Second-Order Perturbative Solutions for 3-D Electromagnetic ...

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Second-Order Perturbative Solutions for 3-D Electromagnetic Radiation and Propagation in a Layered Structure With Multilayer Rough Interfaces Chao Wu and Xiaojuan Zhang

Abstract—We present a fast method for deriving a general explicit closed-form expression of second-order perturbative solutions of the problem of radiation and propagation electromagnetic (EM) fields in a hybrid layered structure with an arbitrary number of rough interfaces and planar stratified medium based on the classical small perturbation method (SPM). The second-order SPM expression of the electric and magnetic fields in each region of the rough-layered structure is provided by theoretical analysis and fully formula derivation instead of considering any other approximation and equivalent process. In addition, we give the complete understanding of physical radiation and propagation mechanism of double-bounce second-order SPM solutions. Finally, the secondorder SPM solutions derived by employing the proposed method are verified with existing methods and numerical results. What is more, we investigate completely the influence of various parameters of layered structure and configuration of radar system on EM scattering fields. Index Terms—Bistatic scattering, high order, multilayer rough interfaces, radiation and propagation, remote sensing, small perturbation method (SPM).

I. INTRODUCTION

T

HE problem of electromagnetic (EM) scattering from layered structure with random rough surfaces has been a subject of numerous studies due to its applications in the microwave remote sensing of soil moisture [1], biomass estimation, glacier monitoring, medical imaging, infrastructure defect detection, component quality control, and so on. In order to model properly the EM propagation of the radar signal in natural settings and manmade objects such as soil with or without vegetation, planetary exploration [2], multiyear ice, civil infrastructure, and solid state components deposited in metal or dielectrics, the model should be layered with dielectric structures with multilayer rough interfaces. The forward model which is more close to the natural scenes can obtain more accurate simulation results of the response wave received by the synthetic aperture radar (SAR) system.

Manuscript received October 13, 2013; revised April 23, 2014; accepted April 23, 2014. This work was supported in part by the National High Technology Research and Development Program of China (863 Program 2009AA12Z132) and in part by the National Natural Science Foundation of China under Grant 61172017 and Grant 60890071. The authors are with the Key Laboratory of Electromagnetic Radiation and Sensing Technology, Chinese Academy of Sciences, Beijing 100190, China (e-mail: [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/JSTARS.2014.2320506

Till date, much work has been dedicated to modeling and prediction of rough surface EM scattering. The existing modeling methods can be generally categorized into empirical regression techniques and theoretical scattering models. The empirical methods [3], [4] are based on large experimental data at specific locations and with specific sensors, and, therefore, have a generally small domain of validity. The theoretical approaches are based on physical model and have widely applied area. The theoretical methods can be further classified into analytical, numerical, and semi-numerical solutions. The analytical methods [5]–[9] may have restraint of surface roughness and can obtain closed-form expression. Therefore, they usually have the highest computational efficiency and give clear physical meanings. The numerical methods [10]–[13] are not restricted by properties of rough interface. However, it will bring more computational cost. The semi-numerical methods [14], [15] such as the extend boundary condition method (EBCM) is a good tradeoff between the validity domain and the computational cost, but its solution was found to be unstable for larger roughness [16]. The analytical methods have advantages on solving the problem of complex model with multilayer rough interfaces. The small perturbation method (SPM) [5], [8] is the most broadly used analytical method to investigate the EM scattering from slightly rough surfaces. As of now, numerous previous studies and utilizing methods are limited to investigate the problem of layered structure with less than two rough interfaces [6], [7], [18]–[24], as well as extending to more rough layers becomes not tractable and have complex processing procedures. In addition, most prior works are restricted to only study the first-order [17]–[23] perturbative solution as higher-order solutions bringing much more complex process based on the classical SPM. Scattered fields up to high order were considered in literature [24]–[26] with only two rough-interface geometry, but its solutions increase in complexity as order or layer increases, so the explicit expression for highorder perturbative solution of the problem of layered structure with multilayer rough interfaces has not been previously presented. Furthermore, all prior work and utilizing methods cannot give the explicit expressions for the radiation and propagation EM fields inside layered medium with multilayer rough interfaces, which issues can be solved by employing our proposed method and will be highly desirable. Although the first-order perturbative solutions should dominate scattering fields in many situations, high-order perturbative solution is significant for some quantities of interests such as

