Reciprocity in electromagnetics: application to modelling marine magnetometric resistivity data

The marine magnetometric resistivity (MMR) can be used to obtain resistivity structure beneath the ocean floor. Because of logistical reasons, many more transmitters than receivers are deployed in a survey. This makes it difficult to carry out 3-D forward modelling of magnetometric resistivity data, since each transmitter source requires a separate solution of Maxwell's equations in order to generate the fields. Two methods are presented to overcome this difficulty. The first is based upon the Lorentz reciprocity theorem. With this theorem, the magnetic field at a receiver, generated by a long vertical electrical bipole, is exactly same as the normalized electromotive force induced in the transmitter wire generated by an artificial magnetic dipole located at the receiver position. The second is the adjoint method in which the magnetic field can be obtained by solving an adjoint equation with an artificial source at each receiver. We show that these two methods are eventually identical: the artificial source in both methods is a steady current in a loop, and the “measurement” is the voltage along the transmitter wire. However the adjoint algorithm is computationally more efficient and we use it in the 3-D marine MMR forward modelling. We verify the code with a synthetic 3-D example. Use of the reciprocity significantly reduces the computational load, making the practical marine MMR problem tractable.