The domain Ω, on which potentials and fields are dealt with, is depicted on Fig. However, as a result of the adopted approach, which makes use of information on the field and not on the potential, the modelling proposed here is suitable only for a restricted class of fields. We finally succeed in reconstructing the corresponding local field by these relationships at any altitude and colatitude within Ω. 1, enables us to establish a relationship between global and local Gauss coefficients. The proper resolution of the BV problem on the conical domain, depicted on Fig. In this paper, we propose a new SCHA expansion that removes this drawback. Therefore, it is impossible, using this formalism, to solve properly a boundary value (BV) problem where the boundary conditions are a global field derived from an SH model (either a complete model or a model containing only the highest degrees in the expansion). None of the basis functions proposed by Haines is appropriate for adjusting fields decreasing as r − n where n is a real integer. However, the second and most serious problem, stressed by De Santis & Falcone (1995) and also mentioned by Lowes ( Langel & Hintz 1998), comes from the failure of modelling correctly the radial dependence. This is especially true for the regional secular variation and explains why it is meaningless to model it over small caps with SCHA ( De Santis et al. When dealing with a small cap, a larger expansion is required, most of the time larger than permitted by the number of available data. Numerical experiments demonstrate that the convergence of SCHA is dramatically slow for relatively small caps and an insufficient expansion produces unrealistic oscillations in the results when interpolating on a dense grid. First, it is broadly admitted that its behaviour goes worst as the cap goes smaller. Nevertheless, SCHA practitioners encounter two kind of difficulties. 1989), field modelling ( Hwang & Chen 1997), or even regional secular variation modelling ( Korte & Haak 2000). On the basis of these assertions, SCHA has been widely used for crustal anomalies modelling ( De Santis et al. The method is claimed to be valid over any spherical cap at any altitude above the surface of the Earth. Its formalism looks like a natural extension of the spherical harmonic analysis.
The spherical cap harmonic analysis (SCHA) proposed by Haines (1985) is an attractive regional modelling.
Common techniques, like polynomial modelling in latitude and longitude or rectangular harmonic analysis ( Alldredge 1981), have been used successfully before the availability of satellite data but the resulting models could not be properly upward or downward continued ( Haines 1990). For a limited portion of the Earth, spherical harmonics (SH), well known for global modelling, are no longer suitable, because they are no longer orthogonal over the restricted area.
The advent of satellite magnetic measurements brought a major breakthrough for the reconstruction of a regional magnetic field at short wavelengths. Regional modelling is in theory a powerful method for detailed description of potential fields over areas where an appropriate dense set of data is available. Magnetic field, regional modelling, spherical cap harmonic analysis 1 Introduction Although the example worked out here applies only to a limited class of fields, which verifies some special flux condition, the ideas behind this formalism are quite general and should offer a new way of processing data in a bounded region of space. In this paper, we show that these difficulties are overcome if the SCHA modelling is formulated as a boundary value (BV) problem in a cone bounded radially by the surface of the Earth and an upper surface suitable for satellite data, and bounded laterally in order to encompass a specific region of study. Such a relationship would be useful, for instance, for introducing prior constraint on an inverse problem dealing with the estimation of local Gauss coefficients based upon a local data set. With the SCHA adopted so far, difficulties arise in upward continuation and in establishing a relationship between global and local Gauss coefficients.
The spherical cap harmonics analysis (SCHA) is an attractive regional modelling tool having close relationship with global SH. For regions with a fairly dense coverage of data at different altitudes, a regional model ought to offer a better spatial resolution of the regional field over the volume under study than a global field expanded in spherical harmonics (SH). The geomagnetic field above the surface of the Earth in the current-free region may be expressed as the gradient of a scalar potential solving Laplace's equation.