Marine Controlled-source Electromagnetic Modeling Research Articles

LibEMMI_MGFD: A program of marine controlled-source electromagnetic modelling and inversion using frequency-domain multigrid solver

We develop a software package libEMMI_MGFD for 3D frequency-domain marine controlled-source electromagnetic (CSEM) modelling and inversion. It is the first open-source C program tailored for geometrical multigrid (GMG) CSEM simulation. An volumetric anisotropic averaging scheme has been employed to compute effective medium for modelling over uniform and nonuniform grid. The computing coordinate is aligned with acquisition geometry by rotation with the azimuth and dip angles, facilitating the injection of the source and the extraction of data with arbitrary orientations. Efficient nonlinear optimization is achieved using quasi-Newton scheme assisted with bisection backtracking line search. In constructing the modularized Maxwell solver and evaluating the misfit and gradient for 3D CSEM inversion, the reverse communication technique is the key to the compaction of the software while maintaining the computational performance. A number of numeric tests demonstrate the efficiency of the modelling while preserving the solution accuracy. A 3D marine CSEM inversion example has been examined for resistivity imaging. Program summaryProgram Title: libEMMI_MGFDCPC Library link to program files:https://doi.org/10.17632/mpk7xrd38m.1Developer's repository link:https://github.com/yangpl/libEMMI_MGFDLicensing provisions: GNU General Public License v3.0Programming language: C, ShellSoftware dependencies: MPI [1]Nature of problem: 3D Controlled-source electromagnetic modelling and inversion.Solution method: Frequency-domain modelling on staggered grid; Geometrical multigrid (GMG) method as an iterative solver; Quasi-Newton method for nonlinear minimization; Bisection-based backtracking line search.

Finite-Element Scheme With Total-Field for 3-D Marine Controlled-Source Electromagnetic Modeling With Bathymetry and Multisources

Finite-Element Scheme With Total-Field for 3-D Marine Controlled-Source Electromagnetic Modeling With Bathymetry and Multisources

3D marine controlled-source electromagnetic modeling using an edge-based finite element method with a block Krylov iterative solver

Abstract An edge-based finite element method for the numerical modeling of electromagnetic fields in complex media is presented. We used the analytical solution on an electric field in a homogeneous half space to develop a source correct factor to reduce the influence of source singularity and boundary conditions on the numerical accuracy, so that we can minimize the time required to construct the field source term in the scattered field formula. The modeling domain was discretized using an unstructured tetrahedral mesh. We transformed the complex equations of the electrical field into two real-valued equations by decomposing the field into real and imaginary components. Thereafter, we adopted a block conjugate orthogonal conjugate gradient (BL_COCR) iterative solver with an incomplete LU decomposition preconditioner, which was robust for ill-conditioned systems and efficient for multiple source electromagnetic modeling to solve the real-valued equation systems. Using the analytical solution on an electric field in a homogeneous layer model, we evaluated the accuracy of our numerical forward solution and the results showed that the source correct factor can reduce forward modeling errors associated with boundary effects and source singularities. We also applied the developed algorithm to compute the CSEM responses for typical 3D marine geo-electric models with different number of sources and compared with different iterative solvers, and the results showed that the BL_COCR solver has high computational efficiency when solving multiple right-hand term problems.

Goal-Oriented 3-D Time-Domain Marine CSEM Modeling With Anisotropy and Topography

We have proposed a goal-oriented iteration algorithm with adaptive error estimation for the 3-D modeling of the time-domain marine controlled-source electromagnetic (CSEM) method. The algorithm is based on finite-element time-domain (FETD) formulations, and it updates both the time step and the mesh size in a staggered manner. With this scheme, the computational error is well suppressed, and the efficiency of the FETD simulation is guaranteed by adaptive time-stepping within each iteration. The verification results of an electric dipole radiation model and a synthetic 1-D marine CSEM model have shown that the proposed algorithm achieves a fine balance between computational accuracy and efficiency. Through numerical examples incorporated with stratum anisotropy and seafloor topography, we further conclude that both background anisotropy and topography impede the identification of submarine high-resistivity bodies with the electric field response. Contrarily, the time derivative of the magnetic field is sensitive to high-resistivity reservoirs over an extensive period of time in both anisotropic and topographic models. Based on this, we suggest that the magnetic field should be collected along with the electric field over the whole target area to reveal the anisotropic structure of submarine sediments and the existence of oil and gas reservoirs.

