^{①}中国科学院电子学研究所 北京 100190;

^{②}中国科学院电磁辐射与探测技术重点实验室 北京 100190;

^{③}中国科学院大学 北京 100049;

^{④}中国科学院 北京 100864

**摘要**：传统的频率-波数域成像方法能有效地重建均匀介质中的目标图像,但对于分层介质,不能生成聚焦的图像,而且目标也无法准确定位.考虑到各层介质介电常数的差异和层间的不连续性,该文推导了适用于分层介质的相移偏移成像方法.并由分层介质中点目标的散射传递函数,分析了成像方法所做的假设和数学近似.通过仿真模拟和试验,验证了所提的方法适用于分层介质实时成像.

^{①}Institute of Electronics, Chinese Academy of Sciences, Beijing 100190, China;

^{②}Key Laboratory of Electromagnetic Radiation and Sensing Technology, Chinese Academy of Sciences, Beijing 100190, China;

^{③}University of Chinese Academy of Sciences, Beijing 100049, China;

^{④}Chinese Academy of Sciences, Beijing 100864, China

**Abstract**: The classical Frequency-Wavenumber (F-K) imaging algorithm can efficiently reconstruct the image for homogeneous medium; however, it cannot generate focused and properly-located images for layered medium. Considering the electrical properties of individual layer and the discontinuity between layers, the phase shift migration for layered medium is derived in this paper. The analysis on the backscattered transfer function shows that some assumptions and mathematical approximations are applied in the proposed method. The numerical and experimental results are presented to show the feasibility of the proposed method for real-time imaging of layered medium.

Microwave imaging technique for evaluation
of hidden objects in a structure has been widely
applied in the areas of breast tumor imaging,Ground
Penetrating Radar (GPR),through-wall
imaging and concealed weapon detection^{[1, 2, 3]}. It
has advantages over other Non-Destructive Evaluation
(NDE) methods regarding low cost,portability,
good penetration in nonmetallic materials,
good resolution,and contactless detection. The
imaging algorithms for microwave imaging technique
can be classified into quantitative and qualitative
methods^{[1, 4]}. Quantitative methods retrieve
the shapes and the electromagnetic parameters of
unknown targets by solving a nonlinear inverse
scattering problem. The disadvantage of quantitative
methods is time consuming.

Qualitative methods usually make some assumptions
and mathematical approximations for
lower computational complexity^{[5]}. The classical
method is Frequency-Wavenumber (F-K) migration,
also known as Stolt migration^{[3, 6]}. F-K migration
can be computed efficiently by Fast Fourier
Transform (FFT). The algorithm works well for
homogeneous background medium. However,in
the case of layered medium,the image becomes
unfocused and the hidden objects are incorrectly
located. Another classical qualitative method involving
Back-Projection (BP) algorithm is based
on delay-and-sum beamforming. The image is reconstructed
by summation of the backscattered
data with proper phase compensation. The phase
of the scattered signal in lossless medium is proportional
to the round-trip distance in homogenous
medium. For layered medium,the electromagnetic
wave refracts across the discontinuity at the
interface. The phase for two-layered medium can
be solved numerically^{[7, 8]}. But for three or more
layers,the computational complexity will increase
drastically. An efficient tomographic algorithm
was proposed for 2-D microwave imaging
through layered media in Ref. ^{[9]}. In this paper,
an algorithm based on Phase Shift Migration
(PSM) is presented for microwave imaging of
layered medium.

PSM can handle velocity variations with
depth^{[10]}. F-K migration can be regarded as a special
case of PSM. PSM originates from seismic engineering
and geophysics^{[10, 11]}. It recursively extrapolates
the phase shift along depth in steps and
uses the result of each step as input to the next
iteration.

In this paper,the phase shift for layered medium is derived without recursion using the phase matching at the interface. So the image can be reconstructed in a fairly straight forward way. We also compensate the discontinuity of the interface using Fresnel transmission coefficients. The analysis on backscattered transfer function for layered medium makes clear to understand the assumptions and mathematical approximations in PSM.

This paper is organized as follows. Section 2 gives the derivation of PSM for layered medium. Section 3 analyzes the backscattered transfer function for layered medium and achieves the identical imaging equation by making some assumptions and approximations. In Section 4,numerical and experimental results are provided to show the effectiveness of PSM for layered medium. Finally,Section 5 gives a brief conclusion. 2 PSM for Layered Medium

PSM method was first proposed by Gazdag in
1978^{[10]}. It is based on the concept of exploding
source model^{[11]},which assumes that a fictitious
point source explodes at a reference time of $t=0$
at the target locationand radiates EM wave to the
receiver. The velocity in the medium is replaced
by half of the true propagation velocity for exploding
source model.

