# High-Accuracy Quasi-Geoid Determination Using Molodensky’s Series Solutions and Integrated Gravity/GNSS/Leveling Data

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Datasets

^{2}within a hilly terrain, as depicted in Figure 1. To determine the quasi-geoid of the Shangyu area, a combination of topography; terrestrial gravity; Earth Gravitational Model 2008 (EGM2008), representing the global gravity field; and GNSS/leveling data from multiple sources was employed.

#### 2.1. Topographic Data

#### 2.2. Ground Gravity Anomaly

^{−3}mGal (1 mGal = 10

^{−5}m∙s

^{−2}).

^{−3}mGal.

^{−3}mGal. Next, the normal gravity field and free-air corrections are applied to the adjusted terrestrial gravity observations, yielding the free-air gravity anomaly.

#### 2.3. GGM

#### 2.4. GNSS/Leveling Data

^{−8}·d in the north–south direction, 2.35 mm + 0.09 × 10

^{−8}·d in the east–west direction and better than 4.67 mm + 0.16 × 10

^{−8}·d in the vertical direction. For the baseline length direction, the accuracy was found to be 1.10 mm + 0.06 × 10

^{−8}·d, where d represents the distance between adjacent points.

## 3. Method for Local Quasi-Geoid Determinations

#### 3.1. Linearized Molodensky’s Solution

_{p}= R + h

_{p}; and h

_{p}is the normal height of the computation point, P; h denotes the normal height of the running point; R is the average radius of the Earth; and ψ is the spherical distance between the computation point and the integration element.

#### 3.2. Combination of Heights

_{j}is the parameter to be determined. The resulting quasi-geoid height is denoted by $\zeta \left(\varphi ,\lambda \right)$.

## 4. Results and Discussion

#### 4.1. Computation of Hybrid Quasi-Geoid Model

_{2}term on the height anomaly within the test area was found to be negligible, measuring less than 1 mm, the contributions of the G

_{2}term as well as the higher-order terms were disregarded. The computational procedure adopted for the gravimetric quasi-geoid model was as follows:

- Computation of the zero-order term of the height anomaly: The residual height anomaly was computed using Stokes’ theory by taking the difference (residual gravity anomaly) between the free-air and reference gravity anomalies derived from EGM2008. Adding the reference geoidal undulation to the residual height anomaly yielded the zero-order term of the height anomaly, ζ
_{0}; - Computation of the G
_{1}term: Utilizing Equation (13), the G_{1}term was determined based on the DTM, the gridded free-air anomaly, and the zero-order term of the height anomaly, ζ_{0}; - Computation of the first-order term of the height anomaly: The effect of the first-order term on height anomaly, denoted as ζ
_{1}, was obtained using the spherical Stokes’ integral given by

_{1}term to both the gravity and height anomalies are illustrated in Figure 9a,b, respectively. In particular, the effect of the G

_{1}term on the height anomalies exhibited a maximum value of 1.2 cm in the southeastern part of the study area. This outcome aligns with expectations as this region experiences higher and more variable topographic heights. However, in the northern region, where the terrain is relatively lower and smoother, the effect of the G

_{1}term was comparatively small.

- 4.
- Computation of the gravimetric quasi-geoid model: The gravimetric quasi-geoid model, denoted as ζ
^{grav}is expressed by