3-D marine controlled-source electromagnetic modelling in an anisotropic medium using a Wavelet–Galerkin method with a secondary potential formulation

SUMMARY This paper presents a new algorithm for solving 3-D MCSEM modelling problems in an anisotropic medium using a Wavelet–Galerkin method (WGM) based on compactly supported Daubechies wavelets which are differentiable according to the requirement. In order to avoid the source singularity, we adopted a secondary potential formulation for the quasi-static Maxwell's equation. The primary field on the modelling domain is calculated using fast Hankel transform. The domain can be discretized by locally intensive nodes to deal with the model's complexity, which is observed to improve the accuracy of the solution. The sparse system of the WGM equations is solved using the direct solver MUMPS. This study's algorithm is then applied to calculate the response of the MCSEM in isotropic and anisotropic mediums. The results generated are verified against with solution obtained by FE method and confirmed the performance of the algorithm presented in this study.

A block rational Krylov method for 3-D time-domain marine controlled-source electromagnetic modelling

We introduce a novel block rational Krylov method to accelerate three-dimensional time-domain marine controlled-source electromagnetic modeling with multiple sources. This method approximates the time-varying electric solutions explicitly in terms of matrix exponential functions. A main attraction is that no time stepping is required, while most of the computational costs are concentrated in constructing a rational Krylov basis. We optimize the shift parameters defining the rational Krylov space with a positive exponential weight function, thereby producing smaller approximation errors at later times and reducing iteration numbers. Furthermore, for multi-source modeling problems, we adopt block Krylov techniques to incorporate all source vectors in a single approximation space. The method is tested on two examples: a layered seafloor model and a 3D hydrocarbon reservoir model with seafloor bathymetry. The modeling results are found to agree very well with those from 1D semi-analytic solutions and finite-element time-domain solutions using a backward Euler scheme, respectively. Benchmarks of the block rational Krylov method demonstrate that it can be as up to 10 times faster than backward Euler. The block method also benefits from better memory efficiency, resulting in considerable speedup compared to non-block methods.

2.5D marine CSEM modeling in the frequency-domain based on an improved interpolation scheme at receiver positions

In the marine controlled-source electromagnetic (CSEM) survey, the receivers are usually placed at the seafloor. The resistivity contrast between the seawater and seafloor sediments is large, which can cause difficulties in numerical modeling of CSEM fields at receiver locations. In this paper, we present an improved interpolating method for calculating electric and magnetic fields at the seafloor with a resistivity contrast. This method is applied to the 2. 5 dimensional (2. 5D) frequency-domain CSEM modeling with towed transmitters and receivers located at the seafloor. Considering the discontinuity of the normal electric fields, we use the normal current electric density for interpolation. We simulate the 2. 5D marine CSEM responses by the staggered finite-difference (SFD) method with Fourier transform to the strike direction. The final SFD equations are solved by the direct solver MUMPS (MUltifrontal Massively Parallel Sparse direct Solver). To avoid the source singularities, the secondary-field approach is used and the primary fields excited by the electric dipole source can be calculated quasi-analytically for the one-dimensional (1D) layered background model. We focus on interpolating of electric and magnetic fields in the wavenumber domain to the receiver locations at the seafloor interface between the conductive seawater and resistive seafloor formation. The secondary electric and magnetic fields are used for interpolation instead of the total fields for high numerical accuracy. After performing the inverse Fourier transform to the wavenumbers, the electric and magnetic fields in the space domain are obtained. To check the accuracy of our 2. 5D marine CSEM SFD modeling algorithm with the improved receiver interpolating technique, we compare our results with both the 1D analytical results and the adaptive finite element results. The SFD numerical results are approved to be accurate. We also compare the numerical accuracy between our improved interpolation scheme and others, i.e., the conventional linear interpolation and the rigorous interpolation. The proposed interpolation only utilizes the nodes below/above the seafloor interface, and is proved to be much more accurate than the other two interpolating methods used.

