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

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 (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 (del x del) structure and another one has a Laplace (del(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.