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 In this paper, polarimetric microwave emissions from wind-generated foam-covered ocean surfaces are investigated. The foam is treated as densely packed air bubbles coated with thin seawater coating. The absorption, scattering and extinction coefficients are calculated by Monte Carlo simulations of solutions of Maxwell equations of a collection of coated particles. The effects of boundary roughness of ocean surface are included by using the second-order small perturbation method (SPM) describing the reflection coefficients between foam and ocean. An empirical wavenumber spectrum is used to represent the small-scale wind-generated sea surfaces. The iterative method is employed to solve dense media radiative transfer (DMRT) equations, and is applied to calculate results of all four Stokes parameters of rough ocean surfaces. The theoretical results of four Stokes brightness temperatures with typical parameters of foam in passive remote sensing at 10.8 GHz, 19 GHz and 36.5 GHz are illustrated. The azimuth variations of polarimetric brightness temperature are calculated. Emission with various wind speed and foam layer thickness is studied. The results are also compared with those based on Quasi-Crystalline Approximation (QCA).
 Fully polarimetric microwave remote sensing means that all four Stokes parameters are measured [Tsang, 1991]. There has been an increasing interest in the applications of polarimetric microwave radiometers for ocean wind remote sensing [Yueh et al., 1994; Johnson et al., 1994; Yueh, 1997; Kunkee and Gasiewski, 1997]. It is known that foam on the ocean surface can affect the brightness temperatures measured by microwave radiometers. Although foam typically cover only a few percent of sea surfaces, increasing foam coverage on the sea surface can substantially increase the sea surface emissivity [Stogryn, 1972; Smith, 1988; Huang and Jin, 1995]. However, there is relatively little known definitively concerning the impact of foam on the retrieval of the ocean surface wind vector from satellite-mounted microwave instruments. This gap in knowledge is due in large part to the difficulty in making measurements at high wind speeds, when significant foam coverage is present. In the past, empirical microwave emissivity models [Stogryn, 1972; Smith, 1988; Williams, 1971; Wilheit, 1979; Pandey and Kakar, 1982; Yueh, 1997] were used to estimate the effect of the foam above the ocean surface on the passive microwave remote sensing measurements. These are empirical fitting procedures using experimental data. Recently, a physically based approach taking into account the microstructure of foam was developed [Guo et al., 2001; Chen et al., 2003]. The model treats the foam as densely packed air bubbles coated with thin seawater coating. It was shown that the polarization and frequency of the brightness temperatures depend on the physical microstructure properties of foam and the foam layer thickness. Guo et al.  used the Quasi-Crystalline Approximation (QCA) model for foam, while Chen et al.  used the Monte Carlo Simulations. Controlled experiments of radiometric measurements of foam microwave emissions have also been made [Rose et al., 2003].
 In this paper, the polarimetric microwave emissions from wind-generated foam-covered rough ocean surfaces are studied. The dense media model of foam is applied to calculate the values of the complex effective propagation constants, the extinction coefficients and the albedo. These are used to describe the characteristics of the foam layer. The effects of boundary roughness of ocean surface are included in the boundary conditions of DMRT by using the second-order small perturbation method (SPM) [Tsang et al., 1985; Yueh et al., 1994; Tsang and Kong, 2001] describing the bistatic reflection coefficients between foam and ocean. The small-scale wind-generated sea surfaces are generated by using an empirical wavenumber spectrum [Durden and Vesecky, 1985]. The iterative method is employed to solve dense media radiative transfer (DMRT) equations. The method is summarized in Section 2. The parameters of foam model are described in Section 3. The results with those based on Quasi-Crystalline Approximation (QCA) are compared in Section 4. Theoretical results of four Stokes brightness temperature with typical parameters of foam based on Monte Carlo simulations [Chen et al., 2003] in passive remote sensing at 10.8 GHz, 19 GHz and 36.5 GHz are illustrated in Section 5. Four Stokes parameters of emission with various wind speed and their azimuthal variations are studied.
2.1. Foam Model
 We modeled the foam as densely packed spherical air bubbles with thin seawater coating [Guo et al., 2001; Chen et al., 2003]. In dense media theory, we calculate the extinction, absorption and scattering coefficients by considering the collective effects of a large number of N particles in a volume V. The extinction, absorption and scattering coefficients are defined as the extinction, absorption and scattering cross sections per unit volume. The results converge for large N and V. In the present Monte Carlo simulations, the coated particles are arranged with face-centered-cubic (fcc) structure to achieve high density packing. Then, Monte Carlo simulations of solutions of Maxwell equations are applied. 7 realizations are used in the calculation.
 Consider one of the six unit-cubic faces, the cubic side length is a, and the diagonal length is 4r, where r is the sphere radius. Thus the cubic side length a is 2r, and the volume Vc of the unit-cubic is 16r3. The unit-cubic contains 6 semispheres at the center of each face; and 8 one-eighth-spheres at each zenith. So, all the unit-cubic has 4 spheres. The fraction volume is , which is equal to π/. Thus, the face-centered-cubic lattice has a fractional volume of π/ ≈ 74% occupied by the particles.