#### 4.2. Accuracy Assessment

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Brennecke, J.; Groten, E. The deviations of the sea surface from the geoid and their effect on geoid computation. Bull. Geod.
**1977**, 51, 47–51. [Google Scholar] [CrossRef] - Sansò, F.; Sideris, M.G. Geoid Determination: Theory and Methods, Lecture Notes in Earth System Sciences; Springer: Berlin/Heidelberg, Germany, 2013. [Google Scholar]
- Steinbergerk, B.; Rathnayake, S.; Kendall, E. The Indian Ocean Geoid Low at a plume-slab. Tectonophysics
**2021**, 817, 229037. [Google Scholar] [CrossRef] - Falcão, A.P.; Matos, J.; Gonçalves, A.; Casaca, J.; Sousa, J. Preliminary Results of Spatial Modelling of GPS/Levelling Heights: A Local Quasi-Geoid/Geoid for the Lisbon Area. In Gravity, Geoid and Earth Observation; International Association of Geodesy, Symposia; Mertikas, S., Ed.; Springer: Berlin/Heidelberg, Germany, 2010; Volume 135. [Google Scholar]
- Fotopoulos, G. Calibration of geoid error models via a combined adjustment of ellipsoidal, orthometric and gravimetric geoid height data. J. Geod.
**2005**, 79, 111–123. [Google Scholar] [CrossRef] - Grigoriadis, V.N.; Vergos, G.S.; Barzaghi, R.; Carrion, D.; Koc, O. Collocation and FFT-based geoid estimation within the Colorado 1 cm geoid experiment. J. Geod.
**2021**, 95, 52. [Google Scholar] [CrossRef] - Sansò, F.; Rummel, R. Geodetic Boundary Value Problems in View of the One Centimeter Geoid; Springer: Berlin/Heidelberg, Germany, 1977. [Google Scholar]
- Kotsakis, C.; Tsalis, I. Combination of Geometric and Orthometric Heights in the Presence of Geoid and Quasi-geoid Models. In Gravity, Geoid and Height Systems; International Association of Geodesy, Symposia; Marti, U., Ed.; Springer: Cham, Switzerland, 2014; Volume 141. [Google Scholar]
- Featherstone, W.E.; Olliver, J.G. A method to validate gravimetric-geoid computation software based on Stokes’s integral formula. J. Geod.
**1997**, 71, 571–576. [Google Scholar] [CrossRef] - Sjöberg, L.E. Topographic effects by the Stokes-Helmert method of geoid and quasi-geoid determinations. J. Geod.
**2000**, 74, 255–268. [Google Scholar] [CrossRef] - Ardestani, V.E.; Martinec, Z. Geoid determination through ellipsoidal Stokes boundary-value problem. Stud. Geophys. Et Geod.
**2000**, 4, 353–363. [Google Scholar] [CrossRef] - Molodensky, M.S.; Eremeev, V.F.; Yurkina, M.I. Methods for the Study of the External Gravitational Field and Figure of the Earth; Israeli Program for Scientific Translations: Jerusalem, Israel, 1962. [Google Scholar]
- Brovar, V.V. On the solution of Molodensky’s boundary value problem. Bull. Geod.
**1964**, 72, 167–173. [Google Scholar] [CrossRef] - Ardalan, A.A.; Grafarend, E.W.; Ihde, J. Molodensky potential telluroid based on a minimum-distance map. Case study: The quasi-geoid of East Germany in the World Geodetic Datum 2000. J. Geod.
**2002**, 76, 127–138. [Google Scholar] [CrossRef] - Kuroishi, Y.; Ando, H.; Fukuda, Y. A new hybrid geoid model for Japan, GSIGE02000. J. Geod.
**2002**, 76, 428–436. [Google Scholar] [CrossRef] - Nahavandchi, H.; Soltanpour, A. Improved determination of heights using a conversion surface by combining gravimetric quasi-geoid/geoid and GPS-levelling height differences. Stud. Geophys. Geod.
**2006**, 50, 165–180. [Google Scholar] [CrossRef] - Tziavos, I.N.; Vergos, G.S.; Grigoriadis, V.N.; Andritsanos, V.D. Adjustment of Collocated GPS, Geoid and Orthometric Height Observations in Greece. Geoid or Orthometric Height Improvement? In Geodesy for Planet Earth; International Association of Geodesy, Symposia; Kenyon, S., Pacino, M., Marti, U., Eds.; Springer: Berlin/Heidelberg, Germany, 2012; Volume 136. [Google Scholar]
- Hwang, C.; Hsu, H.; Featherstone, W.; Cheng, C.; Yang, M.; Huang, W.; Wang, C.; Huang, J.; Chen, K.; Huang, C.; et al. New gravimetric-only and hybrid geoid models of Taiwan for height modernisation, cross-island datum connection and airborne LiDAR mapping. J. Geod.
**2020**, 94, 83. [Google Scholar] [CrossRef] - Yang, Y. Robust estimation of geodetic datum transformation. J. Geod.
**1999**, 73, 268–274. [Google Scholar] [CrossRef] - Awange, J.L.; Paláncz, B. Robust Estimation. In Geospatial Algebraic Computations; Springer: Cham, Switzerland, 2016. [Google Scholar]
- Denker, H. Evaluation of SRTM3 and GTOPO30 terrain data in Germany. In International Association of Geodesy Symposia; Springer: Cham, Switzerland, 2005; Volume 129, pp. 218–223. [Google Scholar]
- Amante, C.; Eakins, B.W. ETOPO1, 1 Arc-Minute Global Relief Model: Procedures, Data Sources and Analysis; NOAA Technical Memorandum NESDIS; National Geophysical Data Center, NOAA: Boulder, CO, USA, 2000; Volume 24, p. 19.
- Yang, M.; Hirt, C.; Rexer, M.; Pail, R.; Yamazaki, D. The tree canopy effect in gravity forward modelling. Geophys. J. Int.
