Skip to main content
Log in

First-order derivatives of principal and main invariants of gravity gradient tensor of the tesseroid and spherical shell

  • Original Article
  • Published:
Journal of Geodesy Aims and scope Submit manuscript

Abstract

The invariants of gravity (or gravitational) gradient tensor can be applied as the additional internal parameters for the gravity gradient tensor, which have been widely used in the recovery of the global gravity field models in geodesy, interpretation of geophysical properties in geophysics, and gravity matching in navigation and positioning. In this contribution, we provide the general formulae of the first-order derivatives of principal and main invariants of gravity gradient tensor (FPIGGT and FMIGGT), where their physical meaning is the change rate of the invariants of gravity gradient tensor, and their expressions consist entirely of gravity gradient tensor and gravitational curvatures (i.e. the third-order derivatives of gravitational potential). Taking the mass bodies (i.e. tesseroid and spherical shell) in spatial domain as examples, the expressions for the FPIGGT and FMIGGT are derived, respectively. The classic numerical experiments with the summation of gravitational effects of tesseroids discretizing the entire spherical shell are performed to investigate the influences of the geocentric distance and latitude using different grid resolutions on the FPIGGT and principal invariants of gravity gradient tensor (PIGGT). Numerical experiments confirm the occurred very-near-area problem of the FPIGGT and PIGGT. The FPIGGT and PIGGT of the tesseroid using the Cartesian integral kernels can avoid the polar-singularity problem. Meanwhile, the finer the grid resolution, the smaller the relative approximation errors of the FPIGGT. The grid resolution lower than (or including) \(1^{\circ }\times 1^{\circ }\) at the satellite height of 260 km provides satisfactory results with the relative approximation errors of the FPIGGT and PIGGT in \(\mathrm{Log}_{10}\) scale less than zero. The proposed first-order derivatives of principal and main invariants of gravity gradient tensor will provide additional knowledge of the gravity field for geodesy, geophysics, and related geoscience applications.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

Data Availability Statement

The data that support the findings of this study are available from the corresponding author on reasonable request.

References

Download references

Acknowledgements

We are very grateful to Prof. Kusche, the associate editor, and three anonymous reviewers for their valuable comments and suggestions, which greatly improved the manuscript. This study is supported by China Postdoctoral Science Foundation (Grant No. 2021M691402) and National Natural Science Foundation of China (Grant No. 41974094).

Author information

Authors and Affiliations

Authors

Contributions

XLD derived the formulae, designed the analysis, performed the experiments, and wrote the manuscript. MY checked the derived formulae. WBS, MY, and JR reviewed the manuscript and provided comments and suggestions for improvements. All authors have reviewed and approved the final submitted version of the manuscript.

Corresponding author

Correspondence to Jiangjun Ran.

Appendices

Appendix A: Derivation of first-order derivatives of principal invariants of gravity gradient tensor

The principal invariants of a symmetric second-order tensor (denoted as \(I_i\) with \(i\in \{1,2,3\}\)) can be given as (Gurevich 1964; Rummel 1986; Baur et al. 2008):

$$\begin{aligned} I_{1}= & {} V_{11} + V_{22} + V_{33} \end{aligned}$$
(19)
$$\begin{aligned} I_{2}= & {} V_{11} V_{22} + V_{11} V_{33} + V_{22} V_{33} \nonumber \\&- V_{12}^2 - V_{13}^2 - V_{23}^2 \end{aligned}$$
(20)
$$\begin{aligned} I_3= & {} V_{11} V_{22} V_{33} + 2 V_{12} V_{13} V_{23} \nonumber \\&- V_{11} V_{23}^2 - V_{22} V_{13}^2 - V_{33} V_{12}^2 \end{aligned}$$
(21)

where \(V_{ij}\) with \(i, j \in \{1,2,3\}\) is the component of gravity gradient tensor, which is the symmetric second-order tensor. When \(V_{ij}\) is the trace-free tensor, \(I_1=0\) in Eq. (19), which is Laplace’s equation for the second-order derivatives of the potential.

