Numerical Algorithm And Realization Of Ellipsoidal Harmonic Expansion Of The Earth Gravity Field

The modeling of gravity field is the basis of the study of gravity field,which is widely used in the fields of Geoscience,space science,military and geodesy.With the development of satellite altimetry,gravity gradient measurement and other technologies,the resolution and accuracy of the available observation data of the Earth's gravity field are getting higher and higher,which greatly promotes the research of the Earth's gravity field.At the same time,the computer ability is also increasing,which makes the gravity field model more precise.At present,the degree of the international mainstream spherical harmonic model is higher than 2160,we can predict that in the future gravity field model will reach a higher degree.Spherical harmonic expansion is the most commonly used method for gravity field modeling.When using the spherical harmonic model to restore the earth's gravity field,if the calculation point is near the surface of Earth,the harmonic series may diverge,which has little influence on the 2160 degree model.However,the degree of gravity field model may be 10800 or higher in the future,so this problem needs to be paid attention.In order to solve the above problem,the ellipsoidal harmonic expansion is studied in this paper.The main work and research results are as follows:(1)The basic theory of ellipsoidal harmonic expansion of the earth's gravity field is summarized and derived.The expansion formulas of gravitational potential,gravity potential,normal gravity potential and disturbance potential are obtained.The relationship between the first-degree ellipsoidal harmonic coefficients and the physical characteristics of the Earth is deduced;(2)The convergence characteristics of the spherical harmonic model and the ellipsoidal harmonic model are compared on the reference ellipsoid.The results show that the convergence of the high-degree spherical harmonic model has problems.Its convergence is related to latitude.The closer to the pole,the worse the convergence is.The divergence of the 1400 degree spherical harmonic model at the pole is obvious.At60°N,the divergence of the spherical harmonic model is not very remarkable,and no divergence is observed at 30°N.And there is no problem in the convergence of higher degree ellipsoidal harmonic model.Gravity field recovery using 10800 degree model should adopt ellipsoidal harmonic synthesis;(3)The calculation principle of the first and the second kind of associated Legendre functions(ALFs)is summarized:the first kind of ALFs is calculated by the standard forward column algorithm,and the overflow problem is solved by the X-number,and the ratio of the second kind of ALFs is calculated by the recurrence method proposed by Fukushima;(4)The basic principle of ellipsoidal harmonic analysis of the Earth's gravity field is given.The ellipsoidal harmonic analysis and calculation are carried out with the help of SHTOOLS package.The accuracy of the two analysis methods of SHTOOLS is compared.The results show that the accuracy of Guass-Legendre quadrature is better than Driscoll/Healy quadrature,and the error of the latter is about 1.5 times of the former;(5)The reliability of the gravity anomaly data used to calculate the ellipsoidal harmonic coefficient is verified.The accuracy of model A calculated from disturbance potential data and model B calculated from gravity anomaly data are tested by the EGM2008 model.The RMS error of model A is 0.376 m~2/s~2,and the RMS error of model B is 0.363 m~2/s~2,which shows that the accuracy of the two models is the same;(6)The ellipsoid harmonic expansion of three components of gravity,gravity disturbance,normal gravity,vertical deflection and gravity anomaly are derived.A set of Fortran program is compiled to calculate the above parameters and gravity potential.Based on Open MP parallel technology and FFT technology,the program realizes the fast calculation of regular grid and scattered point.The fast calculation of regular grid can greatly improve the calculation speed,and its calculation time is less than 1%of the calculation of scattered points.When the resolution of the calculation grid is low,FFT technology does not significantly improve the calculation efficiency.With the improvement of the resolution,the effect of FFT technology becomes obvious;(7)The accuracy of model B is evaluated by using the measured vertical deflection DEFLEC96 and the gravity anomaly data NGS99 given by NGS.According to the verification results of DEFLEC96,the accuracy of the vertical deflection calculated by model B in Continental United States is generally good,but there is still a large error in the local area.The RMS error of East-West vertical deflection of model B is 1.324",and RMS error of the North-South vertical deflection of model B 1.135",according to the verification results of NGS99,the accuracy of gravity anomaly of model B in America is good on the whole,but the local error is large,and its RMS error is 13.539m Gal.(8)The trial calculation of 10800 degree ellipsoidal harmonic model of gravity field is carried out.The results show that the algorithm in this paper fully supports the analysis and synthesis of 10800 degree ellipsoidal harmonic model.