**2019**, 219, 271–289. [Google Scholar] [CrossRef] - Scherneck, H.G.; Rajner, M.; Engfeldt, A. Superconducting gravimeter and seismometer shedding light on FG5’s offsets, trends and noise: What observations at Onsala Space Observatory can tell us. J. Geod.
**2020**, 94, 80. [Google Scholar] [CrossRef] - GB/T 20256-2019; Specifications for the Gravimetry Control. Standards Press of China: Beijing, China, 2019.
- Francis, O. Correction to: Performance assessment of the relative gravimeter Scintrex CG-6. J. Geod.
**2022**, 96, 2. [Google Scholar] [CrossRef] - GB/T 17944-2018; Specifications for the Dense Gravity Measurement. Standards Press of China: Beijing, China, 2019.
- Dziewonski, A.M.; Anderson, D.L. Preliminary reference earth model. Phys. Earth Planet. Inter.
**1981**, 25, 297–356. [Google Scholar] [CrossRef] - Merriam, J.B. Atmospheric pressure and gravity. Geophys. J. Int.
**1992**, 190, 488–500. [Google Scholar] [CrossRef] - Spratt, R.S. Modelling the effect of atmospheric pressure variations on gravity. Geophys. J. R. Astron. Soc.
**1982**, 71, 173–186. [Google Scholar] [CrossRef] - Wei, S.C.; Xu, J.Q.; Zhou, J.C. Piece-wise linear dynamic adjustment for gravity network. Acta Geod. Cartogr. Sin.
**2016**, 45, 511–520. [Google Scholar] - Wijaya, D.D.; Muhammad, N.A.; Prijatna, K.; Sadarviana, V.; Sarsito, D.A.; Pahlevi, A.; Variandy, E.D.; Putra, W. pyGABEUR-ITB: A free software for adjustment of relative gravimeter data. Geomatika
**2019**, 25, 95–102. [Google Scholar] [CrossRef] - Hwang, C.; Wang, C.G.; Lee, L.H. Adjustment of relative gravity measurements using weighted and datum-free constraints. Comput. Geosci.
**2002**, 28, 1005–1015. [Google Scholar] [CrossRef] - Lagios, E. A Fortran IV program for a least-squares gravity base-station network adjustment. Comput. Geosci.
**1983**, 10, 263–276. [Google Scholar] [CrossRef] - Fullea, J.; Fernàndez, M.; Zeyen, H. FA2BOUG-A FORTRAN 90 code to compute Bouguer gravity anomalies from gridded free-air anomalies: Application to the Atlantic-Mediterranean transition zone. Comput. Geosci.
**2008**, 34, 1665–1681. [Google Scholar] [CrossRef] - Shepard, D. A Two-Dimensional Interpolation Function for Irregularly Spaced Data. In Proceedings of the 23rd National Conference ACM, New York, NY, USA, 27–29 August 1968; pp. 517–523. [Google Scholar]
- Pavlis, N.K.; Holmes, S.A.; Kenyon, S.C.; Factor, J.K. An earth gravitational model degree 2160: EGM2008. In Proceedings of the 2008 General Assembly of the European Geosciences Union, Vienna, Austria, 13–18 April 2008. [Google Scholar]
- Sánchez, L.; Ågren, J.; Huang, J.; Wang, Y.M.; Forsberg, R. Basic Agreements for the Computation of Station Potential Values as IHRS Coordinates, Geoid Undulations and Height Anomalies within the Colorado 1 cm Geoid Experiment. In Proceedings of the International Symposium on Gravity, Geoid and Height Systems 2018 (GGHS2018), Copenhagen, Denmark, 17–21 September 2018. [Google Scholar]
- Pail, R.; Bruinsma, S.; Migliaccio, F.; Förste, C.; Goiginger, H.; Schuh, W.-D.; Höck, E.; Reguzzoni, M.; Brockmann, J.M.; Abrikosov, O.; et al. First GOCE gravity field models derived by three different approaches. J. Geod.
**2011**, 85, 819–843. [Google Scholar] [CrossRef] - Vaľko, M.; Mojzeš, M.; Janák, J.; Papčo, J. Comparison of two different solutions to Molodensky’s G
_{1}term. Stud. Geophys. Geod.**2008**, 52, 71–86. [Google Scholar] [CrossRef] - Tiron, M. Some problems regarding the way of solving Molodensky’s integral equation for the earth considered as a plane. Stud. Geophys. Et Geod.
**1965**, 9, 137–144. [Google Scholar] [CrossRef] - Moritz, H. Series solutions of Molodensky’s problem. Publ. Deut. Geod. Komm.
**1971**, A, 70. [Google Scholar] [CrossRef] - Guo, J.Y. Physical Geodesy. Springer Textbooks in Earth Sciences, Geography and Environment; Springer: Cham, Switzerland, 2023. [Google Scholar]
- Sideris, M.G.; Schwarz, K.P. Solving Molodensky’s series by fast Fourier transform techniques. Bull. Geod.
**1986**, 60, 51–63. [Google Scholar] [CrossRef] - Müssle, M.; Heck, B.; Seitz, K.; Grombein, T. On the effect of planar approximation in the Geodetic Boundary Value Problem. Stud. Geophys. Geod.
**2014**, 58, 536–555. [Google Scholar] [CrossRef] - Heiskanen, W.A.; Moritz, H. Physical Geodesy; Freeman: San Francisco, CA, USA, 1967. [Google Scholar]
- Yu, B.; Guan, Z.; Xu, X.; Yang, L.O. The Application of an Improved Multi-surface Function Based on Earth Gravity Field Model in GPS Leveling Fitting. In Geo-Informatics in Resource Management and Sustainable Ecosystem GRMSE; Communications in Computer and Information, Science; Bian, F., Xie, Y., Cui, X., Zeng, Y., Eds.; Springer: Berlin/Heidelberg, Germany, 2013; Volume 399. [Google Scholar]
- Klees, R.; Prutkin, I. The combination of GNSS-levelling data and gravimetric (quasi-) geoid heights in the presence of noise. J. Geod.
**2010**, 84, 731–749. [Google Scholar] [CrossRef] - Watson, D.F.; Philip, G.M. A refinement of inverse distance weighted interpolation. Geo-Processing
**1985**, 2, 315–327. [Google Scholar] - Kubik, K. An error theory for the Danish method. In Proceedings of the Symp Math Models, Accuracy Aspects and Quality Control, ISPRS Commission III Symposium, Helsinki, Finland, 7–11 June 1982. [Google Scholar]

