The effect of numerical differentiation methods on the earth’s gravity field recovery

The recent dedicated satellite gravimetry missions have provided huge amount of high quality gravity data with global coverage. From computational point of view, estimation of the unknown gravity field parameters is a highly demanding task due to the sheer number of observations and the unknown coefficients. Different computational schemes have been proposed to tackle the problem. Since the early days of satellite geodesy, energy balance based methods for gravity field determination have been considered. If non-conservative forces are known the Hamiltonian along the orbit will be a constant function of the motion. Thus the gravity field can be determined if position and velocity of the satellite are known and accelerometer measurements are available to model the non-conservative part. A satellite mission dedicated to the improvement of our knowledge of the earth’s gravitational field with a direct (in situ) measurement system has been in the proposal stages for a long time and at several agencies. Of course, gravitational field knowledge comes also by tracking satellites from ground stations, and many long wavelength models of the field have been deduced from such data. But, these models derive from the observations of a large collection of satellites that have been tracked over various periods during the long history of earth-orbiting satellites, where none of these was launched for the expressed purpose of providing a global and detailed model of the gravitational field. The method has been applied in a close-loop simulation to the Gravity Recovery and Climate Experiment (GRACE) data and the achieved results show high performance of the proposed method. This article focuses on the development of new techniques for global gravity field recovery from high-low (hl) and low-low (ll) satellite-to-satellite tracking (SST) data. There are a number of approaches to global gravity field recovery known from literature, including the variational equations approach, short arc approach, energy balance approach and acceleration approach. The focus of the article is the energy balance approach with an aim to produce high-quality global gravity field models using simulated data from GRACE satellite missions. The GRACE mission has substantiated the low–low satellite-to-satellite tracking (LL-SST) concept. The LL-SST configuration can be combined with the previously realized high–low SST concept in the CHAMP mission to provide a much higher accuracy.
A new, rigorous model is developed for the difference of gravitational potential between two close earth-orbiting satellites in terms of measured range-rates, velocities and velocity differences, and specific forces. It is particularly suited to regional geopotential determination from a satellite-to-satellite tracking mission. Based on energy considerations, the model specifically accounts for the time variability of the potential in inertial space, principally due to earth’s rotation. Analysis shows the latter to be a significant (~1m2/s2) effect that overshadows by many orders of magnitude other time dependencies caused by solar and lunar tidal potentials. Also, variations in earth rotation with respect to terrestrial and celestial coordinate frames are inconsequential. Results of simulations contrast of the new model to the simplified linear model (relating potential difference to range-rate) and delineate accuracy requirements in velocity vector measurements needed to supplement the range-rate measurements. The numerical analysis is oriented toward the scheduled Gravity Recovery and Climate Experiment (GRACE) mission and shows that accuracy in the velocity difference vector of 2~10?5 m/s would be commensurate within the model to the anticipated accuracy of 10?6 m/s in range-rate. A fast iterative method for gravity field determination from low Earth satellite orbit coordinates has been developed and implemented successfully. As the method is based on energy conservation and it avoids problems related to orbit dynamics and initial state. In addition, the particular geometry of a repeating orbit is exploited using a very efficient iterative estimation scheme, in which a set of normal equations is approximated by a sparse block-diagonal equivalent. Recovery experiments for spherical harmonic gravity field models up to degree and order 70 were conducted based on a 29-day simulated data set of orbit coordinates. The method was found to be very flexible and could be easily adapted to include observations of non-conservative accelerations, such as (to be) provided by satellites like CHAMP, GRACE, and GOCE.
So, calculation of velocity and acceleration vectors is a necessary stage in Earth’s gravity field recovery using GRACE observations. Different numerical differentiation methods have been proposed to compute the acceleration vector. In this paper, Newton, spline and Taylor methods have been implemented. The effect of outliers has also been investigated in different differentiation techniques. The numerical analysis of the recovered solutions shows that the Newton method yields the optimal solution. The comparison is performed based on the difference in the simulated and recovered gravity anomalies and the geoidal heights.