Mise-à-la-masse modelling by integral equation with the continuity equation as a constraint

Abstract

Mise-à-la-masse anomalies are calculated by solving the potential problem for steady current flow. For a linear, piecewise homogeneous medium, the problem can be solved using the Fredholm integral equation of the second kind for the surface charge density. The location of the primary source in the conductor makes the numerical solution of the mise-à-la-masse problem difficult with this formulation because the primary term in the equation diminishes as the conductivity contrast increases and the equation tends to a homogeneous equation. The solved surface charge density and the potential may be too small even by an order of magnitude. The continuity equations can however be added as supplementary equations of the Fredholm equation of the second kind. Then the number of equations exceeds the number of unknowns. The solution can be sought in the least-squares sense. Convergence of the solution takes place rapidly because continuity equations guarantee the right amount of total surface charge on the boundaries of the bodies. Satisfactory solution is then achieved using relatively sparse discretisation. The present method permits us to calculate flexibly the mise-à-la-masse anomalies of bodies of varying conductivity contrast.