Can we derive the current density from three point LEO measurements of the magnetic field perturbations?

In this paper, we perform critical test of the well-known curlometer technique. The curlometer technique allows a derivation of the current density using measurements of the magnetic field at spatially separated points. At LEO altitudes this generally three dimensionally problem is simplified to a two dimensions by the fact that the current is almost solely flowing along the magnetic field lines. Strictly speaking the current density is derived from integration around a closed loop but in the curlometer technique this integration is simplified to a summation over just three points. In this paper we present a critical analysis of when and to what extend this approximation is valid. We show some simple theoretical considerations, results from a series of simulations, and finally apply the knowledge to ESA SWARM measurements.