**Figure 2.**Gravity measurements (red stars indicate the gravity control-points, blue dots indicate the terrestrial gravity points, and purple dots indicate the grav_32.1).

**Figure 6.**Statistical results of the difference between the reference and terrestrial gravity anomalies.

**Figure 7.**GNSS/Leveling stations (blue dots indicate the control points, and red dots indicate the check points).

**Figure 10.**(

**a**) Differences between the hybrid quasi-geoid model and observed heights at the control points, and (

**b**) differences between the hybrid quasi-geoid model and observed heights at the check points.

Model | Max | Min | Mean | STD |
---|---|---|---|---|

GNSS/leveling heights (m) | 11.72 | 9.82 | 10.72 | 0.44 |

topographic data (m) | 960.38 | −34.81 | 108.80 | 179.74 |

terrestrial gravity (mGal) | 61.26 | −12.43 | 4.38 | 11.02 |

reference gravity anomalies (mGal) | 50.97 | −18.51 | 0.99 | 11.93 |

**Table 2.**Comparisons of the hybrid quasi-geoid model with reference height anomaly from EGM2008 and gravimetric quasi-geoid (unit: m).

Model | Max | Min | STD |
---|---|---|---|

compared with reference height anomalies | 0.117 | −0.111 | 0.049 |

compared with gravimetric quasi-geoid | 0.049 | −0.035 | 0.018 |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Guo, D.; Chen, X.; Xue, Z.; He, H.; Xing, L.; Ma, X.; Niu, X.
High-Accuracy Quasi-Geoid Determination Using Molodensky’s Series Solutions and Integrated Gravity/GNSS/Leveling Data. *Remote Sens.* **2023**, *15*, 5414.
https://doi.org/10.3390/rs15225414

**AMA Style**

Guo D, Chen X, Xue Z, He H, Xing L, Ma X, Niu X.
High-Accuracy Quasi-Geoid Determination Using Molodensky’s Series Solutions and Integrated Gravity/GNSS/Leveling Data. *Remote Sensing*. 2023; 15(22):5414.
https://doi.org/10.3390/rs15225414

**Chicago/Turabian Style**

Guo, Dongmei, Xiaodong Chen, Zhixin Xue, Huiyou He, Lelin Xing, Xian Ma, and Xiaowei Niu.
2023. "High-Accuracy Quasi-Geoid Determination Using Molodensky’s Series Solutions and Integrated Gravity/GNSS/Leveling Data" *Remote Sensing* 15, no. 22: 5414.
https://doi.org/10.3390/rs15225414