Research ArticleReduction of Electromagnetic Reflections in 3D AirborneTransient Electromagnetic Modeling: Application of the CFS-PML in Source-Free Media

Yanju Ji ,1,2 Xuejiao Zhao ,1 Jiayue Gu,1 Dongsheng Li ,1 and Shanshan Guan 1,2

1College of Instrumentation and Electrical Engineering, Jilin University, Changchun, China2Key Laboratory of Earth Information Detection Instrumentation, Ministry of Education, Jilin University, Changchun, China

Correspondence should be addressed to Shanshan Guan; guanshanshan@jlu.edu.cn

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, whichpermits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

To solve the problem of electromagnetic reflections caused by the termination of finite-difference time-domain (FDTD) grids, weapply the complex frequency-shifted perfectly matched layer (CFS-PML) to airborne transient electromagnetic (ATEM) modelingin 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 ofMaxwell’s third equation is derived with regard to the CFS-PML form. Second, because our method adopts an inhomogeneoustime-step, a recursive formula composed of convolution items based on a nonuniform time-step is proposed. The proposedapproach is verified via a calculation of the electromagnetic response using homogeneous half-space models with differentconductivities. The results show that the CFS-PML can reduce a 60 dB relative errors in late times. Moreover, this approach isalso applied to 3D anomalous models; the results indicate that the proposed method can reduce reflections and substantiallyimprove the identification of anomalous bodies. Consequently, the CFS-PML has good implications for ATEM modeling in asource-free medium.

1. Introduction

The airborne transient electromagnetic (ATEM) system,which is an economic alternative for acquiring electromag-netic data with highly efficient detection capabilities and asubstantial depth of investigation, has been widely appliedto problems associated with hydrogeological surveying,mineral exploration, and environmental monitoring [1–3]. Furthermore, high-accuracy ATEM modeling can pro-vide a theoretical basis for subsequent data inversion andprospective instrument design. Numerous studies havebeen conducted using the three-dimensional (3D) TEMmodeling approach developed by Oristaglio and Hohmannbased on finite-difference time-domain (FDTD) grids [4].More recently, Commer and Newman improved the accel-erated simulation scheme for 3D TEM modeling usinggeometric multigrid concepts [5–7]. Subsequently, Guancalculated the 3D ATEM response based on the graphics

processing unit (GPU) [8], and Sun et al. improved uponthe 3D FDTD modeling of TEM in consideration of theramp time [9]. However, boundary reflections still consti-tute one of the most important challenges for the accuracyof TEM modeling. The most widely used Dirichlet bound-ary condition (DBC) exhibits better effects only at earliertimes before the diffusion field has arrived at the boundary[10]. With an increase in the computing time, the model-ing accuracy is progressively affected by electromagneticreflections at the boundaries. Therefore, it is necessary todevelop a more effective boundary condition. Berengerproposed a perfectly matched layer (PML), which canabsorb electromagnetic waves with any incident angleand frequency [11]. Subsequently, several approaches, suchas the uniaxial perfectly matched layer (U-PML), multiax-ial perfectly matched layer (M-PML), and complexfrequency-shifted perfectly matched layer (CFS-PML),have been developed with many absorbing boundary

HindawiInternational Journal of Antennas and PropagationVolume 2018, Article ID 1846427, 14 pageshttps://doi.org/10.1155/2018/1846427

conditions based on PML theory [12–14]. In theseimprovements, the CFS-PML absorbing boundary condi-tion has shown the best performance with regard to theabsorption of low-frequency induction fields and latereflections [15, 16]. Roden and Gedney more efficientlyimplemented the CFS-PML condition by utilizing it withthe FDTD method based on recursive convolution; thiscondition, known as the convolutional perfectly matchedlayer (C-PML), is highly absorptive of evanescent modes,and it is independent of the host medium [17]. Drossaertand Giannopoulos effectively reduced the computationtime and memory by using the complex frequency-shifted stretching function while applying the CFS-PMLwith the velocity-stress wave equations. Based on theFDTD technique [18], Giannopoulos proposed an electro-magnetic modeling method using the CFS-PML with thecomplex frequency-shifted stretching function and ana-lyzed the effects of the absorption coefficient in differentmedia [19]. Gedney and Zhao applied the CFS-PMLboundary condition to the simulation of two-dimensional(2D) electromagnetic waves based on an auxiliary differen-tial equation (ADE), thereby improving the absorbingproperties of the CFS-PML in their research [20]. More-over, Li and Huang used the CFS-PML to calculate theTEM response in a source medium while discretizing theair [21]. Feng et al. reduced the memory requirements inan implementation of the CFS-PML based on thememory-minimized method (Tri-M) and adopted the dis-crete Zernike transform (DZT) to guarantee the optimalaccuracy [22]. Yu et al. improved the C-PML parametersto absorb low-frequency electromagnetic waves in boththe ground and the air and observed good absorption[23]. Furthermore, Hu et al. applied the CFS-PML withina fictitious wave domain and discussed the selection ofvarious CFS parameters [24]. Zhao et al. improved thediscretization of Maxwell’s divergence equation in theCFS-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 appliedit to 3D low-frequency subsurface electromagnetic sensingproblems [26].