Based on Eq. (1), the formulae of the FPIGGT (\(I_{ij}\) with \(i,j\in \{1,2,3\}\)) can be presented with respect to the adopted coordinate system (denoted as {\(x_1\), \(x_2\), \(x_3\)}) as:

$$\begin{aligned} I_{11}= & {} \frac{\partial I_1}{\partial x_1} = V_{111} + V_{221} + V_{331} \end{aligned}$$
(22)
$$\begin{aligned} I_{12}= & {} \frac{\partial I_1}{\partial x_2} = V_{112} + V_{222} + V_{332} \end{aligned}$$
(23)
$$\begin{aligned} I_{13}= & {} \frac{\partial I_1}{\partial x_3} = V_{113} + V_{223} + V_{333} \end{aligned}$$
(24)
$$\begin{aligned} I_{21}= & {} \frac{\partial I_2}{\partial x_1} = V_{111}(V_{22}+V_{33})\nonumber \\&+V_{221}(V_{11}+V_{33})+V_{331}(V_{11}+V_{22}) \nonumber \\&- 2V_{112}V_{12} - 2V_{113}V_{13}-2V_{123}V_{23} \end{aligned}$$
(25)
$$\begin{aligned} I_{22}= & {} \frac{\partial I_2}{\partial x_2} = V_{112}(V_{22}+V_{33})\nonumber \\&+ V_{222}(V_{11}+V_{33})+V_{332}(V_{11}+V_{22}) \nonumber \\&-2V_{221}V_{12}-2V_{123}V_{13}-2V_{223}V_{23} \end{aligned}$$
(26)
$$\begin{aligned} I_{23}= & {} \frac{\partial I_2}{\partial x_3} = V_{113}(V_{22}+V_{33}) \nonumber \\&+ V_{223} (V_{11} + V_{33}) + V_{333}(V_{11}+V_{22}) \nonumber \\&- 2 V_{123} V_{12} - 2 V_{331} V_{13} - 2 V_{332} V_{23} \end{aligned}$$
(27)
$$\begin{aligned} I_{31}= & {} \frac{\partial I_3}{\partial x_1} = V_{111}(V_{22}V_{33}-V_{23}^2)+2V_{113}(V_{12}V_{23}-V_{13}V_{22})\nonumber \\&+2V_{112}(V_{13}V_{23}-V_{12}V_{33}) \nonumber \\&+ V_{221}(V_{11}V_{33}-V_{13}^2)+ 2 V_{123}(V_{12}V_{13}-V_{11}V_{23}) \nonumber \\&+V_{331}(V_{11}V_{22}-V_{12}^2) \end{aligned}$$
(28)
$$\begin{aligned} I_{32}= & {} \frac{\partial I_3}{\partial x_2} = V_{112}(V_{22}V_{33}-V_{23}^2) + 2 V_{123}(V_{12}V_{23}-V_{13}V_{22})\nonumber \\&+ 2 V_{221}(V_{13}V_{23}-V_{12}V_{33}) \nonumber \\&+ V_{222}(V_{11}V_{33}-V_{13}^2) + 2 V_{223}(V_{12}V_{13}-V_{11}V_{23}) \nonumber \\&+ V_{332}(V_{11}V_{22}-V_{12}^2) \end{aligned}$$
(29)
$$\begin{aligned} I_{33}= & {} \frac{\partial I_3}{\partial x_3} = V_{113}(V_{22}V_{33}-V_{23}^2) + 2V_{331}(V_{12}V_{23}-V_{13}V_{22})\nonumber \\&+ 2V_{123}(V_{13}V_{23}-V_{12}V_{33}) \nonumber \\&+ V_{223}(V_{11}V_{33}-V_{13}^2)+2V_{332}(V_{12}V_{13}-V_{11}V_{23})\nonumber \\&+ V_{333}(V_{11}V_{22}-V_{12}^2) \end{aligned}$$
(30)

where \(V_{ijk}=\partial V_{ij}/\partial x_{k}\) with \(i, j, k \in \{1,2,3\}\) is the component of gravitational curvatures, which is the symmetric third-order tensor. When \(V_{ijk}\) is the trace-free tensor, \(I_{11}=I_{12}=I_{13}=0\) in Eqs. (22)–(24), which is Laplace’s equation for the third-order derivatives of the potential.

Appendix B: Derivation of first-order derivatives of main invariants of gravity gradient tensor

The main invariants of a symmetric second-order tensor (\(J_i\) with \(i\in \{1,2,3\}\)) can be given as (Gurevich 1964; Baur et al. 2008):

$$\begin{aligned} J_1= & {} V_{11} + V_{22} + V_{33} \end{aligned}$$
(31)
$$\begin{aligned} J_2= & {} V_{11}^2 + V_{22}^2 + V_{33}^2 +2V_{12}^2 \nonumber \\&+ 2V_{13}^2 + 2V_{23}^2 \end{aligned}$$
(32)
$$\begin{aligned} J_3= & {} V_{11}^3 + V_{22}^{3} + V_{33}^3 + 3 V_{11} (V_{12}^2 +V_{13}^2) \nonumber \\&+ 3 V_{22} (V_{12}^2 + V_{23}^2) + 3 V_{33} ( V_{13}^2 + V_{23}^2 )+ 6 V_{12} V_{13} V_{23} \end{aligned}$$
(33)

when \(V_{ij}\) is the trace-free tensor, \(J_1=0\) in Eq. (31), which is Laplace’s equation for the second-order derivatives of the potential as well.

