Abstract
The harmonic correction (HC) is one of the key quantities when using residual terrain modelling (RTM) for high-frequency gravity field modelling. In the RTM technique, high-frequency topographic gravitational signals are obtained through removing gravitational effects of a long-wavelength reference surface, e.g., MERIT2160. There might be points located below the reference surface. In such cases, the RTM gravity field is calculated in the non-harmonic condition, HC is therefore required. Over past decades, though various methods have been proposed to handle the HC issue for the RTM technique, most of them were focused on the HC for RTM gravity anomaly rather than for other gravity functionals, such as RTM geoid height. In practice, the HC for RTM geoid height was generally assumed to be negligible, but a detailed quantification was missing for present-day RTM computations. This might cause large errors in the regional geoid determination over rugged areas. In this study, we derive HC expressions for the RTM geoid height in the framework of the classical condensation method. The HC terms are derived under four different assumptions separately: residual masses approximated by an unlimited Bouguer plate, residual masses approximated by a limited Bouguer plate which overcomes the mass inconsistency effect, residual masses approximated by a Bouguer shell which overcomes the effect of planar approximation, and residual masses approximated by a limited Bouguer shell which overcomes the errors induced by both planar approximation and mass-inconsistency. The errors due to various approximations in HC terms are investigated through comparison among various terms. Besides, HC terms are computed using an expansion up to degree and order 2159. Our results show that HC for RTM geoid height is less 1 mm and could be ignored over \(\sim 99\)% of continental areas, but be of great significance for regional geoid determination over mountain areas, e.g., more than 10 cm effect over very rugged areas. The validation through comparison with terrestrial measurements and a baseline solution of the RTM technique proves that the HC terms provided in this study can improve the accuracy of RTM geoid heights and are expected to be useful for applications of the RTM technique in regional and global gravity field modelling.
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Abbreviations
- BS:
-
Bouguer shell
- DEM:
-
Digital elevation model
- DL:
-
Disc layer
- RTM:
-
Residual terrain modelling
- GGM:
-
Global gravity field model
- HC:
-
Harmonic correction
- LBP:
-
Limited Bouguer plate
- LBS:
-
Limited Bouguer shell
- LSL:
-
Limited Spherical layer
- ML:
-
Mass layer
- NI:
-
Numerical integration
- SGM:
-
Spectral gravity forward modelling
- SHC:
-
Spherical harmonic coefficients
- SL:
-
Spherical layer
- UBP:
-
Unlimited Bouguer plate
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Acknowledgements
This work was supported by the Strategic Priority Research Program of Chinese Academy of Sciences (Grant No. XDB 41000000), the National Natural Science Foundation of China (Grant No. 42104083), and the Natural Science Foundation of Guangdong Province, China (Grant No. 2021A1515011425).
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Appendices
Appendix A: Derivation of Harmonic Correction for the Geoid Height under Unlimited Bouguer Plate Approximation
As is introduced in Sect. 2.1, the HC-UBP with unlimited Bouguer plate approximation is value of HC-LBP with limited Bouguer plate approximation when \(R\rightarrow \infty\),
Let \(\kappa =\frac{1}{R}\), then
It follows from L’Hospital’s rule (Taylor 1952) that
When \(\kappa \rightarrow 0\), the first term above tends to \(h^2\). The limit of the second term is concluded by applying L’Hospital’s rule again, that is
By direct computation of derivatives, we yield
Therefore,
Combing Eqs. (23), (24) and (27), we obtain
Therefore,
Appendix B: Derivation of Harmonic Correction for Geoid Height under Limited Bouguer Shell Approximation
The geometry of the limited Bouguer shell is represented by a spherical cap at the left panel of Fig. 14. The inner radius of spherical cap is \(r_1\), outer radius \(r_2\), and density \(\rho\). \(\psi _0\) is the half-angle subtended at the Earth’s centre. \(P_+ (0,0,r)\) denotes the computation point when it is located below the reference surface and the subscript \('+'\) indicates that the point adheres to or just above the Earth’s surface. The magnitude of height difference between \(P_+\) and its respective point \(P^{\mathrm {R}}\) on the reference surface is h. Its generated gravitational potential is (Tenzer et al. 2007; Kadlec 2011)
with \((\psi ',\alpha ',r')\) indicating the coordinates of integration point and
Its analytical solution was widely discussed by LaFehr (1991), Hensel (1992), Heck and Seitz (2007), Tenzer et al. (2007), and Kadlec (2011). Here, the general solution in Kadlec (2011) was adopted. It adapts for various cases when computation points are located out or in the spherical cap.
where
In the framework of the condensation method under limited Bouguer shell approximation, the spherical cap is compressed into a mass layer with a constant radius \(r_1\), infinitesimal thickness, and shares the same mass with the spherical cap. The spherical cap layer is moved down to just below the computation point \(P_+\) (right panel at Fig. 14). Its generated gravitational potential could be derived following (Tenzer et al. 2007; Kadlec 2011)
with \(\rho _c\) indicating the density of compressed masses, \((\psi ',\alpha ',r_1)\) the coordinates of integration point and
After integration over \(\alpha '\), we get
After integration over \(\psi '\), we get
For computation point \(P_+\), the HC for geoid height is
The comparison between two kinds of HC under LBS is implemented with an integration radius varying from 0 to 550 km. In this experiment, \(N^{\text {HC-LBS}}\) is calculated following the method introduced in Sect. 2.3. The tesseroid is bounded by surfaces defined by \(\lambda =-\psi\) and \(\lambda =\psi\), \(\varphi =-\psi\) and \(\varphi =\psi\), and \(r=r_1\) and \(r=r_2\). The HC has calculated through diving the integration masses into a series of the prism at a resolution of \(15^{\prime \prime }\). \(N^{\text {HC-LBS}}_{\text {Spherical Cap}}\) is calculated with Eq.(38). The differences are shown in Fig. 15. It is obvious that the differences between \(N^{\text {HC-LBS}}\) and \(N^{\text {HC-LBS}}_{\text {Spherical Cap}}\) reduce with increasing integration radius. When the integration radius is larger than \(\sim 20\) km, the differences would be less than 0.05 cm, which is able to be ignored in the most practical applications.
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Yang, M., Hirt, C., Wu, B. et al. Residual Terrain Modelling: The Harmonic Correction for Geoid Heights. Surv Geophys 43, 1201–1231 (2022). https://doi.org/10.1007/s10712-022-09694-4
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DOI: https://doi.org/10.1007/s10712-022-09694-4