In this paper, we focus on the problem of electromag-netic reflections at boundaries in source-free media andapply the CFS-PML boundary condition to ATEM model-ing based on a nonuniform time-step. Maxwell’s curlequation with a source was used as the governing equationin previous studies about CFS-PML, as the source func-tions are difficult to approximate and the analytical solu-tion in the source medium is unavailable. In contrast toprevious investigations, our research adopts the source-free Maxwell’s equations as the governing equations andthe analytical solutions as the initial conditions to performthe iterative calculations; therefore, we can obtain model-ing results with a high accuracy. Because Maxwell’s diver-gence equation must be incorporated into the governingequations to ensure the uniqueness of the solution [10],one of the fundamental aspects of our approach is thediscretization of Maxwell’s divergence equation in thestretched coordinate space. Another crucial component of

the proposed method is the derivation of a recursiveconvolution formula based on nonuniform time-steps inthe discretized stretched coordinate formulation. Finally,the proposed method is verified via a calculation of theelectromagnetic response using homogeneous half-spacemodels and anomalous models with 3D bodies, therebyeffectively demonstrating the performance of the proposedmethod with regard to the absorption of electromagneticreflections during the modeling of ATEM.

2. Methods

Maxwell’s divergence equation is employed as the govern-ing equation in FDTD-based ATEM modeling in asource-free medium to advance the magnetic field of Hz,while the other fields are advanced using Maxwell’s curlequations. Following the method proposed by Wangand Hohmann [10], the initial conditions are calculatedbased on a homogeneous half-space. The staggered Yeegrid is used to discretize the earth model inhomogen-eously. In addition, the time-steps are advanced using amodified Du Fort-Frankel method [4, 27]. The ATEMmodel contains a ground-air boundary and a computa-tional domain boundary; therefore, we adopt an upwardcontinuation at the ground-air boundary and the CFS-PML as the computational domain boundary condition.However, two problems are encountered during the appli-cation of the CFS-PML: we must derive the discrete formof Maxwell’s divergence equation in the stretched coordi-nate, and the recursive convolution formulation based ona nonuniform time-step must be derived. Consequently,these key issues in the application of the CFS-PMLboundary 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 withinthe governing equations during the modeling of ATEMin a source-free medium to ensure the uniqueness andstability of the solution. Therefore, Maxwell’s divergenceequation (equation (3)) is adopted to advance the mag-netic field Bz in the stretched coordinate space [28–32],while the other electromagnetic fields are advanced usingMaxwell’s curl equations ((1) and (2)).

∇s ×H = εjωE + σE, 1

∇s × E = −μjωH, 2

∇s ·H = 0, 3

where E is the electric field, H is the magnetic field, jω isthe complex frequency variable, μ is the permeability, ε isthe permittivity, and σ is the electrical conductivity.

In the stretched coordinate space, the Hamilton operator(∇s) can be expressed as follows [33]:

∇s = x1sx

∂∂x

+ y1sy

∂∂y

+ z1sz

∂∂z

, 4

2 International Journal of Antennas and Propagation

where si = ki + σpi/ αi + jωμ , ki and αi are positive and lessthan 1 and σpi is the conductivity of the CFS-PML layers(i = x, y, z).

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

Inserting (4) into (3) leads to

1sx

∂Hx

∂x+ 1sy

∂Hy

∂y+ 1sz

∂Hz

∂z= 0 5

Equation (5) is next transformed into the time domain toobtain the renewal equation ofHz, after which the expressionof Si (i = x, y, z) is inserted into (5), which can consequentlybe expressed as

1kx

∂Hx

∂x+ 1ky

∂Hy

∂y+ 1kz

∂Hz

∂z+ ψhzx + ψhzy + ψhzz = 0, 6

where ψhzi is the auxiliary expression of a convolution itemand is implemented as

