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

Abstract

Abstract. 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.

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

“…For the sake of simplicity small-scale fluctuation errors are not considered here (see De Keyser et al, 2007). We also do not address spacecraft position or timing errors.…”

“…This section summarizes the least-squares technique developed by De Keyser et al (2007) for computing the gradient. We slightly generalize the description of the approximation error as this will turn out to be useful later on.…”

“…Many of these difficulties can be succesfully addressed by least-squares gradient computation (LSGC), as recently described by De Keyser et al (2007). The rationale is that, if the gradient remains constant over a given time interval (thus relaxing the requirement of simultaneity), the information content from a larger set of data points can be exploited for computing the gradient.…”

“…De Keyser et al (2007) assume that the homogeneity length and time scales are given. While they point out that a suitable value can be chosen based on physical considerations, this may not always be easy to do in practice.…”

“…However, intercalibration is difficult as the instruments and their operating environments are never identical. Systematic gradient computations with ESA's CLUSTER multi-spacecraft mission have been limited to magnetometer data (FGM instrument, Balogh et al, 1997Balogh et al, , 2001) with their high precision and good calibration Dunlop and Balogh, 2005;Vallat et al, 2005;Dunlop et al, 2006), and to electron density data derived from the plasma frequency (WHISPER instrument, Décréau et al, 1997Décréau et al, , 2001Trotignon et al, 2003) because of their absolute calibration De Keyser et al, 2007).…”