Similarly, the formulae of the FMIGGT (\(J_{ij}\) with \(i,j\in \{1,2,3\}\)) can be presented as:

$$\begin{aligned} J_{11}= & {} \frac{\partial J_1}{\partial x_1} = V_{111} + V_{221} + V_{331} \end{aligned}$$
(34)
$$\begin{aligned} J_{12}= & {} \frac{\partial J_1}{\partial x_2} = V_{112} + V_{222} + V_{332} \end{aligned}$$
(35)
$$\begin{aligned} J_{13}= & {} \frac{\partial J_1}{\partial x_3} = V_{113} + V_{223} + V_{333} \end{aligned}$$
(36)
$$\begin{aligned} J_{21}= & {} \frac{\partial J_2}{\partial x_1} = 2 V_{111} V_{11} + 2 V_{221} V_{22} \nonumber \\&+ 2 V_{331} V_{33} + 4 V_{112} V_{12} + 4 V_{113} V_{13} + 4 V_{123} V_{23} \end{aligned}$$
(37)
$$\begin{aligned} J_{22}= & {} \frac{\partial J_2}{\partial x_2} = 2 V_{112}V_{11} + 2 V_{222}V_{22} \nonumber \\&+ 2 V_{332} V_{33}+ 4 V_{221} V_{12} + 4 V_{123} V_{13} + 4 V_{223} V_{23} \end{aligned}$$
(38)
$$\begin{aligned} J_{23}= & {} \frac{\partial J_2}{\partial x_3} = 2 V_{113} V_{11} + 2 V_{223} V_{22} \nonumber \\&+ 2 V_{333} V_{33} + 4 V_{123} V_{12} + 4 V_{331} V_{13} + 4 V_{332} V_{23} \end{aligned}$$
(39)
$$\begin{aligned} J_{31}= & {} \frac{\partial J_3}{\partial x_1} = 3 V_{111}(V_{11}^2+V_{12}^2 +V_{13}^2) \nonumber \\&+ 6 V_{112}(V_{11}V_{12}+V_{12}V_{22}+V_{13}V_{23}) \nonumber \\&+ 6 V_{113}(V_{11}V_{13}+V_{12}V_{23}+V_{13}V_{33})\nonumber \\&+ 6 V_{123}(V_{12}V_{13}+V_{22}V_{23}+V_{23}V_{33}) \nonumber \\&+ 3 V_{221} (V_{12}^2+V_{22}^2+V_{23}^2)\nonumber \\&+ 3 V_{331}(V_{13}^2+V_{23}^2+V_{33}^2) \end{aligned}$$
(40)
$$\begin{aligned} J_{32}= & {} \frac{\partial J_3}{\partial x_2} = 3 V_{112}(V_{11}^2+V_{12}^2+V_{13}^2) \nonumber \\&+ 6 V_{221} (V_{11}V_{12}+V_{12}V_{22}+V_{13}V_{23}) \nonumber \\&+6 V_{123}(V_{11}V_{13}+V_{12}V_{23}+V_{13}V_{33})\nonumber \\&+ 6 V_{223} (V_{12}V_{13}+V_{22}V_{23}+V_{23}V_{33}) \nonumber \\&+ 3 V_{222}(V_{12}^2+V_{22}^2+V_{23}^2)\nonumber \\&+ 3 V_{332}(V_{13}^2+V_{23}^2+V_{33}^2) \end{aligned}$$
(41)
$$\begin{aligned} J_{33}= & {} \frac{\partial J_3}{\partial x_3} = 3 V_{113}(V_{11}^2+V_{12}^2+V_{13}^2) \nonumber \\&+ 6 V_{123} (V_{11}V_{12}+V_{12}V_{22}+V_{13}V_{23}) \nonumber \\&+ 6 V_{331} (V_{11}V_{13}+V_{12}V_{23}+V_{13}V_{33})\nonumber \\&+ 6 V_{332} (V_{12}V_{13}+V_{22}V_{23}+V_{23}V_{33}) \nonumber \\&+ 3 V_{223}(V_{12}^2+V_{22}^2+V_{23}^2) \nonumber \\&+ 3 V_{333}(V_{13}^2+V_{23}^2+V_{33}^2) \end{aligned}$$
(42)

when \(V_{ijk}\) is the trace-free tensor, \(J_{11}=J_{12}=J_{13}=0\) in Eqs. (34)–(42), which is Laplace’s equation for the third-order derivatives of the potential.

