**摘要**: 本文提出一种新的双尺度(Modified Two-Scale Model, MTSM)粗糙度地表微波散射模型,采用小波包变换的方法将面上的粗糙度(

*z*(

*x, y*))变换到频域进行处理,粗糙度频谱的低频部分代表大尺度分量,高频部分代表小尺度分量,然后分别用基尔霍夫近似和小扰动模型来分别模拟大小尺度的粗糙度,考虑到小尺度的粗糙度是叠于大尺度粗糙度之上,模型在计算小尺度粗糙度的后向散射贡献时还考虑了大尺度起伏造成的几何倾斜。MTSM的解为大尺度粗糙度的基尔霍夫解加上几何倾斜修正后的小尺度粗糙度的小扰动解。最后采用改进的积分方程模型(AIEM)对MTSM做了初步的验证,结果表明在入射角

*θ*

_{i}<30°,粗糙度较小(

*ks*=0.354)的时候,MTSM有比较好的精度。

**Abstract**: In this paper, we present a Modified Two-Scale Microwave (MTSM) scattering model to describe the scattering coefficient of naturally rough surfaces. The surface roughness is assumed to be Gaussian in the proposed model so that the surface height

*z*(

*x, y*) can be split into large- and small-scale components by the wavelet packet transform according to electromagnetic wavelength. We used the Kirchhoff Model(KM) and Small Perturbation Method (SPM) to estimate the backscattering coefficient of large- and small-scale roughness, respectively. The tilting effect caused by the slope of large-scale roughness was corrected when calculating the contribution of backscattering to small-scale roughness. The backscattering coefficient of the MTSM comprised the total backscattering contributions of surfaces with both scales of roughness. The MTSM was tested and validated using the Advanced Integral Equation Model (AIEM) for dielectric randomly rough surfaces. The accuracy of the MTSM showed favorable agreement with AIEM, both when the incident angle was less than 30° (

*θ*

_{i}<30°) and when the surface roughness was small (

*ks*=0.354).

Research on scattering models for randomly rough surfaces is an important part of microwave remote sensing theory. Several theoretical scattering models for bare soil surfaces have been proposed over the past decades^{[1,2,3]}. However,the scattering from a randomly rough surface is a very complex problem that does not have a closed-formed solution from the Maxwell equations. Therefore,some approximate analytic solutions can only be used when the geometric scales of the surface are restricted within the valid ranges. Three representative approximate models have been developed,namely,the Kirchhoff Model (KM)^{[4]},the Small Perturbation Model (SPM)^{[5]},and the Integral Equations Model (IEM)^{[6]}. In the broad sense,KM is most suitable for relatively rough surfaces (large-scale roughness surface); however,it is only valid when the electromagnetic wavelength is much smaller than the rough surface curvature radius. Here,the root mean square height (*s*) and the associated correlation length (*l*),which define the surface roughness statistically,are both larger than the wavelength. SPM requires a relatively smooth surface (small-scale roughness surface) and is only effective when both *s* and *l* are smaller than the electromagnetic wavelength. However,natural surfaces contain various proportions of roughness and may not satisfy the validity conditions of KM or SPM. To improve the applicability of the scattering models,Fung^{[6]} proposed IEM and demonstrated its favorable accuracy. As its basic theory,IEM divides the surface scattering field into two parts: the first part keeps the Kirchhoff field (the near field of tangent plane) as the initial solution,and the other part introduces the compensation field that is used to correct the Kirchhoff field. IEM extends the valid range of the theoretical scattering model and fills the gap between KM and SPM. Therefore,IEM has become the most popular method for calculating the electromagnetic scattering of natural surfaces. Wu^{[7, 8]} introduced two important modifications on IEM that improved the accuracy of the model in simulating randomly rough surfaces. This upgraded version is called Advanced Integral Equations Model (AIEM).

Natural surfaces may comprise complex mixtures of large- and small-scale roughnesses,such as an ocean surface (on which small waves cover the large waves) or some blocks of clod with small granular soil. The electromagnetic scattering properties of this two-scale rough surface can be described by calculating the scattering effect of both sizes of roughness. Ulaby,*et al*.^{[9]} attempted to solve such problem by introducing a Two-Scale Model (TSM). In this model,the randomly rough surface is assumed to be a combination of a slowly undulating surface,which satisfies the KM condition,and a small perturbation surface for which SPM is valid. When the incident angle is near normal (0°<q* _{i}*<25°),the scattering is dominated by the large-scale surface roughness. In contrast,at large incident angles (q

*>25°),the scattering is dominated by small-scale surface roughness,which is tilted by the slope of the large-scale surface roughness. SPM accounts for the tilting effect by introducing a local coordinate system; the scattering coefficient is then calculated by considering both the large- and small-scale roughnesses. Brown proposed another TSM*

_{i}^{[10]},which is a kind of Fourier function. This model is used for transforming the surface height into a spectrum domain for a perfectly Gaussian surface,which is filtered in its height spectral domain by a low pass filter. The small-scale roughness,represented by the high-frequency spectral region and a first-order perturbation solution of Burrow’s model

^{[11]},adequately describes the scattering properties of the surface. Meanwhile,the large-scale roughness,represented by the low-frequency spectral region,is assumed to be sufficiently smooth to form the unperturbed surface and to propose physical optics approaches for determining the scattering field. The final result is the sum of the contributions of both types of scale surface roughness.

These two-scale models,which are often used in ocean surfaces and ocean-like grounds,have been shown to yield excellent results^{[12]}. However,these models have their respective shortcomings. Although Ulaby’s TSM is simple,it cannot easily measure the large- and small-scale roughnesses in the field measurements. Brown’s TSM does not need to measure the two roughness sizes. Moreover,his model adopts a low pass filter with a rectangle window function as its spectral response; however,the rectangle-window filter is not suitable and its validity is related to the spectral shape of the surface,thereby presenting a constraint.

