Applied comparisons between SCHA and R‐SCHA regional modeling techniques

Spherical cap harmonic analysis (SCHA) has become a common tool for the regional modeling of potential fields since its introduction by Haines (1985). The fact that SCHA satisfies Laplace equation and the possibility of representing high‐frequency fields with a small number of coefficients (compared to the global spherical harmonic analysis) made SCHA the preferred choice for the development, for example, of magnetic field models at national scale. However, Thébault et al. (2006a) demonstrated that the traditional SCHA presented some deficiencies, in particular related to the inversion of multilevel data sets. The authors presented the R‐SCHA technique as an alternative method in which the introduction of a new set of basis functions and boundary conditions solved this issue. In this paper we present some numerical comparisons between the SCHA and R‐SCHA techniques applied with different synthetic vector data sets, from near‐surface main field, main difference, and crustal field data simulating a World Digital Magnetic Anomaly Map subset. Other analyses are carried out with synthetic vector data set that mimics the expected data distribution from a multisatellite mission like the forthcoming European Swarm mission. No regularization, weighting, or ad hoc procedures are applied to the synthetic vector data, and a cap of 7° aperture is considered. The numerical analyses show that SCHA is a satisfying approximation in a band‐limited spectral region that depends on the cap's size. It does not work correctly either for main field or for the short‐scale crustal field modeling. These aspects are supported by equations illustrating why SCHA may fail. On the contrary, R‐SCHA converges more slowly than SCHA but is valid in all cases. It gives a consistent set of regional coefficients and fits the radial variation of the field in a realistic way. At last, the special case of data incompatibility shows that R‐SCHA does not fit incompatible data while SCHA assimilates most of them. These results should help the scientific community to evaluate the level of approximation needed for the development of regional magnetic field models in the era of the European Space Agency Swarm mission.