SUMMARY We present a parallel and high-order Nédélec finite element solution for the marine controlled-source electromagnetic (CSEM) forward problem in 3-D media with isotropic conductivity. Our parallel Python code is implemented on unstructured tetrahedral meshes, which support multiple-scale structures and bathymetry for general marine 3-D CSEM modelling applications. Based on a primary/secondary field approach, we solve the diffusive form of Maxwell’s equations in the low-frequency domain. We investigate the accuracy and performance advantages of our new high-order algorithm against a low-order implementation proposed in our previous work. The numerical precision of our high-order method has been successfully verified by comparisons against previously published results that are relevant in terms of scale and geological properties. A convergence study confirms that high-order polynomials offer a better trade-off between accuracy and computation time. However, the optimum choice of the polynomial order depends on both the input model and the required accuracy as revealed by our tests. Also, we extend our adaptive-meshing strategy to high-order tetrahedral elements. Using adapted meshes to both physical parameters and high-order schemes, we are able to achieve a significant reduction in computational cost without sacrificing accuracy in the modelling. Furthermore, we demonstrate the excellent performance and quasi-linear scaling of our implementation in a state-of-the-art high-performance computing architecture.

We present the capabilities and results of the Parallel Edge-based Tool for Geophysical Electromagnetic modeling (PETGEM), as well as the physical and numerical foundations upon which it has been developed.PETGEM is an open-source and distributed parallel Python code for fast and highly accurate modeling of 3D marine controlled-source electromagnetic (3D CSEM) problems. We employ the Nédélec Edge Finite Element Method (EFEM) which offers a good trade-off between accuracy and number of degrees of freedom, while naturally supporting unstructured tetrahedral meshes. We have particularised this new modeling tool to the 3D CSEM problem for infinitesimal point dipoles asumming arbitrarily isotropic media for lowfrequencies approximations. In order to avoid source-singularities, PETGEM solves the frequency-domain Maxwell's equations of the secondary electric field, and the primary electric field is calculated analytically for homogeneous background media. We assess the PETGEM accuracy using classical tests with known analytical solutions as well as recent published data of real life geological scenarios. This assessment proves that this new modeling tool reproduces expected accurate solutions in the former tests, and its flexibility on realistic 3D electromagnetic problems. Furthermore, an automatic mesh adaptation strategy for a given frequency and specific source position is presented. We also include a scalability study based on fundamental metrics for high-performance computing (HPC) architectures. contributed to the parallel implementation and its documentation.2 Josep de la Puente contributed to the math background and to the meshes generation. 3 José María Cela carried out simulations and its scalability analysis.

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Abstract. This paper presents an edge-based parallel code for the data computation that arises when applying one of the most popular electromagnetic methods in geophysics, namely, the controlled-source electromagnetic method (CSEM). The computational implementation is based on the linear Edge Finite Element Method in 3D isotropic domains because it has the ability to eliminate spurious solutions and is claimed to yield accurate results.The framework structure is able to exploit the embarrassingly-parallel tasks and the advantages of the geometric flexibility as well as to work with three different orientations for the dipole, or excitation source, on unstructured tetrahedral meshes in order to represent complex geological bodies through a local refinement technique. We demonstrate the performance and accuracy of our tool on the Marenostrum supercomputer (Barcelona Supercomputing Center) through scaling tests and canonical tests, respectively.

We use machine learning algorithms (artificial neural networks, ANNs) to estimate petrophysical models at seismic scale combining well-log information, seismic data and seismic attributes. The resulting petrophysical images are the prior inputs in the process of full-waveform inversion (FWI). We calculate seismic attributes from a stacked reflected 2-D seismic section and then train ANNs to approximate the following petrophysical parameters: P-wave velocity ($$V_\mathrm{{p}}$$ V p ), density ($$\rho $$ ρ ) and volume of clay ($$V_\mathrm{{clay}}$$ V clay ). We extend the use of the $$V_\mathrm{{clay}}$$ V clay by constraining it with the well lithology and we establish two classes: sands and shales. Consequently, machine learning allows us to build an initial estimate of the earth property model ($$V_\mathrm{{p}}$$ V p ), which is iteratively refined to produce a synthetic seismogram that matches the observed seismic data. We apply the 1-D Kennett method as a forward modeling tool to create synthetic data with the images of $$V_\mathrm{{p}}$$ V p , $$\rho $$ ρ and the thickness of layers (sands or shales) obtained with the ANNs. A nonlinear least-squares inversion algorithm minimizes the residual (or misfit) between observed and synthetic full-waveform data, which improves the $$V_\mathrm{{p}}$$ V p resolution. In order to show the advantage of using the ANN velocity model as the initial velocity model for the inversion, we compare the results obtained with the ANNs and two other initial velocity models. One of these alternative initial velocity models is computed via P-wave impedance, and the other is achieved by velocity semblance analysis: root-mean-square velocity (RMS). The results are in good agreement when we use $$\rho $$ ρ and $$V_\mathrm{{p}}$$ V p obtained by ANNs. However, the results are poor and the synthetic data do not match the real acquired data when using the semblance velocity model and the $$\rho $$ ρ from the well log (constant for the entire 2-D section). Nevertheless, the results improve when including $$\rho $$ ρ , the layered structure driven by the $$V_\mathrm{{clay}}$$ V clay (both obtained with ANNs) and the semblance velocity model. When doing inversion starting with the initial $$V_\mathrm{{p}}$$ V p model estimated using the P-wave impedance, there is some gain of the final $$V_\mathrm{{p}}$$ V p with respect to the RMS initial $$V_\mathrm{{p}}$$ V p . To assess the quality of the inversion of $$V_\mathrm{{p}}$$ V p , we use the information for two available wells and compare the final $$V_\mathrm{{p}}$$ V p obtained with ANNs and the final $$V_\mathrm{{p}}$$ V p computed with the P-wave impedance. This shows the benefit of employing ANNs estimations as prior models during the inversion process to obtain a final $$V_\mathrm{{p}}$$ V p that is in agreement with the geology and with the seismic and well-log data. To illustrate the computation of the final velocity model via FWI, we provide an algorithm with the detailed steps and its corresponding GitHub code.