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

annales geophysicae

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