A comparison of $A\! -\! \phi $ formulae for three-dimensional geo-electromagnetic induction problems

Abstract Geo-electromagnetic forward modeling problems are ill-posed due to the low signal frequencies being used and electrically insulating air space. To overcome this numerical issue, the $A - \phi $ formula using the magnetic vector potentials ($\bf A$) and electric scalar potentials ($\phi $) was developed. At present, there are two sets of $A - \phi $ formulae being used: one has a curl–curl ($\nabla \times \nabla $) structure and another one has a Laplace (${\nabla ^2}$) structure where the Coulomb gauge is enforced. The question as to which of the two approaches have superior performance for 3D geo-electromagnetic induction problems still remains open. In this study, we systemically compared the performances of these two $A - \phi $ systems in terms of both numerical accuracy and convergence rate. Numerical experiments suggest that for both magnetotelluric and controlled-source electromagnetic problems, the $A - \phi $ system with Laplace structure has better performance than the variant with curl–curl structure in terms of convergence rates.