The microwave vector radiative transfer (VRT) equation of a coated spherical bubble layer is derived by means of the second-order Rayleigh approximation field when the microwave wavelength is larger than the coated spherical particle diameter. Meanwhile, the perturbation method is developed to solve the second-order Rayleigh VRT equation for the small ratio of the volume scattering coefficient to the extinction coefficient. As an example, the emissive properties of a sea surface foam layer, which consists of seawater coated bubbles, are investigated. The extinction, absorption, and scattering coefficients of sea foam are obtained by the second-order Rayleigh approximation fields and discussed for the different microwave frequencies and the ratio of inner radius to outer radius of a coated bubble. Our results show that in the dilute limit, the volume scattering coefficient decreases with increasing the ratio of inner radius to outer radius and decreasing the frequencies. It is also found that the microwave emissivity and the extinction coefficient have a peak at very thin seawater coating and its peak value decreases with frequency decrease. Furthermore, with the VRT equation and effective medium approximation of densely coated bubbles, the mechanism of sea foam enhancing the emissivity of a sea surface is disclosed. In addition, excellent agreement is obtained by comparing our VRT results with the experimental data of microwave emissivities of sea surface covered by a sea foam layer at L-band (1.4 GHz) and the Camps' model.
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 In the remote sensing of environment, the various shapes and components of nature composites, e.g., porous rocks, colloidal suspensions, clays, trees, leaves and crop, always give rise to a large influence on retrieving environmental physical and geometrical parameters [Rosenkranz and Staelin, 1972; Reul and Chapron, 2003; Di Vittorio, 2009]. Composite materials classified as coated structures are very common in natural environment, e.g., living cells, fruits, stalks and sea foam. Because of the effects of coating matter, the microwave properties of coated composites have attracted much attention in both theoretical and experimental studies [Dombrovskiy, 1982; Anguelova, 2008]. For example, in studies of thermal emission from crop canopies (rice, soya beans, corn), the empirical mean dielectric constant of crop canopy was used as a random mixture of fruit and stalks in the vector radiative transfer (VRT) equation [Jin et al., 1992]. In remote sensing of the ocean, because the sea surface foam (aggregates of seawater coated bubbles) produced by wave breaking plays a key role in estimating the sea surface temperature, wind speed, sea surface salinity (SSS) and chlorophyll concentrations, various empirical methods were proposed to investigate the influence of foam on the remotely sensed data. Some examples are least squares fit of foam emissivity or brightness temperature as a function of incidence angle and frequency [Stogryn, 1972; Smith, 1988; Rose et al., 2002], wind speed model of foam emissivity [Wilheit, 1979], emissivity model from foam structures [Militskii et al., 1978] and effective medium model [Droppleman, 1970]. Although the empirical models are useful in some cases, they cannot disclose the physical mechanism by which a foam layer enhances the sea surface emissivity or brightness temperature in the passive remote sensing. To investigate this mechanism, the VRT theory was applied to the sea surface covered by a foam layer employing the quasi-crystalline approximation, and then the effect of foam layer thickness on the sea surface emissivity was discussed using the distorted Born approximation [Guo et al., 2001]. Furthermore, for estimating the effect of a foam layer on SSS retrieval from the Soil Mositure and Ocean Salinity (SMOS) Earth Explorer Opportunity mission, the emissivity of foam-covered ocean was analyzed at L-band (1.4 GHz) with the model and data of Foam, Rain, Oil Slicks and GPS Reflectometry (FROG) 2003 field experiment [Camps et al., 2005]. However, there are few analytical studies discussing the effects of particle microstructures on the microwave emissivity. Here we shall investigate theoretically the enhanced emissivity of sea surface covered by different foam microstructures on the basis of microwave VRT theory.
