Forward modelling of direct current and low-frequency electromagnetic fields using integral equations

We present a semi-analytical, unifying approach for modelling the electroma gnetic response of 3-D bodies excited by low-frequency electric and magneti c sources. We write the electric and magnetic fields in terms of power seri es of angular frequency, and show that to obey Maxwell's equations, the fie lds must be real when the exponent is even, and imaginary when it is odd. T his leads to the result that the scattering equations for direct current fi elds and for fields proportional to frequency can both be explicitly formul ated using a single, real dyadic Green's function. Although the underground current flow in each case is due to different physical phenomena, the inte raction of the scattering currents is of the same type in both cases. This implies that direct current resistivity, magnetometric resistivity and elec tric and magnetic measurements at low induction numbers can all be modelled in parallel using basically the same algorithm. We make a systematic deriv ation of the quantities required and show that for these cases they can all be expressed analytically. The problem is finally formulated as the soluti on of a system of linear equations. The matrix of the system is real and do es not depend on the type of source or receiver. We present modelling resul ts for different arrays and apply the algorithm to the interpretation of fi eld data. We assume the standard dipole-dipole resistivity array for the di rect current case, and vertical and horizontal magnetic dipoles for inducti on measurements. In the case of magnetometric resistivity we introduce a mo ving array composed of an electric dipole and a directional magnetometer, T he array has multiple separations for depth discrimination and can operate in two modes, The mode where the predominant current flow runs along the pr ofile is called MMR-TM. This mode is more sensitive to lateral variations i n resistivity than its counterpart, MMR-TE, where the mode of conduction is predominantly perpendicular to the profile.