Hybrid-precision arithmetic for numerical orbit integration towards future satellite gravimetry missions

Abstract

The precision of numerical orbit integration is crucial in precise orbit determination and gravity field recovery, especially in the context of future satellite gravimetry missions where nanometer-level inter-satellite measurements are provided by laser ranging interferometer. In this paper, we propose a hybrid-precision arithmetic in both Cowell’s and Encke’s formulations for numerical orbit integration after comprehensive analyses on the computational time costs and numerical errors by dividing the integration process into two parts, namely the increment calculation and the orbit propagation. Based on such a separation, we find that the round-off errors basically occur in the least time-consuming orbit propagation part while the more time-demanding increment calculation part is less sensitive to the round-off due to its small numerical magnitude. Therefore, it is natural in the hybrid-precision arithmetic to use the double- and quadruple-precision arithmetic in the increment calculation part and orbit propagation part, respectively. Numerical results demonstrate that the hybrid-precision arithmetic is as efficient as the double-precision method, while far better than its double-precision counterparts in terms of both accuracy and relative precision. In consideration of the ultra-high precision inter-satellite range and range rate measurements of future satellite gravimetry missions, the currently widely-used double-precision arithmetic cannot meet the numerical requirements of one-day arc orbit integration for the inter-satellite range measurement, in which numerical errors should be at least the same level of the measurement precision and had better be one-order magnitude smaller to fully avoid their contaminations, while for the inter-satellite range rate the double-precision Gauss-Jackson integrator is qualified. Instead, by implementation of hybrid-precision arithmetic the range rate precision is easily achieved at 10−12 m/s in terms of the maximum absolute error of the one-day arc by either Cowell’s or Encke’s formulation. Furthermore, the sub-nanometer-level range precision is obtainable in the Encke’s formulation with reference orbit selected as the best-fit one. Therefore, it is recommended to adopt the hybrid-precision arithmetic for orbit integration to fully exploit the ultra-high precise measurement from the pure numerical data processing point of view.