors would like to thank Prof. Kefei Zhang and Dr. Suqin Wu for their assistance with the preparation of this paper.
References
[1] Heiskanen W. A. and Moritz H., 1967. Physical Geodesy. San Francisco: Freeman and Company.10.1007/BF02525647Search in Google Scholar
[2] Engelis T., Rapp R. H., Tscherning C. C., 1984. The precise computation of geoid undulation differences with comparison to results obtained from the global positioning system. Geophysical research letters, 1(9): 821–824.10.1029/GL011i009p00821Search in Google Scholar
[3] Ahmed A. E. M., 2013. Geoid undulation of Sudan using orthometric heights compared with EGM96 and EGM2008. International Journal of Advanced Research in IT and Engineering, 2(11): 43–53.Search in Google Scholar
[4] Turgut B, Inal C, Corumluoglu O, 2004. Comparison of the geoid undulations obtained by EGM96, TG99 and GPS. Turkey. Acta Geod. Geoph. Hung. 39(4): 403–410.10.1556/AGeod.39.2004.4.8Search in Google Scholar
[5] Haines G. V., 1985. Spherical cap harmonic analysis. Journal of Geophysical Research, 90(B3): 2583–2591.10.1029/JB090iB03p02583Search in Google Scholar
[6] Haines G. V., 1988. Computer programs for spherical cap harmonic analysis of potential and general fields. Computers & Geosciences, 14(4): 413–447.10.1016/0098-3004(88)90027-1Search in Google Scholar
[7] Nevanlinna H., Ryno J., Haines G. V., Borg K., 1988. Spherical Cap Harmonic Analysis Applied to the Scandinavian Geomagnetic Field 1985.0. Deutsche Hydrografische Zeitschrift, 41(3): 177–186.10.1007/BF02225927Search in Google Scholar
[8] Li J., Chao D., Ning J., 1995. Spherical cap harmonic expansion for local gravity field representation. Manuscr Geod, 20(4): 265–277.Search in Google Scholar
[9] Zhao J., Wang S. H., Liu H., Li D., 2010. Study one establishing local geomagnetic model using spherical cap harmonic analysis. Science of Surveying and Mapping, 35(1): 9, 50–52.Search in Google Scholar
[10] Liu J, Chen R., An J., Wang Z., Hyyppa J., 2014. Spherical Cap Harmonic Analysis of the Arctic Ionospheric Tec for One Solar Cycle. J. Geophys. Res. Space Physics, 119(1): 601–619.10.1002/2013JA019501Search in Google Scholar
[11] An J., Ning X., Wang Z., Zhang X., 2015. Antarctic Ionospheric Prediction Based on Spherical Cap Harmonic Analysis and Time Series Analysis. Geomatics and Information Science of Wuhan University, 40(5): 112–116.Search in Google Scholar
[12] Wang J., Chen H., Chen Y., 2012. The Analysis of the Associated Legendre Functions with Non-integral Degree. Applied Mechanics and Materials, 1503(130): 3001–3005.10.4028/www.scientific.net/AMM.130-134.3001Search in Google Scholar
[13] Thebault E., Schott J. J., Mandea M., 2006. Revised spherical cap harmonic analysis (R-SCHA): Validation and properties. Journal of Geophysical Research, 111(B00102): 1–17.10.1029/2005JB003836Search in Google Scholar
[14] De-Santis A., 1992. Conventional spherical harmonic analysis for regional modeling of geomagnetic field. Geophysical Research Letter, 19(10): 1065–1067.10.1029/92GL01068Search in Google Scholar
[15] De-Santis A., Torta J. M., Lowes F. J., 1999. Spherical cap harmonics revisited and their relationship to ordinary spherical harmonics. Physics and Chemistry of the Earth (A), 24(11): 935–941.10.1016/S1464-1895(99)00138-6Search in Google Scholar
[16] Younis G., Jäger R, Becker M., 2013. Transformation of Global Spherical Harmonic Models of the Gravity Field to a Local Adjusted Spherical Cap Harmonic Model. Arabian Journal of Geosciences, 6(2): 375–381.10.1007/s12517-011-0352-1Search in Google Scholar
[17] Cao Y., Wang J., 2008. Application of Spherical Cap Harmonic Analysis to Fit GPS Level Height. Geomatics and Information Science of Wuhan University, 33(5): 740–743.Search in Google Scholar
[18] Guo J., Wang S., Li G., Mao W., Ji Y., 2012. Local Quasi-Geoid Refinement based on Spherical Cap Harmonic Model. Applied Mechanics and Materials, 226(228): 1947–1950.10.4028/www.scientific.net/AMM.226-228.1947Search in Google Scholar
[19] Lebedev N. N., 1972. Special functions and their application. New York, NY, Dover.Search in Google Scholar
[20] Hwang C., Chen S., 1997. Fully normalized spherical cap harmonics: application to the analysis of sea-level data from TOPEX/POSEIDON and ERS-1. Geophysical Journal International, 129(2): 450–460.10.1111/j.1365-246X.1997.tb01595.xSearch in Google Scholar
[21] Wang J., Chen H., Chen Y., 2012. The analysis of the associated Legendre functions with non-integral degree. Applied Mechanics and Materials, 130(134): 3001–3005.10.4028/www.scientific.net/AMM.130-134.3001Search in Google Scholar
[22] Wang J., Chu W., Dai B, Liu S., 2017. Geoid numerical calculation and structural analysis. Science of Surveying and Mapping, 42(04): 1–7.Search in Google Scholar
© 2019 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- Research Articles
- Predicting orbit and clock corrections during their outage in real-time positioning using GPS, GLONASS and QZSS for natural hazard warning systems
- Vertical ionospheric delay estimation for single-receiver operation
- The stochastic model for Global Navigation Satellite Systems and terrestrial laser scanning observations: A proposal to account for correlations in least squares adjustment
- Robust external calibration of terrestrial laser scanner and digital camera for structural monitoring
- System identification of a robot arm with extended Kalman filter and artificial neural networks
- Construction of regional geoid using a virtual spherical harmonics model
Articles in the same Issue
- Frontmatter
- Research Articles
- Predicting orbit and clock corrections during their outage in real-time positioning using GPS, GLONASS and QZSS for natural hazard warning systems
- Vertical ionospheric delay estimation for single-receiver operation
- The stochastic model for Global Navigation Satellite Systems and terrestrial laser scanning observations: A proposal to account for correlations in least squares adjustment
- Robust external calibration of terrestrial laser scanner and digital camera for structural monitoring
- System identification of a robot arm with extended Kalman filter and artificial neural networks
- Construction of regional geoid using a virtual spherical harmonics model