A General Algorithm for the Linear and Quadratic Gradients of Physical Quantities Based on 10 or More Point Measurements. Issue 6 (22nd June 2021) - Shen, Chao; Zhou, Yufei; Ma, Yonghui; Wang, Xiaogang; Pu, Zuyin; Dunlop, M.
Abstract: A novel algorithm for estimating both the linear and quadratic gradients of physical quantities based on multiple spacecraft observations using the least squares method is put forward. Using 10 or more spacecraft constellation measurements as input, this new algorithm can yield both the linear and quadratic gradients at the barycenter of the constellation. Iterations were used in the algorithm. Tests on cylindrical flux ropes, dipole magnetic field, and modeled geomagnetospheric field were carried out. The results of these tests indicate that the linear gradient obtained is of second‐order accuracy, while the quadratic gradient is of first‐order accuracy. The test on the modeled geomagnetospheric field showed that, the greater the number of spacecraft in the constellation, the greater the accuracy of the quadratic gradient calculated. However, the accuracy of the linear gradient obtained was independent of the number of spacecraft. The feasibility, reliability, and accuracy of this algorithm have been successfully verified. This algorithm could find wide applications in the design of future multiple spacecraft missions as well as in the analysis of multiple‐point measurement data. Plain Language Summary: As space exploration develops, constellations of 10 or more spacecraft may become a reality in the near future. However, there is still no general mathematical algorithm for calculating the second‐order or quadratic gradient of various physical quantities using 10 or more point measurements. In this paper, we present a universal approach that can estimate both the linear and the quadratic gradients of physical quantities based on 10 or more point measurements. This algorithm has been tested and its reliability verified. The results of these tests show that the linear gradient obtained is of second‐order accuracy, while the quadratic gradient is of first‐order accuracy. This new algorithm will aid the design of future multiple spacecraft missions and will have wide application in analyzing multiple‐point measurement data. Key Points: A general algorithm for linear and quadratic gradients based on 10 or more spacecraft measurements is presented for the first time The characteristic matrix of the constellation affecting the determination of the quadratic gradient is found and its features demonstrated The algorithm is confirmed with tests on magnetic fields, indicating that the linear magnetic gradient obtained is of second‐order accuracy