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depolarized investigation. In addition, the high-order solutions can give insight into the accuracy of the first-order solution. What is more, high-order solutions can improve the accuracy and give a guarantee to obtain a successful inversion process. A general closed-form explicit high-order perturbative expression for the problem of radiation and propagation EM fields in a hybrid layered structure with an arbitrary number of rough interfaces and planar stratified medium is not available in the literature yet, and it would be highly desirable. Therefore, the objective of this work is to develop a fast method for deriving high-order perturbative solutions for the problem of radiation and propagation EM fields inside a layered structure with multilayer rough interfaces based on the classical SPM. The new method (named Discrete Superposition Multilayer SPM, DSM-SPM) is developed by theoretical analysis and fully formula derivation instead of considering any other approximation and equivalent current method [17], [19] in the framework of classical SPM. The calculation intrinsically considers multiple scattering processes between each rough boundary. The demonstration of the consistency of the presented method is analytically provided and validated with existing methods. The key point is that the proposed method can give explicit expression of the high-order SPM solutions for the problem of 3-D radiation and propagation of EM fields in a hybrid layered structure with an arbitrary number of rough interfaces and planar stratified medium, which is highly desirable and not considered by other authors. The proposed method can be applied to microwave imaging and target detection due to the obtainment of inside EM field distributions of inhomogeneous stratified medium, as well as give a new approach for dealing with the inverse problem [27] based on the proposed forward model with less requirements of the remote sensing system by employing microwave sensors to obtain the inside EM fields of layered structure. In addition, the presented method has advantage on deriving the accurate solutions of EM multiple scattering from 3-D buried discrete random media in layered structure with rough interfaces [28]. What is more, our presented method can give a complete and clear understanding of the physical scattering mechanism based on the obtained explicit expression for the high-order SPM solutions in order to better use in practice. Furthermore, the new method proposes a relevant strategy that decomposes complex problem into some tractable problems, which is of great value and meaningful. This paper is organized as follows. Section II provides a brief introduction of the SPM for scattering fields. In Section III, the fully formulas derivation of the presented method is provided analytically to derive the high-order solutions based on the SPM. In Section IV, a general explicit expression for the second-order perturbative solutions of the problem of the inside radiation and propagation EM fields in multilayer geometry is provided. Section V gives physical insight into the double-bounce scattering component due to a rough interface inside a layered structure. In Section VI, the comparison and numerical validation are provided, and investigating the influence of various parameters on radar cross-section. Summary and conclusion are presented in Section VII.

Fig. 1. Geometry for 3-D multilayer rough interfaces.

II. PROBLEM GEOMETRY AND DEFINITION A. Geometry The studied system (see Fig. 1) is made up of a 3-D layered structure with an arbitrary number of rough interfaces. The th , rough interface profile is denoted by where is zero-mean stationary random process with known statistical properties. The quantity is the small perturbation parameter used to match the boundary condition equations to specific orders. The quantity is the mean separation between the boundaries and is referred to as the th layer thickness. The parameters with subscript belong to region , . The layer’s permittivity is , and the media is assumed to be nonmagnetic , which assumption does not impact analytical processes and can be applied to inhomogeneous medium. The root-meansquare (rms) heights of the rough interfaces are assumed to be small numbers compared with the incident wavelength and the rms slopes of the rough interfaces are small compared to the unit in order to ensure that we can employ the SPM to derive the highorder perturbative solutions. The region of validity of SPM [8] has been known to be < , < for a single dielectric rough surface and reported in literature [29] for a two-layer dielectric structure with rough interfaces. However, the range of validity of SPM for layered structure with multilayer rough interfaces is out of the scope of this paper and will be addressed in a separate paper by the authors. B. Problem Definition In the following, the symbol denotes the projection of the corresponding vector on the plane . The subscript , , or denotes the projection of the corresponding vector on , , or axis, respectively. Assume that the incident plane wave on layered structure is

where

is the position vector in Cartesian space, ,

is the unit

is the vector of the incident direction, and wave number of the top layer. Since the incident wave is a plane

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wave, it means that is perpendicular to , then we can decompose the incident electric-field intensity into two perpendicular unit vectors in the plane perpendicular to the direction of

with

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,(

), where sub-

scripts and indicate the partial derivative with respect to and , respectively. The SPM suggests that these unknown coefficients should be represented by their asymptotic expansions in with

where

,

and are the and where the unit vectors polarized components of the incident plane wave. The time factor is understood. We can derive the EM fields in the spectral domain as the linear superposition of the infinite number of up- and down-going plane waves in each infinite region by employing the Huygens principle [30]. The electric and magnetic fields at the point in the region can be expressed as follows:

new unknown coefficients , where the superscript ( ) represents the order of the coefficients. The coefficients can be expanded by

Assume that the roughness is small expand items into the Taylor series

, then we can

The coefficients and are the known intensities of the incident EM field components and can be written as

where is the Dirac function. In Section III, we derive the other unknown coefficients by employing the SPM and matching boundary conditions. Furthermore, we can obtain the radiation and propagation EM fields and scattering cross-section according to the definitions of scattering fields and scattering coefficients.