Application of the perfectly matched layer in 3-D marine controlled-source electromagnetic modelling

In this study, the complex frequency-shifted perfectly matched layer (CFS-PML) in stretching Cartesian coordinates, is successfully applied to three-dimensional (3D) frequency-domain marine controlled-source electromagnetic (CSEM) field modelling. The Dirichlet boundary, which is usually used within the traditional framework of EM modeling algorithms, assumes the electric or magnetic field values are zero at the boundaries. This requires the boundaries be sufficiently far away from the sources in the area of interest. To mitigate the boundary artifacts, a large modelling area may be necessary even though cell sizes are allowed to grow toward the boundaries due to the diffusion of the electromagnetic wave propagation. Compared with the conventional Dirichlet boundary, the PML boundary is preferred as the modelling area of interest could be restricted to the target region and only a few absorbing layers surrounding can effectively depress the artificial boundary effect without losing the numerical accuracy. Furthermore, for joint inversion of seismic and marine CSEM data, if we used the PML for CSEM field simulation instead of the conventional Dirichlet, the modeling area for these two different geophysical data collected from the same survey area could the same, which is convenient for joint inversion grid matching. We apply the CFS-PML boundary to 3D marine CSEM modelling by using the staggered finite-difference (SFD) discretization. Numerical test indicates that the modeling algorithm using the CFS-PML also shows good accuracy compared to the Dirichlet. Furthermore, the modeling algorithm using the CFS-PML shows advantages in computational time and memory saving than that using the Dirichlet boundary. For the 3D example in this study, the memory saving using the PML is nearly 42 % and the time saving is around 48% compared to using the Dirichlet.

Application of the perfectly matched layer in 2.5D marine controlled-source electromagnetic modeling

For the traditional framework of EM modeling algorithms, the Dirichlet boundary is usually used which assumes the field values are zero at the boundaries. This crude condition requires that the boundaries should be sufficiently far away from the area of interest. Although cell sizes could become larger toward the boundaries as electromagnetic wave is propagated diffusively, a large modeling area may still be necessary to mitigate the boundary artifacts. In this paper, the complex frequency-shifted perfectly matched layer (CFS-PML) in stretching Cartesian coordinates is successfully applied to 2.5D frequency-domain marine controlled-source electromagnetic (CSEM) field modeling. By using this PML boundary, one can restrict the modeling area of interest to the target region. Only a few absorbing layers surrounding the computational area can effectively depress the artificial boundary effect without losing the numerical accuracy. A 2.5D marine CSEM modeling scheme with the CFS-PML is developed by using the staggered finite-difference discretization. This modeling algorithm using the CFS-PML is of high accuracy, and shows advantages in computational time and memory saving than that using the Dirichlet boundary. For 3D problem, this computation time and memory saving should be more significant.

A tetrahedral mesh generation approach for 3D marine controlled-source electromagnetic modeling

3D finite-element (FE) mesh generation is a major hurdle for marine controlled-source electromagnetic (CSEM) modeling. In this paper, we present a FE discretization operator (FEDO) that automatically converts a 3D finite-difference (FD) model into reliable and efficient tetrahedral FE meshes for CSEM modeling. FEDO sets up wireframes of a background seabed model that precisely honors the seafloor topography. The wireframes are then partitioned into multiple regions. Outer regions of the wireframes are discretized with coarse tetrahedral elements whose maximum size is as large as a skin depth of the regions. We demonstrate that such coarse meshes can produce accurate FE solutions because numerical dispersion errors of tetrahedral meshes do not accumulate but oscillates. In contrast, central regions of the wireframes are discretized with fine tetrahedral elements to describe complex geology in detail. The conductivity distribution is mapped from FD to FE meshes in a volume-averaged sense. To avoid excessive mesh refinement around receivers, we introduce an effective receiver size. Major advantages of FEDO are summarized as follow. First, FEDO automatically generates reliable and economic tetrahedral FE meshes without adaptive meshing or interactive CAD workflows. Second, FEDO produces FE meshes that precisely honor the boundaries of the seafloor topography. Third, FEDO derives multiple sets of FE meshes from a given FD model. Each FE mesh is optimized for a different set of sources and receivers and is fed to a subgroup of processors on a parallel computer. This divide and conquer approach improves the parallel scalability of the FE solution. Both accuracy and effectiveness of FEDO are demonstrated with various CSEM examples.

Enhancing the detectability of a high-resistivity target by using a synthetic aperture source for 3D marine CSEM modelling of a rugged seafloor

When processing marine controlled-source electromagnetic (CSEM) data from a rugged seafloor, enhancing the reservoir response and suppressing the topographic effect and other interference are significant issues, especially in shallow water. We simulated the CSEM responses specific to these issues using an efficient finite-difference (FD) code. The synthetic aperture technique was applied to steer the EM field toward a high-resistivity target on the seafloor. A weighted 2D synthetic aperture source was constructed by imposing a real weighting factor on each source point. Numerical experiments showed that using the weighted 2D synthetic aperture source significantly enhanced the effective CSEM signals. Because of the destructive interference between bathymetric distortion and the airwave effect in shallow water, the synthetic aperture technique is useful for dealing with seafloor topography. Better results can be obtained before steering the distorted response with a bathymetric correction. However, the detectability results may exhibit a huge difference in numerical values if the background resistivity of the bathymetric model is estimated incorrectly.

