![]() ![]() Madsen & Ziolkowski 1990 Jahandari & Farquharson 2014) methods. 2013 Grayver & Bürg 2014) or finite volume (FV see e.g. Wang & Hohmann 1993 Streich 2009 Cherevatova et al. 2006 Kruglyakov & Kuvshinov 2018 Liu et al. Most developed 3-D EM forward codes use one of the following numerical approaches to obtain a solution to the forward problem, namely the integral equation (IE see e.g. In other words, accurate, efficient and robust forward modelling routines are crucial. One of the key factors for the high computational costs is the expensive solution of the 3-D EM forward modelling problem of large-scale and geologically complex settings, which has to be solved numerous times throughout the inversion process. In spite of these advances, 3-D inverse modelling still remains a challenging and computationally demanding task. The availability of high-performance computing facilities has enabled the processing of increasingly larger data sets (Newman 2014). review papers by Avdeev 2005 Commer & Newman 2008 Börner 2010 Streich 2016 Miensopust 2017) due to improvements in computational resources, especially in terms of speed and memory capabilities and development in numerical methods. Reliable interpretation of that data relies on fast, efficient and robust discretization and solution techniques.įorward and inverse modelling of EM data has undergone steady progress ( cf. Large amounts of high-quality 3-D EM data have been acquired in various complex geological settings, such as surveying areas with rough terrains. Thus, EM methods have been more widely used in geophysical surveys related to hydrocarbon (Constable & Srnka 2007 Streich 2016 Patro 2017), mineral (Oldenburg & Pratt 2007 Farquharson & Craven 2009 Kalscheuer et al. ![]() In the last few decades, the increasing importance of electromagnetic (EM) methods has been substantially driven by huge improvements in both instrumentation and data processing techniques. To our knowledge, our final example that includes pronounced surface topography and two geometrically complicated conductive anomalies represents the first successful attempt at using 2nd order hexahedral elements supporting curved edges and non-planar faces in controlled-source EM geophysics.Ĭontrolled source electromagnetics (CSEM), Electromagnetic theory, Numerical modelling 1 INTRODUCTION The presented numerical experiments give evidence that 2nd and 3rd degree polynomials in combination with moderately discretized meshes provide better trade-offs in terms of computational resources and accuracy than lowest and higher order spectral element methods. A convergence study illuminates the relation between high order polynomial approximation and mesh size and their effects on accuracy and computational cost revealing that high-order approximation yields accurate modelling results for very coarse meshes but is accompanied by high computational cost. Comparisons to semi-analytical and finite element results confirm accurate representation of the EM responses and indicate low dependency on mesh discretization for the spectral element method. Five numerical examples comprehensively study the benefits of this algorithm. The resulting complex-valued linear system of equations is solved using the direct solver MUMPS, and, subsequently, the magnetic field is computed at the points of interest by Faraday’s law. The total electric field on the elements is expanded in terms of high-order Lagrangian interpolants, and element-wise integration in the weak form of the boundary value problem is accomplished by Gauss–Legendre–Lobatto quadrature. As a further improvement over existing spectral element methods, our approach does not only allow for arbitrary distributions of conductivity, but also of magnetic permeability and dielectric permittivity. Combining unstructured grids and a total field formulation provides advantages in dealing with topography, in particular, when the transmitter is located on rough surface topography. Our code is the first spectral element algorithm in EM geophysics that uses the total field formulation (here that of the electric field). Complex-shaped structures and topography are accommodated by using unstructured hexahedral meshes, in which the elements can have curved edges and non-planar faces. For forward modelling of realistic 3-D land-based controlled-source electromagnetic (EM) problems, we develop a parallel spectral element approach, blending the flexibility and versatility of the finite element method in using unstructured grids with the accuracy of the spectral method. ![]()
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