Modelling of 3-D electromagnetic responses using the time-wavenumber method

The diffusion of electromagnetic fields in time and the three spatial dimensions can be modelled using a new numerical algorithm that is tailored for geophysical applications. The novel feature of the algorithm is that a large part of the computation is done in the wavenumber domain. Here, the spatial Fourier transforms of the vertical magnetic field and the vertical current density are used to define two scalar potentials. For either a vertical electric or a vertical magnetic dipole source at the subsurface these wavenumber potentials can be represented by a simple Gaussian distribution function. In the air, the fields satisfy the Laplace equation. The flow of this algorithm is as follows: the potentials are defined in the wavenumber domain as an initial condition depending on the source configuration, the vector current density J in space is obtained from the potentials using the inverse Fourier transform, the vector electric field E is obtained by multiplying J by resistivity, and the updated potentials are then obtained from the forward Fourier transform of E. Using the updated potential as a subsequent initial condition these steps are repeated until the solution reaches the final time. Since spatial derivatives can be exactly evaluated in the wavenumber domain by simple multiplications, this algorithm requires far less memory than the conventional finite difference (FD) method. The conventional FD method needs finer discretization in space in order to minimize the numerical dispersion caused by numerical differentiation in space. The conductivity distribution for this algorithm is piece-wise continuous and bounded in the wavenumber domain.