ψhzi = −σpi

εk2iexp −

σpiεki

+ αiε

t ∗∂H i

∂i7

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

2.2. The Discrete Form of Maxwell’s Divergence Equation inthe Stretched Coordinate Space. During ATEM modeling,the time-step of an FDTD method usually increases grad-ually according to the attenuation characteristics of TEMresponses (i.e., the TEM field exhibits relatively sharp var-iations at earlier times and gradually becomes smooththereafter). Here, the time-step is advanced using a modi-fied Du Fort-Frankel method. The traditional recursiveconvolution formula is deduced based on a uniformtime-step; therefore, we need to modify the derivation pro-cedure based on inhomogeneous time-steps which isshown in the appendix. In the stretched coordinate space,ψhzx at tn+1/2 can be expressed as

ψhzx n + 12

= −σpx

εk2xexp −

σpx

εkx+ αx

εtn+1/2 ∗

∂Hn+1/2x

∂x

= ξhx tn+1/2 ∗∂Hn+1/2

x

∂x8

To obtain the discrete form of the convolution formula-tion, we assume that Δtn = tn+1/2 − tn−1/2 (n = 0, 1, 2,… , n),and thus, equation (8) can be rewritten as

ψhzx n + 12 = ξhx tn+1/2 ∗

∂Hn+1/2x

∂x

=tn+1/2

0ξhx τ

∂Hx tn+1/2 − τ

∂xdτ,

9

where ξhx t = −Axe−Bxt , Ai = σpi/εk2i , Bi = σpi/εki + αi/ε

(i = x, y, z).Then ψhzx n + 1/2 can be performed recursively:

ψhzx n + 12 = 1

2Ax

Bxe−BΔtn −

Ax

Bx

∂Hn+1/2x

∂x+ ∂Hn−1/2

x

∂x

+ e−BxΔtnψhzx n −12

10

We can obtain ψhzz n + 1/2 and ψhzx n + 1/2 in thesame manner:

ψhzz n + 12 = 1

2Az

Bze−BzΔtn −

Az

Bz

∂Hn+1/2z

∂z+ ∂Hn−1/2

z

∂z

+ e−BzΔtnψhzz n −12 ,

11

ψhzx n + 12 = 1

2Ax

Bxe−BxΔtn −

Ax

Bx

∂Hn+1/2x

∂x+ ∂Hn−1/2

x

∂x

+ e−BxΔtnψhzx n −12

12

As shown in (6) and (11), ψhzz contains Hz at the currenttime in the discretization of Maxwell’s divergence equation toobtain the iterative formula of Hz at the time tn+1/2. There-fore, the iterative formula of Hz requires further derivation.Equation (6) can be discretized in both space and timeaccording to the staggered Yee scheme as

1kx

Hn+1/2x i + 1, j + 1/2, k + 1/2 −Hn+1/2

x i, j + 1/2, k + 1/2Δx

+ 1ky

Hn+1/2y i + 1/2, j + 1, k + 1/2 −Hn+1/2

y i + 1/2, j, k + 1/2Δy

+ 1kz

Hn−1+1/2z i + 1/2, j + 1/2, k + 1 −Hn−1+1/2

z i + 1/2, j + 1/2, kΔz

+ ψn+1/2hzx + ψn+1/2

hzy + ψn+1/2hzz = 0

13

3International Journal of Antennas and Propagation

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

Hn+1/2z i + 1

2 , j +12 , k

=Hn+1/2z i + 1

2 , j +12 , k + 1 + CzDz

Δz i + 1/2, j + 1/2, k

Hn−1+1/2z i + 1

2 , j +12 , k + 1

−Hn−1+1/2z i + 1

2 , j +12 , k

+ Dz

kyΔy i + 1/2, j, k + 1/2

Hn+1/2y i + 1

2 , j + 1, k + 12

−Hn+1/2y i + 1

2 , j, k +12

+ Dz

kxΔx i, j + 1/2, k + 1/2

Hn+1/2x i + 1, j + 1

2 , k +12

−Hn+1/2x i, j + 1

2 , k +12

+Dzψn+1/2hzx +Dzψ

n+1/2hzy + e−BzΔtnDzψ

n−1/2hzz ,

14

where Cz = 1/2 Az/Bz e−BzΔtn − Az/Bz , Dz = kz/ 1 + kzCz Δz i + 1/2 , j + 1/2 , k .

Note that Hz at tn+1/2 is related to the value of ψhzz of theprevious time-step, and thus, we must refresh ψhzz at the endof the calculation for the next iteration.

