Research On Theory And Methodology Of Earth Gravity Field Recovery Based On GOCE Gravity Gradient Data

The development of satellite gravimetry technology plays an important role in refining the medium and long wave of the Earth gravity field.The launching of GOCE satellite delivered a wealth of data to bring about a whole new level of understanding of the Earth gravity field undoubtedly.In this paper,some problems in the recovery of the Earth gravity field using GOCE gravity gradient data are studied,and the theory and method of space-wise method are deeply studied and analyzed.The main work and innovations of this paper are as follows:1.In order to analyze the computational characteristics of different recursive formulas for computing fully normalized Associated Legendre Functions,the computational accuracy and efficiency of standard forward recursion method,Holmes method,X-number method and Belikov method are analyzed.The reasons for the occurring of overflow problem when test Belikov method beyond degree 3200 are studied.We present a fully normalized recursive formula for Belikov method,which can be used to compute the values of fnALFs up to complete degree and order 64800 at least with a precision better than 10-11 in the IEEE double precision environment without the overflow problem in the global scope.The fully normalized recursive formula can fully meet the needs of the current and future construction of ultra-high degree and order Earth gravity field model.2.In order to recover the Earth gravity field model using the grid mean data,we deduced a group of recursive algorithms for computing the definite integrals of spherical harmonic functions,which are needed in spherical harmonic analysis and synthesis.The symmetrical characteristics and accuracy check formulas of these integral formulas were investigated.We propose a new recursive formula for computing the definite integrals of(cos(θ)sin θ(abbreviated to Inm),which is independent of Pnm(cosθ)in the case of m>1 without the IEEE underflow problem.This new formula can also be transformed to compute the definite integrals of(cosθ)cosθ.Other definite integrals can be expressed as a function of Inm.The Belikov method,X-number method and the new method for calculating the definite integrals of fnALFs are compared and studied.Experiments show that these three methods can be test up to degree 21600 at least,but compared with X-number method,the new method can improve the computational accuracy by 10-1000 times,the calculation efficiency by 3 times,and the Belikov method can improve the calculation accuracy by 10 times,but the computational efficiency is a little lower than others.3.In order to compute the disturbing gravity gradient tensors in the geocentric Cartesian coordinate system,we deduce the partial derivatives of the spherical harmonic terms in the geocentric Cartesian coordinate system for the three coordinate axes in the real number domain for the first time using the mathematical induction method and get the same recursive form as that in the complex number domain.When computing the gravity gradients at GOCE orbits,the geocentric Cartesian coordinate system can be used directly,without the need of conversion to the geocentric spherical coordinate system,and the singularity at the poles can be effectively solved.4.The theory and method of using gravity gradients to calculate the potential coefficients of Earth gravity field model by harmonic analysis method are studied.Vectorization form,FFT form and Vectorization/FFT combination form of harmonic analysis method are deduced respectively.The deduced conclusion is validated by inversion of Earth gravity field model up to degree 300 from simulated data.The results show that the degree variances of potential coefficients derived by harmonic analysis method deviate greatly after degree 250;Except the combination of grid point values Txz and Tyz,the cumulative geoid errors of other components exceeds 1cm at degree 200 when using gravity gradient data with a resolution of 30’×30’ to recover the Earth gravity field model,if the resolution is up to 20’×20’ and 15’× 15’,the cumulative geoid errors are less than 1 cm for degree 200.This is because harmonic analysis method requires the gravity gradient data to be continuous in theory,but it can not be achieved in practice,the improvement of resolution can only approximate the condition of data continuity,so it will improve the accuracy of inversion results,but this improvement is finite.In terms of computational efficiency,it only takes 0.65 seconds to recover the Earth gravity field model with a degree 300 using the grid data component Tzz.Compared with the traditional FFT algorithm,the efficiency of vectorized harmonic analysis method can be improved by nearly five times,and the combination of the two forms can be increased by 16 times.5.The theory and method of solving potential coefficients of gravity field model by least squares(LS)method using gravity gradient data are explored.We deduced the vectorization form of the block-diagonal LS method and the FFT expressions for solving the free term of the normal equation.The consistency between the least squares method and discrete harmonic analysis method is proved.We simulate studied the accuracy and computational efficiency of inversion Earth gravity field model up to degree 300 using different types of data and with different resolutions.Experiments show that the accuracy of potential coefficients derived by LS method using different components(combinations)are identical.The degree error RMS of coefficients relative to the Earth gravity field model Eigen6c4 