International Journal of Antennas and Propagation

Volume 2018, Article ID 1846427, 14 pages

https://doi.org/10.1155/2018/1846427

## Reduction of Electromagnetic Reflections in 3D Airborne Transient Electromagnetic Modeling: Application of the CFS-PML in Source-Free Media

^{1}College of Instrumentation and Electrical Engineering, Jilin University, Changchun, China^{2}Key Laboratory of Earth Information Detection Instrumentation, Ministry of Education, Jilin University, Changchun, China

Correspondence should be addressed to Shanshan Guan; nc.ude.ulj@nahsnahsnaug

Received 9 April 2018; Revised 23 July 2018; Accepted 8 August 2018; Published 12 September 2018

Academic Editor: Luciano Tarricone

Copyright © 2018 Yanju Ji et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

To solve the problem of electromagnetic reflections caused by the termination of finite-difference time-domain (FDTD) grids, we apply the complex frequency-shifted perfectly matched layer (CFS-PML) to airborne transient electromagnetic (ATEM) modeling in a source-free medium. To implement the CFS-PML, two important aspects are improved. First, our method adopts the source-free Maxwell’s equations as the governing equations and introduces the divergence condition, consequently, the discrete form of Maxwell’s third equation is derived with regard to the CFS-PML form. Second, because our method adopts an inhomogeneous time-step, a recursive formula composed of convolution items based on a nonuniform time-step is proposed. The proposed approach is verified via a calculation of the electromagnetic response using homogeneous half-space models with different conductivities. The results show that the CFS-PML can reduce a 60 dB relative errors in late times. Moreover, this approach is also applied to 3D anomalous models; the results indicate that the proposed method can reduce reflections and substantially improve the identification of anomalous bodies. Consequently, the CFS-PML has good implications for ATEM modeling in a source-free medium.

#### 1. Introduction

The airborne transient electromagnetic (ATEM) system, which is an economic alternative for acquiring electromagnetic data with highly efficient detection capabilities and a substantial depth of investigation, has been widely applied to problems associated with hydrogeological surveying, mineral exploration, and environmental monitoring [1–3]. Furthermore, high-accuracy ATEM modeling can provide a theoretical basis for subsequent data inversion and prospective instrument design. Numerous studies have been conducted using the three-dimensional (3D) TEM modeling approach developed by Oristaglio and Hohmann based on finite-difference time-domain (FDTD) grids [4]. More recently, Commer and Newman improved the accelerated simulation scheme for 3D TEM modeling using geometric multigrid concepts [5–7]. Subsequently, Guan calculated the 3D ATEM response based on the graphics processing unit (GPU) [8], and Sun et al. improved upon the 3D FDTD modeling of TEM in consideration of the ramp time [9]. However, boundary reflections still constitute one of the most important challenges for the accuracy of TEM modeling. The most widely used Dirichlet boundary condition (DBC) exhibits better effects only at earlier times before the diffusion field has arrived at the boundary [10]. With an increase in the computing time, the modeling accuracy is progressively affected by electromagnetic reflections at the boundaries. Therefore, it is necessary to develop a more effective boundary condition. Berenger proposed a perfectly matched layer (PML), which can absorb electromagnetic waves with any incident angle and frequency [11]. Subsequently, several approaches, such as the uniaxial perfectly matched layer (U-PML), multiaxial perfectly matched layer (M-PML), and complex frequency-shifted perfectly matched layer (CFS-PML), have been developed with many absorbing boundary conditions based on PML theory [12–14]. In these improvements, the CFS-PML absorbing boundary condition has shown the best performance with regard to the absorption of low-frequency induction fields and late reflections [15, 16]. Roden and Gedney more efficiently implemented the CFS-PML condition by utilizing it with the FDTD method based on recursive convolution; this condition, known as the convolutional perfectly matched layer (C-PML), is highly absorptive of evanescent modes, and it is independent of the host medium [17]. Drossaert and Giannopoulos effectively reduced the computation time and memory by using the complex frequency-shifted stretching function while applying the CFS-PML with the velocity-stress wave equations. Based on the FDTD technique [18], Giannopoulos proposed an electromagnetic modeling method using the CFS-PML with the complex frequency-shifted stretching function and analyzed the effects of the absorption coefficient in different media [19]. Gedney and Zhao applied the CFS-PML boundary condition to the simulation of two-dimensional (2D) electromagnetic waves based on an auxiliary differential equation (ADE), thereby improving the absorbing properties of the CFS-PML in their research [20]. Moreover, Li and Huang used the CFS-PML to calculate the TEM response in a source medium while discretizing the air [21]. Feng et al. reduced the memory requirements in an implementation of the CFS-PML based on the memory-minimized method (Tri-M) and adopted the discrete Zernike transform (DZT) to guarantee the optimal accuracy [22]. Yu et al. improved the C-PML parameters to absorb low-frequency electromagnetic waves in both the ground and the air and observed good absorption [23]. Furthermore, Hu et al. applied the CFS-PML within a fictitious wave domain and discussed the selection of various CFS parameters [24]. Zhao et al. improved the discretization of Maxwell’s divergence equation in the CFS-PML boundary based on a uniform time-step [25]. Feng et al. proposed a Crank–Nicolson cycle-sweep-uniform FDTD method based on CFS-PML and applied it to 3D low-frequency subsurface electromagnetic sensing problems [26].