3. Numerical Tests

To investigate the feasibility of the proposed method, we firstcompare the solutions of our method with the numericalsolutions calculated by the integral method in homogeneous

half-space models [8, 34, 35]. Then, the proposed method isapplied to several models with 3D conductors. With theexception of the slanted model, every model has101× 101× 50 grids [36]. The grids are nonuniform witha minimum spacing of 10m and a maximum spacing of120m. The spacing of the CFS-PML boundary is 120m,and the boundary is not contained within any of theabovementioned grids. A transmitting coil is located atthe center of the earth model with a height of 120m.The magnetic moment is 4π × 10−7A·m2, and the trans-mitter current is 0.7× 107A. The receiving coil is situatedat a height of 60m, and it is 130m away from the trans-mitting coil in the x-direction.

3.1. Validation with Homogeneous Half-Space Models. Theelectromagnetic responses of the homogeneous half-spacemodels are calculated, and the FDTD solutions are comparedwith the numerical solutions to verify the accuracy of ourmethod. To demonstrate the absorbing effect of the proposedmethod in different host media, the conductivities are set as0.1 S/m, 0.01 S/m, and 0.005 S/m; the diagrammatic drawingof the homogeneous half-space model is shown in Figure 1.

The FDTD-based solutions (calculated using the DBCand the CFS-PML with 8 layers) are compared with thenumerical solutions, as shown in Figure 2. In the models withdifferent conductivities, we can discern that these solutionsare in good agreement at earlier times because the diffusionfields have not reached the boundaries yet. With an increasein the time; however, the effectiveness of the CFS-PML grad-ually becomes clearer, especially in the models with a smallerconductivity (as presented in Figure 2 b~c). The average rel-ative errors of the FDTD solutions using the CFS-PML andDBC are shown in Figure 2(d). In the model with a large con-ductivity (0.1 S/m) and a slower propagation speed, theseboundary conditions both demonstrate a better absorptionof electromagnetic waves with average relative errors of1.61% and 0.79% (corresponding to the DBC and CFS-PML, respectively). When the model conductivity is 0.01 S/m, the average relative errors of the DBC and CFS-PML are21.47% and 2.63%, respectively, effectively revealing the bet-ter ability of the CFS-PML to absorb electromagnetic waves.In addition, when the ground conductivity is 0.005 S/m, thestrong electromagnetic wave reflected from the DBC leadsto an enormous deviation in the numerical solution, whereasthe CFS-PML still exhibits good absorption; in this case, theaverage relative errors of the DBC and CFS-PML are 42.9%and 4.17%, respectively.

Except for the decay curves at the center of the receivingcoil, we also investigate the electromagnetic reflections at theboundaries. Therefore, time slices of the electromagneticresponses from a homogeneous half-space model with a con-ductivity of 0.01 S/m are presented in Figure 3. As illustrated,the electromagnetic responses calculated by the DBC andCFS-PML boundaries are compared at three separatemoments. At earlier times, the electromagnetic responses ofthe DBC and CFS-PML boundaries display the same varia-tions; meanwhile, at later times, serious distortions areobserved at both 4.5ms and 9ms when the DBC is used. Incontrast, no distortions are evident when the CFS-PML is

Earth

Air

CFS-PML

120

m

60m

Receivingcoil

x (m)

Transmittingcoil

130 m

�휎 = 0.1 S/m, 0.01 S/m, 0.005 S/m

z (m

)

Figure 1: Diagrammatic drawing of homogeneous half-spacemodels.

4 International Journal of Antennas and Propagation

adopted regardless of the point in time, indicating that theCFS-PML boundary condition demonstrates a better abilityto absorb electromagnetic waves.

To study the reflection errors, the time-dependent field atthe center of the receiving coil is recorded when the conduc-tivity of the host medium is 0.01 S/m. The error relative to thenumerical solution (in dB) is computed using:

ErrordB = 20 log10Vz t −Vzref t

Vzref max, 15

where Vz t represents the voltage received by the receivingcoil, Vzref(t) represents the numerical solutions, and Vzrefmaxrepresents the maximum of Vzref(t). The relative errors (indB) computed via (13) are recorded in Figure 4, from whichwe can see that the relative errors are similar at earlier times.Moreover, compared with the DBC, the CFS-PML canreduce the relative error by nearly 60 dB at later times.

