Journal of Geodesy

, Volume 92, Issue 5, pp 561–572 | Cite as

The gravity field model IGGT_R1 based on the second invariant of the GOCE gravitational gradient tensor

  • Biao Lu
  • Zhicai Luo
  • Bo Zhong
  • Hao Zhou
  • Frank Flechtner
  • Christoph Förste
  • Franz Barthelmes
  • Rui Zhou
Original Article


Based on tensor theory, three invariants of the gravitational gradient tensor (IGGT) are independent of the gradiometer reference frame (GRF). Compared to traditional methods for calculation of gravity field models based on the gravity field and steady-state ocean circulation explorer (GOCE) data, which are affected by errors in the attitude indicator, using IGGT and least squares method avoids the problem of inaccurate rotation matrices. The IGGT approach as studied in this paper is a quadratic function of the gravity field model’s spherical harmonic coefficients. The linearized observation equations for the least squares method are obtained using a Taylor expansion, and the weighting equation is derived using the law of error propagation. We also investigate the linearization errors using existing gravity field models and find that this error can be ignored since the used a-priori model EIGEN-5C is sufficiently accurate. One problem when using this approach is that it needs all six independent gravitational gradients (GGs), but the components \(V_{xy}\) and \(V_{yz}\) of GOCE are worse due to the non-sensitive axes of the GOCE gradiometer. Therefore, we use synthetic GGs for both inaccurate gravitational gradient components derived from the a-priori gravity field model EIGEN-5C. Another problem is that the GOCE GGs are measured in a band-limited manner. Therefore, a forward and backward finite impulse response band-pass filter is applied to the data, which can also eliminate filter caused phase change. The spherical cap regularization approach (SCRA) and the Kaula rule are then applied to solve the polar gap problem caused by GOCE’s inclination of \(96.7^{\circ }\). With the techniques described above, a degree/order 240 gravity field model called IGGT_R1 is computed. Since the synthetic components of \(V_{xy}\) and \(V_{yz}\) are not band-pass filtered, the signals outside the measurement bandwidth are replaced by the a-priori model EIGEN-5C. Therefore, this model is practically a combined gravity field model which contains GOCE GGs signals and long wavelength signals from the a-priori model EIGEN-5C. Finally, IGGT_R1’s accuracy is evaluated by comparison with other gravity field models in terms of difference degree amplitudes, the geostrophic velocity in the Agulhas current area, gravity anomaly differences as well as by comparison to GNSS/leveling data.


Invariants of the gravitational gradient tensor Least squares method Linearization Band-pass filter GOCE 



Thanks for the constructive comments and beneficial suggestions from the anonymous reviewers and editors, which help us a lot for improving this manuscript. We also would like to express appreciation to Dr. X.Y. Wan of Qian Xuesen Laboratory of Space Technology (QLST), Dr. O. Abrykosov of German Research Centre for Geosciences (GFZ) and Dr. J. Bouman of German Geodetic Research Institute (DGFI) for their kind help and discussions. Thanks for the European Space Agency for providing the GOCE data. This study is supported by the Chinese Scholarship Council (No. 201506270158), the Natural Science Foundation of China (Nos. 41104014, 41131067, 41374023, 41474019 and 41504013) and the Key Laboratory of Geospace Environment and Geodesy, Ministry Education, Wuhan University (No. 16-02-07).


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  1. 1.School of Geodesy and GeomaticsWuhan UniversityWuhanPeople’s Republic of China
  2. 2.GFZ German Research Centre for GeosciencesPotsdamGermany
  3. 3.Department of Geodesy and Geoinformation ScienceTechnical University of BerlinBerlinGermany
  4. 4.MOE Key Laboratory of Fundamental Physical Quantities Measurement, School of PhysicsHuazhong University of Science and TechnologyWuhanPeople’s Republic of China
  5. 5.Institute of GeophysicsHuazhong University of Science and TechnologyWuhanPeople’s Republic of China
  6. 6.Key Laboratory of Geospace Environment and Geodesy, Ministry of EducationWuhan UniversityWuhanPeople’s Republic of China
  7. 7.Zhengzhou Information Engineering UniversityZhengzhouPeople’s Republic of China

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