Rigorous interpolation near tilted interfaces in 3-D finite-difference EM modelling

S U M M A R Y We present a rigorous method for interpolation of electric and magnetic fields close to an interface with a conductivity contrast. The method takes into account not only a well-known discontinuity in the normal electric field, but also discontinuity in all the normal derivatives of electric and magnetic tangential fields. The proposed method is applied to marine 3-D controlled-source electromagnetic modelling (CSEM) where sources and receivers are located close to the seafloor separating conductive seawater and resistive formation. For the finite-difference scheme based on the Yee grid, the new interpolation is demonstrated to be much more accurate than alternative methods (interpolation using nodes on one side of the interface or interpolation using nodes on both sides, but ignoring the derivative jumps). The rigorous interpolation can handle arbitrary orientation of interface with respect to the grid, which is demonstrated on a marine CSEM example with a dipping seafloor. The interpolation coefficients are computed by minimizing a misfit between values at the nearest nodes and linear expansions of the continuous field components in the coordinate system aligned with the interface. The proposed interpolation operators can handle either uniform or non-uniform grids and can be applied to interpolation for both sources and receivers.

Fast multimodel finite-difference controlled-source electromagnetic simulations based on a Schur complement approach

We have developed an efficient numerical scheme for fast multimodel 3D electromagnetic simulations by applying a Schur complement approach to a frequency-domain finite-difference method. The scheme is based on direct solvers and developed with constrained inversion algorithms in view. Such algorithms normally need many forward modeling jobs with different resistivities for the target zone and/or background formation. We geometrically divide the computational domain into two subdomains: an anomalous subdomain, the resistivities of which were permitted to change, and a background subdomain, having fixed resistivities. The system matrix is partially factorized by precomputing a Schur complement to eliminate unknowns associated with the background subdomain. The Schur complement system is then solved to compute fields inside the anomalous subdomain. Finally, the background subdomain fields are computed using inexpensive local substitutions. For each successive simulation, only the relatively small Schur complement system has to be solved, which results in significant computational savings. We applied this approach to two moderately sized 3D problems in marine controlled-source electromagnetic modeling: (1) a deepwater model in which the resistivities of the seawater and the air layer were kept fixed and (2) a model in which focused inversion was performed in a scenario in which the resistivities of the background formation, the air layer, and the seawater were known. We found a significant reduction of the modeling time in inversion that depended on the relative sizes of the constrained and unconstrained volumes: the smaller the unconstrained volume, the larger the savings. Specifically, for a focused inversion of the Troll oil field in the North Sea, the gain amounted up to 80% of the total modeling time.

Three-dimensional electromagnetic responses of disk-shaped hydrocarbon reservoir in marine sediments

This paper presents a three-dimensional (3D) marine controlled-source electromagnetic modeling algorithm using primary fields for a homogeneous half-space to accurately account for airwave effects. This algorithm is validated with analytic solutions for 1D two- and four-layer models and numerical results from another 3D model. Using this code, we investigate the 3D electromagnetic responses of a 100 m thick, 5 km disk-shaped hydrocarbon reservoir buried at a depth of 1 km below the seafloor. From the numerical results, we can recognize that a 3D effect of the reservoir typically produces a transition zone in comparison with 1D model responses. The transition zone decreases with the airwave effect as the depth of water becomes shallow. As the source frequency increases, the sensitivity to the reservoir increases, whereas the amplitude decreases and falls at higher than 1 Hz below the current system noise floor. Broadside electric fields for a 10-km diameter disk model are only about 5% of in-line electric fields for the 5-km disk model. The Tequivalence is observed at such a low frequency of 1 Hz for the thin resistive tabular target, whose response varies almost linearly with the target thickness and resistivity even in the transition zone.

