Three-dimensional modeling of Marine Controlled Source Electromagnetic field using a space-wavenumber domain method

Under complex conditions, the three-dimensional modeling of Marine Controlled Source Electromagnetic field requires a significant amount of computation, resulting in slow calculation speed and high storage requirements. To solve these problems, we propose a 3D numerical simulation method of electromagnetic field in the space-wavenumber domain under the Lorenz gauge. Firstly, the new method utilizes the two-dimensional Fourier transform in the horizontal direction to transform the 3D partial differential equations of the Lorenz vector potentials into multiple independent ordinary differential equations. Secondly, the finite element method is used to solve the ordinary differential equations. The fields in the spatial domain are then obtained by using the inverse Fourier transform, and finally, an iterative method with a compact operator is applied to approximate the true solution. The approach requires less computation and storage, and the algorithm is highly parallel, which effectively improves the efficiency of 3D forward modeling of electromagnetic fields. The correctness, effectiveness, and computational efficiency of the method are verified by the design model.


Introduction
The marine controlled source electromagnetic (MCSEM) method is a novel technology with significant commercial value in offshore oil and gas exploration.It effectively distinguishes between oil and water reservoirs by utilizing the electrical differences of materials.As technology advances, the challenge of oil and gas exploration increases.To handle large-scale complex marine electromagnetic field data, precise and efficient 3D inversion tools are necessary, and forward modeling is fundamental.Therefore, the research on high-precision forward modeling algorithms is of critical theoretical and practical significance.
Methods for 3D numerical simulations of ocean electromagnetism can be mainly divided into integral equation methods, finite difference methods, finite element methods, and finite volume methods.The development of these methods mainly focuses on the following four aspects [1]- [3]: ①Grid partition optimization; ②Unit fitting basis function; ③Equation solving technique; and ④Parallel acceleration technique.The development of these techniques has improved the accuracy or efficiency of 3D numerical simulation of marine controlled source electromagnetic field under complex conditions to some extent.However, the high-precision forward algorithms remain a research topic.
The space-wavenumber domain algorithm is a method that combines Fourier transform algorithms with spatial domain equations.This method has been successfully applied in the forward modeling of direct current [4] and magnetotellurics [5].In this paper, we apply it to 3D modeling of the MCSEM with the Lorenz gauge.The vector potentials of the secondary field under Lorenz gauge satisfy the Helmholtz equation and are transformed into some ordinary differential equations of multiple wave numbers by using the 2D Fourier transforms in the horizontal direction.The ordinary differential equations are solved by the finite element method (FEM).A stable and convergent iteration scheme is established by using the compact operator.The correctness of the algorithm is verified by experimental results.

Theory
The time-harmonic factor is i t e  ，and the Maxwell equations with electrical source in the frequency domain can be written as where E and H represent the electric field and the magnetic field, ˆi z  = represents impedance, μ represents the permeability, all values in this paper are set the free-space magnetic permeability μ 0 , ˆi y   =+ represents admittance, σ represents the conductivity, ε represents the dielectric constant, all values in this paper are set as the free-space dielectric constant ε 0 , J e the external electrical source, ρ the free charge.
Introduce vector potential A and scalar potential Φ as follows: Eq. ( 1) can be converted to: (3) Eq. ( 3) is solved based on a secondary field method, the background is set as a uniform layered medium.Substitute eq. ( 2)into eq.( 3), the governing equation satisfied by the secondary field vector potential can be obtained: represents the background admittance, σ b , the background conductivity of the layered medium, A s , the secondary vector potential, and Φ s , the secondary scalar potential, b     = − , the abnormal conductivity.
Introduce the Lorenz gauge between the secondary vector and scalar potential: JE as the secondary field scattering current source, E represents the total electric field, bb represents the wave number of the background electromagnetic field.After derivation, eq. ( 4) can be written as the Helmholtz equation: Perform a 2D Fourier transform on the eq.( 6) in the horizontal direction, set x and k y is the wave number of the Fourier transform in the x and y directions, respectively.Eq. ( 6) can be transformed as follows: where, ,, AAA are the space-wavenumber domain vector potentials in three directions.,, sss x y z JJJ are the space-wavenumber domain scattering currents in three directions.
Combined with the boundary conditions, we can solve the 1D vector potential equation using the finite element method.We utilize the second-order shape function [6] in the vertical direction, and eq.( 7) can be transferred into three five-diagonal linear equations.The chasing method is utilized to solve the matrix equation, and the vector potentials of different wave numbers are obtained.The EM fields are then calculated using the relationship between vector potentials and EM fields, as described in eq. ( 2).Finally, the fields in the spatial domain are recovered using 2D inverse Fourier transform.
The true solution of the electromagnetic field is successively approximated by an iterative method using compact operators [7]: The 3D partial differential equations satisfied by the vector potential are transformed into multiple ordinary differential equations by using a 2D Fourier transform in the horizontal direction.In this case, the computational and storage requirements are greatly reduced.The ordinary differential equation is solved by the efficient chasing method.This new algorithm has a high degree of parallelism and is suitable for efficient computation of large-scale 3D marine electromagnetic models.