 Different microstructures induce different distributions of radiation, inclusion inner and scattering fields due to interaction of the electromagnetic radiation with materials of various structures [Wei et al., 2008]. To study the effects of foam microstructures on ocean surface emission, controlled experiments were performed to measure the coating thickness (or bubble wall thickness), the bubble size distribution, the foam layer thickness and void fraction of foam layers [Rose et al., 2002; Camps et al., 2005]. Recently, theoretical studies focused on the particle positions, coherent wave interaction [Ding et al., 1992; Tsang et al., 1985; Chen et al., 2003; Zhang et al., 2003] and sticky force parameters of dense particle aggregates at different frequencies [Tsang et al., 2000]. Generally, these theoretical studies are based on Maxwell's theory to derive the microwave VRT equation by combining some theoretical and numerical approximations. For complex structures or dense inclusions, the electromagnetic transfer problem is not yet analytically solved from first principles because of complex boundary problem and multiple interactions. For the regular structures of spherical and cylindrical scatterers, in the dilute limit, the Rayleigh and Mie scattering formulas were derived [Kong, 2002]. In the practical applications, Mie scattering formula gives the best result while the particle size is comparable to the wavelength. If the particle size is much smaller than the wavelength, the best way is to take in Rayleigh scattering formula. However, nature particle sizes are always smaller than the probing microwave wavelength, e.g., the earth microwave remote sensing. To improve the application of Rayleigh scattering to small coated particles, it is necessary to derive the higher-order Rayleigh scattering and its corresponding VRT equation including the scattering, absorption and extinction coefficients. In addition, there are few theoretical studies of Rayleigh approximation to discuss the effects of coating thickness of coated particles on the microwave scattering, extinction coefficients and emissivities. In this paper, based on the Maxwell's equation and the microwave VRT theory, the emissive properties of coated spherical particles embedded in an isotropic medium will be investigated for the propose of microwave remote sensing using the second-order Rayleigh approximation fields when the particle sizes are smaller than the incident wavelength.
 In section 2, the inner electric field distribution of a coated spherical structure is exactly derived at microwave frequency. Then, the second-order Rayleigh approximation is applied to obtain scattering field. In section 3, at low particle concentration, the VRT equation is derived using the scattering field, and perturbation method is employed to obtain its solution. An application to a sea foam layer is given in section 4. The scattering, extinction, and absorption coefficients as well as emissivity are discussed in terms of the ratio of the inner and outer radii of coated bubbles. The VRT results are compared with foam layer experimental data. Finally, a brief conclusion is given in section 5.
2. Second-Order Rayleigh Scattering Field
 Considering an isotropic particle with permittivity ɛα and permeability μ0 embedded in an infinite isotropic host with permittivity ɛh and permeability μ0, we rewrite the particle Maxwell's equations of time harmonic electric field and magnetic field at a angular frequency ω as equations (1) and (2) under an incidence wave i() = iE0, where i = ki and i = sin i cos i + sin i sin i + cos i [Kong, 2002; Jin, 1993],
where the normalized permittivity αn = α/h. The equation of particle inner electric field is obtained from equations (1) and (2),
where k2 = ω2hμ0. The source term s() is s() = −iωh(αn − 1) for particle region v0 and s() = 0 for host region, respectively. By introducing the Hertzian vector [Ishimaru, 1978] and assuming a far field approximation consisting of the condition ∣ − ′∣ ≈ r − · ′, where the position vector ′ is in the particle region, and and are a observation position vector and its unit vector in host region, respectively, the scattering electric field s induced by the particle was derived from equation (3) using dyadic Green's function [Kong, 2002; Jin, 1993],
where v0 is the particle region. (′) is the inner electric field of particles. The is the unit vector of a scattering wave vector along scattering direction (, ) in spherical coordinate, i.e., s = sin cos + sin θ sin ϕ + cos θ. Therefore, the scattering electric field induced by a particle of arbitrary shape can be solved from equation (4) using the particle inner field. Furthermore, considering the unit horizontal and vertical polarization vectors, = ( × )/∣ × ∣, = × , we can rewrite equation (4) as
where the exponent · ′ in equation (5) represents the phase variation. When the particle size is very smaller than the incidence wavelength, the term · ′ can be omitted and the classic Rayleigh scattering field is derived. Next, the inner fields of a coated sphere will be derived in the quasi-static limit, and then the scattering field of a coated spherical particle will be approximated by equation (5) up to the second-order of term at microwave frequencies, i.e., second-order Rayleigh approximation.