Assuming that a point object is located at $r(x,y,z)$ and an antenna is located at $r'(x',y',z')$. The antenna illuminates the target with angular frequency . The antenna synthesizes a rectangular aperture on a plane $z=z'$ parallel to xy. If we make $z'=0$,EM field $f (x,y,z,t)$ can be described by the following 3-D scalar wave equation for homogeneous medium:

$\left( {{{{\partial ^2}} \over {\partial {x^2}}} + {{{\partial ^2}} \over {\partial {y^2}}} + {{{\partial ^2}} \over {\partial {z^2}}} - {1 \over {v_m^{^2}}}{{{\partial ^2}} \over {\partial {t^2}}}} \right)f\left( {x,y,z,t} \right) = 0$ | (1) |

Applying 3-D Fourier transform to Eq. (1) over $x,y$,and $z$,we get

$\left( {k_x^2 + k_y^2 + {{{\partial ^2}} \over {\partial {z^2}}} - {k^2}} \right)F\left( {{k_x},{k_y},z,w} \right){\rm{ = }}0$ | (2) |

$F\left( {{k_x},{k_y},z,w} \right){\rm{ = }}F\left( {{k_x},{k_y},z{\rm{ = }}0,w} \right){{\rm{e}}^{jk{z^z}}}$ | (3) |

Doing 3-D inverse Fourier transform to $F({k_x},{k_y},z,w)$ over $x,y$,and $t$,we get:

$\eqalign{ & f\left( {x,y,z,t} \right) = \int\!\!\!\int\!\!\!\int F \left( {{k_x},{k_y},z{\rm{ = }}0,w} \right) \cr & \qquad \qquad \qquad \cdot {{\rm{e}}^{jk{z^z}}}{{\rm{e}}^{jk{x^x}}}{{\rm{e}}^{jk{y^y}}}{{\rm{e}}^{j{w^t}}}d|{k_x}d{k_y}dw \cr} $ | (4) |

According to the imaging principle^{[11]},the
wavefront shape at $t=0$ corresponds to the reflector
shape. Let $t=0$,we have the reflectivity at
$r(x,y,z)$:

$\eqalign{ & I\left( {x,y,z} \right) = f\left( {x,y,z,t = 0} \right) \cr & \, \qquad \qquad = \int {FT_{xy}^{ - 1}\left\{ {F\left( {{k_x},{k_y},z{\rm{ = }}0,w} \right){{\rm{e}}^{jk{z^z}}}} \right\}} dw \cr} $ | (5) |

The multiplication by ${{\rm{e}}^{jk{z^z}}}$ in Eq. (5) represents phase shift factor. It means that the extrapolation along $z$ direction in the frequencywavenumber domain is a phase shift operation.

Consider a point object embedded in an Nlayer medium,as shown in Fig. 1. The electromagnetic properties vary with depth only,$i.e$. the z-direction. For layered medium,the $z$ component of the wavenumber vector is not constant:

${k_{iz}} = \sqrt {4{{\left( {w/{v_i}} \right)}^2} - k_x^2 - k_y^2} ,i = 1,2, \cdot \cdot \cdot ,N$ | (6) |

We can generally define the phase shift factor f or the point target in layer $n$ as:

${P_{\left( z \right)}} = {{\rm{e}}^{j{k_1}{z^{d1}}}}\left( {\prod\limits_{i = 1}^{n - 1} {{{\rm{e}}^{j{k_1}z\left( {{d_i} - {d_{i - 1} \ \ }} \right)}}} } \right){{\rm{e}}^{j{k_n}{z^{\left( {z - {d_{n - 1} \ \ }} \right)}}}}$ | (7) |

$I\left( {x,y,z} \right) = \int {FT_{xy}^{ - 1}\left[ {F\left( {{k_x},{k_y},z{\rm{ = }}0,w} \right){{\rm{e}}^{j{k_1}{z^{d1}}}} \\ \qquad \qquad \quad \cdot \left( {\prod\limits_{i = 1}^{n - 1} {{{\rm{e}}^{j{k_i}z\left( {{d_i} - {d_{i - 1} \ \ }} \right)}}} } \right){{\rm{e}}^{j{k_n}{z^{\left( {z - {d_{n - 1 \ \ } }} \right)}}}}} \ \ \right]} dw$ | (8) |