In this paper, we focus on the problem of electromagnetic reflections at boundaries in source-free media and apply the CFS-PML boundary condition to ATEM modeling based on a nonuniform time-step. Maxwell’s curl equation with a source was used as the governing equation in previous studies about CFS-PML, as the source functions are difficult to approximate and the analytical solution in the source medium is unavailable. In contrast to previous investigations, our research adopts the source-free Maxwell’s equations as the governing equations and the analytical solutions as the initial conditions to perform the iterative calculations; therefore, we can obtain modeling results with a high accuracy. Because Maxwell’s divergence equation must be incorporated into the governing equations to ensure the uniqueness of the solution [10], one of the fundamental aspects of our approach is the discretization of Maxwell’s divergence equation in the stretched coordinate space. Another crucial component of the proposed method is the derivation of a recursive convolution formula based on nonuniform time-steps in the discretized stretched coordinate formulation. Finally, the proposed method is verified via a calculation of the electromagnetic response using homogeneous half-space models and anomalous models with 3D bodies, thereby effectively demonstrating the performance of the proposed method with regard to the absorption of electromagnetic reflections during the modeling of ATEM.

#### 2. Methods

Maxwell’s divergence equation is employed as the governing equation in FDTD-based ATEM modeling in a source-free medium to advance the magnetic field of *H _{z}*, while the other fields are advanced using Maxwell’s curl equations. Following the method proposed by Wang and Hohmann [10], the initial conditions are calculated based on a homogeneous half-space. The staggered Yee grid is used to discretize the earth model inhomogeneously. In addition, the time-steps are advanced using a modified Du Fort-Frankel method [4, 27]. The ATEM model contains a ground-air boundary and a computational domain boundary; therefore, we adopt an upward continuation at the ground-air boundary and the CFS-PML as the computational domain boundary condition. However, two problems are encountered during the application of the CFS-PML: we must derive the discrete form of Maxwell’s divergence equation in the stretched coordinate, and the recursive convolution formulation based on a nonuniform time-step must be derived. Consequently, these key issues in the application of the CFS-PML boundary condition are discussed in this paper.

##### 2.1. Governing Equations in the Stretched Coordinate Space

In the modeling of ATEM in a source-free medium, Maxwell’s divergence equation must be included within the governing equations during the modeling of ATEM in a source-free medium to ensure the uniqueness and stability of the solution. Therefore, Maxwell’s divergence equation (equation (3)) is adopted to advance the magnetic field in the stretched coordinate space [28–32], while the other electromagnetic fields are advanced using Maxwell’s curl equations ((1) and (2)). where is the electric field, is the magnetic field, is the complex frequency variable, is the permeability, is the permittivity, and is the electrical conductivity.

In the stretched coordinate space, the Hamilton operator () can be expressed as follows [33]: where , and are positive and less than 1 and is the conductivity of the CFS-PML layers ().

The difference scheme of Maxwell’s curl equations in the stretched coordinate space was developed by Roden and Gedney similar to the application of the CFS-PML [17]. Thus, in this paper, we mainly deduce the discrete form of Maxwell’s divergence equation (equation (3)).

Inserting (4) into (3) leads to

Equation (5) is next transformed into the time domain to obtain the renewal equation of *H _{z}*, after which the expression of

*S*(

_{i}*i = x, y, z*) is inserted into (5), which can consequently be expressed as where is the auxiliary expression of a convolution item and is implemented as

The governing equations of other fields in the stretched coordinate can be similarly derived.

##### 2.2. The Discrete Form of Maxwell’s Divergence Equation in the Stretched Coordinate Space

During ATEM modeling, the time-step of an FDTD method usually increases gradually according to the attenuation characteristics of TEM responses (i.e., the TEM field exhibits relatively sharp variations at earlier times and gradually becomes smooth thereafter). Here, the time-step is advanced using a modified Du Fort-Frankel method. The traditional recursive convolution formula is deduced based on a uniform time-step; therefore, we need to modify the derivation procedure based on inhomogeneous time-steps which is shown in the appendix. In the stretched coordinate space, at can be expressed as

To obtain the discrete form of the convolution formulation, we assume that (), and thus, equation (8) can be rewritten as where , , ().

Then can be performed recursively:

We can obtain and in the same manner:

As shown in (6) and (11), contains *H*_{z} at the current time in the discretization of Maxwell’s divergence equation to obtain the iterative formula of at the time . Therefore, the iterative formula of requires further derivation. Equation (6) can be discretized in both space and time according to the staggered Yee scheme as

Inserting (10)–(12) into (13), we obtain where , .

Note that at is related to the value of of the previous time-step, and thus, we must refresh at the end of the calculation for the next iteration.

#### 3. Numerical Tests

To investigate the feasibility of the proposed method, we first compare the solutions of our method with the numerical solutions calculated by the integral method in homogeneous half-space models [8, 34, 35]. Then, the proposed method is applied to several models with 3D conductors. With the exception of the slanted model, every model has 101 × 101 × 50 grids [36]. The grids are nonuniform with a minimum spacing of 10 m and a maximum spacing of 120 m. The spacing of the CFS-PML boundary is 120 m, and the boundary is not contained within any of the abovementioned grids. A transmitting coil is located at the center of the earth model with a height of 120 m. The magnetic moment is 4*π* × 10^{−7}A·m^{2}, and the transmitter current is 0.7 × 10^{7}A. The receiving coil is situated at a height of 60 m, and it is 130 m away from the transmitting coil in the x-direction.

##### 3.1. Validation with Homogeneous Half-Space Models

The electromagnetic responses of the homogeneous half-space models are calculated, and the FDTD solutions are compared with the numerical solutions to verify the accuracy of our method. To demonstrate the absorbing effect of the proposed method in different host media, the conductivities are set as 0.1 S/m, 0.01 S/m, and 0.005 S/m; the diagrammatic drawing of the homogeneous half-space model is shown in Figure 1.