2.1. Inner Field of a Coated Sphere
 At microwave frequency, we consider the quasi-static electrical potential Φα equation of an isotropic coated spherical particle with the inner radius a and outer radius b of a linear constitutive relation D = ɛαE in spherical coordinates, where α = i, c, h represent core, coating and host materials, respectively,
If a uniform incident electric field i = iE0 is applied to the composites along the unit vector direction i = αx + αy + αz, the potential at an infinite point is Φr→ ∞ = −(αxx + αyy + αzz)E0. With the continuous boundary conditions of potentials and normal electrical displacements on the phase interfaces, the potential solutions Φh, Φc and Φi of equation (6) are obtained in the host (r ≥ b), coating (a ≤ r < b) and core (r < a) regions, respectively,
where the position vector = x + y + z, and the coefficients are listed as follows:
 The electrical fields are given, respectively, by
where is a unit dyad and the dyad = . Here note that in the quasi-static limit the Rayleigh scattering field induced by a coated sphere is exactly derived from equation (10) in the dilute limit, i.e., h() − iE0. Taking account of the phase effects on a scattering field of coated spherical composites, we shall use equation (5) to approximately derive the scattering field up to the second-order of term when the coated particle sizes are smaller than an incidence wavelength.
2.2. Second-Order Scattering Field of a Coated Sphere
 Based on the above inner fields of a coated sphere, the scattering field is estimated with equation (5), where the phase variation term is approximated up to the second-order term for parameter ∣kb∣ 1. Note that if the term is approximated by setting ≈ 1, the classic Rayleigh scattering field is obtained. Substituting equations (11) and (12) into equation (5), the second-order Rayleigh scattering field can be derived taking the second-order approximation ≈ 1 − is · ′ − (s · ′)2. This approximated field is more suitable for the scattering and emissive problems at microwave frequencies.
 From the core inner electric field, the scattering field induced by the spherical core regions vi(r ≤ a) is calculated,
where Vi = 4πa3/3 is spherical core volume. The i = αvi + αhi is the unit vector of a incidence field, whose unit vertical and horizontal polarization vectors, i and i, are i = ( × i)/∣ × i∣, i = i × i, respectively. In the scattering coordinates (s, s, s), the above core scattering field can be expressed in terms of () = s + Es, where E and E are the vertical and horizontal scattering field components of a scattering wave, respectively,
where E (i.e., αvE0) and E (i.e., ahE0) are the vertical and horizontal polarization components of an incidence wave, respectively. Note that the term k2a2 of equation (13) is the second-order contribution of core phase variation.
 For coating material (a < r ≤ b), the scattering field is calculated with equation (5)
where vc notes the coating material region (a < r ≤ b), a1 = −VbAc (1 − q3), a2 = Vb(kb)2[Bcq3 (1 − q2) + Ac (1 − q5)], q = a/b, Vb = 4πb3/3. Clearly, the term a2 represents phase contribution of coating materials to scattering field. Furthermore, this scattering field can be rewritten as
 Therefore, the total scattering field of a coated sphere is obtained,
where the amplitude functions fαβ of scattering fields are listed as below:
Here we can demonstrate that the classic Rayleigh scattering field of a coated sphere will be derived by setting kb = 0 in the term R of equation (15). Moreover, the Rayleigh scattering field of a sphere is also given from equation (15) with the two conditions ɛc = ɛh and kb = 0. Therefore, we have exactly obtained the second-order scattering fields of Rayleigh approximation for a coated sphere.