3.2. Validation with Anomalous Models. To test the effective-ness of the proposed method, we design three anomalousmodels. In each model, the ground conductivity is 0.01 S/m,and the thickness of the CFS-PML is 10 cells. The first modelis shown in Figure 5; two anomalous bodies with differentsizes, depths, and conductivities are selected. An analyticalsolution exists only in the homogeneous half-space. There-fore, to study the reflection errors due to the boundaries,the mesh is extended by an additional 40 cells in each direc-tion to obtain the reference solution, and the CFS-PML isused to minimize the reflections. To ensure that the electro-magnetic waves cannot reach the boundaries during theobservation time (10ms), the diffusion depth is set as

d = 2 tσμ0

, 16

�휎 = 0.1 S/m

Numerical solutionCFS-PMLDBC

Time (s)10−4

10−4

10−5

10−6

10−3 10−2

�휕B/�휕

t (V

/A)

(a)

�휎 = 0.1 S/m

Numerical solutionCFS-PMLDBC

Time (s)10−4 10−2

10−4

10−6

10−8

�휕B/�휕

t (V

/A)

(b)

Numerical solutionCFS-PMLDBC

�휎 = 0.005 S/m

Time (s)10−6

10−5

10−10

10−4 10−2

�휕B/�휕

t (V

/A)

(c)

0.79 2.634.17

1.61

21.47

42.9

0

10

20

30

40

50

0.1 0.01 0.005

Ave

rage

rela

tive e

rror

(%)

Conductivity (S/m)

CFSDBC-PML

0.79 2.634.17

1.61

21.47

(d)

Figure 2: Comparison of the induced voltage at the center of the receiving coil at 10ms when the conductivity is (a) 0.1 S/m, (b) 0.01 S/m, and(c) 0.005 S/m; (d) shows a comparison of the average relative errors obtained at these conductivities.

5International Journal of Antennas and Propagation

where σ is the conductivity, t is the observation time, and d isthe diffusion depth [37].

Time slices of the anomaly responses are shown inFigure 6. Both at earlier times (3ms) and at later times(10ms); reflected electromagnetic waves exist in the modelwhen the DBC is adopted. Moreover, especially at 10ms,the locations of anomalous bodies cannot be identified accu-rately. In contrast, the reflections are absorbed well in themodel with the CFS-PML, and the results are in agreementwith the reference solution. A comparison of ∂B/∂t at thecenter of the lattice along the x-direction between 3ms and10ms is shown in Figure 7. In earlier times, the DBC solutionis in good agreement with the reference solution at the center

of the model; however, such good agreement is not observednear the boundary. Furthermore, at 10ms, the DBC solutionseverely deviates from the reference solution along the wholeboundary. Meanwhile, the responses in the model using theCFS-PML are consistent with the reference solution regard-less of the time. The relative errors are shown inFigures 7(c) and 6(d). The maximum relative error is lessthan 3% at earlier times and no more than 4.6% at later timeswhen the CFS-PML is adopted. However, the resultsobtained with the DBC are consistent with the reference solu-tion only at the center of the model area at earlier times.

For further verification, an anomalous model is designedand presented in Figure 8. In this model, the anomalous bodyadjoins the boundary. The dimensions of the anomalousbody are 480m× 120m× 180m, and the depth of the bodyis 190m. Time slices of the anomaly responses are shownin Figure 9. Intense reflections of electromagnetic wavesappear when the DBC is utilized, and the information ofthe anomalous body cannot be retrieved accurately. In con-trast, through the implementation of the CFS-PML boundarycondition, the reflections are effectively suppressed, and theresults are approximately equivalent to the reference solu-tion. Consequently, the information of the anomalous bodyis presented clearly. The responses at the boundary at thecenter of the model along the x-direction are shown inFigure 10. The DBC solution evidently coincides with the ref-erence solution at the center of the boundary, although theDBC solution is different near the boundary at earlier timesand performs progressively worse with increasing time.Meanwhile, the responses obtained using the CFS-PML arestill in good agreement. The maximum relative error of the

2000

2000

1000

1000x (m)

y (m

)

00

DBC t = 1 ms ×10−7

15

10

5

0

(a)

y (m

)

20001000x (m)

00

2000

1000

DBC t = 4.5 ms ×10−7

3

4

2

1

0

−1

(b)

y (m

)

20001000x (m)

00

2000

1000

DBC t = 10 ms

×10−9

0

1

−4

−3

−2

−1

(c)

y (m

)

20001000x (m)

00

2000

1000

CFS-PML t = 1 ms ×10−7

15

10

5

0

(d)

y (m

)

20001000x (m)

00

2000

1000

CFS-PML t = 4.5 ms ×10−7

3

4

2

1

0

(e)

y (m

)

20001000x (m)

00

2000

1000

CFS-PML t = 10 ms ×10−9

8

6

4

2

(f)

Figure 3: Time slices of the electromagnetic responses in the air using the DBC at (a) 1ms, (b) 4.5ms, and (c) 10ms in addition to those usingthe CFS-PML at (d) 1ms, (e) 4.5ms, and (f) 10ms. The ground conductivity is 0.01 S/m.