$I\left( {x,y,z} \right) = \int {FT} _{xy}^{ - 1}\left[ {F\left( {{k_x},{k_y},z = 0,w} \right) \\ \qquad \qquad \quad \cdot {{{e^{j{k_{1z}}^{_{^{d1}}}}}\left( {\prod\limits_{i = 1}^{n - 1} \ {{e^{j{k_{_{_{iz\left( {{d_{_i}} - {d_{_i}} - 1} \ \ \right)}}}}}}} } \ \ \ \ \right) \ {{\rm{e}}^{j{k_{nz\left( {z - {d_n} - 1} \right) \ \ }}}}} \over {\prod\limits_{i = 1}^{n - 1} {{T_{i,i + 1}} \ \ {T_{i + 1,i}}} }}} \ \ \right]dw$ | (9) |

Generally,when the antenna plane is $z=z'$, the reflectivity at $r(x,y,z)$ can be expressed as:

$I\left( {x,y,z} \right) = \int {FT} _{xy}^{ - 1}\left[ {F\left( {{k_x},{k_y},z = z',w} \right) \\ \qquad \qquad \quad \cdot {{{e^{j{k_{1z}}^{\left( {_{^{d1}} - z'} \right)}}}\left( {\prod\limits_{i = 1}^{n - 1} {{e^{j{k_{^{_{_{_{iz\left( {{d_{_i}} - {d_{_i}} - 1} \ \ \right)}}}}}}}}} } \ \ \ \ \right){{\rm{e}}^{j{k_{nz\left( {z - {d_n} - 1} \ \ \right)}}}}} \over {\prod\limits_{i = 1}^{n - 1} {{T_{i,i + 1}} \ \ {T_{i + 1,i}}} }}} \ \ \right]dw$ | (10) |

The backscattered transfer function should be figured out to calculate the scattering field. Scattering problem can be typically formulated as the superposition of the scattering from a set of points,assuming that no interaction occurs between points. This leads to the following model:

$f\left( {r',w} \right) = \mathop{\int\!\!\!\int\!\!\!\int}\limits_{\kern-5.5pt V} I \left( r \right){G_{GT}}\left( {r,r',w} \right)dr$ | (11) |

$G_{RT}(r,r',w) = G_1(r,r',w)G_2(r,r',w)$ | (12) |

$I\left( r \right) = \int {dw\int\!\!\!\int\limits_{S'} f } \left( {r',w} \right)G_{_{GT}}^{ - 1}\left( {r,r',w} \right)dr'$ | (13) |

^{[12, 13]}:

$\eqalign{ & {G_1}\left( {r,r',w} \right) = {{ - j} \over {8{\pi ^2}}}\int\!\!\!\int {{{{e^{ - j{k_x}\left( {x - x'} \right) - j{k_y}\left( {y - y'} \right)}}} \over {{k_{1z}}}}} \cr & \qquad \qquad \qquad \cdot {F_ - }\left( {z,z'} \right)d{k_x}d{k_y} \cr} $ | (14) |

$\eqalign{ & {G_2}\left( {r',r,w} \right) = {{ - j} \over {8{\pi ^2}}}\int\!\!\!\int {{{{e^{ - j{k_x}\left( {x - x'} \right) - j{k_y}\left( {y - y'} \right)}}} \over {{k_{1z}}}}} \cr & \qquad \qquad \qquad \cdot {F_ + }\left( {z,z'} \right)d{k_x}d{k_y} \cr} $ | (15) |

Neglecting the upgoing wave below region n, ${F_ - }(z,z')$ and ${F_ + }(z',z)$ can be simplified as:

${F_ - }\left( {z,z'} \right) = {{\tilde T}_{1n}}{e^{^{ - j{k_{1z\left( {{d_1} - z'} \right)}}}}}{e^{^{ - j{k_{nz\left( {z - {d_n} - 1} \right)}}}}}$ | (16) |

${F_ + }\left( {z,z'} \right) = {{\tilde T}_{n1}}{e^{^{ - j{k_{1z\left( {{d_1} - z'} \right)}}}}}{e^{^{ - j{k_{nz\left( {z - {d_n} - 1} \right)}}}}}$ | (17) |

^{[12]}:

${\tilde T_{1n}} = \frac{{\prod\limits_{i = 1}^{n - 1} {{e^{ - j{k_{iz}}\left( {{d_i} - {d_{i - 1}}} \right)}}} {T_{i,i + 1}}}}{{1 - {R_{i + 1,i}}{{\tilde R}_{i + 1,i + 2}}{e^{ - j2{k_{i + 1,z}}\left( {{d_{i + 1}} - {d_i}} \right)}}}}$ | (18) |

