Least-square estimation of GRACE intersatellite and its time derivatives

One of the most valuable and important application of artificial satellites in geodetic sciences is recovery of the earth's gravity field. Orbits of satellites either gravimetric or nongravimetric are analyzed to improve the earth's gravity field. Gravimetric satellites are launched at low altitudes to observe gravity field in more detail. GRACE twin satellites are the second spacecraft of gravity field dedicated missions, realizing the high-low (HL) and low-low (LL) satellite-to satellite tracking (SST) concepts. The onboard GPS receivers collect the HL observations while the k-band ranging system (KBR) realizes the LL configurations. Nevertheless, two data sets have different sampling rates. The positions are recorded every 60 seconds whereas the KBR system observes the range changes every 5 seconds. In this article, we propose a new idea for combining these two data sets. We employ Hermite polynomial approximation in Least-squares mode in order to provide a model to interpolate the position by KBR sampling rate. Hermite interpolation is a method closely related to the Newton method of interpolation in numerical analysis, which allows us to consider given derivatives at data points, as well as the data points themselves. We apply a linear Least-squares method of Hermite approximation which fits a polynomial and its derivatives to observations. First, the method was tested on simulated data and the suitable degree of polynomial selected. The algorithm was applied on real data and the suitable polynomial was found. The proposed algorithm leads to finding the best degree of polynomial that estimates range and its derivatives with reasonable accuracy.