Spatial Gradients and the Volumetric Tensor

All physically observable field parameters, such as particle populations, electric and magnetic fields, vary in both space and time, and the understanding of the physical processes within the medium requires knowledge of both the temporal and the spatial variations. The ISEE-1 and -2 mission, launched in 1977, was the first attempt to separate systematically temporal and spatial gradients; two satellites allow the determination of the component of the spatial gradient in the direction of their separation vector. The determination of all three components of a spatial gradient requires at least four spacecraft. These spacecraft define a polyhedron in space. Clearly the success with which any spatial gradient can be measured depends upon the size and shape of the polyhedron, and for the Cluster mission much thought has been given to the optimum geometry to meet specific scientific objectives. In this chapter we examine from first principles the least squares determination of the spatial gradient using data acquired simultaneously from four or more spacecraft. It is found that the gradient is always expressed in terms of the inverse of a symmetric tensor formed from relative positions of the spacecraft. It is shown that this same tensor describes certain basic geometrical properties of the polyhedron defined by the spacecraft: its characteristic size (mean square thickness) in three mutually orthogonal directions, and the orientation of these directions in space. Conversely, these six geometrical parameters, three characteristic dimensions and three angles, define completely the symmetric tensor; therefore they contain the totality of the geometrical information required to determine the spatial gradient by the least squares method. These results are in agreement with what one intuitively expects: that the quality of the polyhedron for the determination of spatial gradients will involve its size, its anisotropy, and the orientation in space of that anisotropy. This geometric tensor is shown to be closely related to the inertia tensor. In the special case of four spacecraft, the product of the three characteristic dimensions is exactly three times the volume of the tetrahedron; for this reason it is called the “volumetric” tensor. The importance of the volumetric tensor for describing the geometry of a polyhedron was first noted by J. Schœnmækers of ESOC Flight Dynamics Division [private communication], but its fundamental importance lies in the key role it plays in the determination of spatial gradients, for which purpose it must be inverted. The magnitude and the direction of the smallest characteristic dimension define how well the spatial gradient can be determined or, indeed, whether it can be determined at all. The values of the other two