III. ANALYSIS AND EXTENSION where . The superscripts and and denote the up- and down-going waves, respectively. and denote the unknown amplitudes of the up- and downgoing waves of and polarizations in the region . The unknown amplitudes will be solved in Section III by employing the boundary conditions and SPM. is the wave impedance of region , , . The unit and can be written as vectors

. where The boundary conditions which ensure the continuity of the tangential components of electric and magnetic fields at the th interface can be expressed as follows:

In this section, we calculate the unknown coefficients by employing the SPM and boundary conditions. Without loss of generality, we consider the representative stratification with an intermediate layer and two rough interfaces denoted by and , respectively. This model refers to the geometry of Fig. 2. In the following part, we calculate the unknown coefficients , , and by employing the SPM and boundary conditions. The SPM suggests that these coefficients should be represented by their asymptotic expansions in with new unknown coefficients , , and , where the superscript ( ) represents the order of the coefficients. Applying the boundary conditions (9) and (10) and the expansions (11) and (12) into the integral equations (5) and (6) will result in four integral equations. Calculating the integral equations we obtain four linear vector equations with weak form. The linear vector equations set for and directions are independent, then we match the and components of each of these equations to the high-order and obtain eight linear equations. The scalar form of vector equations is presented in Appendix A. The linear scalar equations for direction are not independent, and we will not present them here.



where , . The first element of vector can be expressed as

Fig. 2. Geometry for the 3-D dielectric structure with two lightly rough interfaces problem. The boundaries are the Gaussian rough interfaces. The mean thickness between two rough interfaces is .

Balancing the zeroth-order terms of the each equations set, we can obtain eight linear equations with eight zeroth-order ( ) unknowns, which can be represented as

where . Now, in the same way, we can match the linear equations to the first order and obtain this system of eight linear equations with eight first-order ( ) unknowns as where the dummy variable

,

, and

. The other elements can also be calculated

where , . is the matrix of coefficients for all orders and just has a relationship with the number of layers. Since the elements of contain the spatial function , we should use the Fourier transform formula

similarly and will not be shown here only to save space. Solving these systems of linear equations, we can derive the unknown coefficients of each region ( ) and obtain the high-order SPM solutions of radiation and propagation EM fields in layered structure with two rough interfaces. The first two orders ( ) scattered field in region 0 can be expressed as

The first element of vector can be expressed as below. The other elements also can be calculated similarly and will not be shown here only to save space In the far-field zone, the scattering field can be approximated by the stationary phase method [31] shown as

where , is the angle between the scattering wave and axis. The scattering coefficient is defined as

In the same manner, we can match the linear equations to the second order and obtain this system of eight linear equations with eight second-order ( ) unknowns as

where , is the area illustrated by incident wave. In the remainder of this section, we provide the compact closedform SPM solutions of two representative cases: one case is the problem of layered structure with one bottom rough interface, the other case is the problem of layered structure with one top rough interface, as well as investigate the relationship between these two


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Fig. 3. Geometry for one bottom rough interface (Rough Bot model).

Fig. 4. Geometry for one top rough interface (Rough Top model).

representative problems and the general geometry problem of layered structure with two rough interfaces.

With the same procedure, we can obtain the second-order equations with unknowns as

A. Geometry With One Bottom Rough Interface (Bot Model) Fig. 3 shows the 2-D section of the 3-D geometry with one bottom rough interface, which model can be named Bot model for simplify and convenience to be used in this section. We can derive the second-order closed-form SPM solutions of the problem of layered structure with a slightly rough interface covered by a homogeneous layer according to the processes of problem of layered structure with two rough interfaces for the first time. In other words, we consider the special case of the layered structure with two rough interfaces model, which can become Bot model when . This system of eight linear equations with eight zeroth-order unknowns can be represented as

Since the zero-order SPM solutions do not have relationship with rough interfaces, we can derive the zeroth-order solutions, which are the same as the problem of layered structure with two rough interfaces model. Similarly, we can obtain the first-order equations with unknowns as

of the first element of vector The expression of can be expressed as

Then we can obtain the second-order solutions of Bot model through calculating the equations set (27) which can be expressed as follows:

where . B. Geometry With One Top Rough Interface (Top Model) Fig. 4 illustrates the geometry with one top rough interface, which is similarly named Top model. In [20], Fuks has employed the plane wave expansion of EM fields and an equivalent current method [32] to calculate the first-order scattering from one rough surface on top of a stratified media, instead of Green’s function method [30]. However, in this work, we will derive the secondorder solutions of Top model using the new method in the framework of SPM for the first time. We consider the special case of layered structure with two rough interfaces model which can become Top model when . The train of thought to solve the Top model is the same as Bot model. The zeroth-order solutions are the same as the problem of layered structure with two rough interfaces. By the same token, we can obtain the first-order equations with unknowns as

