# Electromagnetic Nondestructive Testing by Perturbation Homotopy Method

Authors: Liang Ding; Jun Cao
Journal: Mathematical Problems in Engineering
 (2014)

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## Abstract

Now electromagnetic nondestructive testing methods have been applied to many fields of engineering. But traditional electromagnetic methods (usually based on least square and local iteration) just roughly give the information of location, scale, and quality. In this paper we consider inverse electromagnetic problem which is concerned with the estimation of electric conductivity of Maxwell's equations (2D and 3D). A perturbation homotopy method combined with damping Gauss-Newton methods is applied to the inverse electromagnetic problem. This method differs from traditional homotopy method. The structure of homotopy function is similar to Tikhonov functional. Sets of solutions are produced by perturbation for every homotopy parameterλ=λi, i=0,…,L. At each iterative step of the algorithm, we add stochastic perturbation to numerical solutions. The previous solution and perturbation solution are regarded as the initial value in the next iteration. Although the number of solution in set increased, it increased the likelihood of obtaining correct solution. Results exhibits clear advantages over damping Gauss-Newton method and testify that it is an available method, especially on aspects of wide convergence and precision.


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## Body

## 1. Introduction

In this paper, we discuss an inverse problem of electromagnetic field. For a given source, the direct problem is to determine the electromagnetic field for the known coefficient, which has been well studied [1]. Our work is devoted to the numerical solution of the electrical conductivity inversion. This problem is a model problem for many real industrial applications including mathematical physics, atmospheric science, quantum mechanics, telemetry, nondestructive testing, and medical imaging. The first pioneering solution of the fully 3D Maxwell’s equation inverse problem was presented by Eaton more than 15 years ago [2]. Yet in spite of this, until recently, the trial-and-error forward modelling was almost the only available tool to interpret the fully 3D EM dataset. Today the situation has slightly improved, and the methods of unconstrained nonlinear optimization [3] are gaining popularity to address the problem. Although many new theoretical approaches have been applied to this domain [4, 5] successfully and the high level of computer hardware is changing with each passing day, the desired numerical solution of the inverse electromagnetic problems is still difficult to resolve for the following reasons. (1) The rapid, accurate, and stable forward numerical solution [6–9] is generally required by inverse problem; it may accelerate convergence of the inverse problem (especially, for models with low conductivity contrasts) [10, 11], but the general stable and accurate forward solution is still an open problem. (2)  The problem has the property of ill-posedness and nonlinearity. It means that the solution is not unique; any slight noise will lead to huge error. (3) In order to get a unique solution which is dependent on the measured data continuously and stably, it is necessary to investigate a stable regularization theory. Therefore, it is meaningful to design a highly efficient, numerically stable, and globally convergent algorithm to overcome the above-mentioned difficulties.Homotopy method is a global convergence algorithm for nonlinear operator equation. It has been applied to inverse problem in recent decades. Han et al. first applied homotopy method to inverse petroleum well-logging problem in 1991 [12]. Afterwards, they extended this method to general parameter identification. Fu et al. gave the numerical solution of inverse acoustic problem and elastic wave equation with fluid saturated porous media by homotopy method, respectively [13, 14]. Vasco applied homotopy method to geophysics inverse problem in 1994 [15]. He makes use of singularity and bifurcation theory to research inverse problem. Dunlavy applied homotopy optimization method to protein structure prediction in 2005 [16]. In summary, the homotopy method has shown great advantage for solving inverse problem, especially in aspects of enlarging the convergence domain and enhancing stability with noise. But homotopy inversion method is still far away from perfection and bifurcation, and convolution about homotopy curves will appear in the process of computation.As mentioned above, we will introduce a perturbation homotopy method for inverse electromagnetic problem. The model is Maxwell’s equations. In each iterative step of the algorithm, we add stochastic perturbation to numerical solutions. The previous solution and perturbation solution are regarded as the initial value in the next iteration. Although the number of solution in set increased, it increased the likelihood of obtaining correct solution. With the number of elements in set increasing, we require the input valueNmax⁡ (maximum number of solutions in a set) to limit the size of set.The paper is organized as follows. We first briefly review some basic ideas about inverse electromagnetic problem and homotopy. In Section2, we develop perturbation homotopy method and derive the basic coefficient identification algorithm using this method. In Section 3, numerical experiments are presented which demonstrate the performance of the algorithm and indicate the validity of this method. And finally, in Section 4, some conclusions and future directions are given.