Appendix C: Relationship between first-order derivatives of principal and main invariants of gravity gradient tensor

According to Waring’s and Newton’s formulae transform between the principal and main invariants systems in Gurevich (1964), the relationship between principal invariants (i.e. \(I_1\), \(I_2\), and \(I_3\)) and main invariants (i.e. \(J_1\), \(J_2\), and \(J_3\)) of the second-order tensor can be given as (Baur et al. 2008):

$$\begin{aligned} I_1= & {} J_1 \end{aligned}$$
(43)
$$\begin{aligned} I_2= & {} \frac{1}{2} (J_1^2 - J_2) \end{aligned}$$
(44)
$$\begin{aligned} I_3= & {} \frac{1}{3} J_3 - \frac{1}{2} J_1 J_2 + \frac{1}{6} J_1^3 \end{aligned}$$
(45)
$$\begin{aligned} J_1= & {} I_1 \end{aligned}$$
(46)
$$\begin{aligned} J_2= & {} I_1^2 - 2 I_2 \end{aligned}$$
(47)
$$\begin{aligned} J_3= & {} I_1^3 - 3 I_1 I_2 +3 I_3 \end{aligned}$$
(48)

The relationship from the FMIGGT (\(J_{ij}\) with \(i,j\in \{1,2,3\}\)) to the FPIGGT (\(I_{ij}\) with \(i,j\in \{1,2,3\}\)) can be derived with respect to the chosen coordinate system ({\(x_{1}\), \(x_{2}\), \(x_{3}\)}) as:

$$\begin{aligned} I_{11}= & {} \frac{\partial I_1}{\partial x_1} = \frac{\partial J_1}{\partial x_1} = J_{11} \end{aligned}$$
(49)
$$\begin{aligned} I_{12}= & {} \frac{\partial I_1}{\partial x_2} = \frac{\partial J_1}{\partial x_2} = J_{12} \end{aligned}$$
(50)
$$\begin{aligned} I_{13}= & {} \frac{\partial I_1}{\partial x_3} = \frac{\partial J_1}{\partial x_3} = J_{13} \end{aligned}$$
(51)
$$\begin{aligned} I_{21}= & {} \frac{\partial I_2}{\partial x_1} = \frac{1}{2} \left( \frac{\partial (J_1^2)}{\partial x_1} - \frac{\partial J_2}{\partial x_1}\right) \nonumber \\= & {} J_{11} J_{1} - \frac{1}{2} J_{21} \end{aligned}$$
(52)
$$\begin{aligned} I_{22}= & {} \frac{\partial I_2}{\partial x_2} = \frac{1}{2} \left( \frac{\partial (J_1^2)}{\partial x_2} - \frac{\partial J_2}{\partial x_2}\right) \nonumber \\= & {} J_{12} J_{1} - \frac{1}{2} J_{22} \end{aligned}$$
(53)
$$\begin{aligned} I_{23}= & {} \frac{\partial I_2}{\partial x_3} = \frac{1}{2} \left( \frac{\partial (J_1^2)}{\partial x_3} - \frac{\partial J_2}{\partial x_3}\right) \nonumber \\= & {} J_{13} J_{1} - \frac{1}{2} J_{23} \end{aligned}$$
(54)
$$\begin{aligned} I_{31}= & {} \frac{\partial I_3}{\partial x_1} = \frac{1}{3} \frac{\partial J_3}{\partial x_1} - \frac{1}{2} \frac{\partial (J_1 J_2)}{\partial x_1}\nonumber \\&+ \frac{1}{6} \frac{\partial (J_1^3)}{\partial x_1} = \frac{1}{3} J_{31} + \frac{1}{2}(J_{11}J_{1}^2 - J_{11}J_2 - J_{21}J_1) \end{aligned}$$
(55)
$$\begin{aligned} I_{32}= & {} \frac{\partial I_3}{\partial x_2} = \frac{1}{3} \frac{\partial J_3}{\partial x_2} - \frac{1}{2} \frac{\partial (J_1 J_2)}{\partial x_2} \nonumber \\&+ \frac{1}{6} \frac{\partial (J_1^3)}{\partial x_2} = \frac{1}{3} J_{32} + \frac{1}{2}(J_{12}J_{1}^2-J_{12}J_2 - J_{22}J_1) \end{aligned}$$
(56)
$$\begin{aligned} I_{33}= & {} \frac{\partial I_3}{\partial x_3} = \frac{1}{3} \frac{\partial J_3}{\partial x_3} - \frac{1}{2} \frac{\partial (J_1 J_2)}{\partial x_3} \nonumber \\&+ \frac{1}{6} \frac{\partial (J_1^3)}{\partial x_3} = \frac{1}{3} J_{33} + \frac{1}{2}(J_{13}J_{1}^2 - J_{13}J_2 - J_{23}J_1) \end{aligned}$$
(57)