${\tilde T_{n1}} = \frac{{\prod\limits_{i = 1}^{n - 1} {{e^{ - j{k_{iz}}\left( {{d_i} - {d_{i - 1}}} \right)}}{T_{i + 1,i}}} }}{{1 - {R_{i,i + 1}}{{\tilde R}_{i,i - 1}}{e^{ - j2{k_{iz}}\left( {{d_i} - {d_{i - 1}}} \right)}}}}$ | (19) |

$\eqalign{ & {G_{RT}}(r,r',w) \cr & \qquad = \left( {\prod\limits_{i = 1}^{n - 1} {{T_{i,i + 1}}{T_{i + 1,i}}} } \right) \cr & \qquad \quad \cdot {\left( {\int\!\!\!\int {{e^{ - j{k_x}\left( {x - x'} \right) - j{k_y}\left( {y - y'} \right)}}\left( {\prod\limits_{i = 1}^{n - 1} {{e^{ - j{k_{iz}}\left( {{d_i} - {d_i} - 1} \right)}}} } \right) \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \\ \cdot } {e^{^{ - j{k_{1z\left( {{d_1} - z'} \right)}}}}}{e^{^{ - j{k_{nz\left( {z - {d_n} - 1} \right)}}}}}d{k_x}d{k_y}} \right)^2} \cr} $ | (20) |

^{[12, 14, 15]}. The backscattered transfer function can be approximated by:

$\eqalign{ & {G_{RT}}(r,r',w) \cr & \qquad = \left( {\prod\limits_{i = 1}^{n - 1} {{T_{i,i + 1}}{T_{i + 1,i}}} } \right) \cr & \qquad \quad \cdot \int\!\!\!\int {{e^{ - j{k_x}\left( {x - x'} \right) - j{k_y}\left( {y - y'} \right)}}\left( {\prod\limits_{i = 1}^{n - 1} {{e^{ - j{k_{iz}}\left( {{d_i} - {d_i} - 1} \right)}}} } \right)} \cr & \qquad \quad \cdot {e^{^{ - j{k_{1z\left( {{d_1} - z'} \right)}}}}}{e^{^{ - j{k_{nz\left( {z - {d_n} - 1} \right)}}}}}d{k_x}d{k_y} \cr} $ | (21) |

$I\left( {x,y,z} \right) = \int {FT} _{xy}^{ - 1}\left[ {F{T_{xy}}\left[ {f\left( {x',y',z',w} \right)} \right] \\ \qquad \qquad \quad \cdot {{{e^{j{k_{1z}}^{\left( {_{^{d1}} - z'} \right)}}}\left( {\prod\limits_{i = 1}^{n - 1} {{e^{j{k_{^{_{_{_{iz\left( {{d_{_i}} - {d_{_i}} - 1} \ \right)}}}}}}}}} } \ \ \ \right){{\rm{e}}^{j{k_{nz\left( {z - {d_n} - 1} \right)}}}}} \over {\prod\limits_{i = 1}^{n - 1} {{T_{i,i + 1}} \ \ \ {T_{i + 1,i}}} }}} \ \ \ \right]dw$ | (22) |

The PSM for layered medium can be viewed as an approximate inversion of the inverse scattering problem. The assumption of the proposed algorithm can be summarized as follows. (1) Under first-order Born approximation,multiple scattering within the target is ignored. (2) For the object buried in layer n,the upgoing wave below region $n$ is ignored. That means the generalized reflection coefficient for the layered medium below region $n$ is set to zero. (3) The multiple reflections in layered medium are not considered. (4) The propagation decay is ignored. (5) The imaging area is in the far field of the antenna. 4 Results

In order to show the high computational efficiency and accurate image formation of the algorithm, both numerical simulation and experimental measurement are demonstrated in this section. Matlab is used to perform the simulation and imaging on a 64 bit PC with 16 GB RAM and four-core 2.5 GHz CPU. 4.1 Numerical results

Firstly,we present a numerical example for the imaging of the targets buried in a layered medium shown in Fig. 2. The antenna is placed on the plane $z=0$ with a standoff distance of 2 cm. The area is scanned along x direction from $x=-15$ cm to 15 cm with a step size of 0.5 cm. The first layer is wood with dielectric constant $ε_1=2$. The second layered is concrete with dielectric constant $ε_2=6$. Thicknesses of wood and concrete are chosen to be 5 cm and 7.5 cm,respectively. The buried objects are metallic cylinders with 1.5 cm radius located at (-5 cm,5 cm) and (5 cm,11 cm). The measured data is generated in time domain using 2-D Finite Difference Time Domain (FDTD) method and is transformed to frequency domain by Fourier transform. The frequency range covers from 3 GHz to 8 GHz with a step size of 25 MHz.

