Bayesian approach to ionospheric elementary current systems using differentiable basis functions

Spherical elementary current systems (SECS) provide a method for modeling ionospheric currents and other ionospheric vector fields. The original formulation of the SECS assumes point-like sources for the divergence and curl of the fields, which lead to singularities. Here we formulate a general differentiable alternative for the original SECS and show that any function with a finite Legendre series can be used as the basis of the analysis. We also present how a particular choice of the Legendre coefficients leads to closed-form expressions for the magnetic field and electric currents, making the differentiable SECS no more complicated than the original SECS. A common application of SECS is solving the currents from magnetic field measurements. We demonstrate how to regularize the system with Bayesian tools with intuitive, physically meaningful parameters for the currents. In particular, we show how prior knowledge about the amplitudes and correlation lengths of the currents is transformed to prior information on SECS amplitudes. The differentiable SECS and the inversion method are verified with a test case built using the Average Magnetic field and Polar current System (AMPS) model. In addition, we apply our method to data from a ground magnetometer network and compare our results with results obtained with the original SECS method.