## 2. Electromagnetic Forward Model

The time domain Maxwell equations has the form(1)∇×E+μ∂H∂t=0,∇×H-σE-ε∂E∂t=sr(t),
over the domain Ω×[0,T], where E and H are the electric and magnetic fields, σ, ε, and μ are conductivity, dielectric constant, and permeability, respectively, and sr(t) is transmitting source. Discretizing (1) over time grid [tn,tn+1], we get the equations
(2)∇×En+1+τnμHn+1=τnμHninΩ,∇×Hn+1-(σ+τnε)En+1=srn+1-τnεEninΩ,
where τn=1/(tn+1-tn). With discretization we get matrix equation
(3)A(σ)u=q.In the inverse problem we want to recover the modelσwith the measured data. Measured data can be written as
(4)dobs=Qu+η,
where Q is a projection operator which projects the electromagnetic fields onto the observation locations and η is the measurement noise. From (3) and (4), we can obtain
(5)dobs=QA-1q+η.


## 3. Inversion Framework

### 3.1. Damping Gauss-Newton Method

For every homotopy parameterλ=λi, i=1,…,L, we must solve the unconstrained optimization problem
(6)min⁡σH(σ,λ)=min⁡σλ∥F(σ)-dobs∥2+(1-λ)∥W(σ-σref)∥2.
This can be transformed to the Euler-Lagrange system
(7)∂H∂σ=g(σ)=(1-λ)WTW(σ-σref)+λJ(σ)T(F(σ)-dobs).
Denoting β=(1-λ)/λ, we obtain
(8)βWTW(σ-σref)+J(σ)T(F(σ)-dobs)=0,
where β played a regularization parameter role. In order to avoid the calculation of the second Fréchet derivative of F to σ, F is linearized by
(9)F(σ+δσ)=F(σ)+J(σ)δσ+R(σ,δσ).
Then the minimization problem is solved iteratively. At the kth iteration, we solve
(10)(J(σkT)J(σk)+βWTW)δσ=J(σkT)(dobs-F(σk)+J(σk))-βWTW(σk-σref)
to find the perturbation δσ. At each iteration, we have an option of solving directly for a perturbation or solving for an updated model. To formulate the latter option, we write σk+1=σk+δσ and substitute into (10) to obtain
(11)(J(σkT)J(σk)+βWTW)σk+1=J(σkT)(dobs-F(σk)+J(σk)σk)-βWTWσref.


