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

Abstract

We present a semi‐analytical, unifying approach for modelling the electromagnetic response of 3‐D bodies excited by low‐frequency electric and magnetic sources. We write the electric and magnetic fields in terms of power series of angular frequency, and show that to obey Maxwell’s equations, the fields must be real when the exponent is even, and imaginary when it is odd. This leads to the result that the scattering equations for direct current fields and for fields proportional to frequency can both be explicitly formulated using a single, real dyadic Green’s function. Although the underground current flow in each case is due to different physical phenomena, the interaction of the scattering currents is of the same type in both cases. This implies that direct current resistivity, magnetometric resistivity and electric and magnetic measurements at low induction numbers can all be modelled in parallel using basically the same algorithm. We make a systematic derivation of the quantities required and show that for these cases they can all be expressed analytically. The problem is finally formulated as the solution of a system of linear equations. The matrix of the system is real and does not depend on the type of source or receiver. We present modelling results for different arrays and apply the algorithm to the interpretation of field data. We assume the standard dipole–dipole resistivity array for the direct current case, and vertical and horizontal magnetic dipoles for induction measurements. In the case of magnetometric resistivity we introduce a moving array composed of an electric dipole and a directional magnetometer. The array has multiple separations for depth discrimination and can operate in two modes. The mode where the predominant current flow runs along the profile is called MMR‐TM. This mode is more sensitive to lateral variations in resistivity than its counterpart, MMR‐TE, where the mode of conduction is predominantly perpendicular to the profile.

Time-dependent magnetometric resistivity anomalies of groundwater contamination: Synthetic results from computational hydro-geophysical modeling

“…Three transmitter confi gurations are considered when computing MMR anomalies, these include single source mode (Figure 5a), MMR-TE mode (Figure 5b) where the predominant current flows perpendicular to the profile, and MMR-TM mode (Figure 5c) where the current runs along the profile (Méndez-Delgado et al, 1999). The following results are presented in terms of MMR anomaly (%) (or anomalous magnetic field) defined in equation (5), where B i n denotes the i-component of the magnetic field for the uniform half-space (also called the normal magnetic field) and B i a is the anomalous magnetic fi eld that is equal to the actual magnetic fi eld subtracting the 'normal' fi eld.…”

“…Various geo-electrical techniques have proven the most suitable for mapping and monitoring contaminant transport in groundwater systems (Fitterman and Stewart, 1986;Buselli et al, 1988;Frangos, 1997;Greenhouse and Monier-Williams, 1985;Greenhouse and Slaine, 1986;Méndez-Delgado et al, 1999;Moore and Barker, 2001;Meads et al, 2003;Elders and Asten, 2004) since the contamination causes an increase in dissolved salts in the groundwater and hence the consequent increase in its electrical conductivity.…”

“…Since its invention by Edwards in early 1970s, the MMR method has been widely employed in various geological and geophysical applications (Edwards, 1974;Wolfgram et al, 1986;Nobes et al, 1992;Méndez-Delgado et al, 1999;Elders and Asten, 2004;Yang, 2005) as well as in imaging near-surface conductors associated with subsurface solute fl ow (Kulessa et al, 2002;LaBrecque et al, 2004). More recently, review papers by Yaramanci et al (2005) and Zonge et al (2005) further emphasized the possible use of the MMR method in groundwaterrelated hydrogeophysics.…”

“…Electrical conductivity governing the electrical field is the most critical physical parameter used in geoelectrical exploration. The majority of current forward modeling theoretical studies (Méndez-Delgado et al, 1999;Ogilvy et al, 2002;Chen et al, 2002) have only emphasized its space-dependence and ignored the transient features. In fact, as a contaminant travels with groundwater flow, hydrodynamic dispersion causes the spreading of the contaminant plumes in a direction transverse to the fl ow path as well as in the longitudinal fl ow direction.…”

1-D laterally constrained inversion of EM34 profiling data

“…Two sets of synthetic data generated by a 3-D forward modelling program (Méndez-Delgado et al, 1999) have been used to test the algorithm. According to their authors (Méndez-Delgado et al, 1999), the forward algorithm used to generate the synthetic data is a semianalytical approach for modelling the electromagnetic response of 3-D media excited by low frequency electromagnetic fields.…”