Theoretical And Applicational Study On Gravitational Curvatures

Proper understanding of how the Earth’s mass distributions and redistributions influence the Earth’s gravity field-related functionals is crucial for numerous applications in geodesy,geophysics and related geosciences.In the theoretical part,the concept of the topographic and crustal gravitational curvatures is illustrated,and the 2D/3D formulas of spherical integral kernels and Cartesian integral kernels for the gravitational curvatures of a tesseroid are derived in spatial domain.Meanwhile,the gravitational curvatures formulas of the spherical shell are derived as the reference models.The fourth-order radial components of gravitational potential,in the expressions of spherical harmonic series in spectral domain and tesseroids in spatial domain,are derived as well.In the part of numerical approaches,the general Taylor series expansion expres-sions of the different gravitational parameters are given both in 2D/3D forms,and the formulas are given up to sixth order.The generally used numerical approaches for computing the topographic and crustal effects（e.g.,GP,GV,GGT,GC）of a tesseroid are studied,including the Taylor Series Expansion（TSE）,Gauss–Legendre Quadra-ture（GLQ）,Opened or Closed Newton–Cotes Quadrature（NCQ） approaches and the adaptive discretization stack-based algorithm by Gauss-Legendre quadrature approach.In the numerical experiment apart,for the non-zero GC functionals,the preci-sion levels of the different Taylor order terms are about 10^-16m^-1s^-2 for zero-order,10^-24 or 10^-23m^-1s^-2 for second-order,10^-29m^-1s^-2for fourth-order and 10^-35 or 10^-34m^-1s^-2 for sixth-order.Whereas,for the zero GC functionals,the precision levels of the different Taylor order terms are about 10^-33～10^-28m^-1s^-2 for zero-order,10^-34～10^-31m^-1s^-2 for second-order,10^-37～10^-33m^-1s^-2 for fourth-order and 10^-39～10^-35m^-1s^-2 for sixth-order.The computational burdens of Cartesian integral kernels are fewer than those of spherical integral kernels with GC functionals(V_ijk^m)by 3%,54% and 528% for the zero-order,second-order and fourth-order,respec-tively.When estimating the total sum of the average time costs of GP,GV,GGT and GC functionals,5%,82% and 813% are required for Cartesian integral kernels for the zero-order,second-order and fourth-order,respectively,which are respectively fewer than those of spherical integral kernels by 1%,18% and 170%.For the evalua-tion of the 2D and 3D GC functionals with the TSE,GLQ and NCQ methods,the 2D form of the GC components is more efficient than the 3D form in Cartesian integral kernel at the same accuracy.Among the numerical approaches（e.g.,TSE,GLQ and NCQ）,the 2D/3D GLQ approach is recommended for practical calculation in terms of the computational efficiency,time and stability.Among the gravitational effects,the relative approximation errors of the GC functionals are largest under the same condi-tion in the very-near-area,and the range of δ?V₂for the GGT Laplace parameters is about 10^-25～10^-14s^-2 and δ?V₃ for the GC Laplace parameters is approximately 10^-30～10^-20m^-1s^-2.The approximation errors are larger for the GP,GV,GGT and GC evaluation with thicker spherical shell.When the computation point approaches the surface of the spherical shell,the changes of the relative errors for the GC functionals are more sensitive than the GP,GV and GGT functionals.For the 3D GC formulas with spherical integral kernels,there exits polar singularity problem in the polar region.However,for the 3D GC formulas with Cartesian integral kernels,the polar singularity problem can be avoided with different numerical approaches.Different distance-size ratio values D=6,7,14,30,35,41 and 50 for the GC component are recommended to reach the 0.1% threshold error at corresponding computational heights 260 km,150km,50 km,10 km,8 km,6 km and 4 km.To reach the 0.1% threshold error for the fourth order radial component,different D values are required for different heights,for instance,D=15 for the 260 km height,D=22 for the 150 km height and D=50 for the 50 km height,which are larger than those of the GC component.The selection of different gravity field models has little effects on the values of radial component of the gravitational curvatures and the fourth radial component of the gravitational potential in the spectral domain.The application with the topographic gravitational curvatures in China region re-veals that,the values of the GC component V_zzz may better demonstrate the topogra-phy variation of the northeast and southeast Tibet region,the southeast coastal area of China,and regions near China,than those of the GP,GV and GGT components(i.e.,V,V_z,V_zz).For the 19 different gravitational components,the variations with different crustal layers of the ETOPO1 and CRUST1.0 show that with the increased order of the derivatives from the zero-order component to the fourth-order component,the fineness level of different crustal effects for the gravitational components increases accordingly.Meanwhile,the GGT Laplace parameter?GGT^χfor the different crustal effects in spatial domain is about 10^-8 E?tv?s,and in spatial domain for the GC Laplace parameters ?δGC₁^χ、?δGC₂^χ and ?δGC₃^χ of the different crustal effects,the approx-imation errors are at the levels of 10^-29,10^-28,10^-27 or 10^-26m^-1s^-2.Furthermore,the range of Laplace parameter ?GGT^ref for the refined GGT components after the stepwise stripping crustal corrections is about 10^-8E?v?s.And for the refined GC Laplace parameters?δGC₁^ref、?δGC₂^ref and ?δGC₃^ref,the precision levels are about 10^-21m^-1s^-2 with STD about 10^-22m^-1s^-2.Among the 19 refined gravitational func-tionals,the Pearson correlation coefficient value of the refined disturbing radial gravity vector δT_z^ref is largest as 0.928.For the different refined gravitational parameters,the optimized d/o are 10 for δT^ref with the PCC value 0.708,17 for δT_z^ref with the PCC value 0.928,45 for δT_zz^refwith the PCC value 0.865,160 for δT_zzz^ref with the PCC value0.544 and 160 for δT_zzzz^ref with the PCC value 0.296,respectively.These refined gravi-tational components could help to get better understanding of the internal structures（i.e.,Moho）of the Earth.