3. Vector Radiative Transfer Equation and Its Solution
3.1. Vector Radiative Transfer Equation
 For an elliptically polarized monochromatic plane wave = (Ev + Eh) propagating in a medium with intrinsic impedance η = , the modified Stokes parameter [Fung, 1994] that represents the microwave radiation intensity will be used to derive the VRT equation of a coated spherical composite from the scattering field induced by a coated sphere given in section 2. In passive sensing, the emission intensity can be estimated using only the first two Stokes parameters Iv and Ih (which are related to the Poynting vectors) of vertical and horizontal directions. For a spherical particle, because of its symmetry, the Stokes parameters are independent of azimuth i, and we can integrate the scattering phase matrix about the azimuth i considering multiparticles scattering. Here three-layer structure of a sea surface foam layer shown in Figure 1 is considered. Layer 0 is air. Layer 1 consists of bubbles coated with seawater embedded in air, and layer 2 includes air bubbles in seawater. For a layer of thickness d, the VRT equations of a coated spherical composite are built for upward and downward Stokes vectors, (, z) and (π − , z), respectively,
where z is the vertical variable with a original point at the middle depth of the layer 1, and is the radiation angle between the microwave emission direction and the vertical direction . The radiation intensity vector is = (Iv, Ih) with upward and downward intensities (θ, z) and (π − θ, z) within the range 0 ≤ θ ≤ π/2 and −d/2 ≤ z ≤ d/2. The temperature vector of layer 1 is 1 = (T1, T1). The constant C = Bɛ′h/(λ2ɛ0) is calculated by using the Boltzmann constant B and the real part ɛ′h of host dielectric constant. The scattering phase matrix (θ, θ′) is derived from the scattering field amplitudes of section 2 [Fung, 1994; Jin, 1993].
where is the volume scattering coefficient matrix.
 The scattering, extinction and absorption coefficient matrixes , and can also be derived by the particle scattering fields, inner field and dielectric constants. Because of the spherical symmetry and homogenous particle distribution, these coefficient matrixes are isotropic,
where n0 denotes the number of particles in a unit volume.
 The absorption coefficient κa consists of the particle and host medium absorption coefficients, κas and κah, respectively, i.e., κa = κas + κah. The particle absorption coefficient κas is given by equation (20),
where i and (′) are the incidence and particle inner fields, respectively. The f3 = n0Vb denotes the volume fraction of particles in foam layer. The angular frequency is ω; ɛ″i and ɛ″c are the imaginary parts of spherical core and coating dielectric constants, respectively. The absorption coefficient κah of a background medium is calculated,
where hr = h/0, k0 = ω and = Re + i Im. Thus, with above coefficients, the extinction coefficient κe is given,
 The boundary conditions of equations (16) and (17) are given at the interfaces of three layers,
where and are diagonal reflectivity and transmittivity matrices from the layer n to layer m. 2 = (T2, T2) and is the temperature vector in the layer 2 of seawater. C2 = Bɛ′w/(ɛ0λ2) is a constant related to the real part ɛ′w of layer 2 composite (i.e., seawater and air bubbles) permittivity.
 Letting x = cosθ, x′ = cosθ′, the upward and downward Stokes vectors are (, z) = + (x, z) and (π − θ, z) = −(−x, z), respectively. The VRT equations are rewritten as below:
with the boundary conditions in the range 0 ≤ x ≤ 1,
3.2. Perturbation Method
 It is assumed that the ratio ς = κs/κe is a small parameter to solve equations (23) and (24). Note that this assumption is valid for a strong absorption medium and weak scattering particles. For example, the seawater is a strong absorption medium, and the coated seawater bubble has weak microwave scattering. Therefore, the perturbation method is adopted to solve the integral equations. Letting the vertical and horizontal components of upward and downward Stokes parameters, Ip+ (x, z) = iIp+(i)(x, z) and Ip− (−x, z) = ςiIp−(i)(−x, z), where Ip+(i) (x, z) and Ip−(i) (−x, z) are Stokes parameters of the i − th order perturbation for p = v, h polarization, we have
Substituting equations (25) and (26) into equations (23) and (24), and comparing the power of the small parameter ς, we obtain a set of differential equations of vertical and horizontal Stokes parameters.
 For the 0-th order Stokes parameter, the upward and downward transfer equations are, respectively,
where p = v, h. The corresponding perturbation boundary conditions are given as follows:
where Rnmp and tnmp note the p-component of reflectivity and transmittivity, respectively.