Therefore,a Modified Two-Scale Model (MTSM) is introduced in the current paper. This model presents two essential improvements. First,the randomly rough surface is analyzed in the spectrum domain,whereas the wavelet packet transform,which can freely choose wavelet packet bases and is more flexible than the Fourier transform,is introduced to split the surface height spectral and to calculate the two scale roughness. Second,we derive the formula of MTSM,in which KM is used to simulate the scattering of the large-scale roughness,while the first-order SPM is applied to calculate the small-scale roughness. The total scattering is the sum of the Kirchhoff and first-order SPM scattering.

This paper is arranged as follows: Section 2 discusses the method of splitting the spectral by using the wavelet packet,Section 3 derives the formula of the total scattering,Section 4 presents the validation results,and Section 5 concludes the paper.

2 Surface Roughness Decomposition by Wavelet Packet 2.1 Surface description
The randomly rough surface is assumed to be a Gaussian stationary stochastic process. For any point (*x*,*y*) on the surface,the surface height (*z*(*x*,*y*)) is a statistical variable with the following statistical characteristics:

$\quad\quad E\langle z(x,y)\rangle = 0$ | (1) |

$\quad\quad E\langle {z^2}(x,y)\rangle = {s^2}$ | (2) |

\[\quad \quad p(z(x,y)) = \frac{1}{{\sqrt {2\pi } s}}\exp \left\{ { - \frac{{z{{(x,y)}^2}}}{{2{s^2}}}} \right\}\] | (3) |

where *E*<> is the ensemble average,*s* is the RMS of surface height,*p*(*z*(*x*,*y*)) is the probability distribution function,and *z*(*x*,*y*) follows the Gaussian distribution.

According to composite surface scattering^{[13]} and stochastic process theories^{[14, 15]},a Gaussian stationary stochastic process can be decomposed into two independent Gaussian stationary stochastic processes. Therefore,the surface height *z*(*x*,*y*) may be rearranged (Fig. 1) as

$z(x,y) = {z_k}(x,y) + {z_s}(x,y)$ | (4) |

where *z _{k}*(

*x*,

*y*) is the height of large-scale roughness at (

*x*,

*y*),and

*z*(

_{s}*x*,

*y*) is the height of small-scale roughness.

Given that *z _{k}*(

*x*,

*y*) and

*z*(

_{s}*x*,

*y*) are independent,the two-scale surface height spectrum

*S*(

*k*,

_{x}*k*) may be expressed as

_{y}$S({k_x},{k_y}) = {S_s}({k_x},{k_y}) + {S_k}({k_x},{k_y})$ | (5) |

where *S _{k}*(

*k*,

_{x}*k*) and

_{y}*S*(

_{s}*k*,

_{x}*k*) are the height spectra for

_{y}*z*(

_{k}*x*,

*y*) and

*z*(

_{s}*x*,

*y*). In addition,

*k*and

_{x}*k*are the wave numbers of the electromagnetic field incident upon the rough surface along the

_{y}*x*and

*y*directions,respectively.

Eqs. (4) and (5),along with the fact that *z _{k}*(

*x*,

*y*) and

*z*(

_{s}*x*,

*y*) are zero-mean Gaussian,are crucial to the development of the model. If

*z*(

*x*,

*y*) is non-Gaussian,the densities of

*z*(

_{k}*x*,

*y*) and

*z*(

_{s}*x*,

*y*) and their derivatives cannot be uniquely defined

^{[16]}. More importantly,the spectral dichotomy,as expressed by Eq. (5),may be invalid.

The solid line represents the surface of *z*(*x*,*y*),and the dotted line *S*_{0 }represents the large-scale surface [*z _{k}*(

*x*,

*y*)] in the

*x-z*plane.

Wavelet is a time-frequency representation that has been used successfully in a broad range of applications,particularly in signal analysis^{[17, 18]}. Wavelet packets resulted from the development of wavelet Multi-Resolution Analysis (MRA),a generalization of the wavelet transform that allows for arbitrary tree-shape bandpass filtering,and can be adapted to the characteristics of the particular signal that is being analyzed.

Details of the wavelet transform can be found in several studies^{[17, 19, 20]}. In wavelet MRA,the Hilbert space [*L*^{2}(*R*)] is separated to the orthogonal sum of all subspaces given by

${L^2}(R) = \oplus {W_j} \,,\,\,j \in Z$ | (6) |

where $W_j$ represents the subspaces of the wavelet basis function j(*t*). To develop the frequency resolution,we further divide the wavelet subspace *W _{j }*according to the binary frequency segment. Therefore,the wavelet scaling subspaces

*V*and

_{j}*W*can be jointed and represented by a new space ${U_j}^n\left( {n \in N} \right)$. If we suppose

_{j}*U*

_{j}^{0}=

*V*and

_{j}*U*

_{j}^{1 }=

*W*,as the orthogonal decomposition formula of

_{j}*L*

^{2}(

*R*),${V_{j + 1}} = {V_j} \oplus {W_j}$ can be written as the decomposition of

*U*,which is expressed as

_{j}^{n}$U_{j + 1}^0 = U_j^0 \oplus U_j^1$ | (7) |

In Eq. (7),*U _{j}^{n}* and

*U*

_{j}^{2n}are defined as the wavelet subspaces of

*u*(

_{n}*t*) and

*u*

_{2n}(

*t*),respectively,and

*u*(

_{n}*t*) must satisfy the equations given by

$\quad\quad{u_{2n}}(t) = \sqrt 2 \sum\limits_{k \in Z} {g(k){u_n}(2t - k)} $ | (8) |