### 3.2. A Perturbation Homotopy Method

As above mentioned, we can formulate the inverse electromagnetic problem as the following nonlinear operator equation:(12)F(σ)=dobs,
where F=QA-1q is the forward nonlinear operator. It is very difficult to estimate σ in the problem (12) due to the presence of local minima. Therefore we design a globally convergent algorithm which can overcome the local minima, namely, perturbation homotopy method, to solve the nonlinear equation (12). For inverse electromagnetic problem, we want to solve the minimization problem. Given ϕ:X⊆Rn→R, find σ*∈X⊆Rn such that
(13)ϕ(σ*)=min⁡σ∈X∥F(σ)-dobs∥2.
In order to apply homotopy to PDE optimization problem, we denote
(14)ϕ0=∥W(σ-σref)∥2,ϕ1=∥F(σ)-dobs∥2,
where ∥·∥ is 2-norm, W is positive weight function, and σref is the known reference model. So continuous homotopy functional H:Rn+1→R can be written as
(15)H(σ,λ)=λϕ1+(1-λ)ϕ0,
where λ∈[0,1] are homotopy parameters, and the homotopy step length Δi, i=0,…,L-1, is satisfied with ∑i=0L-1Δi=1. A general sketch of the homotopy algorithm is as follows.(i)
Input: given minimum norm solution ofϕ0, namely, σ0=σref, λ0=0, and Δi>0, i=1,…,L.(ii)
Update: forλi=λi-1+Δi-1, we use damping Gauss-Newton iteration method, compute an approximate solution σi to minimize min⁡σ∈XH(σ,λi) by initial value σi-1.(iii)
Output: the iteration will be stopped if the solutionσi satisfied stopping criterion; output: σ1=σm.The homotopy algorithm’s greatest advantage lies in increasing region of convergency. But in the process of computation, the nonconvergent phenomenon often happens to homotopy curve, for example, standstill, convolution, and bifurcation. These difficulties can make the homotopy method quite complicated. In order to make the homotopy more efficient, sets of solutions are produced by perturbation for every homotopy parameter λ=λi, i=0,…,L. Although the calculation increased, it ensures the likelihood of obtaining accurate solution. With different homotopy parameter λi, the minima is in the sequence, and σi is solved by damping Gauss-Newton method with the initial value σi-1. The next set of local minima (λ=λi+1) is solved with the initial values in the previous set (λ=λi) which has one or more perturbations of each of those points in the set. We denote the perturbation of solution per(σ). The perturbation stochastically disturbs one or more of the solutions σ in set. We use a single initial value σ0 for λ=λ0; the number of solution solved at each homotopy parameter λ increases exponentially. Without limit on the size of the set, the number of solutions in the set would be 2i when homotopy iteration stops. In order to constraint on computational complexity, it is necessary to limit the size of the set. We denote by  Nmax⁡ the maximum number of solutions in a set. If the number of local minima in the set at the current iteration is less than the maximum set size Nmax⁡, then every solution in the set is used in the next iteration; otherwise we will choose the preferable solution. An ideal measure of what constitutes the preferable solutions is regularization functional value of (15); solution with the lower regularization functional values is considered the preferable. The perturbation homotopy algorithm is as follows.(i)
Input: given minimum norm solution ofϕ0, namely, σ0=σref, λ0=0, Nmax⁡≥1, and Δi>0, i=1,…,L.(ii)
Update: forλi=λi-1+Δi-1, we use damping Gauss-Newton iteration method, compute an approximate solution σik, k=1,…,2i, to minimize min⁡σ∈XH(σ,λi) by initial value σi-1j and (per(σi-1))j, j=1,…,2i.(iii)
Order the distinct solutions amongσik, k=1,…,2i, from best to worst as σi1, σi2,… and discard any worst solution when the number of elements in X larger than Nmax⁡.(iv)
Output: the iteration will be stopped if the solutionσim, 1≤m≤2i, is the best solution in the set and satisfied stopping criterion; output: σ1=σm.The sketch map of the algorithm is shown in Figure1.Figure 1
The sketch map of perturbation homotopy method.


## 3.1. Damping Gauss-Newton Method

For every homotopy parameterλ=λi, i=1,…,L, we must solve the unconstrained optimization problem
(6)min⁡σH(σ,λ)=min⁡σλ∥F(σ)-dobs∥2+(1-λ)∥W(σ-σref)∥2.
This can be transformed to the Euler-Lagrange system
(7)∂H∂σ=g(σ)=(1-λ)WTW(σ-σref)+λJ(σ)T(F(σ)-dobs).
Denoting β=(1-λ)/λ, we obtain
(8)βWTW(σ-σref)+J(σ)T(F(σ)-dobs)=0,
where β played a regularization parameter role. In order to avoid the calculation of the second Fréchet derivative of F to σ, F is linearized by
(9)F(σ+δσ)=F(σ)+J(σ)δσ+R(σ,δσ).
Then the minimization problem is solved iteratively. At the kth iteration, we solve
(10)(J(σkT)J(σk)+βWTW)δσ=J(σkT)(dobs-F(σk)+J(σk))-βWTW(σk-σref)
to find the perturbation δσ. At each iteration, we have an option of solving directly for a perturbation or solving for an updated model. To formulate the latter option, we write σk+1=σk+δσ and substitute into (10) to obtain
(11)(J(σkT)J(σk)+βWTW)σk+1=J(σkT)(dobs-F(σk)+J(σk)σk)-βWTWσref.


