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.
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Data Availability Statement
The data that support the findings of this study are available from the corresponding author on reasonable request.
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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).
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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.
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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):
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:
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):
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:
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):
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:
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:
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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
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DOI: https://doi.org/10.1007/s00190-021-01547-z