The elements of

can be expressed as follows:

Further, we can obtain the second-order equations with unknowns as where we can get other elements

. In the same way, The elements of

can be expressed as follows:



Then, we can derive the closed-form second-order SPM solutions of Top model by calculating the equations set (30) which can be expressed as follows:

where . C. Conclusion Based on the classical SPM, combining Bot with Top model to investigate their connection with the two rough interfaces model, the consistent matrix of coefficients is the nonsingular matrix, so we can calculate that

which means

,

,

, . Substituting (32) and (33) into (22), we can obtain the first two orders scattering fields of the two rough interfaces problem in region 0 as below, which are expressed by the Bot and Top models. In addition, we can also calculate the radiation and propagation of the EM fields in each region ( ) of the layered structure

If we assume that the boundaries are independent random processes, then we can derive the first two orders bistaticscattering coefficients of the problem of layered structure with two rough interfaces, which can be expressed by superposition of the Bot and Top model bistatic-scattering coefficients. Without loss of generality, the analysis of the two rough layers can be extended to more rough layers, so we can use the same procedure to calculate the problem of radiation and propagation EM fields in the layered structure with N-layer rough interfaces. Therefore, the general problem of the scattering from the layered structure with all rough interfaces can be expressed by superimposing the cases in which one boundary is rough, whereas the other one is flat. What is more, the new proposed method can also calculate the radiation and propagation of the EM fields in a hybrid layered structure with an arbitrary number of rough interfaces and planar stratified medium. The train of thought of calculating the problem of layered structure with N-rough interfaces is to divide this problem into N equivalent problems in which one interface is rough, whereas the other one is flat, then the first-order global bistatic scattering crosssection of the N-rough interfaces layered media can be expressed as

In the same way, we can give the second-order overall bistatic scattering cross-section of the N-rough interfaces layered medium, which can be expressed as

According to the definition of the bistatic-scattering coefficients, the first two orders bistatic-scattering coefficients of the layered structure with two rough interfaces can be expressed by the Bot and Top models as

R

R

where is the joint density of the rough interfaces between Bot and Top models.

where denote the incident and the scattered polarization states, respectively, and may stand for horizontal polarization ( ) or vertical polarization ( ), and the coefficient is relative to the -polarized incident wave impinging on the structure from the upper half-space zero and to the -polarized scattering contribution from structure into the upper half-space, originated from the rough interface between the layer and . Therefore, according to the scattering coefficients definition (23), the global bistatic scattering coefficient in the upper halfspace zero can be approximated by superimposing the first two order scattering coefficients as follows:

In Section IV, we will calculate the problem of layered structure with one rough interface embedded into the multilayer planar medium, in order to obtain the global bistatic scattering coefficients of the problem of layered structure with N-rough interfaces by employing the new proposed method (Discrete Superposition Multilayer SPM, DSM-SPM).


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IV. SCATTERING FROM LAYERED STRUCTURE WITH ONE ROUGH INTERFACE The proposed new method can give the explicit expression of the high-order perturbative solutions for the problem of radiation and propagation EM fields in a hybrid layered structure with an arbitrary number of rough interfaces and planar medium, which is highly desirable and not considered by other authors. When we derive each high-order perturbative solution for the problem of layered structure with N-layer rough interfaces, we can decompose the N-rough layered model into N equivalent one rough layer models in which one interface is rough, whereas the other one is flat, which not only can give the clear analysis process and physical radiation and propagation mechanism in the N-rough layered medium, but also can bring more efficiency and compact explicit expression. Therefore, we just need to derive the high-order SPM solutions for the problem of radiation and propagation EM fields in layered structure with one single rough interface embedded into the multilayer planar interfaces. The strategy that decomposes complex problem into some tractable problems is of great value and meaningful. The high-order terms can be interpreted as multiple scattering: the first-order scattering is due to the unperturbed wave field alone, second-order one comprises rescattering of the first-order waves, and so on. We employ the generalized reflection and transmission coefficients [33] shown in Appendix B to specify the reflection from planar medium. The waves scattering from the embedded rough interface are expanded into perturbation series with unknown coefficients. The expansion coefficient of each order can be solved by applying boundary conditions and the propagation law. The detailed processes of calculating the high-order SPM solutions are the same as previous analysis of two rough interfaces geometry in Section III. We will give the explicit expression of the second-order perturbative solution for the general problem of layered structure with one rough interface embedded into the multilayer inhomogeneous planar medium. The geometry problem refers to Fig. 5. For obtaining the unitary formalism, we use the following formulism (41) for the second-order scattering coefficient relevant to the double-bounce contribution of the th rough interface

Fig. 5. Physical interpretation for the double-bounce scattering from an arbitrary layered structure with an embedded rough interface; solid line represents the ordinary reflection and transmission and dotted line represents the generalized reflection and transmission.