$\quad\quad{u_{2n + 1}}(t) = \sqrt 2 \sum\limits_{k \in Z} {h(k){u_n}(2t - k)} $ | (9) |

where *g*(*k*) and *h*(*k*) are the lowpass and highpass filter coefficient,respectively,and are related by

$h(k) = {( - 1)^k}g(1 - k)$ | (10) |

When *n*=0,Eqs. (8) and (9) can be rewritten as

${u_0}(t) = \sqrt 2 \sum\limits_{k \in Z} {g(k){u_0}(2t - k)} $ | (11) |

${u_1}(t) = \sqrt 2 \sum\limits_{k \in Z} {h(k){u_0}(2t - k)} $ | (12) |

The scaling function j(*t*) and basis function f(*t*) in wavelet MRA are respectively presented as

$\varphi (t) = \sqrt 2 \sum\limits_{k \in Z} {g(k)} \varphi (2t - k)$ | (13) |

$\phi (t) = \sqrt 2 \sum\limits_{k \in Z} {h(k)} \varphi (2t - k)$ | (14) |

By comparing Eqs. (11)–(14),we find that *u*_{0}(*t*) and *u*_{1}(*t*) are degenerated into j(*t*) and f(*t*),respectively. In addition,{*u*_{n}(*t*)} is the orthogonal wavelet packet that is determined by *u*_{0}(*t*) =

f(*t*). Therefore,the wavelet decomposition is the special column of the wavelet packet. When *n*=0,we use *u*_{0}(*t*) and *u*_{1}(*t*) as the subspaces of the wavelet packet,and the wavelet packet transform changes into the wavelet transform. However,if we continue decomposing the high-frequency spectral by MRA,the wavelet transform becomes the wavelet packet transform.

The surface roughness is decomposed by wavelet packet through the process described below.

**Step 1** The random surface height *z*(*x*,*y*) is transformed into spectral domain using the wavelet packet transform. The height spectral function is given by *S*(*k _{x}*,

*k*).

_{y}
**Step 2** A free wave number *k _{d}* is chosen to split the roughness. Therefore,the height spectrum for

*z*(

_{s}*x*,

*y*) in (|

*k*≥

_{x}*k*|∪|

_{d}*k*≥

_{y}*k*|) is

_{d}*S*(

_{s}*k*,

_{x}*k*),while that for

_{y}*z*(

_{k}*x*,

*y*) in |

*k*<

_{x}*k*|∩|

_{d}*k*<

_{y}*k*| is

_{d}*S*(

_{k}*k*,

_{x}*k*). The surface height spectrum can be computed using Eq. (5).

_{y}
**Step 3** An inverse transform of the wavelet packet is used to transform *S _{s}*(

*k*,

_{x}*k*) into the height function

_{y}*z*(

_{s}*x*,

*y*) of small-scale roughness. Next,we judge whether the RMS height

*s*and correlation length

_{s}*l*of

_{s}*z*(

_{s}*x*,

*y*) are consistent with the ranges of validity of the SPM (

*ks*0.3,

_{s}<*s*/

_{s}*l*<0.21)

_{s}^{[21]}. Otherwise,we repeat Step 2 and reselect the

*k*.

_{d}

**Step 4** When *k _{d}* is determined,

*S*(

_{s}*k*,

_{x}*k*) and

_{y}*S*(

_{k}*k*,

_{x}*k*) can be calculated. Given that both scales of the roughness can be obtained by the inverse transform of the wavelet packet,the surface roughness is successfully divided into two parts.

_{y}Fig. 2 illustrates the surface roughness decomposition by wavelet packet.

The far-zone equation of electromagnetic scattering is used to derive the formula of the MTSM. Following the Stratton-Chu equation^{[9]},the far-zone scattered electric field is computed in terms of tangential electric and magnetic fields expressed as

\[{E^s} = K{n_s} \times \int [(n \times E) - {\eta _s}{n_s} \times (n \times {\text{ }}H)]{{\text{e}}^{{\text{j}}{k_s}r \times {n_s}}}{\text{d}}S\] | (15) |

$K = - {\rm j}{k_s}{{\rm e}^{ - {\rm j}{k_s}{R_0}}}/(4 {π} {R_0}) \hspace{95pt}$ | (16) |

where * E* and

*are the tangential electric and magnetic fields,respectively;*

**H**

**n**_{s}is the unit vector in the scattering direction;

*is the surface normal unit vector; h*

**n***is the intrinsic impedance in the upper medium;*

_{s}*S*is the area of scattering cross section; j is the imaginary unit;

*k*is the wave number in the upper medium;

_{s}*r*is the position vector of

*; and*

**r***R*

_{0}is the distance from the center of the illuminated area to the receiver. The scattered electric field in the upper medium can be expressed as

${E^{s}} \!=\! K{{n}_s} \! \times \!\! \int \!\! {[({{n}_1} \!\! \times \!\! E) \!-\! {\eta _1}{{n}_s} \! \times \! ({{n}_1} \!\! \times \!\! H)]} {{\rm{e}}^{{\rm j}{k_1}({{n}_s} - {{n}_i}){r}'}}{\rm{d}}S'$ | (17) |

where **n**_{1} is the normal unit vector of the upper medium,* n_{i}* is the unit vector in the incident direction,

