Analysis of numerical differentiation methods applied for determination of kinematic velocities for LEOs

Abstract

Kinematic orbits provide a time series of independent positions, which are a good base for gravity field recovery. Gravity field recovery using the energy integral requires numerical differentiation in order to get velocity information for kinetic energy. This paper deals with numerical differentiation methods to test the most effective method for velocity determination of a LEO (Low Earth Orbiter).

Spectral analysis of CHAMP kinematic velocities determined by applying smoothing cubic splines

“…10 to the power −3 mm 2 /s 2 /Hz signal, which is at this frequency equivalent to 0.003-0.01 mm/s. This is consistent in magnitude with the RMS differences of the kinematic velocities in Table 7 of [4].…”

“…These were: dynamic orbit using EIGEN-1S model [7], dynamic orbit using TEG-4 model (both up to degree and order of 120), reduced-dynamic orbit based EIGEN-2 gravity model [8]. In [4] the dynamic and reduced-dynamic positions were compared to the kinematic positions; the RMS of the positions were 1.1233 m, 1.1091 m and 0.0311 m for the dynamic EIGEN-1S, dynamic TEG-4 and reduced-dynamic EIGEN-2 orbits, respectively. A kind of comparison also has been done for the velocities: the EIGEN-1S and the TEG-4 velocities differ from the reduced-dynamic velocities with an RMS of 1.5826 mm/s and 1.3562 mm/s, respectively.…”

“…The method has not been mentioned yet: the analysed method is the smoothing by cubic spline functions In [4] this method (among other 5) has provided an impressive result for determination of kinematic velocity. The smoothing could effectively be applied only on data with a nearly white spectrum.…”

“…In [4] the need of kinematic velocity determination has been introduced, which is not repeated here. Only certain aspects, which are important for the recent analysis, are addressed here again.…”

A Comparison Between Numerical Differentiation and Kalman Filtering for a Leo Satellite Velocity Determination

“…However, velocity vector is required for more applications e.g. in gravity field recovery using the energy integral [8]. The numerical differentiation is a mathematical process for computing the numerical value of derivative of a function [9].…”