A discussion of the uniqueness of a Laplacian potential when given only partial field information on a sphere

Abstract

For a vector field defined by a scalar potential outside a surface enclosing all the sources, it is well known that the potential is defined uniquely if either the potential itself, or its derivative normal to the surface, is known everywhere on the surface. For a spherical surface, the normal derivative is the radial component of the field; the horizontal (vector) component of the field also gives uniqueness (except for any monopole contribution). This paper discusses the way other partial information of the field on the spherical surface can give a unique, or almost unique, knowledge of the external potential/field, bringing together and correcting previous work. For convenience the results are given in the context of the geomagnetic field B. This is often expressed in terms of its local Cartesian components (X, Y, Z), equivalent to (-Bθ, BΦ, -Br); it can also be expressed in terms of Z and the vector horizontal component H = (X, Y). Alternatively, local "spherical polar" components (F, I, D) are used, where F = |B|, the inclination I is the angle in the vertical plane downward from H to B, and the declination D is the angle in the horizontal plane eastward from north to H. Knowledge of X over the sphere gives a complete knowledge of the potential, apart from that of any monopole (which is zero in geomagnetism), and Y gives the potential except for any axially symmetric part (which can be provided by a knowledge of X along a meridian, or of H along any path from pole to pole). In terms of (F, I, D) the situation is more complicated; either F or the total angle (I, D) needs to be known throughout a finite volume; for the latter, this paper shows how, in principle, the actual potential can be determined (except for an unknown scaling factor). Similarly D on the sphere also needs a knowledge of |H| on a line from (magnetic) pole to pole. We also discuss how these various properties affect the determination, by surface integration, of the Gauss coefficients of the field representation in terms of spherical harmonics.