## 3.2. A Perturbation Homotopy Method

As above mentioned, we can formulate the inverse electromagnetic problem as the following nonlinear operator equation:(12)F(σ)=dobs,
where F=QA-1q is the forward nonlinear operator. It is very difficult to estimate σ in the problem (12) due to the presence of local minima. Therefore we design a globally convergent algorithm which can overcome the local minima, namely, perturbation homotopy method, to solve the nonlinear equation (12). For inverse electromagnetic problem, we want to solve the minimization problem. Given ϕ:X⊆Rn→R, find σ*∈X⊆Rn such that
(13)ϕ(σ*)=min⁡σ∈X∥F(σ)-dobs∥2.
In order to apply homotopy to PDE optimization problem, we denote
(14)ϕ0=∥W(σ-σref)∥2,ϕ1=∥F(σ)-dobs∥2,
where ∥·∥ is 2-norm, W is positive weight function, and σref is the known reference model. So continuous homotopy functional H:Rn+1→R can be written as
(15)H(σ,λ)=λϕ1+(1-λ)ϕ0,
where λ∈[0,1] are homotopy parameters, and the homotopy step length Δi, i=0,…,L-1, is satisfied with ∑i=0L-1Δi=1. A general sketch of the homotopy algorithm is as follows.(i)
Input: given minimum norm solution ofϕ0, namely, σ0=σref, λ0=0, and Δi>0, i=1,…,L.(ii)
Update: forλi=λi-1+Δi-1, we use damping Gauss-Newton iteration method, compute an approximate solution σi to minimize min⁡σ∈XH(σ,λi) by initial value σi-1.(iii)
Output: the iteration will be stopped if the solutionσi satisfied stopping criterion; output: σ1=σm.The homotopy algorithm’s greatest advantage lies in increasing region of convergency. But in the process of computation, the nonconvergent phenomenon often happens to homotopy curve, for example, standstill, convolution, and bifurcation. These difficulties can make the homotopy method quite complicated. In order to make the homotopy more efficient, sets of solutions are produced by perturbation for every homotopy parameter λ=λi, i=0,…,L. Although the calculation increased, it ensures the likelihood of obtaining accurate solution. With different homotopy parameter λi, the minima is in the sequence, and σi is solved by damping Gauss-Newton method with the initial value σi-1. The next set of local minima (λ=λi+1) is solved with the initial values in the previous set (λ=λi) which has one or more perturbations of each of those points in the set. We denote the perturbation of solution per(σ). The perturbation stochastically disturbs one or more of the solutions σ in set. We use a single initial value σ0 for λ=λ0; the number of solution solved at each homotopy parameter λ increases exponentially. Without limit on the size of the set, the number of solutions in the set would be 2i when homotopy iteration stops. In order to constraint on computational complexity, it is necessary to limit the size of the set. We denote by  Nmax⁡ the maximum number of solutions in a set. If the number of local minima in the set at the current iteration is less than the maximum set size Nmax⁡, then every solution in the set is used in the next iteration; otherwise we will choose the preferable solution. An ideal measure of what constitutes the preferable solutions is regularization functional value of (15); solution with the lower regularization functional values is considered the preferable. The perturbation homotopy algorithm is as follows.(i)
Input: given minimum norm solution ofϕ0, namely, σ0=σref, λ0=0, Nmax⁡≥1, and Δi>0, i=1,…,L.(ii)
Update: forλi=λi-1+Δi-1, we use damping Gauss-Newton iteration method, compute an approximate solution σik, k=1,…,2i, to minimize min⁡σ∈XH(σ,λi) by initial value σi-1j and (per(σi-1))j, j=1,…,2i.(iii)
Order the distinct solutions amongσik, k=1,…,2i, from best to worst as σi1, σi2,… and discard any worst solution when the number of elements in X larger than Nmax⁡.(iv)
Output: the iteration will be stopped if the solutionσim, 1≤m≤2i, is the best solution in the set and satisfied stopping criterion; output: σ1=σm.The sketch map of the algorithm is shown in Figure1.Figure 1
The sketch map of perturbation homotopy method.


## 4. Numerical Simulation

In this section, we exhibit clear advantages of perturbation homotopy method over traditional damping Gauss-Newton method in the previous three experiments. In the fourth experiment we analysis the influence of homotopy parameter on inversion results. In the fifth experiment we apply our algorithm to more complicated model and confirm that this algorithm is stable. In the last experiment we solve a simple practice data. In the first three 3D experiments we select homotopy parameterλi=i/10, i=0,…,10, Nmax⁡=2, and internal Gauss-Newton iteration 5 with each homotopy step in our numerical experiments. The space domain is Ω. The inverse problem is considered on a uniform 32×32×16grid. The transmitter source is s(t)=t2e-αtsin(ω0t), where α=ω0/3, ω0=2πf0, and f0=100MHz is the central frequency. In the last three 2D experiments Nmax⁡=4 and internal Gauss-Newton iteration is 10. The inversion space Ω=2.5m×2.5m, space step Δx=Δy=0.05m, sampling time t=36ns, time step τ=18ns, and the transmitter source is Ricker wavelet with 900 MHz central frequency.