 For the i-th order Stokes parameter (i > 0), the transfer equations are, respectively,
where the function Fp(i − 1)(x, z) is related to the (i − 1)-th order Stokes parameters,
The corresponding boundary conditions are derived as follows:
 With above perturbation equations, it is easy to obtain the Stokes parameter solutions up to an arbitrary order. The 0-th order Stokes parameter solution is given in equations (31) and (32),
 The i-th order Stokes parameter solution is derived from equations (29) and (30) by means of the (i − 1)-th order solutions.
Thus, with the 0-th order Stokes parameter solutions of equations (31) and (32), the arbitrary order solutions of Stokes parameters can be calculated by the iterative calculation of equations (33) and (34).
4. Application to Sea Foam
 Based on the solutions of VRT equations, the radiation brightness temperatures of a sea surface covered by a foam layer, which is described as the three-layers structure shown in Figure 1, is obtained in the passive remote sensing,
where x0 = cosθ1 and the incidence angle θ1 of layer 1 is determined with a refraction angle θ0 in air medium (i.e., layer 0) using Snell's law n0sinθ0 = n1sinθ1. Thus, for a given incidence angle θ0 in layer 0, the emissivity ep(θ0) (p = v, h) of sea surface coved by foam-layer can be estimated using equation (36),
4.1. Effects of the Seawater Coating
 To calculate Stokes emission vector, we first investigate the properties of scattering, extinction and absorption coefficients with varying seawater coating thickness δ = b − a of sparse coating bubbles because the formulas of section 3 are derived in low concentrations of coated spheres. Note that the low concentrations limit of coated sphere is no larger than 0.1.
 For the volume fraction f3 = 0.1 of sweater coated bubbles embedded in air and the given outer radius b = 540 μm of coated bubbles, Figure 2 and Figure 3 show the effects of seawater coating thickness (i.e., δ = b(1 − a/b)) on the scattering and extinction coefficients. Clearly, the scattering coefficient decreases with the coating thickness decrease. It indicates that the decreasing seawater content, i.e., seawater coating thickness δ, can reduce the foam scattering coefficients since the nonlinear interactions between the bubble structure parameter q = a/b and the inner electrical fields of coated spheres (see equation (19)). However, the scattering coefficient increases with the microwave frequency principally due to the increase of the radiation microwave wave number and the seawater permittivity changing with frequencies [Klein and Swift, 1977]. In Figure 3, it is not difficult to understand the extinction coefficient increases with decreasing the coating thickness δ (i.e., increasing the structure parameter q) because the increase of structure parameter q will enhance the modulus of polarization dipole factors Ac and Bc in a coated structure. From equation (20), it is seen that the increasing radiation frequency will also enhance the extinction coefficient. Moreover, for each microwave frequency, there exists a peak of extinction coefficient at a very thin coating of seawater, and this peak results from the nonlinear interactions between the structure parameter q and the dipole factor of seawater coating electrical field (see equations (11) and (20)). Thus it is possible that a peak of extinction coefficient appears at a value of structure parameter q because the scattering coefficient takes a very small contribution to the extinction properties.
 Furthermore, we have given the ratio of the scattering to absorption coefficients of seawater coating bubbles in Figure 4. Note that the scattering coefficient is much smaller than absorption coefficient for all of structure parameter q even if the ratio κs/κa of scattering to absorption coefficients increases with frequency. It can be seen from the equations (19) and (20) that the scattering coefficient κs of sweater coating bubbles decreases with the term (1 − q3)2 and the absorption coefficient κa decreases with the term (1 − q3) when the structure parameter q tends to 1. Thus, the ratio κs/κa will be zero when we take the limit q → 1. Figure 5 shows the extinction coefficient varying with the sea surface temperature (SST) (or the seawater coating temperature) and the sea surface salinity (SSS) at 1.4 GHz. Clearly, if the extinction coefficient of a foam layer is detected by a measurement way at 1.4 GHz, the SSS can be retrieved from it. Therefore, with above discussions, it is found that the ratio of the inner and outer radii plays an important role in the VRT equation.
