Numerical Error in Interplanetary Orbit Determination Software

The core of every orbit determination process is the comparison between the measured observables and their predicted values, computed using the adopted mathematical models, and the minimization, in a least square sense, of their differences, known as residuals. In interplanetary orbit determination, Doppler observables, obtained by measuring the average frequency shift of the received carrier signal over a certain count time, are compared against their predicted values, usually computed by differencing two round-trip light-times. This formulation is known to be sensitive to round-off errors, caused by the use of finite arithmetic in the computation, giving rise to an additional noise in the residuals, called numerical noise, that degrades the accuracy of the orbit determination solution. This paper presents a mathematical model for the expected numerical errors in two-way and three-way Doppler observables, computed using the differenced light-time formulation. The model was validated by comparing its prediction to the actual noise in the computed observables, obtained by NASA/Jet Propulsion Laboratory's Orbit Determination Program. The model proved to be accurate within $3 \times 10^{-3} \,\text{mm/s}$ at $60 \,\text{s}$ integration time. Then it was applied to the case studies of Cassini's and Juno's nominal trajectories, proving that numerical errors can assume values up to $6 \times 10^{-2} \,\text{mm/s}$ at $60 \,\text{s}$ integration time, and consequently that they are an important noise source in the Doppler-based orbit determination processes. Three alternative strategies are proposed and discussed in the paper to mitigate the effects of numerical noise.