The electromagnetic response in a layered vertical transverse isotropic medium: A new look at an old problem

Abstract

We determined that the electromagnetic vertical transverse isotropic response in a layered earth can be obtained by solving two equivalent scalar equations, which were for the vertical electric field and for the vertical magnetic field, involving only a scalar global reflection coefficient. Besides the complete derivation of the full electromagnetic response, we also developed the corresponding computer code called EMmod, which models the full electromagnetic fields including internal multiples in the frequency-wavenumber domain and obtains the frequency-space domain solutions through a Hankel transformation by computing the Hankel integral using a 61-point Gauss-Kronrod integration routine. The code is able to model the 3D electromagnetic field in a 1D earth for diffusive methods such as controlled source electromagnetics as well as for wave methods such as ground penetrating radar. The user has complete freedom to place the source and the receivers in any layer. The modeling is illustrated with three examples, which aim to present the different capabilities of EMmod, while assessing its correctness.

Inversion of controlled-source electromagnetic reflection responses

“…Depending on which component needs to be modeled, a different part of this function separated by an if statement is executed. This function is very similar to the corresponding function used in the electromagnetic forward-modeling package mentioned earlier (Hunziker et al, 2015) with the exception that the so-called Fortran routines not only compute the electromagnetic fields but also the gradient of them. Variables starting with the letter d represent variables related to derivatives, and thus related to the computation of the gradient.…”

“…For a real or a synthetic data set with noise, no stabilization would be required because the data would never be zero. For forward modeling, we use our publicly available modeling code (Hunziker et al, 2015). A smaller misfit F means that the individual is fitter than another individual with a larger misfit F. As in evolution theory, the fitter the individual, the larger the chances that this guess is able to transmit its properties to the next generation.…”

“…For this inversion algorithm, not only the forward problem but also the gradient of the forward problem needs to be computed. This is done analytically by using a modified version of the before-mentioned forward-modeling code (Hunziker et al, 2015). As developing such a code with analytical gradients is rather time-consuming, we include the complete conjugate-gradient inversion scheme implemented in C and Fortran in this paper as an opensource package.…”

Time evolution of the electric field: Part 1 — Using the rapid expansion method (REM) with pseudospectral evaluation of spatial derivatives

“…Next we compare a reflectivity method (Slob et al, 2010) using the program EMmod described in Hunziker et al (2015) to the pseudo-spectral REM method. The model is defined on a 400 by 400 by 400 computational grid with a sampling interval of 10 m in each direction.…”

The electromagnetic response of a horizontal electric dipole buried in a multi‐layered earth

“…We will refrain from a detailed description of these variants but a common feature that they all share is that spatial or temporal derivatives must be taken in order to convert the potentials into electric and magnetic fields. The second group, where the fields are directly calculated without the use of potentials, includes works by Kong (1972), Ursin (1983), Chew (1999), Løseth and Ursin (2007), Streich and Becken (2011), and Hunziker, Thorbecke and Slob (2015). A unique hybrid approach can be found in Chave (2009), which expresses the source current density in terms of toroidal, consoidal, and vertical parts, the former two being associated with derivatives of potentials and the latter having no such relationship.…”