### 4.1. Homogeneous Model

We first test a simple homogeneous model, where permittivityε=56064×10-12F/m, permeability μ=0, and relative conductivity σ=-5.2983. Both damping Gauss-Newton method and perturbation homotopy method select the same relative initial value σ=-4.8993. Iterations of damping Gauss-Newton method are five times, elapsed time is 1230.426 seconds, homotopy algorithm’s outer iterations λi, i=1,…,10, are ten times, inner local iterations (damping Gauss-Newton method) are three times, and elapsed time is 1405.536 seconds. Figure 2 gives results of true model, damping Gauss-Newton, and perturbation homotopy method. The relative error ∥σinv-σtrue∥/∥σtrue∥ of (a) and (b) in Figure 2 is 0.0516 and of (a) and (c) is 0.0830, where σinv and σtrue are inversion solution and true model, respectively. From Figure 2 we can see that when initial value is close to original conductivity, both methods have satisfactory result, but perturbation homotopy is not more efficient.(a) Original model; (b) inversion results of damping Gauss-Newton; (c) inversion results of perturbation homotopy method.
(a)
(b)
(c)


### 4.2. One Circle Cavity Model

The relative conductivity of cavity is −3.2775; other parameters have the same value as in the above model. Both methods select the same initial valueσ=-1.6094. Elapsed time of damping Gauss-Newton method and perturbation homotopy method is 11120.536 seconds and 12578.436 seconds, respectively. The relative error ∥σinv-σtrue∥/∥σtrue∥ of (a) and (b) in Figure 3 is 25.35% and of (a) and (c) is 9.61%. Although elapsed time of perturbation homotopy is ten times more than damping Gauss-Newton method, when initial value is far away from true value, the latter is inefficient compared to the former.(a) True model; (b) results of damping Gauss-Newton; (c) results of perturbation homotopy method; (d) cross-section of (a) atx=0; (e) cross-section of (b) at x=0; (f) cross-section of (c) at x=0.
(a)
(b)
(c)
(d)
(e)
(f)


### 4.3. Two Cavities Model

The conductivity of two cavities is equal; other parameters have the same value as in Section4.2. Both methods select the same relative initial value σ=-0.8775. Elapsed time of damping Gauss-Newton method and perturbation homotopy method is 11080.546 seconds and 15458.768 seconds, respectively. Figure 4 gives results of original model, damping Gauss-Newton, and perturbation homotopy method. The relative error ∥σinv-σtrue∥/∥σtrue∥ of (a) and (b) in Figure 4 is 49.35% and of (a) and (c) is 12.85%. When the initial value is further away from original value, damping Gauss-Newton method is invalid.(a) True model; (b) results of damping Gauss-Newton method; (c) results of perturbation homotopy method; (d) cross-section of (a) atZ=8; (e) cross-section of (b) at Z=8; (f) cross-section of (c) at Z=8.
(a)
(b)
(c)
(d)
(e)
(f)


### 4.4. One Square Cavity 2D Model

The vertex coordinate of square cavity is 1.2 m, 0.4 m; 1.3 m, 0.4 m; 1.2 m, 0.5 m; 1.3 m, 0.5 m, respectively. The original model structure scheme is shown in Figure5. Inversion results by Gauss-Newton method are shown in Figure 7. Measured data is shown in Figure 6. With the influence of local minima Figure 7 roughly describes the information of location and scale of square cavity, but the quality information error is huge. Because traditional Tikhonov regularization would smooth the inversion solution, the solution always appears blurring phenomenon at boundary. Inversion results by perturbation homotopy method with different homotopy parameter are shown in Figure 8; the relative error between inversion and original model is 4.55%. We select homotopy parameter as 1/50001, 1/5001, 1/2001, 1/1501, 1/1001, 1/501, 1/51, 1/6, 2/3, 100/105, 1000/1005, and 100000/100005. From Figure 9 we can conclude that the rate of convergence is fast, but when homotopy parameter is close to 1, numerical solution is diverging, so the optimal solution is not at λ=1.Figure 5
Square cavity model.Figure 6
Measured data.Figure 7
Inversion results of Gauss-Newton method.Figure 8
Inversion results of perturbation homotopy method.Figure 9
Inversion results with different homotopy parameters (1/50001, 1/5001, 1/2001, 1/1501, 1/1001, 1/501, 1/51, 1/6, 2/3, 100/105, 1000/1005, and 100000/100005).


