This paper describes a general-purpose algorithm for computing the gradients in space and time of a scalar field, a vector field, or a divergence-free vector field, from in situ measurements by one or more spacecraft. The algorithm provides total error estimates on the computed gradient, including the effects of measurement errors, the errors due to a lack of spatio-temporal homogeneity, and errors due to small-scale fluctuations. It also has the ability to diagnose the conditioning of the problem. Optimal use is made of the data, in terms of exploiting the maximum amount of information relative to the uncertainty on the data, by solving the problem in a weighted least-squares sense. The method is illustrated using Cluster magnetic field and electron density data to compute various gradients during a traversal of the inner magnetosphere. In particular, Cluster is shown to cross azimuthal density structure, and the existence of field-aligned currents in the plasmasphere is demonstrated.

/pdf/least-squares-gradient-calculation-from-multi-point-3r9xeqmyah.pdf

Least-squares gradient calculation from multi-point observations of scalar and vector fields: methodology and applications with Cluster in the plasmasphere

Ground-based instruments and a number of space missions have contributed to our knowledge of the plasmasphere since its discovery half a century ago, but it is fair to say that many questions have remained unanswered. Recently, NASA’s Image and ESA’s Cluster probes have introduced new observational concepts, thereby providing a non-local view of the plasmasphere. Image carried an extreme ultraviolet imager producing global pictures of the plasmasphere. Its instrumentation also included a radio sounder for remotely sensing the spacecraft environment. The Cluster mission provides observations at four nearby points as the four-spacecraft configuration crosses the outer plasmasphere on every perigee pass, thereby giving an idea of field and plasma gradients and of electric current density. This paper starts with a historical overview of classical single-spacecraft data interpretation, discusses the non-local nature of the Image and Cluster measurements, and emphasizes the importance of the new data interpretation tools that have been developed to extract non-local information from these observations. The paper reviews these innovative techniques and highlights some of them to give an idea of the flavor of these methods. In doing so, it is shown how the non-local perspective opens new avenues for plasmaspheric research.

CLUSTER and IMAGE: New Ways to Study the Earth’s Plasmasphere

Multi-spacecraft missions allow the gradient of important physical quantities in the terrestrial environment to be determined. The gradient can be computed from four simultaneous measurements in a straightforward way, but this computation does not produce proper error estimates, making it hard to assess the meaningfulness of the result. Recently developed least-squares gradient computation techniques offer the possibility to obtain more precise results with all-inclusive error estimates, provided that information about the non-linearity of the space and time variations of the observed quantity is given. The present paper describes several heuristics for estimating these variations, thereby enabling a fully automatic computation of the gradient and the associated error estimates. The performance of these heuristics is illustrated with synthetic data corresponding to 4- and 10-spacecraft configurations.

/pdf/least-squares-multi-spacecraft-gradient-calculation-with-26empprpk9.pdf

Least-squares multi-spacecraft gradient calculation with automatic error estimation

CLUSTER est la premiere mission spatiale permettant l’etude a trois dimensions de la magnetosphere terrestre. Son orbite polaire et sa configuration de quatre satellites en tetraedre lui permettent d’effectuer des mesures in situ dans plusieurs regions de la magnetosphere, notamment dans la plasmasphere. Cette these rassemble une serie de travaux sur les structures de plasma rencontrees sur l’orbite de CLUSTER durant la periode 2001-2004. La quantite physique etudiee est essentiellement la densite electronique, mais aussi le champ magnetique continu. Dans une premiere partie, deux outils d’analyse multipoints, le gradient spatial et la methode des delais, sont decrits, testes et discutes sur des jeux de donnees synthetiques, puis sur des donnees reelles, pour diverses situations typiques. Dans le cadre de conditions favorables, ces analyses revelent le mouvement, l’orientation de frontieres, ainsi que les dimensions des structures rencontrees, impossibles a estimer par une mission composee d’un seul satellite. Une seconde partie s’interesse plus globalement a la plasmasphere, notamment a la topologie et a la dynamique de ses structures de densite, a petite et grande echelle. Cette etude s’appuie sur les analyses multipoints vues precedemment, mais elle utilise egalement des donnees d’autres instruments de CLUSTER, des images globales de la plasmasphere fournies par la mission IMAGE, ainsi que des resultats d’etudes statistiques et de simulations numeriques. Enfin, la these decrit une analyse statistique menee sur un autre jeu de donnees: les densites de puissance spectrales des ondes electromagnetiques mesurees a basse altitude par le satellite Viking durant l’annee 1986.

Etude de la magnétosphère terrestre par l'analyse multipoint des données de la mission CLUSTER : contributions à la caractérisation des frontières et de la magnétosphère interne

Etude des plasmas de l’espace à l’Institut d’Aéronomie Spatiale de Belgique

The four-spacecraft, magnetic field measurements on Cluster can produce an accurate determination of the electric current point by point in time (the Curlometer technique). For example, for planar events, the thickness of the current layer can be accurately estimated from its magnetic profile at each spacecraft and the corresponding boundary crossing times. The latter give a determination of boundary motion relative to the Cluster array. For a range of spacecraft separation distances, the estimate of electric current density can be representative even when the configuration of Cluster spacecraft approaches the thickness of the current layer. The magnitude of the current is often accurately represented and in principle can be tracked through any structure. Minimum variance analysis on the curlometer measurements can be used to estimate the normal to a current layer. Other methods exist which are based on estimation of the magnetic gradients, curvature, or temporal derivative. These methods can be used to calculate a number of other properties, such as the dimensionality of the structure and the corresponding velocity, the field curvature, or boundary normals, but have the commonality of providing a quantity at every moment in time, like the curlometer.

The Curlometer and other gradient measurements with Cluster

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The Curlometer and other gradient measurements with Cluster