The Python-code empymod computes the 3D electromagnetic field in a layered earth with vertical transverse isotropy by combining and extending two earlier presented algorithms in this journal. The bottleneck in frequency- and time-domain calculations of electromagnetic responses derived in the wavenumber-frequency domain is the transformations from the wavenumber to the space domain and from the frequency to the time domain, the so-called Hankel and Fourier transforms. Three different Hankel transform methods (quadrature, quadrature-with-extrapolation [QWE], and filters) and four different Fourier transform methods (fast Fourier transform [FFT], FFTLog, QWE, and filters) are included in empymod, which allows us to compare these different methods in terms of speed and precision. The best transform in terms of speed and precision depends on the modeled frequencies. Published digital filters for the Hankel transform are very fast and precise for frequencies in the range of controlled-source electromagnetic data, but they fail in the frequency range of ground-penetrating radar. Conventional quadrature, on the other hand, is in comparison very slow but can model any frequency. Examples comparing empymod with analytical solutions and with existing electromagnetic modelers illustrate the capabilities of empymod.

The open-source code fdesign makes it possible to design digital linear filters for the Hankel and Fourier transforms used in potential, diffusive, and wavefield modeling. Digital filters can be derived for any electromagnetic (EM) method, such as methods in the diffusive limits (direct current, controlled-source EM [CSEM]) as well as methods using higher frequency content (ground-penetrating radar [GPR], acoustic and elastic wavefields). The direct matrix inversion method is used for the derivation of the filter values, and a brute-force minimization search is carried out over the defined spacing and shifting values of the filter basis. Included or user-provided theoretical transform pairs are used for the inversion. Alternatively, one can provide layered subsurface models that will be computed with a precise quadrature method using the EM modeler empymod to generate numerical transform pairs. The comparison of the presented 201 pt filter with previously presented filters indicates that it performs better for some standard CSEM cases. The derivation of a longer 2001 pt filter for a GPR example with a 250 MHz center frequency proves that the filter method works not only for diffusive EM fields but also for wave phenomena. The presented algorithm provides a tool to create problem specific digital filters. Such purpose-built filters can be made shorter and can speed up consecutive potential, diffusive, and wavefield inversions.

In practice, the complication behind modeling EM induction in geomagnetic observations is twofold. First, one needs to assume a subsurface conductivity model. A number of regional and global conductivity models exist. This study will not focus on how these models are constructed and whether they represent the subsurface accurately, even though inaccurate conductivity models may bias results. It is expected that our knowledge about the electrical structure of the subsurface will continuously improve, allowing for the construction of more accurate models at different scales (Kelbert, 2020). Second, even if a distribution of

We developed a methodology to estimate resistivities from seismic velocities. We applied known methods, including rock physics, depth trends, structural information, and uncertainty analysis. The result is the range of background resistivity models that is consistent with the known seismic velocities. We successfully tested the methodology with real data from the North Sea. These 2D or 3D background resistivity models yield a detailed insight into the background resistivity, and they are a powerful tool for feasibility studies. They could also serve as starting models or constraints in (iterative) forward modeling of electromagnetic data for the determination of subsurface resistivities.

We developed a methodology to estimate resistivities from seismic velocities. We applied known methods, including rock physics, depth trends, structural information, and uncertainty analysis. The result is the range of background resistivity models that is consistent with the known seismic velocities. We successfully tested the methodology with real data from the North Sea. These 2D or 3D background resistivity models yield a detailed insight into the background resistivity, and they are a powerful tool for feasibility studies. They could also serve as starting models or constraints in (iterative) forward modeling of electromagnetic data for the determination of subsurface resistivities.