V. PHYSICAL INTERPRETATION Since we have investigated analytically the problem of the layered structure with N-layer rough interfaces through full closed-form formulas derivation in the framework of the classical SPM, we find that the second-order solutions of the problem of the layered structure with N-layer rough interfaces are consisted of the superposing of the solutions of the problem of layered structure with an embedded rough interface. Therefore, we focus on the physical scattering mechanism for the secondorder SPM solution for the double-bounce scattering in layered structure with an embedded rough interface. Since the analysis of each polarization combination has the same procedure, without loss of generality, the considered polarization is case only. The formulas (42) can be rearranged for appropriately expression, in order to look for the role of the interference phenomena that take place inside the stratification

is associated with the local second-order scattering where which are the classical scattering coefficients of a rough interface between two half-infinite homogenous media and . This equation can be thought of as a ray series or a geometrical optics series, and each term of this equation can be readily identified. The local scattering coefficient can be expressed as follows:

with

The physical mechanism is expressed clearly in Fig. 5. The physical phenomena can be divided into three parts. The first part is associated with the unperturbed local incident field on the rough interface. This can be considered, as the wave undergoes a coherent transmission through the layers zeroth to The expressions

and

are presented in Appendix C.

th in the incident plane (

). The second part

shows an



incoherent (local) scattering with rough interface in the observation plane ( ). The third part expresses an equivalent coherent transmission through the layers th to zeroth in the observation plane ( ). In the remainder of this section, we will explain the three parts, respectively, in details. The term can be formulated as

We consider the series expansion (geometric power series)

EM fields are generated by superposing all the double-bounce scattering components.

repreThe factor sents that the scattering wave undergoes infinite complete roundtrips in the media with coherent reflections at the scattering angle ( ). The factor represents that the wave undergoes ordinary transmission through th to th layer, and the represents that the factor wave undergoes infinite complete roundtrips in the th layer. Each complete roundtrip consists of downward and upward waves; however, the downward wave undergoes ordinary reflection (

corresponds to a complete The term roundtrip in the media . The factor represents that the wave undergoes transmission through th repreinto th, and the factor sents that the wave undergoes infinite complete roundtrips in the th layer. In addition, the downward wave undergoes general) and the upward wave undergoes ordiized reflection ( ) in the th media. Therefore, the factor nary reflection ( indicates that the incident wave undergoes a coherent transmission through each layer from zeroth to th in the incident plane. The factor represents that the wave undergoes the local double-bounce scattering between media and the under th layer (media ). The local EM wave scattering depends on the upward wave generated by the reflection and transmission of the unperturbed local incident wave impinging on the th rough interface, which occurs among the under th layer (media ) in the incidence plane ( ) and observation plane ( ). stands for the double-bounce scatterThe factor ing with dummy wave vector ( ) under the given incidence and observation direction. The second-order scattering consists of scattering from incident direction into an intermediate direction, then another scattering from intermediate direction to scattering direction in spectral domain. The local second-order scattering

can be formulated as follows:

The term

reflection ( factor

) and the upward wave undergoes generalized ) in the

th layer. Therefore, the

shows that the scattering wave under-

goes infinite complete roundtrips in the media and then undergoes a coherent transmission through each layer from th to zeroth layer in the observation plane ( ). The global scattering field can be considered as the superposing of the local scattering diffraction wave from each rough interface in the region 0. From the physical interpretation point of view, we can conclude that the global EM scattering consists of EM wave projection in the incidence plane ( ) and observation plane ( ), the roundtrips and the transmission through the layers are all constrained in the incidence plane or in the observation plane, respectively. This shows that the contributions contemplated by the second-order SPM approximation are restricted within these two planes by projection. VI. VALIDATION AND NUMERICAL RESULTS As we calculate, the high-order SPM solutions of the layered structure with N-layer rough interfaces based on the theoretical analysis and fully formula derivation in the framework of the classical SPM, therefore, the obtained explicit expressions of the first-order SPM solutions, are consistent straightforwardly with the existing models [17], [20]–[23] according to the formulism and numerical results, respectively.


