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, $A - \phi $ formula using magnetic vector potentials ($\bf A$) electric scalar ($\phi $) was developed. At present, there two sets of formulae used: one has a curl–curl ($\nabla \times \nabla structure another Laplace (${\nabla ^2}$) where Coulomb gauge is enforced. The question as which approaches have superior performance for 3D geo-electromagnetic induction still remains open. In study, we systemically compared performances these systems in terms both accuracy convergence rate. Numerical experiments suggest that magnetotelluric controlled-source electromagnetic problems, system with better than variant rates.