−50

−100

−150

−200

0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01Time (s)

CFS-PMLDBC

Relat

ive e

rror

(dB)

Figure 4: Relative errors at the center of the receiving coil in dBusing the CFS-PML and DBC at 10ms when the conductivity is0.01 S/m.

6 International Journal of Antennas and Propagation

results acquired using the CFS-PML is less than 5% at latertimes. Furthermore, the DBC solutions exhibit serious reflec-tions, especially near the anomalous body. From Figures 8and 9, we can conclude that the CFS-PML is very stableand can be established very close to the anomalous body.

Finally, we design an obliquemodel. The anomalous bodyis oblique with an angle of 45°, and it is established at the cen-ter of the x-y plane, as shown in Figure 11. The dimensions ofthe anomalous body are 400m× 610m× 90m, and thedepth is 350m. Time slices of the anomaly responses are

shown in Figure 12. Reflections of electromagnetic wavesevidently occur along the boundary at earlier times whenthe DBC is adopted, while the responses at the centerare basically equivalent to those in the reference solution.At 10ms, the reflections at the DBC boundaries are sointense that the diffusion is completely distorted. In con-trast, the responses with the CFS-PML boundary conditionare highly consistent with the reference solution, even atlater times. The received ∂B/∂t at the boundary acrossthe center of the receiving coil in the x-direction is shown

600 S/m

480

0

0.01 S/m

420 m × 420 m × 420 m

120 m × 60 m × 130 m

Earth

Air

90

550

2080x (m)

z (m

)

970

900

220

1960

200 S/mDepth = 550 m

Depth = 90 m

(a)

0

20201960

0.01 S/m

Depth = 550 m

x (m)

y (m

)

480 900

480

900600 S/m

420 m × 420 m × 420 m

20801960

120 m × 60 m × 130 m200 S/m

Depth = 90 m

(b)

Figure 5: Diagrammatic drawing of model 1 in (a) x-z plane and (b) x-y plane.

2500

2000

1500

1000

500

00 1000 2000

x (m)

3210−1−2−3−4

×10−9

Reference t = 3 ms

y (m

)

(a)

2500

2000

1500

1000

500

00 1000 2000

x (m)

3210−1−2−3−4

×10−9

CFS-PML t = 3 ms

y (m

)

(b)

2500

2000

1500

1000

500

00 1000 2000

x (m)

86420−2−4

×10−9DBC t = 3 ms

y (m

)

(c)

2500

2000

1500

1000

500

00 1000 2000

x (m)

2

0

−2

−4

−6×10−10

Reference t = 10 ms

y (m

)

(d)

2500

2000

1500

1000

500

00 1000 2000

x (m)

2

0

−2

−4

−6×10−10

CFS-PML t = 10 ms

y (m

)

(e)

2500

2000

1500

1000

500

00 1000 2000

x (m)

420−2−4−6−8

×10−10

DBC t = 10 ms

y (m

)

(f)

Figure 6: Comparison of the responses in model 1 in the plane of receiving coil at 3ms (a) (b) (c) and 10ms (d) (e) (f).

7International Journal of Antennas and Propagation

CFS-PMLDBCReference

10−9

�휕B/�휕

t (V

/A)

500 1000 1500x (m)

2000 2500

(a)

CFS-PMLDBCReference

�휕B/�휕

t (V

/A)

4

3

2

1

500 1000 1500x (m)

2000 2500

×10−10

(b)

CFS-PML t = 3 msCFS-PML t = 10 ms

5

4

3

2

1

0

Rela

tive e

rror

(%)

500 1000 1500x (m)

2000 2500

(c)

DBC t = 3 msDBC t = 10 ms

300

200

100

0

Rela

tive e

rror

(%)

500 1000 1500x (m)

2000 2500

(d)

Figure 7: Comparison of the induced voltage at the line cross the center of receiving coil along x-direction at (a) 3ms and (b) 10ms and therelative errors using (c) the CFS-PML and (d) DBC of model 1.

500 S/m

2760

0

0.01 S/m

480 m × 120 m × 180 m

Depth = 190 mEarth

Air

190370

2280x (m)

z (m

)

(a)

500 S/m

27602280

12701390

0.01 S/m

480 m × 120 m × 180 m

Depth = 190 m

x (m)

y (m

)

(b)

Figure 8: Diagrammatic drawing of model 2 in (a) x-z plane and (b) x-y plane.