### 4.5. Three Square Cavities 2D Model

The original model structure scheme is shown in Figure10. Measured data is shown in Figure 11. In order to demonstrate that the method is stable with noise, we add different Gauss white noise to measured data. The SNR (signal-to-noise ratio) is 1 dB, 3 dB, 5 dB, 8 dB, 10 dB, 12 dB, 15 dB, 20 dB, and 30 dB, respectively. From Figure 12 we can see that, with noise decrease, numerical solution is convergent to original model. So our method is stable.Figure 10
Three square cavities model.Figure 11
Measured data.Figure 12
Inversion results with different SNR measured data (1 dB, 3 dB, 5 dB, 8 dB, 10 dB, 12 dB, 15 dB, 20 dB, and 30 dB).


### 4.6. Practice Data Inversion

The practice measured data is provided by Wuhan Changjiang Engineering Geophysical Exploration & Testing Co., Ltd.China. The testing object is a concrete member with three circle rebar in it. The centre of circle is 1.4 m, 0.6 m; 2.2 m, 0.4 m; 3.4 m, 0.7 m, respectively, radius is 0.1 m. The original model structure scheme is shown in Figure13. Practice measured data is shown in Figure 14. Inversion result is shown in Figure 15. The relative error is 15.75%. Before the inversion, it is necessary to preprocess the data because practice measured data has large noise.Figure 13
Original model.Figure 14
Practice measured data.Figure 15
Inversion results.


## 4.1. Homogeneous Model

We first test a simple homogeneous model, where permittivityε=56064×10-12F/m, permeability μ=0, and relative conductivity σ=-5.2983. Both damping Gauss-Newton method and perturbation homotopy method select the same relative initial value σ=-4.8993. Iterations of damping Gauss-Newton method are five times, elapsed time is 1230.426 seconds, homotopy algorithm’s outer iterations λi, i=1,…,10, are ten times, inner local iterations (damping Gauss-Newton method) are three times, and elapsed time is 1405.536 seconds. Figure 2 gives results of true model, damping Gauss-Newton, and perturbation homotopy method. The relative error ∥σinv-σtrue∥/∥σtrue∥ of (a) and (b) in Figure 2 is 0.0516 and of (a) and (c) is 0.0830, where σinv and σtrue are inversion solution and true model, respectively. From Figure 2 we can see that when initial value is close to original conductivity, both methods have satisfactory result, but perturbation homotopy is not more efficient.(a) Original model; (b) inversion results of damping Gauss-Newton; (c) inversion results of perturbation homotopy method.
(a)
(b)
(c)


## 4.2. One Circle Cavity Model

The relative conductivity of cavity is −3.2775; other parameters have the same value as in the above model. Both methods select the same initial valueσ=-1.6094. Elapsed time of damping Gauss-Newton method and perturbation homotopy method is 11120.536 seconds and 12578.436 seconds, respectively. The relative error ∥σinv-σtrue∥/∥σtrue∥ of (a) and (b) in Figure 3 is 25.35% and of (a) and (c) is 9.61%. Although elapsed time of perturbation homotopy is ten times more than damping Gauss-Newton method, when initial value is far away from true value, the latter is inefficient compared to the former.(a) True model; (b) results of damping Gauss-Newton; (c) results of perturbation homotopy method; (d) cross-section of (a) atx=0; (e) cross-section of (b) at x=0; (f) cross-section of (c) at x=0.
(a)
(b)
(c)
(d)
(e)
(f)