The reflected signal from interfaces of layers is relatively stronger compared with the scattered signal from targets. To clearly show the image of targets,we can estimate the background by averaging collected data over the scanning area and subtract the average from the collected data.

$f'\left( {x,w} \right) = f\left( {x,w} \right) - \sum\limits_x {{{f\left( {x,w} \right)} \over {{N_x}}}} $ | (23) |

The collected data was processed by the proposed
algorithm and $F-K$ migration for comparison.
For $F-K$ migration,the effective dielectric
constant can be approximately obtained by Root
Mean Square (RMS)^{[16]}:

$\varepsilon_{RMS} = {\left( {{{\sum\limits_{i = 1}^N {\varepsilon _i^2\Delta {t_i}} } \over {\sum\limits_{i = 1}^N {\Delta {t_i}} }}} \right)^{{1 \over 2}}}$ | (24) |

Fig. 3(a) shows the reconstructed image using
the proposed algorithm. The pixel size of the
image is 256×256. It takes only 1 s to get the image.
However,more than 2 min is needed to reconstruct
the image processed by $BP$ algorithm.
The horizontal lines are added to show the interfaces
of layers. The targets are correctly located.
In Fig. 3(a),the shadows at the interfaces of
layered medium are introduced by the interaction
between the objects and the layered medium.
The image processed by F-K migration with RMS
dielectric constant is shown in Fig. 3(b). As can
be seen from Fig. 3(b),the targets are displaced
away from the target locations. The image entropy
is employed to quantitatively evaluate the
focus quality of reconstructed images. The better
focus corresponds to smaller entropy. The entropy
is defined as^{[17]}:

$\xi = {{{{\left( {\sum {\sum {{{\left| {{x_{ij}}} \right|}^2}} } } \right)}^2}} \over {\sum {\sum {{{\left| {{x_{ij}}} \right|}^4}} } }}$ | (25) |

The proposed imaging algorithm has been validated by experimentally for detecting an object embedded in a three-layered medium. The experimental setup consists of Agilent N5230A (10 MHz~20 GHz)VNA and AEL H-1498 doubleridge horn antenna mounted on a 1-D scanner. The scanner moves the antenna parallel to the layered medium with a standoff distance of 3 cm. A 30.48 cm aperture is synthesized with 25 measurements spaced 1.27 cm in $x$ direction. The aperture center is set to $x=0$,$z=0$,where $z=0$ corresponds to the measurement plane. Data is collected at 1001 points in the frequency range from 2 GHz to 12 GHz with a step of 10 MHz. The complex S-parameter S11 is used as the radar response for layered medium imaging. The scenario is shown in Fig. 4(a). The three layered medium is comprised by two wood blocks and concrete block. The thicknesses of wood block and concrete block are 1.85 cm and 4.46 cm,respectively. Two metallic disks with 1.3 cm radius are placed at the interface between the second layer and third layer,as shown in Fig. 4(b). The locations of the embedded object are (-2 cm,9.3 cm) and (3 cm,9.3 cm).

The dielectric constants of wood block and
concrete block are estimated as 2.12 and 7.25 by
time domain reflectometry^{[17]}. Fig. 5 shows the
resulting image of embedded targets processed by
6 Journal of Radars Vol. x
the proposed algorithm. The pixel size of the image
is 256×256. The processing time is about 2.5
second. The indications of metallic disks in the
image are focused and properly located. The interaction
of the two objects leads to the ghost located
at the center of the object locations. Faint
artifacts at the left and right sides exist because
the concrete block is not homogeneous.

In this paper,phase shift migration for imaging of layered medium is proposed. Based on the analysis of backscattered transfer function,phase shift migration for layered medium can be viewed as an approximation of inverse problem. The proposed algorithm for layered medium is efficient due to the following reasons. Firstly,the coherent summation in BP algorithm is computed efficiently with FFT. Secondly,the solving of nonlinear equation in order to find the refraction points in BP algorithm is avoided. Only the phase shift factor is updated along with depth. Thirdly,instead of the pixel-by-pixel reconstruction in BP algorithm,the proposed algorithm efficiently reconstructs the image with IFFT. Numerical and experimental results are presented to show the effectiveness of the proposed algorithm for real-time imaging of layered medium.

**Acknowledgement**The authors would like to thank ElectroScience Laboratory,the Ohio State University for providing the experimental data.

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