Gradient estimation using configurations of two or three spacecraft

Abstract

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.

Multipoint Field‐Aligned Current Estimates With Swarm

“…In this context Vogt et al (2013) have shown that estimates of current density from the LS method and from other available dual-s/c methods, like the FD method (Ritter & Lühr, 2006;Shen et al, 2012), and BI method (Ritter et al, 2013), are algebraically identical for nearly parallel orbits and small differences in the orbital phases of swA and swC (i.e., up to the first order in the parameters and from Figure A1 in Appendix A). The slightly different amplitude and frequency composition of the L2 estimate, on one hand, and of the LS/BI estimates, on the other hand, can be explained by the different low-pass filter used to preprocess the magnetic data.…”

“…One can therefore conclude that the Vogt et al (2013) method estimates the FAC density pretty accurately if one uses configurations based on simultaneous satellite positions, at least for the typical values of orbital time lag. However, considering a level of magnetic noise of B = 0.5 nT and no data filtering, the uncertainty in current density estimation, evaluated with equation (B1), assumes relative small values even for the configurations that use simultaneous satellite positions, that is, 60 and, respectively, 75 nA/m 2 , to be compared with less than 15 nA/m 2 in the case of rectangular quad configuration.…”

“…Judging in terms of line integral, the quantity at numerator represents the dB line integral for the four-point configuration, since according to our choice of apexes, the contributions along the diagonal Cm-Ap cancel each other being covered in opposite senses. From the same panel one can observe that the Vogt et al (2013) dual-s/c method, which provides a solution based on the least squares approach to the above mentioned overdetermined problem (blue line), presents a more stable evolution in the critical area near the geographic poles. From the same panel one can observe that the Vogt et al (2013) dual-s/c method, which provides a solution based on the least squares approach to the above mentioned overdetermined problem (blue line), presents a more stable evolution in the critical area near the geographic poles.…”

“…The dual-s/c method from Vogt et al (2013), to be used in this paper, offers some advantages over the traditional L2 FAC algorithm. One is the least squares approach, which yields more stable solutions for quad configurations that are far from regular (e.g., the extremely elongated configurations close to the geographic poles), when small perturbations in the magnetic field (noise) have a dramatic influence.…”

“…Since the Vogt et al (2013) method allows for a flexible geometry of the spacecraft array, one could be tempted to optimize the configuration at higher latitude, for example, to reduce the along-track separation (parameter M in Figure A1) in order to probe the EZ region with more regular quads. Since the Vogt et al (2013) method allows for a flexible geometry of the spacecraft array, one could be tempted to optimize the configuration at higher latitude, for example, to reduce the along-track separation (parameter M in Figure A1) in order to probe the EZ region with more regular quads.…”