行星际轨道确定软件中的数值误差
Numerical Error in Interplanetary Orbit Determination Software
摘要：
每一个轨道确定过程的核心是测量的观测值和它们的预测值之间的比较，使用所采用的数学模型计算，并在最小二乘意义上最小化它们的差异，称为残差。在行星际轨道确定中，通过测量在一定计数时间内接收到的载波信号的平均频移而获得的多普勒观测值与它们的预测值进行比较，通常通过对两个往返光时求差来计算。已知该公式对舍入误差敏感，舍入误差由计算中使用有限算术引起，在残差中产生额外的噪声，称为数值噪声，其降低了轨道确定解的精度。本文提出了一个数学模型的预期数值误差的双向和三向多普勒观测，计算使用差分光时公式。该模型进行了验证，通过比较其预测的实际噪声在计算的观测值，获得了美国宇航局/喷气推进实验室的轨道确定计划。证明该模型在$60 \,\text{s}$积分时间的$3 \times 10^{-3} \,\text{mm/s}$内是准确的。然后将其应用于Cassini和Juno的标称轨道的案例研究，证明了数值误差在$60 \,\text{s}$积分时间可以假设值高达$6 \times 10^{-2} \,\text{mm/s}$，因此它们是基于多普勒的轨道确定过程中的重要噪声源。本文提出并讨论了三种可供选择的策略，以减轻数值噪声的影响。
英文摘要：
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.
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