A proposal for regional modelling at the Earth's surface, R-SCHA2D

Modelling the magnetic field at a regional scale has always been challenging. Several techniques were proposed in the past but few solve Laplace equation and meet the requirements of potential theory. Among the regional techniques, the Spherical Cap Harmonic Analysis (SCHA) that solve Laplace equation has become widespread. However, recognizing the inherent difficulties of SCHA to represent the main field over a small spherical cap, a revision of this technique the R-SCHA was proposed to deal with multilevel magnetic measurements. For the reasons explained in this paper, R-SCHA is not suitable either when data are available at one surface only. The present paper proposes an alternative to SCHA and R-SCHA, named R-SCHA2D, that is suitable for modelling geomagnetic data available at a quasi-constant altitude. The efficiency of R-SCHA2D is successfully tested on a set of repeat station data. The magnetic field over France and its secular variation are reliably represented between 1965 and 2007.5. The regional parameters are obtained from the joint inversion in space and time of 29 metropolitan repeat stations and 1 observatory vector data using time power series as a temporal basis function. This approach does not require additional constraints like adding synthetic data, regularizing the inverse problem, or removing a trend prior to the inversion to be stable. It is shown that the geomagnetic jerks over France are detected in the regional parameters themselves and that it is possible to estimate the crustal bias. Considering the real data distribution in space and time, and the parameters used for this modelling, the R-SCHA2D model compared with the CM4 comprehensive model showed no indisputable evidence proving the existence of a secular variation anomaly over France. The main field and the secular variation are sketched for various epochs, in particular for epoch 2007.5. The present approach does neither address the problem of internal/external fields separation nor the upward continuation of the model to satellite altitudes.