Applicability of 1D and 2.5D marine controlled source electromagnetic modelling

ABSTRACTWe present two‐and‐a‐half dimensional (2.5D) and three‐dimensional (3D) integral equation modelling of the marine controlled source electromagnetic method. We implement 2.5D modelling using a point source and a two‐dimensional reservoir and compare the results with point source responses from one‐dimensional and three‐dimensional reservoir models. These methods are based on an electric field domain integral equation formulation. We show how the 2.5D method performs in terms of both accuracy and computing speed with different configurations. We compare the results from 1D, 2.5D and 3D modelling, for a symmetrically placed reservoir and the in‐line acquisition configuration, as a function of different reservoir sizes in the cross‐line direction, thickness and for different frequencies and depths. Depending on the model’s parameters 2.5D modelling can be considered as an accurate and fast method for marine controlled source electromagnetic acquisition optimization and interpretation. If the thickness of the reservoir is less than one fifth of the skin depth of the embedding and if the depth of the reservoir is two or more times the skin depth of the embedding, the largest amplitude difference between two‐dimensional and three‐dimensional reservoirs is less than 10% when the source is above the centre of the reservoir. In this paper we discuss supporting examples with different configurations, where the 2.5D results lead to an optimistic detection estimate. Phase differences between 2.5D and 3D modelling are even smaller and the 2.5D solution can be used to assess the ability to detect the reservoir with a given acquisition configuration.

A 2.5D finite-element-modeling difference method for marine CSEM modeling in stratified anisotropic media

We have developed a 2.5D finite-element modeling (FEM) method for marine controlled-source electromagnetic (CSEM) applications in stratified anisotropic media. The main feature of the method is that delta sources are used to solve the governing partial differential equations for cases with and without a resistive target and to obtain the difference of these two solutions as the scattered field from the target. The total field is then the sum of the analytical background field calculated with a 1D modeling method and the difference or scattered field mentioned above. Compared with a conventional direct solution (using delta sources directly in a 2.5D formulation), the new method has smaller near-field error as a result of the source singularity and smaller boundary reflections. The new method does not require a dense mesh in the source region, which thereby reduces the total number of variables to be solved. In this way, the modeling time can be kept within a few minutes for some cases. We show that the maximum relative error of the calculation can be kept within 2% for targets at depths of approximately [Formula: see text]. The method is valid for stratified anisotropic media. The anisotropic modeling examples show that (1) marine CSEM is predominantly sensitive to target vertical resistivity and not to target horizontal resistivity, provided that the targets are thin, horizontal, high-resistivity layers and (2) marine CSEM is sensitive to the horizontal resistivity of the conductive sediments surrounding the target (e.g., the overburden).

2D marine controlled-source electromagnetic modeling: Part 2 — The effect of bathymetry

Marine controlled-source electromagnetic (CSEM) data are strongly affected by bathymetry because of the conductivity contrast between seawater and the crust below the seafloor. We simulate the marine CSEM response to 2D bathymetry using our new finite element (FE) code, and our numerical modeling shows that all electric and magnetic components are influenced by bathymery, but to different extents. Bathymetry effects depend upon transmission frequency, seabed conductivity, seawater depth, transmitter-receiver geometry, and roughness of the seafloor topography. Bathymetry effects clearly have to be take into account to avoid the misinterpretation of marine CSEM data sets.

2D marine controlled-source electromagnetic modeling: Part 1 — An adaptive finite-element algorithm

In Part 1 of this work, we develop an adaptive finite-element algorithm for forward modeling of the frequency-domain, marine controlled-source electromagnetic (CSEM) response of a 2D conductivity structure that is excited by a horizontal electric dipole source. After transforming the governing equations for the secondary electromagnetic fields into the wavenumber domain, the coupled system of two partial differential equations for the strike-parallel electric and magnetic fields is approximated using the finite-element method. The model domain is discretized using an unstructured triangular element grid that readily accommodates arbitrarily complex structures. A numerical solution of the system of linear equations is obtained using the quasi-minimal residual (QMR) method, which requiresmuch less storagethan full matrix inversion methods. We implement an automatedadaptive grid refinement algorithm in which the finite-element solution is computed iteratively on successively refined grids. Grid refinement is guided by an a posteriori error estimator based on a recently developed gradient recovery operator. The error estimator uses the solution to a dual problem in order to bias refinement toward elements that affect the solution at the electromagnetic (EM) receiver locations and enables the computation of asymptotically exact solutions to the 2.5D partial differential equations. We validate the finite element formulation against the canonical 1D reservoir model and study the performance of the adaptive refinement algorithm. An example model study of a complex offshore structure of interest for petroleum exploration illustrates the utility of the adaptive finite-element method for CSEM modeling.