Physics Department, University of Newcastle Upon Tyne, NE1 7RU, U.K.

When considering functions on the Earth's (spherical) surface, mean-square values are often used to indicate their (relative) magnitude. If a function is separated into it, (essentially) spherical harmonic components then, provided these individual harmonic components are orthogonal over the surface, the concept of spatial power spectrum can be introduced, with each harmonic contributing separately to the total mean square value; this is true for the geomagnetic field vector B, its horizontal component vector H, and its vertical component z. However, because of the lack of orthogonality this concept is not applicable to the horizontal X and y components individually; problems which arise from this are discussed.

The "global" representation of the geomagnetic field in terms of ordinary spherical harmonics (SHs) and its corresponding set {g,h} of coefficients has been studied extensively, but the "local" representation in terms of spherical cap harmonics (SCHs) and its corresponding set {G,H} of coefficients is not yet well understood. This paper clarifies some of the main properties of the SCHs and their proper use along with their relationship with the SHs. In particular, it shows that for the spherical cap part of a global field specified by spherical harmonics there is a strict relation between the ordinary Legendre functions of the global representation and the fractional functions of the local expansion; hence we can express the set of coefficients {G,H} in terms of the set {g,h}. Finally, some attention will be given to the role of the leading (n=0, m=0) term of the SCH expansion.

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.