Gradient estimation using configurations of two or three spacecraft

Abstract. The forthcoming three-satellite mission Swarm will allow us to investigate plasma processes and phenomena in the upper ionosphere from an in-situ multi-spacecraft perspective. Since with less than four points in space the spatiotemporal ambiguity cannot be resolved fully, analysis tools for estimating spatial gradients, wave vectors, or boundary parameters need to utilise additional information such as geometrical or dynamical constraints. This report deals with gradient estimation where the planar component is constructed using instantaneous three-point observations or, for quasi-static structures, by means of measurements along the orbits of two close spacecraft. A new least squares (LS) gradient estimator for the latter case is compared with existing finite difference (FD) schemes and also with a three-point LS technique. All available techniques are presented in a common framework to facilitate error analyses and consistency checks, and to show how arbitrary combinations of planar gradient estimators and constraints can be formed. The accuracy of LS and FD planar gradient estimators is assessed in terms of prescribed and adjustable discretization parameters to optimise their performance along the satellite orbits. Furthermore, we discuss the implications of imperfect constraint equations for error propagation, and address the effects of sub-scale structures. The two-spacecraft LS scheme is demonstrated using Cluster FGM measurements at a planar and essentially force-free plasma boundary in the solar wind where all three different types of constraints to construct out-of-plane derivatives can be applied.