Frequency‐domain modelling of airborne electromagnetic responses using staggered finite differences1

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Abstract
A 3D frequency‐domain EM modelling code has been implemented for helicopter electromagnetic (HEM) simulations. A vector Helmholtz equation for the electric fields is employed to avoid convergence problems associated with the first‐order Maxwell's equations when air is present. Additional stability is introduced by formulating the problem in terms of the scattered electric fields. With this formulation the impressed dipole source is replaced with an equivalent source, which for the airborne configuration possesses a smoother spatial dependence and is easier to model. In order to compute this equivalent source, a primary field arising from dipole sources of either a whole space or a layered half‐space must be calculated at locations where the conductivity is different from that of the background.The Helmholtz equation is approximated using finite differences on a staggered grid. After finite‐differencing, a complex‐symmetric matrix system of equations is assembled and preconditioned using Jacobi scaling before it is solved using the quasi‐minimum residual (QMR) method. The modelling code has been compared with other 1D and 3D numerical models and is found to produce results in good agreement.We have used the solution to simulate novel HEM responses that are computationally intractable using integral equation (IE) solutions. These simulations include a 2D conductor residing at a fault contact with and without topography. Our simulations show that the quadrature response is a very good indicator of the faulted background, while the in‐phase response indicates the presence of the conductor. However when interpreting the in‐phase response, it is possible erroneously to infer a dipping conductor due to the contribution of the faulted background.