Fig. 6. Reproducing the first two orders total bistatic scattering coefficients for two rough-interface layered structures on [24, Fig. 5] and simulation of the , second-order depolarized scattering cross-section with , , , , and and . ,

To validate the expressions of the bistatic scattering coefficients of the second-order SPM solutions derived by the proposed method, we reproduce Fig. 5 in [24] for the simple case of two rough interfaces in order to validate with existing classical SPM and exactly numerical method (method of moments, MoM), which is shown in Fig. 6. Furthermore, we reproduce the curve of 2-2 term of Fig. 1 in [26] for the simple case of single rough interface, which is shown in Fig. 7. The comparison with the reference papers and numerical method shows a very good agreement for second-order SPM solutions when our proposed method is applied to special cases considered by other authors. However, the key point of the proposed method is that we can derive the general explicit expression of the second-order perturbative solution of the problem of layered structure with an arbitrary number of rough interfaces. In addition, the proposed method can give the EM radiation and propagation fields in the rough layered structure and bring more information of the natural scene, in order to detect the 3-D buried targets and analyze the inner medium properties, which is not considered by other authors. In the remainder of this section, we emphasize on analysis of the double-bounce scattering properties of layered structure with four rough interfaces and three intermediate layers. The considered case is a representative of many realistic natural scenes (for instance, air-snow-clayed soil-sandrock), which case has not been considered by other authors yet. In addition, we will give several illustrative examples with different parameters in order to obtain the comprehensive analysis. We assume that the rough interfaces have properties of Gaussian function whose spectral density is given by

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Fig. 7. Reproducing the second-order copolarization HH bistatic scattering coefficients for one rough layered structure on [26, Fig. 1], with , , and and . ,

where and are the correlation length and standard deviation of the th rough interface height, respectively. Fig. 8 illustrates radar cross-section of the four rough-interface layered structure as well as the significance of the second-order contribution at some observation angles. The first two orders total bistatic scattering returns decrease when the scattering angles move away from the specular direction. Fig. 9 shows that along with the azimuthal scattering direction, moves away from ; then we can find that when the observation plane ( ) is close to the incidence plane ( ), the cross-polarization scattering response will decrease until down to zero when the observation plane is coplanar with the incidence plane or specular plane. In addition, the cross-polarization scattering will enhance when the observation plane is close to the plane, which is perpendicular to the incidence plane. Fig. 10 plots the first two orders SPM solutions of bistatic scattering from layered structure with four rough interfaces. The results show that the cross-polarized components have approximately equal status with copolarized components when the angle between observation plane and incidence plane is . Fig. 11 illustrates the total radar cross-section including firstand second-order SPM solutions of scattering field distribution in three-dimensional space, as well as the only second-order double-bounce components are shown in Fig. 12. The plot formats are a projection of the hemisphere onto a horizontal plane, in which the horizontal and vertical axes represent and , respectively. Fig. 13 illustrates the backscattered fields contributed by both first- and second-order SPM solutions. The backscattering response will decrease as the correlation length increase. In addition, the first-order scattering response will reduce much faster than second-order scattering fields along with increasing rough correlation length in copolarization case.



Fig. 8. Simulation of the first two orders bistatic scattering coefficients for four , rough-interface layered structure, with , , , , , , , , , , , and .

Fig. 9. Same simulation as in Fig. 8, but with an azimuth angle

.

In the cross-polarization HV and VH scattering case, the second-order scattering fields dominate the overall scattering response and the contributions from first-order scattering become zero. The influence of depolarization on the global scattering response power is reflected in the cross-polarization components as shown in Fig. 13. A trend of increased backscatter levels as the correlation lengths increase is observed for very small correlation lengths, with a maximum achieved in this case around a correlation length of 0.25 wavelengths. The second-order backscattering fields can give more contributions to total radar cross-section than first-order contributions, as the increasing correlation length as well as second-order effects can dominate the total scattering fields at some larger correlation length. Fig. 14 presents the influence of rough properties on the backscattering scattering fields. The results show typical

Fig. 10. Same simulation as in Fig. 8, but with an azimuth angle

.