are less than 10-20,the cumulative geoid errors are less than 10-10 cm,and the cumulative gravity anomaly errors is no more than 10-14 mGal.This reflects that the errors produced by the adjustment model are very small.The LS method is not sensitive to data resolution and the differences of potential coefficients computed by LS method using data with different resolutions are very small.By combining vectorization operation with FFT technology,the gravity field model with a maximum degree 300 can be recovered within 5 seconds from the grid point values of full gravity gradient tensor and within 11 seconds from the grid mean values of the full gravity gradient tensor.6.The theory and method of constructing gravity field model with gravity gradient tensor invariants are deeply studied and analyzed.We deduced the vectorization forms of two types of gravity gradient tensor invariants using block-diagonal LS method to compute potential coefficients.The FFT formula for calculating the free term of normal equation is given,and the influence of different reference Earth gravity fields on the accuracy of gravity field model are compared and analyzed.In order to adapt to the characteristics of block-diagonal LS,we propose that to construct the coefficient matrix of normal equation by normal gravitational gradients.This can not only suitable for the spherical approximation conditions—Only use the component Trr to recover the gravity field model,but also can be used to compute the coefficients of gravity field model using the combination of Txx,Tyy,Tzz,and Txz in the GRF taking into account of the term J2,J4,J6 and J8 et al.We use the simulated data to validate the derived results.The results show that the accuracies of potential coefficients in the case of considering terms J2,J4,J6 and Js are identcial,which is obviously better than that of spherical approximation after degree 50.The recovery accuracy of the second type of invariant is obviously better than that of the third type of invariant after degree 50.There is no difference between the inversion results of grid point values and grid average values,and the inversion results of data with different resolutions are similar.The cumulative geoid errors relative to the Earth gravity field model Eigen6c4 are less than 10-2 cm,and the cumulative gravity anomaly errors is no more than 10-4 mGal when considering the influence of term J2.The precision of reference gravity field model has important effect on the recovered potential coefficients.Therefore,it is important to chose a reference gravity field model with high precision when using the measured data to construct the gravity field model.7.Aiming at the "remove-restore" method in the processing of GOCE gravity gradient data,we use the vectorization technique and FFT technique to calculate the grid gravity gradient tensor of multi-layer spherical containing the orbit of GOCE first.Then using the space interpolation method to interpolate the gravity gradient tensor of the orbit.In order to select the optimal interpolation method,the interpolation accuracy of linear interpolation method,nearest point method,cubic interpolation method and cubic spline interpolation method are compared.Experiments show that when the number of spherical layers reaches 4,the maximum absolute residual error is much less than the measurement error of gravity gradiometer,and the accuracy is better than four orders of magnitude.The computational efficiency of presented method is nearly 600 times higher than that of point-wise computation method,which can fully meet the need of practical applications.8.In the preprocessing of gravity gradient data,aiming at the outlier detections,we compared capabilities of tracelessness condition method,gravity gradient anomaly method,Overhauser spline interpolation method and Median absolute deviation method.Experiments show that the Median absolute deviation method has excellent "robust" characteristics.The differences between five-parameter method and two-parameter method in the calibrations are also analyzed.Experiments show that the deviation terms calculated by these two methods are quite different,the deviation drift terms are very small,and outliers in the observations have little influence on the calibration process,which only cause 0.3mE deviation on the trace.In order to eliminate the low-frequency colored noise and keep the signals of measurement bandwidth,the FIR filter are applied,and the validity of the filtering algorithm is verified by setting different passbands to the filter.9.We use methods presented in this paper to compute the Earth gravity field models based on GOCE gravity gradient data for seven months.The degree error RMS,geoid cumulative error and gravity anomaly cumulative error are used to evaluate the accuracies of different models derived by different methods.Finally,two Earth gravity field models with complete degree and order 210 are obtained:EGM_GOCE_P-based on the LS method using full tensor grid point value data and EGM_GOCE_M-based on the second type of gravity gradient tensor invariant method using grid mean data taking into account the term J2.10.We use 1007 GPS/leveling points of the national GPS A/B network and 914 astronomical geodetic vertical deflections to evaluate the precisions of these two Earth gravity field models.Compared with EGM2008,the accuracy of quasi-geoid computed using EGM_GOCE_M is improved by 1.5cm.There is no accuracy improvement for the vertical deflections in the east areas,and the improvement in the west areas is finite.All in all,the accuracy improvement for vertical deflections is not obvious.