Estimating gradients of physical fields in space

This study focuses on the development of a multi-point technique for future constellation missions, aiming to measure gradients at various orders, in particular the linear and quadratic gradients, of a general field. It is well established that, in order to estimate linear gradients, the spacecraft must not lie on a plane. Through analytical exploration within the framework of least squares, it is demonstrated that at least 10 spacecraft that do not lie on any quadric surface are required to estimate both linear and quadratic gradients. The spatial arrangement of the spacecraft can be characterized by a set of quality factors. In cases where there is poor temporal synchronization among the spacecraft leading to non-simultaneous measurements, temporal gradients must be included. If the spacecraft have multiple velocities, by incorporating temporal gradients it is possible to reduce the number of required spacecraft. Furthermore, it is proved that the accuracy of the linear gradient is of second order and that of the quadratic gradient is of first order. Additionally, a method for estimating errors in the calculation is also illustrated.