decreasing backscatter returns with incidence angle. The total scattering fields will increase as the interface roughness increase. The second-order scattering fields are sensitive to the interface roughness, because the second-order contribution reflects the double-bounce scattering from the same rough interface. Fig. 15 shows the effect of the first surface roughness on backscattering cross-section. It is observed that the second-order depolarized scattering cross-section increases with the first surface roughness. Furthermore, as the first interface roughness increase, the first surface itself produces more nonspecular copolarized scattered power than crosspolarization for second-order depolarization in the incident plane. Fig. 16 illuminates the influence of layer depth on the secondorder bistatic-scattering coefficients. The results show that the layer depths have small effect on the total radar cross-section. The fluctuation behaviors are observed with larger layer depth. The effect of the first layer thickness on the second-order bistatic-scattering coefficients is shown in Fig. 17. It is noted that the fluctuation behaviors [23] become more significant with larger layer depth. With the observation scattering angles increasing, the oscillation behavior becomes more acute. The effect of the lower layers depth on bistatic scattering coefficients can be analyzed similarly and will not be shown here only to save space. The preceding numerical simulations have all employed the fixed dielectric constants in the four rough-interface layered structure. The relationship between relative permittivities and back-scattering response power can be illustrated clearly as shown in Fig. 18. The results for this geometry show that each layer medium has mutual and interactive effects and give an integrated influence on radar backscattering cross-section. The cross-polarization scattering is contributed by second-order SPM solutions. In addition, the image of backscattering coefficients as a function of the dielectric constants of the first two layers medium in the form of strips are observed clearly. The


Fig. 11. Simulation as in Fig. 8 for total scattering, but with an azimuth angle ranging and observation angle ranging .

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Fig. 13. Back-scattering coefficients for four rough-interface layered structure as a function of rough interface correlation length with same configuration as Fig. 8.

Fig. 14. Simulation of the influence of all rough interfaces properties on radar backscattering cross-section with the same configuration as Fig. 8. Fig. 12. Same simulation as in Fig. 8 for second-order SPM solutions, but with an ranging and observation angle ranging . azimuth angle

backscattering cross-section scattering power increases with the first two layers dielectric constants. The depolarized scattering cross-section increases with the magnitude of the complex dielectric constant of the first two rough interfaces [24]. For the copolarized backscatter cross-section, the largest scattered power occurs when the first layer permittivity is lower than the second layer permittivity as shown in the diagonal of image. In the cross-polarized HV and VH case, as the first two layers permittivities are equal corresponding to the diagonal, which first two layers can be seen as single layer, the cross-polarized components are more sensitive to the variation of permittivity and become larger with the increasing permittivity along the diagonal.

VII. SUMMARY AND FUTURE DEVELOPMENT The presented new method gives the general explicit closedform expressions of the zeroth-, first-, and second-order SPM solutions for the problem of the inside radiation and propagation EM fields of layered structure with an arbitrary number of rough interfaces and planar stratified medium. The proposed method gives a new idea of discrete elements for the scattering object. The core of the new method for calculating the high-order SPM solutions of the problem of layered structure with N-rough interfaces is to convert this problem into N equivalent problems in which one interface is rough, whereas the other one is flat, which can bring more computational efficiency and accuracy. The strategy that decomposes complex problem into some tractable problems is of great value and meaningful. An accurate and fast-forward model is necessary to insure a successful



Fig. 15. Simulation of the influence of the first rough surface roughness on radar backscattering cross-section with the same configuration as Fig. 8.

Fig. 17. Effect of the first layer thickness on radar bistatic scattering coefficients with same configuration as Fig. 8.

Fig. 16. Bistatic scattering coefficients with same configuration as Fig. 8, but with different layer depth. Fig. 18. Image of full polarization backscattering coefficients as a function of the relative permittivities of the first two layers, with same configuration as Fig. 8.

inversion process. The obtained explicit expression of high-order SPM solutions is very suitable to calculate the inversion problem, which can give more accurate inversion results and higher efficiency. The complete physical radiation and propagation mechanism in layered structure with multilayer rough interfaces is illustrated in order to be in better use in practice. The expressions of first- and second-order SPM solutions are validated by comparing with classical SPM and numerical exact moment of method (MoM). We investigate the contributions of zeroth-, first-, and second-order SPM solutions to the scattering pattern of the layered structure with four rough interfaces. The scattering effects of various parameters on radar cross-sections are also carried out to investigate the performance of the fully polarimetric EM wave scattering. The proposed method can be used in retrievals of subsurface soil moisture [34], planetary exploration,

and other natural scenes, as well as for providing an important tool for radar system design. The second-order SPM solution is significant to characterize the cross-polarized backscattering, which plays an important role in EM wave numerical simulation and parameters inversion. Therefore, the high-order solutions are necessary and important for some situations. The proposed method has advantages on obtaining the explicit expressions of high-order SPM solutions for the problem of radiation and propagation EM fields in layered structure with multilayer rough interfaces. Our future work will focus on higher order SPM solutions for a layered structure with an arbitrary number of rough interfaces in order to investigate the multiplebounce component contributed to total scattering fields.