*S'*is the scattering cross section of

*r'*,

*k*

_{1}is the wave number in the upper medium,and

*is the position vector of*

**r**'*r'*.

The phase factor of Eq. (17) can be written as

${k_1}({{n}_s} - {{n}_i}) \cdot {r}' = {q_x}x' + {q_y}y' + {q_z}z'$ | (18) |

where (*x'*,*y'*,*z'*) is the Cartesian coordinate of *r'*. Next,*q _{x}*,

*q*,and

_{y}*q*are respectively computed as

_{z}$\left\{ \begin{array}{l} {q_x} = {k_1}(\sin {\theta _s}\cos {\varphi _s} - \sin \theta_i\cos \varphi_i)\\ {q_y} = {k_1}(\sin {\theta _s}\sin {\varphi _s} - \sin \theta_i\sin \varphi_i)\\ {q_z} = {k_1}(\cos {\theta _s} + \cos \theta_i) \end{array} \right.$ | (19) |

where q* _{i}* and q

*are the incident and scattering angles,respectively,and j*

_{s}*and j*

_{i}*are the incident and scattering azimuth angles,respectively.*

_{s}
If we substitute *k*_{1}(**n**_{s}–* n_{i}*)

*r'*by Eq. (18),Eq. (17) can be rewritten as

$\begin{aligned} {E^{s}} = & K{{n}_s} \times \int {\left[{{{n}_1} \times {E} - {\eta _1}{{n}_s} \times \left( {{{n}_1} \times {H}} \right)} \right]} \\ & .\exp \left( {{q_x}x' + {q_y}y' + {q_z}z'} \right){\rm{d}}S' \end{aligned}$ | (20) |

We then suppose $z' = {z'\!_k} + {z'\!_s}$ and compute *E ^{s}* as

$\begin{aligned} E_{}^s = & K{{n}_s} \times \int {\left[{{{n}_1} \times E - {\eta _1}{{n}_s} \times \left( {{{n}_1} \times {H}} \right)} \right]} \\ & .\exp \left( {{\rm j}{q_x}x' + {\rm j}{q_y}y'} \right)\exp \left[{{\rm j}{q_z}\left( {{{z'\!\!_k}} + {{z'\!\!_s}}} \right)} \right]{\rm d}S'\\ = & K{{n}_s} \times \int {B \cdot \exp \left[{{\rm j}{q_z}\left( {{{z'\!\!_k}} + {{z'\!\!_s}}} \right)} \right]{\rm d}S'} \end{aligned}$ | (21) |

$\hspace{5pt} B \! = \! \left[{{{n}_1} \! \times \! {E} \! - \! {\eta _1}{{n}_s} \! \times \! \left( {{{n}_1} \! \times \! {H}} \right)} \right]\exp \left( {{\rm j}{q_x}x' \!\! + \! {\rm j}{q_y}y'} \right)$ | (22) |

Obviously,*B* is irrelevant with *z'*. According to Taylor’s law,exp[j*qz*$\left( {{{z'\!\!_k}} + {{z'\!\!_s}}} \right)$] can be expressed as

$\exp \left[{{\rm j}{q_z}\left( {{{z'\!\!_k}} + {{z'\!\!_s}}} \right)} \right] = 1 + \sum\limits_{n{\rm{ = 1}}}^\infty {\frac{1}{n}{{\left[{{\rm j}{q_z}\left( {{{z'\!\!_k}} + {{z'\!\!_s}}} \right)} \right]}^n}} $ | (23) |

Thus,the scattering electric field *E ^{s}* is rewritten as

$E_{}^s = K{{n}_s} \! \times \! \int \! {B \! \cdot \! \left(\!1 \!+\! \sum\limits_{{n \rm{= 1}}}^\infty {\frac{1}{n}{{\left[{{\rm j}{q_z}\left( {{{z'\!\!_k}} \!+\! {{z'\!\!_s}}} \right)} \right]}^n}}\!\right) {\rm d}S'} $ | (24) |

Meanwhile,the backscattering coefficient is computed as

$\sigma _{pq}^o({\theta _i}) = \frac{{4 {π} {R_0^2}{\mathop{\rm Re}\nolimits} \left\{ {{\left\langle {{{\left| {E_{}^sE{{_{}^s}^*}} \right|}^2}} \right\rangle } \mathord{\left/ {\vphantom {{\left\langle {{{\left| {E_{}^sE{{_{}^s}^*}} \right|}^2}} \right\rangle } {{\eta ^ * }_s}}} \right. } {{\eta ^ *_s }}}\right\} }}{{{A_0}{\mathop{\rm Re}\nolimits} \left\{ {{{{\left| {{E_0}} \right|}^2}} \mathord{\left/ {\vphantom {{{{\left| {{E_0}} \right|}^2}} {{\eta ^ * }_1}}} \right. } {{\eta ^ *_1 }}}\right\} }}$ | (25) |

where *pq* is the polarization manner,*A*_{0} is the illuminated area,Re{} is the real part operator,* denotes the complex conjugate,and $\left\langle {{{\left| {E_{}^sE{{_{}^s}^*}} \right|}^2}} \right\rangle $ is expressed as

