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

Abstract

Abstract. 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.

Cluster as current sheet surveyor in the magnetotail

“…It is complementary to the discontinuity analyzer (e.g., Dunlop and Balogh, 2005) that associates the timing information with the geometry and the motion. Our method uses the least-square method, as in the gradient computation (De Keyser, 2008;Hamrin et al, 2008), but minimization is applied to the deviation from the hyperbolic tangent profile of the Harris model. In contrast, the least-square gradient computation minimizes higher-order terms in Taylor expansion.…”

“…The potential of multi-spacecraft measurements was fully used to develop, for example, the curlometer technique (e.g., Dunlop et al, 2002), the least-square gradient methods (De Keyser, 2008;Hamrin et al, 2008), and the combination of the curlometer with the discontinuity analyzer (Dunlop and Balogh, 2005) in the studies of current sheet structure and motion. Various methods are also proposed to characterize the current sheet thickness, including minimum variance analysis, the constant velocity approach and the constant thickness approach (Haaland et al, 2004).…”

Gradient estimation using configurations of two or three spacecraft

“…The multi-spacecraft gradient estimation problem is overdetermined when the number of spacecraft S > 4. A LS solution still exists (Harvey, 1998;De Keyser et al, 2007;De Keyser, 2008) and can be written in a form similar to Eq. (2), namely,…”

“…The problem of estimating spatial gradients from multispacecraft measurements has been addressed through several approaches including discretized boundary integrals (Dunlop et al, 1988), spatial interpolation , LS minimisation (Harvey, 1998;De Keyser et al, 2007;De Keyser, 2008;Hamrin et al, 2008;Vogt et al, 2008Vogt et al, , 2009, and FD schemes (Vogt and Paschmann, 1998;Ritter and Lühr, 2006;Shen et al, 2012). Most of these methods yield linear estimators, i.e., gradient reconstruction formulas were the data enter linearly.…”

Analysis of three-spacecraft data using planar reciprocal vectors: methodological framework and spatial gradient estimation

“…Note that without such constraints, the problem of linear gradient estimation from four-point measurements is expected to yield a unique solution (Vogt et al, 2008b). Assuming a certain degree of homogeneity in both space and time, De Keyser et al (2007) presented a comprehensive method also based on a least squares formulation that allows to carry out detailed error analysis and an assessment of the quality of the gradient estimates, see also De Keyser (2008). A related approach was taken by Hamrin et al (2008) to construct the so-called GALS scheme that is able to resolve convecting structures on spatial scales smaller than the typical spacecraft separation distance.…”