Numerical example
As shown in Fig. 1, the 3D marine model is designed.The conductivity of seawater and seabed is 3.33 S/m and 1 S/m, respectively.The electric dipole source along the x direction is (-1000,1,350), the dipole moment is 100, and the thickness of seawater is 0.4 km.The model region is 1×1×0.8km 3 and discretized into 101×101×81 nodes.The volume of a conductive anomaly is 0.2×0.2×0.2 km 3 , and the top of the anomaly body is located at 0.44 km depth.The volume of a conductive anomaly is 0.8×0.4×0.4 km3, and the top of the anomaly body is located at 0.2 km depth.The conductivity of the anomaly is 0.01 S/m.The frequency is 0.1 Hz and the observation surface is on the seabed.The termination condition for this iteration is that the relative error in the sum of the three components of the electric field for all nodes in two adjacent iterations is less than 10 -4 .Fig. 2 shows the comparison of the six components of the electromagnetic field between the proposed algorithm and the IE algorithm [7].Fig. 3 shows the relative errors of the two.As we can see, the two numerical solutions are basically consistent with each other, and the relative error is less than 3%.The algorithm is correct.   1 shows the time and memory consumption of the proposed algorithm with different nodes.As can be seen from the table, for the model with tens of millions of nodes, the proposed algorithm cost 13 s and requires about 8.6 GB of memory, which is small in memory consumption and fast in calculation speed.The results demonstrate the computational efficiency of the proposed method for large-scale models.We compare the computational efficiency of the SWDIE method with the Wavelet-Galerkin method [8].With similar computing resources, the calculation time of the SWDIE method cost 1.75 s with 520251 unknows, while the Wavelet-Galerkin method of Chen et al. takes 663.85 s with 475665 unknows.The result demonstrates the high performance of the SWDIE algorithm and its competitiveness with some advanced FE-based codes.The amplitudes of the 3D electromagnetic field responses of different anomalous bodies with different conductivities of the seafloor prism, set as 100, 10, 0.1, and 0.01 S/m, are shown in Figure 4.The background conductivity is 1 S/m.It can be seen that the electromagnetic field above the anomalous body is significant, indicating the position of the anomaly.And the greater the difference between the anomalous conductivity and the background, the larger the abnormal response.It indicates that marine controlled-source electromagnetic (MCSEM) has significant implications for seafloor resource exploration.

Conclusion
In this paper, present a space-wavenumber domain EM method under Lorenz gauge for 3D forward modeling of marine controlled source electromagnetic field.In this approach, the 3D Helmholtz equation satisfied by the Lorenz vector potential is transformed into a 1D ordinary differential equation with multiple independent wavenumbers by a horizontal 2D Fourier transform.The ordinary differential equation is solved by the finite element method, and after obtaining the electromagnetic field in the space-wavenumber domain, the inverse Fourier transform is used to obtain the electromagnetic field in the spatial domain.Finally, an iterative method is used to approximate the exact solution of the electromagnetic field.The experimental results show that the new method is well suited for large-scale 3D ocean EM numerical simulations due to its high computational accuracy, small memory consumption and high computational efficiency.

Figure 1 .
Figure 1.Synthetic conductivity model with a prism anomalous body in the seabed.(a) Horizontal plane of the model along z = 0 m, (b) Profile section of the model along y = 0 m.

Figure 2 .
Figure 2. The EM fields at the frequency of 0.1Hz calculated via the proposed algorithm and the IE algorithm along the plane of z = 0.

Figure 3 .
Figure 3. Relative errors of the EM fields between the IE algorithm solution and the proposed algorithm solution Table1shows the time and memory consumption of the proposed algorithm with different nodes.As can be seen from the table, for the model with tens of millions of nodes, the proposed algorithm cost 13 s and requires about 8.6 GB of memory, which is small in memory consumption and fast in calculation speed.The results demonstrate the computational efficiency of the proposed method for large-scale models.We compare the computational efficiency of the SWDIE method with the Wavelet-Galerkin method[8].With similar computing resources, the calculation time of the SWDIE method cost 1.75 s with 520251 unknows, while the Wavelet-Galerkin method of Chen et al. takes 663.85 s with

Figure 4 .
Figure 4.The amplitudes of the 3D electromagnetic field responses of different anomalous bodies with different conductivities, including the following five cases: 100 S/m (cyan line), 10 S/m (dashed blue line), no anomaly (red line), 0.1 S/m (green line) and 0.01 S/m (manganese line).

Table 1 .
Time and memory usage of algorithms of different sizes (12th Gen Intel(R)