$\begin{array}{l} \left\langle {{{\left| {E_{}^sE{{_{}^s}^*}} \right|}^2}} \right\rangle = K{{n}_s} \times \int {B \cdot \exp \left[{{\rm j}{q_z}\left( {z'} \right)} \right]{\rm d}S'} \cdot K{{n}_s} \times \int {B \cdot \exp \left[{ - {\rm j}{q_z}\left( {z''} \right)} \right]{\rm d}S''} \\ \quad \quad \quad \quad \quad = \iint C\left\langle {\exp \left[{{\rm j}{q_z}\left( {{{z'\!\!_k}} - {{z''\!\!\!_k}} + {{z'\!\!_s}} - {{z''\!\!\!_s}}} \right)} \right]} \right\rangle {\rm d}S'{\rm d}S''\\ \quad \quad \quad \quad \quad =\iint C \left\langle {\left(1 + \sum\limits_{n = 1}^\infty {\frac{1}{{n!}}{{\left[{{\rm j}{q_z}\left( {{{z'\!\!_k}} - {{z''\!\!\!_k}}} \right) + {\rm j}{q_z}\left( {{{z'\!\!_s}} - {{z''\!\!\!_s}}} \right)} \right]}^n}} \right)} \right\rangle {\rm d}S'{\rm d}S''\\ \quad \quad \quad \quad \quad =\iint C \left(1 + \sum\limits_{n = 1}^\infty {\frac{1}{{n!}}\left\langle {{{\left[{{\rm j}{q_z}\left( {{{z'\!\!_k}} - {{z''\!\!\!_k}}} \right) + {\rm j}{q_z}\left( {{{z'\!\!_s}} - {{z''\!\!\!_s}}} \right)} \right]}^n}} \right\rangle }\right) {\rm d}S'{\rm d}S'' \end{array}$ | (26) |

In Eq. (26),$z'' = z''\!\!\!_{k} + z''\!\!\!_{s}$,$S''$ is the scattering cross section of *r''*,and *C* is independent of *z'* and *z''*. If we expand the *n*-order terms in Eq. (26),then we obtain the equation

$\begin{aligned} \left\langle {{{\left| {E_{}^sE{{_{}^s}^*}} \right|}^2}} \right\rangle = & \iint C \left(1 + \sum\limits_{n = 1}^\infty {\frac{1}{{n!}}[\left\langle {{{\left[{{\rm j}{q_z}\left( {{{z'\!\!_k}} - {{z''\!\!\!_k}}} \right)} \right]}^n}} \right\rangle + \cdots + C_n^k\left\langle {{{\left[{{\rm j}{q_z}\left( {{{z'\!\!_k}} - {{z''\!\!\!_k}}} \right)} \right]}^{n - k}}{{\left[{{\rm j}{q_z}\left( {{{z'\!\!_s}} - {{z''\!\!\!_s}}} \right)} \right]}^k}} \right\rangle + } \cdots \right. \\ & + \left\langle {{{\left[{{\rm j}{q_z}\left( {{{z'\!\!_s}} - {{z''\!\!\!_s}}} \right)} \right]}^n}} \right\rangle] \Biggr){\rm d}S'{\rm d}S'' \end{aligned}$ | (27) |

Given that the large-scale roughness surface *z _{k}* is independent of the small-scale roughness surface

*z*,the ensemble average of the cross-term is 0 in Eq. (27)

_{s}^{[9]}. Thus,we obtain the equation

$\left\langle {{{\left[{{\rm j}{q_z}\left( {{{z'\!\!_k}} - {{z''\!\!\!_k}}} \right)} \right]}^{n - k}}{{\left[{{\rm j}{q_z}\left( {{{z'\!\!_s}} - {{z''\!\!\!_s}}} \right)} \right]}^k}} \right\rangle = 0,k = 1,2,\cdots ,n - 1$ | (28) |

$\begin{aligned} \left\langle {{{\left| {E_{}^sE{{_{}^s}^*}} \right|}^2}} \right\rangle & = \iint C \left(1 + \sum\limits_{n = 1}^\infty {\frac{1}{{n!}}[\left\langle {{{\left[{{\rm j}{q_z}\left( {{{z'\!\!_k}} - {{z''\!\!\!_k}}} \right)} \right]}^n}} \right\rangle + } \left\langle {{{\left[{{\rm j}{q_z}\left( {{{z'\!\!_s}} - {{z''\!\!\!_s}}} \right)} \right]}^n}} \right\rangle]\right){\rm d}S'{\rm d}S''\\ & = \iint C \left(1+ \sum\limits_{n = 1}^\infty {\frac{1}{{n!}}\left\langle {{{\left[{{\rm j}{q_z}\left( {{{z'\!\!_k}} - {{z''\!\!\!_k}}} \right)} \right]}^n}} \right\rangle } \right){\rm d}S'{\rm d}S'' + \iint C\sum\limits_{n = 1}^\infty {\frac{1}{{n!}}\left\langle {{{\left[{{\rm j}{q_z}\left( {{{z'\!\!_s}} - {{z''\!\!\!_s}}} \right)} \right]}^n}} \right\rangle } {\rm d}S'{\rm d}S''\\ & = \iint C \exp \left[{{\rm j}{q_z}\left( {{{z'\!\!_k}} - {{z''\!\!\!_k}}} \right)} \right]{\rm d}S'{\rm d}S'' + \iint C\sum\limits_{n = 1}^\infty {\frac{1}{{n!}}\left\langle {{{\left[{{\rm j}{q_z}\left( {{{z'\!\!_s}} - {{z''\!\!\!_s}}} \right)} \right]}^n}} \right\rangle } {\rm d}S'{\rm d}S'' \end{aligned}$ | (29) |

The first term in Eq. (29) is the scattering field of large-scale roughness,which is calculated by KM. The second term represents the scattering field of small-scale roughness,which can be simulated by SPM without the zero-order solutions. Given that the first-order solutions of SPM represent the surface scattering,the second-order and the other higher-order solutions represent the complex volume scattering. We skip the volume scattering and simply calculate the first-order solutions of SPM. As a result,we substitute $\left\langle {{{\left| {E_{}^sE{{_{}^s}^*}} \right|}^2}} \right\rangle $ into Eq. (25) by Eq. (29). The scattering coefficient of the two-scale surface can be written as

$\sigma _{pq}^o({\theta _i}) = {\sigma _{pq}}{( {\theta _i})^{\rm{KA}}} + {\sigma _{pq}^{(1)}}{({\theta _i})^{\rm{SPM}}}$ | (30) |

where${\sigma _{pq}}{( {\theta _i})^{\rm{KA}}}$ is the scattering contribution of large-scale roughness at q* _{i}*,and ${\sigma _{pq}^{(1)}}{({\theta _i})^{\rm{SPA}}}$ is the first-order solutions of SPM for small-scale roughness. Given that the small-scale roughness lays on the top of the large-scale roughness (

*i*.