4.2. Emissivity of Sea Surface Covered With a Foam Layer
 For natural sea foam, the radii of coated bubbles obey a gamma function distribution with a wide range of radius varying from about one hundred to two thousand micrometers, and the bubbles are densely aggregated through the adhesive force of coating seawater [Camps et al., 2005]. For the dense bubbles, the effective propagation constant and permittivity approximations were employed in modeling the microwave emissivity of a foam layer [Guo et al., 2001; Camps et al., 2005]. Here, to investigate the dense bubbles, we apply a periodic coated sphere assemblage [Milton, 2002] to estimate the effective permittivity of dense coated bubbles.
 The major idea of estimating the effective properties of a periodic coated sphere assemblage is that the inserted coated sphere into a matrix does not disturb the electric displacement outside the coated sphere. In detail, for a host medium with permittivity ɛh, the similar coated spheres with a coating material of permittivity ɛc and a core material of permittivity ɛi can be continue embedded in the host medium without disturbing the displacement and the electric field in the surrounding host because of local dielectric equations. If the coated spheres of various sizes ranging to infinitesimal are added, the all space of host medium is filled completely with a periodic assemblage of coated spheres. In this way, the two-phase composites of coated spheres are obtained. During this process, the electric potential and displacements at the boundary of the unit cell remain unaltered, and the effective permittivity does not change at any stage. So we can identify the initial permittivity ɛh with the final effective permittivity ɛ* of the coated sphere assemblage. Furthermore, Milton derived the effective permittivity ɛ* of the periodic coated sphere assemblage [Milton, 2002],
where q1 = a1/b1, the inner radius a1 and outer radius b1 of a coated sphere. Note that the inserted coated spheres are chosen to have the same ratio of inner and outer radius, and the coated spheres fill all space.
 However, for dense bubbles in the natural foam layer, there exist some gaps among bubbles with different structure parameter qi (i.e., qi = ai/bi with the inner radius ai and outer radius bi). Clearly, equation (37) is an upper bound of a foam layer permittivity because of the effects of air among bubbles and the different bubble sizes. Moreover, in order to apply equation (37), we have to eliminate these effects by multiplying a factor τ on the second term (i.e., the dipole factor) of equation (37), which takes a similar role as the packing coefficient or stickiness parameter in dipole approximation [Vries, 1972; Guo et al., 2001; Camps et al., 2005]. Thus, the modified effective permittivity formula of dense bubbles is
Here we should note that it is difficult in theoretically driving the factor τ and the stickiness parameter. Some discussion of the stickiness parameter is given by Guo et al. .
 To estimate the measured emissivities of sea surface covered with a foam layer in FROG experiment at 1.4 GHz, it is assumed that the foam layer consists of three kinds of bubbles with three different inner and outer radius ratios. Two of three kinds of bubbles are considered as an effective mixed medium, and the other one is regarded as an inclusion embedded in the mixed medium. For instance, the first kind of dense coated bubbles with the structure parameter q1 is regarded as a medium with permittivity ɛe (i.e., equation (38)). The second kind of sparse coated bubbles with the structure parameter q2 is embedded in the medium consisting of the first kind of dense coated bubbles, whose effective permittivity is calculated with equation (38). Thus, the first and the second kinds of bubbles will construct a mixed matrix or a new host. The effective permittivity ɛh of the new host can be calculated by the following formula when the volume fraction f2(≤0.1) of the second kind of bubbles is small,
where f2 is the volume fraction of bubbles with radius ratio q2 in a foam layer.
The third kind of sparse bubbles with the structure parameter q3 and the volume fraction f3 is randomly inserted in the new host (i.e., the mixtures of the first and the second kinds of bubbles) as scattering particles. Therefore, the foam layer is approximated by the three kinds of coated bubbles. Note that when the volume fraction f3(≤0.1) is small in the foam layer, the VRT solutions of section 3 are valid.