The relationship from the FPIGGT (\(I_{ij}\) with \(i,j\in \{1,2,3\}\)) to the FMIGGT (\(J_{ij}\) with \(i,j\in \{1,2,3\}\)) can be derived with respect to the chosen coordinate system ({\(x_{1}\), \(x_{2}\), \(x_{3}\)}) as:

$$\begin{aligned} J_{11}= & {} \frac{\partial J_1}{\partial x_1} = \frac{\partial I_1}{\partial x_1} = I_{11} \end{aligned}$$
(58)
$$\begin{aligned} J_{12}= & {} \frac{\partial J_1}{\partial x_2} = \frac{\partial I_1}{\partial x_2} = I_{12} \end{aligned}$$
(59)
$$\begin{aligned} J_{13}= & {} \frac{\partial J_1}{\partial x_3} = \frac{\partial I_1}{\partial x_3} = I_{13} \end{aligned}$$
(60)
$$\begin{aligned} J_{21}= & {} \frac{\partial J_2}{\partial x_1} = \frac{\partial (I_1^2)}{\partial x_1} \nonumber \\&- 2 \frac{\partial (I_2)}{\partial x_1} = 2 (I_{11} I_1 - I_{21}) \end{aligned}$$
(61)
$$\begin{aligned} J_{22}= & {} \frac{\partial J_2}{\partial x_2} = \frac{\partial (I_1^2)}{\partial x_2} \nonumber \\&- 2 \frac{\partial (I_2)}{\partial x_2} = 2 (I_{12} I_1 - I_{22}) \end{aligned}$$
(62)
$$\begin{aligned} J_{23}= & {} \frac{\partial J_2}{\partial x_3} = \frac{\partial (I_1^2)}{\partial x_3} \nonumber \\&- 2 \frac{\partial (I_2)}{\partial x_3} = 2 (I_{13} I_1 - I_{23} ) \end{aligned}$$
(63)
$$\begin{aligned} J_{31}= & {} \frac{\partial J_3}{\partial x_1} = \frac{\partial (I_1^3)}{\partial x_1} - 3 \frac{\partial (I_1 I_2)}{\partial x_1}\nonumber \\&+ 3 \frac{\partial I_3}{\partial x_1} = 3 (I_{11} I_{1}^2- I_{11}I_2 - I_{21}I_1 + I_{31}) \end{aligned}$$
(64)
$$\begin{aligned} J_{32}= & {} \frac{\partial J_3}{\partial x_2} = \frac{\partial (I_1^3)}{\partial x_2} - 3 \frac{\partial (I_1 I_2)}{\partial x_2}\nonumber \\&+ 3 \frac{\partial I_3}{\partial x_2} = 3 (I_{12} I_{1}^2- I_{12}I_2 - I_{22}I_1 + I_{32}) \end{aligned}$$
(65)
$$\begin{aligned} J_{33}= & {} \frac{\partial J_3}{\partial x_3} = \frac{\partial (I_1^3)}{\partial x_3} - 3 \frac{\partial (I_1 I_2)}{\partial x_3}\nonumber \\&+ 3 \frac{\partial I_3}{\partial x_3} = 3 (I_{13} I_{1}^2- I_{13}I_2 - I_{23}I_1 + I_{33}) \end{aligned}$$
(66)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Deng, XL., Shen, WB., Yang, M. et al. First-order derivatives of principal and main invariants of gravity gradient tensor of the tesseroid and spherical shell. J Geod 95, 102 (2021). https://doi.org/10.1007/s00190-021-01547-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00190-021-01547-z

Keywords

Navigation