*e*.,the tilting effect),we calculate the ensemble average of SPM to correct such effect. Thus,we obtain the equation

$\sigma _{pq}^o({\theta _i}) = {\sigma _{pq}}{( {\theta _i})^{{\rm{KA}}}} + \left\langle {{\sigma _{pq}^{(1)}}{{({\theta _i})}^{{\rm{SPM}}}}} \right\rangle $ | (31) |

Then,we calculate the solutions of ${\sigma _{pq}}{( {\theta _i})^{{\rm{KA}}}}$$\left\langle {{\sigma _{pq}^{(1)}}{{({\theta _i})}^{{\rm{SPM}}}}} \right\rangle $.

3.2 Kirchhoff scattering field
The KM applies tangent plane approximation to calculate the surface fields,which is only valid if the radius of the curvature is larger than a wavelength. When the surface roughness is relatively large (*kl*>6 and*l*^{2}>2.76*s*l),the KM is reduced to the Geometric Optics model by phase approximation,and the backscattering coefficient is computed as Ref. ^{[13]}

${\sigma _{pq}}{( {\theta _i})^{{\rm{KA}}}} = \frac{{{{\left| {{\Gamma _{pq}}(0)} \right|}^2}}}{{2{m^2}{{\cos }^4}{\theta _i}}}\exp \left( - \frac{{{{\tan }^2}{\theta _i}}}{{2{m^2}}}\right)$ | (32) |

where *m* is the average slope,$m = \sqrt 2 {s \mathord{\left/ {\vphantom {s l}} \right. } l}$ for the Gaussian surface,and G* _{pq}*(0) is the Fresnel reflection coefficient in the normal direction (q

*=0). If*

_{i}*p*≠

*q*,${\sigma _{pq}}{( {\theta _i})^{{\rm{KA}}}} = 0$,G

*(0) is computed as*

_{pq}${\Gamma _{{\rm{hh}}}}(0) = {\Gamma _{{\rm{vv}}}}(0) = \left| {\frac{{1 - \sqrt \varepsilon }}{{1 + \sqrt \varepsilon }}} \right|$ | (33) |

where e is a dielectric constant.

When 0.05l<*s*<0.15l,*l*>l and *m*<0.25,the KM is reduced to the Physic Optics model by scalar approximation. The backscattering coefficient is then computed as Ref. ^{[13]}

$\begin{aligned} {\sigma _{pq}}{( {\theta _i})^{\rm{KA}}} = & {k^2}{\cos ^2}{\theta _i}{\Gamma _{pq}}({\theta _i})\exp ( - {(2ks\cos {\theta _i})^2})\\ & \cdot \sum\limits_{n = 1}^\infty {\frac{{{{(2ks\cos ({\theta _i}))}^{2n}}}}{{n!}}} {W^n}(2k\sin ({\theta _i}),0) \end{aligned}$ | (34) |

where ${W^n}(2k\sin {\theta _{i}},0)$ is the roughness spectrum of the surface that is related to the *n*-th power of the surface correlation function by the Fourier transform. For the Gaussian surface,${W^n}(2k\sin {\theta _{i}},0)$ is expressed as

${W^n}(2k\sin ({\theta _i}),0) = \frac{{{l^2}}}{n}\exp \left\{ - \frac{{{{(kl\sin {\theta _i})}^2}}}{n}\right\} $ | (35) |

As stated above,G* _{pq}*(q

*) is the Fresnel reflection coefficient,and is denoted as*

_{i}${\Gamma _{\rm hh}}({\theta _i}) = {\left| {\frac{{\cos {\theta _i} - \sqrt {\varepsilon - {{\sin }^2}{\theta _i}} }}{{\cos {\theta _i} + \sqrt {\varepsilon - {{\sin }^2}{\theta _i}} }}} \right|^2} \hspace{12pt}$ | (36) |

${\Gamma _{{\rm{vv}}}}({\theta _i}) = {\left| {\frac{{\varepsilon \cdot \cos {\theta _i} - \sqrt {\varepsilon - {{\sin }^2}{\theta _i}} }}{{\varepsilon \cdot \cos {\theta _i} + \sqrt {\varepsilon - {{\sin }^2}{\theta _i}} }}} \right|^2}$ | (37) |

In the two-scale surface,given that the large-scale roughness is covered by small-scale roughness,the Fresnel reflection coefficient in KA must be modified as^{[22]}

$\begin{aligned} \left\langle {{\Gamma _{pq}}} \right\rangle = & {\Gamma _{pq}}({\theta _i})\biggr[1 - 2k\sin {\theta _i}\\ & \cdot \int\nolimits_{ - \infty }^\infty \!\! \int\nolimits_{ - \infty }^\infty \! W(u \!-\! k\sin {\theta _i},v) \\ & \times \! {K_{pq}}(u,v){\rm{d}}u{\rm{d}}v \biggr] \end{aligned}$ | (38) |

where G* _{pq}* is the corrected Fresnel reflection coefficient,

*K*(

_{pq}*u*,

*v*) is the KM filed coefficient,and

*W*(

*u*–

*k*sinq

*,*

_{i}*) is the roughness spectrum.*

_{v}
The SPM is valid for the rough surface with a small RMS height and small slope; its effective applying ranges are *ks*<0.3 and *s*/*l*<0.21. According to the former analysis in Eq. (30),the first-order solutions of SPM is expressed as

