FIELD AND SOURCE EQUIVALENCE IN SOURCE RECONSTRUCTION ON 3D SURFACES

This paper describes in detail difierent formulations of the inverse-source problem, whereby equivalent sources and/or flelds are to be computed on an arbitrary 3-D closed surface from the knowledge of complex vector electric fleld data at a specifled (exterior) surface. The starting point is the analysis of the formulation in terms of the Equivalence Principle, of the possible choices for the internal flelds, and of their practical impact. Love's (zero interior fleld) equivalence is the only equivalence form that yields currents directly related to the flelds on the reconstruction surface; its enforcement results in a pair of coupled integral equations. Formulations resulting in a single integral equation are also analyzed. The flrst is the single-equation, two-current formulation which is most common in current literature, in which no interior fleld condition is enforced. The single-current (electric or magnetic) formulation deriving from continuity enforcement of one fleld is also introduced and analyzed. Single-equation formulations result in a simpler implementation and a lower computational load than the dual-equation formulation, but numerical tests with synthetic data support the beneflts of the latter. The spectrum of the involved (discretized) operators clearly shows a relation with the theoretical Degrees of Freedom (DoF) of the measured fleld for the dual-equation formulation that guarantees extraction of these DoF; this is absent in the single-equation formulation. Examples conflrm that single- equation formulations do not yield Love's currents, as observed both with comparison with reference data and via energetic considerations. The presentation is concluded with a test on measured data which shows the stability and usefulness of the dual-equation formulation in a situation of practical relevance.