## 4.3. Two Cavities Model

The conductivity of two cavities is equal; other parameters have the same value as in Section4.2. Both methods select the same relative initial value σ=-0.8775. Elapsed time of damping Gauss-Newton method and perturbation homotopy method is 11080.546 seconds and 15458.768 seconds, respectively. Figure 4 gives results of original model, damping Gauss-Newton, and perturbation homotopy method. The relative error ∥σinv-σtrue∥/∥σtrue∥ of (a) and (b) in Figure 4 is 49.35% and of (a) and (c) is 12.85%. When the initial value is further away from original value, damping Gauss-Newton method is invalid.(a) True model; (b) results of damping Gauss-Newton method; (c) results of perturbation homotopy method; (d) cross-section of (a) atZ=8; (e) cross-section of (b) at Z=8; (f) cross-section of (c) at Z=8.
(a)
(b)
(c)
(d)
(e)
(f)


## 4.4. One Square Cavity 2D Model

The vertex coordinate of square cavity is 1.2 m, 0.4 m; 1.3 m, 0.4 m; 1.2 m, 0.5 m; 1.3 m, 0.5 m, respectively. The original model structure scheme is shown in Figure5. Inversion results by Gauss-Newton method are shown in Figure 7. Measured data is shown in Figure 6. With the influence of local minima Figure 7 roughly describes the information of location and scale of square cavity, but the quality information error is huge. Because traditional Tikhonov regularization would smooth the inversion solution, the solution always appears blurring phenomenon at boundary. Inversion results by perturbation homotopy method with different homotopy parameter are shown in Figure 8; the relative error between inversion and original model is 4.55%. We select homotopy parameter as 1/50001, 1/5001, 1/2001, 1/1501, 1/1001, 1/501, 1/51, 1/6, 2/3, 100/105, 1000/1005, and 100000/100005. From Figure 9 we can conclude that the rate of convergence is fast, but when homotopy parameter is close to 1, numerical solution is diverging, so the optimal solution is not at λ=1.Figure 5
Square cavity model.Figure 6
Measured data.Figure 7
Inversion results of Gauss-Newton method.Figure 8
Inversion results of perturbation homotopy method.Figure 9
Inversion results with different homotopy parameters (1/50001, 1/5001, 1/2001, 1/1501, 1/1001, 1/501, 1/51, 1/6, 2/3, 100/105, 1000/1005, and 100000/100005).


## 4.5. Three Square Cavities 2D Model

The original model structure scheme is shown in Figure10. Measured data is shown in Figure 11. In order to demonstrate that the method is stable with noise, we add different Gauss white noise to measured data. The SNR (signal-to-noise ratio) is 1 dB, 3 dB, 5 dB, 8 dB, 10 dB, 12 dB, 15 dB, 20 dB, and 30 dB, respectively. From Figure 12 we can see that, with noise decrease, numerical solution is convergent to original model. So our method is stable.Figure 10
Three square cavities model.Figure 11
Measured data.Figure 12
Inversion results with different SNR measured data (1 dB, 3 dB, 5 dB, 8 dB, 10 dB, 12 dB, 15 dB, 20 dB, and 30 dB).


## 4.6. Practice Data Inversion

The practice measured data is provided by Wuhan Changjiang Engineering Geophysical Exploration & Testing Co., Ltd.China. The testing object is a concrete member with three circle rebar in it. The centre of circle is 1.4 m, 0.6 m; 2.2 m, 0.4 m; 3.4 m, 0.7 m, respectively, radius is 0.1 m. The original model structure scheme is shown in Figure13. Practice measured data is shown in Figure 14. Inversion result is shown in Figure 15. The relative error is 15.75%. Before the inversion, it is necessary to preprocess the data because practice measured data has large noise.Figure 13
Original model.Figure 14
Practice measured data.Figure 15
Inversion results.


## 5. Conclusion

This paper has addressed the perturbation homotopy method for solving inverse electromagnetic problem. Perturbation homotopy coupled with damping Gauss-Newton methods has been used in the experiments. Results show that when initial value is far away from true model, local convergent Gauss-Newton method just roughly gives the information of location and scale; the quality has huge error with true model. Perturbation homotopy method is a widely convergent algorithm; it could give a more accurate inversion results even if initial value is far away from the original value. When adding Gauss white noise to measured data, we can see that, with noise decrease, numerical solution is convergence to original model, so numerical results support that our method is stable. Traditional homotopy theory tells us that optimum solution is determined atλ=1, but numerical results show that the solution will diverge when λ is close to 1 because of regularization parameter impact. In a word, this method has clear advantages over damping Gauss-Newton method; it is an effective method, especially on aspects of wide convergence, computational efficiency, and precision and has broad application prospects.



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