On the separation of core and crustal contributions to the geomagnetic field

Summary. The widely accepted procedure of separating the contributions to the geomagnetic field arising in the Earth's core from those arising in the Earth's crust on the basis of spherical harmonic degree number has been recently challenged by Carle & Harrison. We argue here that their basic objection is valid only for data over line profiles or flat surfaces. We feel that while Carle & Harrison have addressed a number of significant and fundamental aspects of this problem they appear not to have done so clearly. The contribution contained herein is intended as a clarification. In particular we wish to:1 Acknowledge, as Carle & Harrison asserted, that so‐called ‘linear’ data (see text) over line profiles (or flat two‐dimensional surfaces) are better represented by Fourier components than by Legendre functions in the following sense : Legendre functions do not have regularly spaced zeros over line profiles (or flat surfaces), and hence are not associated with ‘wavelengths’ in the conventional sense. Thus representation and filtering (such as removing low degree components) using Legendre functions does not satisfactorily separate long‐ and short‐wavelength features (defined in the Fourier sense). 2 Emphasize that the above‐mentioned advantage of Fourier analysis over spherical harmonic analysis no longer exists when a (linear) data set over a curved surface is to be represented, since the conventional definition of wavelength breaks down. 3 Demonstrate, by way of squaring a Fourier series and a spherical harmonic series (actually we consider the sample example as given by Carle & Harrison where no φ‐dependence is included), that any non‐linear operations on an originally linear data set, irrespective of the ‘basis functions’ in which the data are expanded, invariably create intermodulations . Carle & Harrison appeared to have confused the issue of intermodulations with the issue of wavelength identification when they employed a non‐linear operation in showing that ‘the use of spherical harmonics in describing the core field and in removing the core field from the total field in order to get the field produced by the crustal magnetization is flawed’. 4 Point out that the power spectrum of the residual field after removal of certain low degree Fourier components from a data set will in general be different from that after removal of corresponding spherical harmonic components from the same data set. This observation may be considered by some to be a valid basis for advocating Fourier analysis over spherical harmonic analysis on line profile data. This argument does not hold, however, for data sets over curved surfaces such as that of the Earth, since there no longer exists the intrinsic property of wavelength attached to a low degree Fourier component vis‐à‐vis a spherical harmonic component of the same degree.By linearizing the problem, a general least squares inversion scheme is sketched through which the magnetic potential may be recovered from a measured geomagnetic intensity field. Limitations and strengths of the proposed scheme are discussed.