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APPENDIX A The expressions of the scalar equations set

APPENDIX B The generalized reflection coefficients for waves at the interface between the region ( the recursive relations

with

and polarized ) and have

,

,

. APPENDIX C The detail expressions of and

are shown as



is the incident angle impinging on the th rough where interface and is the double-bounce scattering angle from the th rough interface. can be derived from based on the Fresnel law. The parameters with superscript refer to the dummy variable . REFERENCES [1] M. Moghaddam et al., “Microwave observatory of sub-canopy and subsurface (MOSS): A mission concept for global deep soil moisture observations,” IEEE Trans. Geosci. Remote Sens., vol. 45, no. 8, pp. 2630–2643, Aug. 2007. [2] H. Huang, B. SanFilipo, and I. J. Won, “Planetary exploration using a small electromagnetic sensor,” IEEE Trans. Geosci. Remote Sens., vol. 43, no. 7, pp. 1499–1506, Jul. 2005. [3] Y. Oh, K. Sarabandi, and F. T. Ulaby, “An empirical model and an inversion technique for radar scattering from bare soil surfaces,” IEEE Trans. Geosci. Remote Sens., vol. 30, no. 2, pp. 370–381, Mar. 1992. [4] P. C. Dubois, J. J. Van Zyl, and E. T. Engman, “Measuring soil moisture with imaging radar,” IEEE Trans. Geosci. Remote Sens., vol. 33, no. 4, pp. 915–926, Jul. 1995. [5] J. A. Kong, Electromagnetic Wave Theory. Hoboken, NJ, USA: Wiley, 1990. [6] N. Pinel, J. Johnson, and C. Bourlier, “A geometrical optics model of three dimensional scattering from a rough surface over a planar surface,” IEEE Trans. Geosci. Remote Sens., vol. 57, no. 2, pp. 546–554, Feb. 2009. [7] A. Voronovich, “Small-slope approximation for electromagnetic wave scattering at a rough interface of two dielectric half-spaces,” Waves Random Media, vol. 4, pp. 337–367, 1994. [8] F. T. Ulaby, R. K. Moore, and A. K. Fung, Microwave Remote Sensing. Reading, MA, USA: Addison-Wesley, 1982. [9] L. Tsang, J. A. Kong, and R. T. Shin, Theory of Microwave Remote Sensing. Hoboken, NJ, USA: Wiley, 1985. [10] D. A. Kapp and G. S. Brown, “A new numerical method for rough-surface scattering calculations,” IEEE Trans. Antennas Propag., vol. 44, no. 5, pp. 711–721, May 1996. [11] V. Jandhyala, E. Michielssen, S. Balasubramaniam, and W. C. Chew, “A combined steepest descent-fast multipole algorithm for the fast analysis of three-dimensional scattering by rough surfaces,” IEEE Trans. Geosci. Remote Sens., vol. 36, no. 3, pp. 738–748, May 1998. [12] F. D. Hastings, J. B. Schneider, and S. L. Broschat, “A Monte-Carlo FDTD technique for rough surface scattering,” IEEE Trans. Antennas Propag., vol. 43, no. 11, pp. 1183–1191, Nov. 1995.


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Chao Wu received the B.S. degree in science and technology of electronic information from Xidian University, Xi’an, China, in 2010, and is currently pursuing the Ph.D. degree at the Institute of Electronics, Chinese Academy of Sciences (CAS), Beijing, China. From 2010 to 2012, he was a Senior System Engineer with the Bell Labs of Alcatel-Lucent, Paris, France, and a Firmware Engineer with the Intel China Research Center (ICRC), Beijing, China. His research interests include computational electromagnetic, SAR data modeling and processing, microwave imaging, electromagnetic wave scattering, echo SAR signal simulation, communication system, and inverse problem.

Xiaojuan Zhang received the B.S. and M.S. degrees from Taiyuan University of Technology, Taiyuan, China, and the Ph.D. degree from the Institute of Electronics, Chinese Academy of Sciences (CAS), Beijing, China, in 1984, 1990, and 2000, respectively. From 1984 to 1997, she was an Associate Professor with the Taiyuan University of Technology. From 2001 to 2002, she was a Postdoctoral Researcher with the University of Illinois at Urbana-Champaign, Illinois, USA. Since 2000, she has been a Professor with the Institute of Electronics, Chinese Academy of Sciences (CAS), Beijing, China. She has authored or coauthored over 100 publications. She undertakes the National Natural Science Foundation of China, major national scientific and technological projects in 863 key projects, Chinese Academy of Sciences projects on important directions. Her research interests include electromagnetic wave scattering in inhomogeneous medium, inverse problem, microwave imaging and application, Antenna technology, SAR system, and computational electromagnetic.