${\sigma _{pq}^{(1)}}{({\theta _i})^{{\rm{SPM}}}} = 8{k^4}{s^2}{\cos ^4}{\theta _i}{\left| {{a_{pq}}({\theta _i})} \right|^2}W(2k\sin {\theta _i},0)$ | (39) |

where *a _{pq}*(q

*) is the polarized amplitude coefficient in SPM,whereas*

_{i}$\quad\quad{a_{{\rm{hh}}}}({\theta _i}) = \frac{{{\varepsilon _s} - 1}}{{{{\left(\cos {\theta _i} + \sqrt {{\varepsilon _s} - {{\sin }^2}{\theta _i}} \right)}^2}}}$ | (40) |

$\quad\quad{a_{{\rm{vv}}}}({\theta _i}) = ({\varepsilon _s} \! - \! 1)\frac{{{{\sin }^2}{\theta _i} \! - \! {\varepsilon _s}(1 \! + \! {{\sin }^2}{\theta _i})}}{{{{\left({\varepsilon _s}\cos {\theta _i} \! + \! \sqrt {{\varepsilon _s} \! - \! {{\sin }^2}{\theta _i}} \right)}^2}}}$ | (41) |

In the two-scale surface,the small-scale roughness lays on the large-scale roughness. Therefore,the tilting effect must be corrected. In the next step,*z _{kx}* and

*z*are assumed to be the slopes of the large-scale roughness (

_{ky}*z*) in the

_{k}*x*-axis and

*y*-axis,respectively,while

*z*,

_{kx}*z*) is the probability distribution function of

_{ky}*z*and

_{kx}*z*. A simple rough surface is used to explain the probability distribution function,as shown in Fig. 3,in which a simple rough surface comprises three lines,namely,

_{ky}*l*

_{1},

*l*

_{2},and

*l*

_{3},in a one-dimensional plane. In addition,

*x*–

*z*,

*z*

_{x}_{2}is the slope of

*l*

_{2},while

*P*(

*z*

_{x}_{2}) is the probability distribution function of

*z*

_{x2}. In the vertical direction

*z*,

*P*(

*z*

_{x2}) can be written as

^{[23]}

$P({z_{x2}}) = \frac{{{{l}_2}{{n}_2} \cdot {z}}}{{({{l}_1}{{n}_1} + {{l}_2}{{n}_2} + {{l}_3}{{n}_3}) \cdot {z}}} = \frac{{{{l}_2}{{n}_2} \cdot {z}}}{L}$ | (42) |

$L = ( {{l}_1}{{n}_1} + {{l}_2}{{n}_2} + {{l}_3}{{n}_3}) \cdot {z} \hspace{73pt}$ | (43) |

In the direction of incident q* _{i}*,

*P*(

*z*

_{x2}) is expressed as

${P_{\theta_i} }({z_{x2}}) = \frac{{{{l}_2}{{n}_2} \cdot {{k}_i}}}{{( {{l}_1}{{n}_1} + {{l}_2}{{n}_2} + {{l}_3}{{n}_3}) \cdot {k_i}}} = \frac{{{{l}_2}{{n}_2} \cdot {{k}_i}}}{{{L_{\theta_i} }}}$ | (44) |

${L_{\theta_i}} = ({{l}_1}{{n}_1} + {{l}_2}{{n}_2} + {{l}_3}{{n}_3}) \cdot {{k}_i}\hspace{73pt}$ | (45) |

where *k _{i}* is the unit vector in the incident direction. From Eqs. (42) to (45),we obtain the following:

${P_{\theta_i}}({z_{x2}}) = \frac{L}{{{L_{\theta_i} }}}\cos {\theta _i}(1 + {z_{x2}}\tan {\theta _i})P({z_{x2}})$ | (46) |

Given that only three roughnesses are observed in Fig. 3,the sum of the probability distribution function is 1,that is,

$\sum\limits_{m = 1}^3 {{P_{\theta_i}}({z_{xm}})} = 1$ | (47) |

Therefore,

$\frac{L}{{L_{\theta_i }}}\cos {\theta _i} = 1$ | (48) |

As a result,in the two-dimensional random surface,${P_{{\theta _{i}}}}$(*z _{kx}*,

*z*) can be written as

_{ky}${P_{\theta_i} }({z_{kx}},{z_{ky}}) = (1 + {z_{kx}}\tan {\theta _i})P({z_{kx}},{z_{ky}})$ | (49) |

where *P*(*z _{kx}*,

*z*) is the probability distribution function of the large-scale roughness and follows a (0,$\sigma _{zk}^2$) Gaussian distribution given by

_{ky}$P({z_{kx}},{z_{ky}}) = \frac{1}{{2\pi \sigma _{Zk}^2}}\exp \left\{ { - \frac{{z_{kx}^2 + z_{ky}^2}}{{2\sigma _{Zk}^2}}} \right\}$ | (50) |

