The magnetic field generated by the lithosphere at satellite altitude, is the gradient of a scalar potential (B = −∇V(θ, ϕ, r, t)). The potential itself is given in terms of (L + 1)(2L + 1) functions FijL(θ, ϕ, r):
where the functions FijL(θ, ϕ, r) are defined by:
where fl ≠ 0, r ≥ a, and a = 6371.2 km is the traditional magnetic reference radius. All the functions FijL(θ, ϕ, r) have the same basic shape and are symmetric relative to the vector pointing from the coordinate system origin in the direction (θi,ϕj) of their centre. Their centre positions are given by:
where xi is the ith zero of the Legendre polynomial PL+1(x). The Ylm(θ, ϕ) are the usual Schmidt semi-normalized spherical harmonics; Negative orders (m < 0) are associated with sin (m θ) terms, whereas zero or positive orders (m ≥ 0) are associated with cos (m θ) terms. The fl in equation (2) are optimised such that the gradient of the functions FijL(θ, ϕ, r) decreases as rapidly as possible with the angular distance from their centre. Figure 1 shows the calculated fl values for a maximum SH degree 90. Also shown, in Figure 2, is the amplitude of the gradient of FijL(θ, ϕ, r) as a function of the angular distance from its centre. Any scalar potential given by equation (1) can be modelled in terms of SH with the usual equation
Conversely, because (2l + 1)(l + 1) functions with their centres defined in equations (3) and (4) are used, any scalar potential given by equation (5) can be parameterised as in equation (1). By equating equations (1) and (5), using the function definitions in equation (2), multiplying both sides by (θ, ϕ) and integrating over the sphere, it is straightforward to find the relation giving the Gauss coefficients as a function of the ij at the reference radius:
The transformation inverse (i.e., a formula giving a possible set of ij as a function of the Gauss coefficients) is presented by Lesur . For a given potential V(θ, ϕ, r), there is more than one possible set of coefficients ij(t). This is simply because (L + 1)(2L + 1) functions are needed to parameterise the potential in equation (1) whereas only L(L + 2) spherical harmonics are needed in equation (5). Clearly, some sort of regularisation has to be introduced to find the ij in equation (1) from a set of magnetic measurements. We usually seek the minimum norm solution by removing zero eigenvalues and the associated eigenvector of the normal equation matrix.