 For layer 2 of air bubbles embedded in seawater as shown in Figure 1, we regard this layer as a homogeneous isotropic effective medium. Its effective permittivity ɛw can be estimated by Maxwell-Garnett formula [Milton, 2002], which is valid up to the middle volume fraction fa of air bubbles in seawater [Guo et al., 2001; Anguelova, 2008]. The effective dielectric constant of seawater having air bubbles is calculated with equation (40).
where ɛsw denotes the seawater permittivity. Thus the effective permittivity ɛw of layer 2 medium is used to determine the constant C2 as an effective medium.
 Without loss of generality, the emissivities of sea surface covered with a foam layer are calculated using equation (36), the upward Stokes parameters of first-order perturbation approximation Ip+ = Ip+(0) + κsIp+(1)/κe and the measured parameters in the FROG foam layer experiment. The measured parameters in FROG foam layer experiment, such as SST, SSS, the most probable radius of coated bubbles rp, the foam layer thickness d, the seawater coating thickness of coated bubbles δ, the air-bubble volume fraction in seawater fa, the measured emissivities ep= v,h of sea surface covered with a foam layer and the incidence angle θ0, are given by Camps et al. [2005, Figure 11] and Table 1. The numerical results of the VRT equation at 1.4 GHz are shown in Figure 6. Good agreement is obtained by comparing with the experimental results and Camps model. In the VRT numerical calculation, the parameters are listed as q1 = (rp − δ)/rp, f2 = f3 = 0.1, q3 = q2 = 0.9 and b3 = rp = 40 μm, where the fa, SST, SSS and d are the same as the values of parameters in FROG experiment. Note that the radius b3 of scattering seawater coating bubbles is chosen as the sum of the most probable radius rp and a constant size 40 μm because the sweater coating bubbles with a larger radius are regarded as scatterers in our model. The added radius size 40 μm is about ten percent of mean most probable radius rp of seawater coating bubbles in FROG experiment. The temperature of layer 1 is equal to that of layer 2 (i.e., T1 = T2 = SST), and the stickiness parameter τ is listed in the legend of Figure 6. Because the microstructure of thin seawater coating is predominant in sea foam according to FROG experiment (q is about 0.98), it is reasonable to regard the bubbles of the most probable radius as the dense bubbles or the first kind of coated bubbles. In the process of the foam dynamics, since the bubble sizes increase and the water content decreases within the bubble's living time, the bubbles with large radius and thick seawater coating are considered as the second and third kinds of sparse bubbles with small volume fraction in our model.
Table 1. Measured Parameters of Foam Layer in FROG 2003 Field Experiment at 1.4 GHz From Camps et al. [2005, Figure 11] and Used in Figures 6a–6h of VRT Results
 Furthermore, the effect of structure parameter q on foam emissivity is investigated qualitatively by VRT equation in Figure 7, where the SST = 18.7°C, SSS = 33.21 psu, d = 16.65 mm, b3 = 544 μm, qi(i = 1,2,3) = q, θ0 = 40°, τ = 1, fa = 0 (i.e., seawater layer without air bubbles) and f2 = f3 = 0. It is found that, with decreasing the seawater coating thickness, the microwave emissivity increases, and the foam layer clearly enhances the emissivity of sea surface when the thickness of seawater coating is small at a given microwave frequency (for example, at 1.4 GHz, the structure parameter q of the peak is about 0.97). At a very thin coating, there is a peak of emissivity in the foam layer changing with frequencies. The thin coating effect, which is similar to the extinction coefficient, is due to the nonlinear interactions between the structure parameter q and the electric field distributions of seawater coating bubbles. The result shows that the coating thickness is very important in estimating emissivity of a foam layer. The small error of the structure parameter q will induce a large difference of emissivities, particularly, at the thin seawater coating. Therefore, the structure parameter q of bubbles will be a new parameter, which should be determined in foam layer experiments, to predict the emissivity of foam layer exactly.
