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_i>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^[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.

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:

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

图 1(Fig.1)

Fig.1 Decomposition of the two-scale roughness surface

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_y) may be expressed as

where S_k(k_x,k_y) and S_s(k_x,k_y) are the height spectra for z_k(x,y) and z_s(x,y). In addition,k_x and k_y are the wave numbers of the electromagnetic field incident upon the rough surface along the 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 S₀ 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 [L²(R)] is separated to the orthogonal sum of all subspaces given by

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_j and W_j can be jointed and represented by a new space ${U_j}^n\left( {n \in N} \right)$. If we suppose U_j⁰ = V_j and U_j¹ = W_j,as the orthogonal decomposition formula of L²(R),${V_{j + 1}} = {V_j} \oplus {W_j}$ can be written as the decomposition of U_jⁿ,which is expressed as

In Eq. (7),U_jⁿ and U_j²ⁿ 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

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

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

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

By comparing Eqs. (11)–(14),we find that u₀(t) and u₁(t) are degenerated into j(t) and f(t),respectively. In addition,{u_n(t)} is the orthogonal wavelet packet that is determined by u₀(t) =
f(t). Therefore,the wavelet decomposition is the special column of the wavelet packet. When n=0,we use u₀(t) and u₁(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_d|) is S_s(k_x,k_y),while that for z_k(x,y) in |k_x<k_d|∩|k_y<k_d| is S_k(k_x,k_y). The surface height spectrum can be computed using Eq. (5).

Step 3 An inverse transform of the wavelet packet is used to transform S_s(k_x,k_y) into the height function z_s(x,y) of small-scale roughness. Next,we judge whether the RMS height s_s and correlation length l_s of z_s(x,y) are consistent with the ranges of validity of the SPM (ks_s<0.3, s_s/l_s<0.21)^[21]. Otherwise,we repeat Step 2 and reselect the k_d.  

Step 4 When k_d is determined,S_s(k_x,k_y) and S_k(k_x,k_y) 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.

Fig. 2 illustrates the surface roughness decomposition by wavelet packet.

图 2(Fig.2)

Fig.2 Decomposition of the two-scale roughness surface

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

where E and H are the tangential electric and magnetic fields,respectively; n_s is the unit vector in the scattering direction; n is the surface normal unit vector; h_s is the intrinsic impedance in the upper medium; S is the area of scattering cross section; j is the imaginary unit; k_s is the wave number in the upper medium; r is the position vector of r; and R₀ 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

where n₁ 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',k₁ is the wave number in the upper medium,and r' is the position vector of r'.

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

where (x',y',z') is the Cartesian coordinate of r'. Next,q_x,q_y,and q_z are respectively computed as

where q_i and q_s are the incident and scattering angles,respectively,and j_i and j_s are the incident and scattering azimuth angles,respectively.

If we substitute k₁(n_s–n_i)r' by Eq. (18),Eq. (17) can be rewritten as

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

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

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

Meanwhile,the backscattering coefficient is computed as

where pq is the polarization manner,A₀ 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

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

Given that the large-scale roughness surface z_k is independent of the small-scale roughness surface z_s,the ensemble average of the cross-term is 0 in Eq. (27)^[9]. Thus,we obtain the equation

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

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

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

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 andl²>2.76sl),the KM is reduced to the Geometric Optics model by phase approximation,and the backscattering coefficient is computed as Ref. ^[13]

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_i=0). If p≠q,${\sigma _{pq}}{( {\theta _i})^{{\rm{KA}}}} = 0$,G_pq(0) is computed as

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]

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

As stated above,G_pq(q_i) is the Fresnel reflection coefficient,and is denoted as

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]

where G_pq is the corrected Fresnel reflection coefficient,K_pq(u,v) is the KM filed coefficient,and W(u–ksinq_i,_v) is the roughness spectrum.

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

where a_pq(q_i) is the polarized amplitude coefficient in SPM,whereas

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_ky are assumed to be the slopes of the large-scale roughness (z_k) in the x-axis and y-axis,respectively,while ${P_{{\theta _{i}}}}$(z_kx,z_ky) is the probability distribution function of z_kx and z_ky. 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,l₁,l₂,and l₃,in a one-dimensional plane. In addition,x–z,z_x2 is the slope of l₂,while P(z_x2) is the probability distribution function of z_x2. In the vertical direction z,P(z_x2) can be written as^[23]

In the direction of incident q_i, P(z_x2) is expressed as

where k_i is the unit vector in the incident direction. From Eqs. (42) to (45),we obtain the following:

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

图 3(Fig.3)

Fig.3 Geometric explanation of the gradient’s probability distribution[23]

Therefore,

As a result,in the two-dimensional random surface,${P_{{\theta _{i}}}}$(z_kx,z_ky) can be written as

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

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

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_i to $\infty$.

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

where Wⁿ(k_sx–k_x,k_sy–k_y) is the roughness spectrum of the surface that is related to the n-th power of the surface correlation function by the Fourier transform, and f_pq and F_pq are the Kirchhoff and complementary field coefficients,respectively. In addition,k_z=kcosq_i,k_sz=kcosq_s,k_x=ksinq_i
·cosj_i,k_sx=ksinq_scosj_s,k_y=ksinq_isinj_i,and k_sy=ksinq_ssinj_s.

The inputs of simulations are set as follows: frequency f=5.63 GHz,complex dielectric constant e_s=19.54+3.08i,and incident angle q_i $ \in $ [10°,70°]. Three different Gaussian surfaces (ks=0.354,1.061,2.352; kl=8.21) are assumed to test the MTSM. The selected wave number k_d for splitting the roughness in MTSM must satisfy the two requirements given by

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

图 4(Fig.4)

Fig.4 Comparison of the backscattering coefficients predicated by the AIEM and MTSM (Here, kl = 8.21, qi Є [10°, 70°] and es = 19.54+3.08i for different roughnesses)

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_d),which is described by a first order SPM. The key parameter k_d 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.

(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.