8 International Journal of Antennas and Propagation

in Figure 13. It is clear that the DBC solution coincideswith the reference solution at the center of the modelarea at earlier times and seriously deviates from the refer-ence solution at later times. However, the CFS-PML caneffectively absorb the reflections from the boundary, evenat later times, and the maximum relative error is lessthan 5%.

4. Conclusions and Perspectives

We applied the CFS-PML boundary condition to the model-ing of ATEM data in a source-free medium and introducedthe source-free Maxwell’s equations as the governing equa-tions. Furthermore, we provided discrete forms of the gov-erning equations in a CFS-PML formulation and deducedthe recursive convolution formula with a nonuniform time-step in the stretched coordinate space. The validated resultsof a homogeneous half-space model demonstrate that theproposed method can absorb incident wave better, and itcan reduce the reflections at later times. The CFS-PML ishighly absorptive and can considerably reduce the relativeerror with a 60dB improvement compared with the tradi-tional boundary condition (i.e., the DBC) at later times.

Finally, we employed the CFS-PML in airborne anoma-lous models and compared the electromagnetic responseswith the reference solution. The results show that the CFS-PML demonstrates better absorptive capabilities and canaccurately retrieve the information of the anomalous bodyregardless of whether the anomalous body is set obliquelyor near the boundary of the model area. Moreover, the

maximum relative error of the improved method is less than5% at later times (10ms). The proposed method also displaysa better recognition precision in the presence of multiple orcomplex 3D conductive bodies, thereby verifying the effec-tiveness of this method for ATEM modeling. Consequently,the proposed method can provide higher-precision simula-tion results for deep geological exploration.

Appendix

To describe Hx accurately, the integral cannot be divided asusual into [t0, t1], [t1, t2], … , [tn, tn+1/2], because the time-steps are inhomogeneous. Here, we propose a new subdivi-sion, as shown in

ψhzx n + 12 =

Δtn

0ξhx τ

∂Hx tn+1/2 − τ

∂xdτ

+Δtn+Δtn−1

Δtn

ξhx τ∂Hx tn+1/2 − τ

∂xdτ

+Δtn+Δtn−1+Δtn−2

Δtn+Δtn−1ξhx τ

∂Hx tn+1/2 − τ

∂xdτ

+⋯ +Δtn+Δtn−1+⋯+Δt2+Δt1

Δtn+Δtn−1+⋯+Δt2ξhx τ

∂Hx tn+1/2 − τ

∂xdτ

+tn+1/2

Δtn+Δtn−1+⋯+Δt1ξhx τ

∂Hx tn+1/2 − τ

∂xd

17

2500

2000

1500

1000

500

00 1000 2000

x (m)

y (m

)Reference t = 3 ms

2520151050−5

×10−10

(a)

2500

2000

1500

1000

500

00 1000 2000

x (m)

y (m

)

CFS-PML t = 3 ms

2520151050−5

×10−10

(b)

2500

2000

1500

1000

500

00 1000 2000

x (m)

y (m

)

DBC t = 3 ms

×10−9

5

0

−5

(c)

2500

2000

1500

1000

500

00 1000 2000

x (m)

y (m

)

Reference t = 10 ms

1

0

−1

−2

−3

−4

−5×10−10

(d)

2500

2000

1500

1000

500

00 1000 2000

x (m)

y (m

)CFS-PML t = 10 ms

1

0

−1

−2

−3

−4

−5×10−10

(e)

2500

2000

1500

1000

500

00 1000 2000

x (m)

y (m

)

DBC t = 10 ms1.5

1

0.5

0

−0.5

−1

−1.5

×10−9

(f)

Figure 9: Comparison of the responses in model 2 in the plane of receiving coil at 3ms (a) (b) (c) and 10ms (d) (e) (f).

9International Journal of Antennas and Propagation

4

3

2

1

0500 1000 1500 2000 2500

x (m)

×10−9�휕

B/�휕

t (V

/A)

CFS-PMLDBCReference

(a)

6

4

2

0500 1000 1500 2000 2500

x (m)

×10−10

�휕B/�휕

t (V

/A)

CFS-PMLDBCReference

(b)

4

2

0500 1000 1500 2000 2500

x (m)

Rela

tive e

rror

(%)

CFS-PML t = 10 msCFS-PML t = 3 ms

(c)

300

200

100

0500 1000 1500 2000 2500

x (m)

Rela

tive e

rror

(%)

DBC t = 10 msDBC t = 3 ms

(d)

Figure 10: Comparison of the induced voltage at the line cross the center of receiving coil along x-direction at (a) 3ms and (b) 10ms and therelative errors using (c) the CFS-PML and (d) DBC of model 2.