 Finally, from the view of sections 4.1 and 4.2, it is noted that the results of the second-order Rayleigh approximation at low frequencies are the same as those for classic Rayleigh theory (i.e., at zero-order) for small particles. For example, with experimental data from foam-covered sea surfaces as presented in Figure 6, we have also calculated the emissivities of sea surfaces using the classic Rayleigh method at 1.4 GHz and found that there is no deviation with the second-order Rayleigh approximation. That is, because the microwave wavelength (i.e., wavelength 21 cm at 1.4 GHz) is much larger than the average diameter (about 1 mm) of foam bubbles, the second-order term of the scattering field can be neglected. To discuss the improvement of the second-order approximation to the classic Rayleigh theory, we have graphed the microwave frequencies dependence of emissivities of foam-covered sea surface in Figure 8, where the sea foam parameters are the same as for Figure 7 with the exception that θ0 = 50°, fa = 0.0559, and τ = 0.45. It is found that the emissivity difference increases with increasing microwave frequency. From 16 GHz to 28 GHz, the induced brightness temperature differences increase for both methods from 0.54 K to 2 K for vertical polarization (from 0.35 K to 1.4 K for horizontal polarization) when the sea water temperature is about 18.7°C. Here we note that the wavelength of 28 GHz microwave radiation is about 1 cm and the bubble diameter is about 1 mm. Because the ratio of wavelength to particle size is larger than 1 (in our example, the ratio is larger than about 10), the Mie scattering formula is not applicable. Therefore, our second-order Rayleigh approximation can improve on the classic Rayleigh at certain microwave frequencies.
 The microwave radiation transfer equation is investigated for a coated spherical composite by the Maxwell's equations and Stokes vector transfer principle. Based on the solutions of inner electrical field of a coated sphere, which is derived in quasi-static limit, the scattering fields are estimated up to the second order of Rayleigh approximation. In low inclusion concentration, the VRT equation of a composite layer having sparse coated spherical particles is derived for the radiation problem in passive remote sensing. Furthermore, the perturbation method is developed to solve the VRT equation at microwave frequencies for small size particles.
 An application of the VRT equation is taken in the sea foam layer. The extinction, scattering and absorption coefficients are discussed by changing the structure parameter q of the inner to the outer radii of a coated bubble. The results show that, for a thin coating structure, these coefficients are very sensitive to a large radius ratio, and a peak appears at the thin seawater coating of coated bubbles for the extinction coefficient due to the nonlinear interaction between the structure parameter q and the coated spherical electrical field. For a sea surface covered by a layer of densely coated bubbles, the effective medium approximation is also employed to estimate the microwave emissivity of sea surface. With VRT equation solutions, the mechanism of a foam layer enhancing the sea surface emissivity is disclosed qualitatively, where the nonlinear interaction of the bubble structure parameter q and the bubble inner electric field will induce a strong microwave radiation power of sea surface, and the largest radiation increment appears at very thin seawater coating of a coating bubble. In addition, to estimate emissivities of sea surface with a foam layer, we have compared the VRT results with the FROG experimental data and Camps et al. model of foam layer. As a conclusion, we note that, to improve the remote sensing data of sea surface covered with a sea foam layer, the distribution of structure parameter q for different bubble radii should be exactly measured in experiments, and as a key role, the structure parameter q should be included in remote sensing models.
 Moreover, the VRT equation can be used to estimate microwave radiation brightness temperature of rough sea surface covered by whitecaps with the modified boundary conditions. For remote sensing of sphere or coated spherical composites, such as rain, porous rocks, colloidal suspensions and sea ice, this VRT equation is also useful. Meanwhile, the VRT equation of the active remote sensing can be derived from our VRT equation when the absorption term is omitted and the initial incidence intensity is added in the boundary conditions. Thus, the VRT equation can be applied in eliminating the effects of foam layer on both active and passive remote sensing models. In addition, the two-dimensional problem of the coated cylindrical composites can be studied for the remote sensing of plant and forest environment.
 This work was supported by the NSFC (grant 40876094) and National 863 Project of China (grants 2009AA09Z102 and 2008AA09A403).