Therefore,the second term in Eq. (31) is expressed as^{[23]}

$\begin{aligned} \! \lt \!\! {\sigma _{pq}^{(1)}}\!({\theta _i}) \!\! { > ^{\rm{SPM}}} \!=\!\! & \int\nolimits_{ - \infty }^\infty \!\! {\int\nolimits_{ - \cot {\theta _i}}^\infty \!\!\!\! {{{(p \! \cdot \! p')}^4}{\sigma _{pq}^{(1)}}({{\theta '}_i})} } \\ & \cdot (1 \!+\! {z_{kx}} \! \tan \! {\theta _i}) P({z_{kx}},{z_{ky}}){\rm d}{z_{kx}}{\rm d}{z_{kx}} \end{aligned}$ | (51) |

where *p* and *p'* are the unit polarized vectors of the basic and local coordinate systems,respectively,and is the local incident angle. To avoid the self-shadowing effect,the integral limit of *z _{kx}* in Eq. (51) is within the range of –cotq

*to $\infty$.*

_{i}The AIEM,which is the improvement of IEM,is adopted to verify the MTSM. The AIEM has the widest range of applicability; thus,it is the most popular method for calculating the surface scattering. In this model,the single scattering term is expressed as

$\begin{aligned} {\sigma _{pqs}} = & \frac{{{k_1^2}}}{2}\exp ( - {s^2}({k_z^2} + {k_{sz}^2}))\\ & \cdot \sum\limits_{n = 1}^\infty {\frac{{{s^{2n}}}}{{n!}}{{\left| {{I_{pq}^n}} \right|}^2}} {W^n}({k_{sx}} - {k_x},{k_{sy}} - {k_y}) \end{aligned} $ | (52) |

$\begin{aligned} I_{pq}^n = & {({k_{sz}} \!+\! {k_z})^n}{f_{pq}}\exp ( - {s^2}{k_z}{k_{sz}})\\ & + \frac{{{{( \!{k_{sz}})}^n}{F_{pq}}(\! - {k_x},- {k_y}) \!+\! {{(\!{k_z})}^n}{F_{pq}}(\! - {k_{sx}},- {k_{sy}})}}{2} \end{aligned} $ | (53) |

where *W ^{n}*(

*k*–

_{sx}*k*–

_{x},k_{sy}*k*) is the roughness spectrum of the surface that is related to the

_{y}*n-*th power of the surface correlation function by the Fourier transform,

*and*

*f*and

_{pq}*F*are the Kirchhoff and complementary field coefficients,respectively. In addition,

_{pq}*k*=

_{z}*k*cosq

*,*

_{i}*k*=

_{sz}*k*cosq

*,*

_{s}*k*=

_{x}*k*sinq

_{i}·cosj

*,*

_{i}*k*=

_{sx}*k*sinq

*cosj*

_{s}*,*

_{s}*k*=

_{y}*k*sinq

*sinj*

_{i}*,and*

_{i}*k*=

_{sy}*k*sinq

*sinj*

_{s}*.*

_{s}
The inputs of simulations are set as follows: frequency *f*=5.63 GHz,complex dielectric constant e* _{s}*=19.54+3.08

*i*,and incident angle q

*$ \in $ [10°,70°]. Three different Gaussian surfaces (*

_{i}*ks*=0.354,1.061,2.352;

*kl*=8.21) are assumed to test the MTSM. The selected wave number

*k*for splitting the roughness in MTSM must satisfy the two requirements given by

_{d}$\quad\quad\quad\quad {s_s} = \frac{s}{5},\quad ks \lt 1.5$ | (54) |

$\quad\quad\quad\quad k{s_s} = 0.3,ks \ge 1.5$ | (55) |

Fig. 4 shows the scatter plots of the backscattering coefficients estimated using the MTSM versus AIEM on three different roughness surfaces. As can be seen,the MTSM is in favorable agreement with the AIEM at small incident angles. When q* _{i}* increases,the difference between these two models becomes larger,reaching the largest value of 2.561 dB at q

*= 60° in VV polarization. When the surface roughness is small (*

_{i}*ks*= 0.354),the MTSM provides very accurate predictions,presenting less than 0.68 dB difference with the AIEM for both VV and HH polarizations. As the surface roughness increases,the results of the MTSM move far from those of the AIEM.

Fig. 4(c) shows that the difference in the results of these two models exceeds 2 dB for both polarizations when q

*>45°. The estimation of the MTSM in VV polarization,instead of HH,is slightly closer to that of the AIEM,indicating that the polarization manner has slightly influenced the MTSM. Therefore,the MTSM can achieve favorable accuracy when both the incident angle and surface roughness are small.*

_{i}
This paper proposed a new model (MTSM) for describing the backscattering for a Gaussian-distributed rough surface. The surface height function *z*(*x*,*y*) is transformed into a frequency domain by wavelet packet,and the surface height spectrum is split into two parts. The large-scale roughness corresponding to the low frequency portion of the spectrum (*k*<*k _{d}*) assumes that the KM adequately describes the scattering from these height excursions. The small-scale roughness is represented by the high frequency portion of the total height spectrum (

*k*≥

*k*),which is described by a first order SPM. The key parameter

_{d}*k*for spectral dichotomy is determined using Eq. (49). The tilting effect caused by the large-scale roughness has been corrected,and the backscattering of the MTSM represents the sum of both scale roughness surfaces. From the numerical experiments,several conclusions are drawn.

_{d}(1) Large- and small-scale roughnesses cannot be easily measured using the traditional TSM. Hence,wavelet packet transform is used to split the surface height spectrum into two parts,thereby increasing the usefulness of the TSM in practice.

(2) The simulations of the AIEM and MTSM are in excellent agreement when both the incident angle and surface roughness are small. The different polarization manner has a slight influence on the MTSM.

(3) The MTSM is established to describe the electromagnetic scattering properties of a typical two-scale land surface. This model has only been tested by numerical simulation using the AIEM model. The application of this model to natural surfaces and the ranges of validity of its surface roughness should be investigated further in future research.

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