500 S/m

0

0.01 S/m

400 m × 610 m × 90 m

Depth = 350 mEarth

Air

350

z (m

)

x (m)

(a)

500 S/m

3760

1130 1530

1025

1635

0.01 S/m

400 m × 610 m × 90 m

Depth = 350 m

x (m)

y (m

)

2760

(b)

Figure 11: Diagrammatic drawing of model 3 in (a) x-z plane and (b) x-y plane.

10 International Journal of Antennas and Propagation

The further calculation of (17) leads to

ψhzx n + 12 = 1

2Ax

Bxe−BΔtn −

Ax

Bx

∂Hn+1/2x

∂x+ ∂Hn−1/2

x

∂x

+ 12

Ax

Bxe−B Δtn+Δtn−1 −

Ax

Bxe−BΔtn

∂Hn−1/2x

∂x+ ∂Hn−1−1/2

x

∂x

+ + 12Ax

Bxe−B Δtn+Δtn−1+⋯+Δt1

−Ax

Bxe−B Δtn+Δtn−1+⋯+Δt2

∂H1/2x

∂x+ ∂H1+1/2

x

∂x

+ 12

Ax

Bxe−B tn+1/2 −

Ax

Bxe−B Δtn+Δtn−1+⋯+Δt1

∂H0x

∂x+ ∂H1/2

x

∂x,

18

where ξhx t = −Axe−Bxt , Ai = σpi/εk2i , Bi = σpi/εki + αi/ε

(i = x, y, z).

From (18), we can obtain ψhzx n − 1/2 as

ψhzx n −12 = 1

2Ax

Bxe−BΔtn−1 −

Ax

Bx

∂Hn−1/2x

∂x+ ∂Hn−1−1/2

x

∂x

+ 12

Ax

Bxe−B Δtn−1+Δtn−2 −

Ax

Bxe−BΔtn−1

∂Hn−1−1/2x

∂x+ ∂Hn−2−1/2

x

∂x

+⋯ + 12

Ax

Bxe−B Δtn−1+Δtn−2+⋯+Δt1

−Ax

Bxe−B Δtn−1+Δtn−2+⋯+Δt2 ∂H1/2

x

∂x+ ∂H1+1/2

x

∂x

+ 12

Ax

Bxe−B tn−1/2 −

Ax

Bxe−B Δtn−1+Δtn−2+⋯+Δt1

∂H0x

∂x+ ∂H1/2

x

∂x

19

From (18) and (19) we can get

ψhzx n + 12 = 1

2Ax

Bxe−BΔtn −

Ax

Bx

∂Hn+1/2x

∂x+ ∂Hn−1/2

x

∂x

+ e−BxΔtnψhzx n −12

20

2500

2000

1500

1500

500

0

y (m

)

0 1000 2000x (m)

Reference t = 3 ms3210−1−2−3−4

×10−8

(a)

2500

2000

1500

1500

500

0

y (m

)

0 1000 2000x (m)

CFS-PML t = 3 ms3210−1−2−3−4

×10−8

(b)

2500

2000

1500

1500

500

0

y (m

)

0 1000 2000x (m)

DBC t = 3 ms3210−1−2−3−4

×10−8

(c)

2500

2000

1500

1500

500

0

y (m

)

0 1000 2000x (m)

Reference t = 10 ms ×10−9

15

10

5

0

(d)

2500

2000

1500

1500

500

0

y (m

)

0 1000 2000x (m)

CFS-PML t = 10 ms ×10−9

15

10

5

0

(e)

2500

2000

1500

1500

500

0

y (m

)

0 1000 2000x (m)

DBC t = 10 ms ×10−9

15

10

5

0

(f)

Figure 12: Comparison of the responses in model 3 in the plane of receiving coil at 3ms (a) (b) (c) and 10ms (d) (e) (f).

11International Journal of Antennas and Propagation

Note that within the integral interval τ ∈ 0, Δtn in equa-tion (17), Hx tn+1/2 − τ has two values at Hn+1/2

x and Hn−1/2x ,

and we approximate ∂Hx tn+1/2 − τ /∂x as 1/2 ∂Hn+1/2x /∂

x + ∂Hn−1/2x /∂x .

Data Availability

The (MATLAB.mat) data used to support the findings ofthis study are available from the corresponding authorupon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

Project supported by the National Natural Science Founda-tion of China (Grant nos. 41674109, 41604084), the

Instrument Developing Project of the Chinese Academyof Sciences (Grant no. ZDYZ2012-1-03), and the Key Lab-oratory of Electromagnetic Radiation and Detection Tech-nology of the Chinese Academy of Sciences.

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