 The internal field of each particle can be calculated by using the volume integral equation derived from Maxwell equation [Tsang et al., 2001] for a collection of particles. The extinction coefficient is κe = κs + κa and the Albedo is = κs/κe. The effective permittivity is calculated as in Chen et al. . The extinction, absorption, scattering coefficients, albedo and effective permittivity are used in DMRT equations. The Rayleigh phase matrix [Tsang et al., 1985, 2000b] is also used.
2.2. Dense Media Radiative Transfer Theory
 Consider the thermal emission problem of a foam layer embedded above the wind-generated rough ocean surfaces, as indicated in Figure 1. The foam layer consists of coated dielectric particles in region 1. The DMRT equations for passive remote sensing in region 1 can be written as follows:
where u and d represent, respectively, upward- and downward-going specific intensities and 4 × 1 column matrices containing the four Stokes parameters; μ = cosθ; μ′ = cosθ′; κes = κe/cosθ; κas = κa/cosθ; κss = κs/cosθ; is the temperature profile in the layer.
in which κa and κs are the absorption and scattering coefficient matrices taken to be diagonal. C is equal to KbK′2/(λ2k2) and Kb is the Boltzman's constant. (μ, ϕ;μ′,ϕ′) is Rayleigh scattering phase matrix.
2.3. Boundary Conditions
 The boundary condition at z = 0 is
where (θ) is a reflection matrix of the flat surface at the air-foam interface with effective propagation constant K and effective relative permittivity εeff for dense media.
 The rough surface boundary condition at z = −d is bistatic and is determined by second-order SPM.
where and are the coherent and incoherent scattering phase matrixes of the rough surface at z = −d.
For , fαα = fαα(0) + fαα(2), fαβ = fαβ(2), and for , fαβ = fαβ(1). Here, both of symbols α and β represent v or h. fvv(0) and fhh(0) are Fresnel reflection coefficients for vertical and horizontal polarizations with the zero-order fields considered. fαβ(1), and fαβ(2) are scattering coefficients of the α-polarized component of the first and second-order scattered fields with β-polarized, respectively, which are given in Tsang and Kong . The zero- and second-order fields give the coherent reflection coefficients of the surfaces. fαβ(1) gives the incoherent polarimetric bistatic scattering coefficient due to the first-order scattered field. They are derived using the second-order small perturbation method (SPM) [Tsang et al., 1985; Tsang and Kong, 2001]. Tg is the temperature of the half space below the foam and is the emissivities of the lower boundary.
2.4. Iterative Method of Solution of DMRT Equations With Rough Surface Boundary Conditions
 The differential equations (1) have standard solutions of the form
Next, we incorporate the rough surface boundary conditions into a form suitable for iterative solutions. Substituting the boundary conditions into (6) and (7), we have
 The source terms in (8) are the upward temperature originating from the layer temperature profile. The contribution from the lower half space is . The source term in (9) is the downward temperature. All other terms in (8) that depend on the upward and downward temperatures can be evaluated using these three source terms. After taking integration over z, we have
Another term in (8) accounting for upward scattering of the downward temperature by the lower boundary at z = −d can be evaluated as
Thus the complete expression for the upward temperature at z = 0 within the layer is
This is the first-order solution of . The second-order solution for the upward temperature at z = 0 is
 The brightness temperatures in the direction (θ0, ϕ0), where θ0 = sin−1(K′sinθ/k) as related to θ by Snells' law, are given by
 We assume that there are Ns species of air bubbles with seawater coating embedded in a background of air, as shown in region 1 of Figure 1. The foam parameters are defined as follows: d, foam layer thickness; aj, outer radius of coated air bubbles of the jth species; bj, inner radius of coated air bubbles of the jth species; fj, fractional volume of coated air bubbles of the jth species
where nsj is the number of the jth species per unit volume. The parameter f is the total fractional volume of air bubbles
and parameter fw is the fractional volume of seawater in foam
 The air region includes the core regions of the coated particles as well as the interstitial space between them. Thus the foam void fractional volume is
By choosing various values for aj and bj, fv can be on the order of 90%, which agrees well with the experimental measurements of artificially generated foam [Rose et al., 2003]. The experimental measurements were conducted at the Chesapeake Bay Detachment at 10.8 GHz and 36.5 GHz. The diameters of air bubbles range from 500 μm to 5000 μm with median between 900 μm and 1000 μm. An average foam layer is 2.8 cm.
4. Comparison With QCA
 In this section, we compare the results with those based on QCA [Guo et al., 2001], in which wave scattering and emission in a medium consisting of densely packed coated particles are solved by using QCA in combination with the dense medium radiative transfer theory.
 Using the parameters of foam model in Table 1, the quantities for foam layer calculated by QCA and by Monte Carlo simulations are shown in Tables 2 and 3, respectively. We assume that there are two species of air bubbles, which have the same outer radius a, but have different coating thickness with inner radii of b1 and b2. Emissivities of horizontal polarization and vertical polarization are simulated at 10.8 GHz and 36.5 GHz. Figure 2 is based on Monte Carlo simulations. We can see that the emissivities at 10.8 GHz and 36.8 GHz are comparable. This feature is consistent with experimental measurements [Rose et al., 2003]. On the other hand, as shown in Figure 3, which is based on QCA, the emissivities at 36.5 GHz are higher than at 10.8 GHz. For dense media consisting of particles densely packed, there are two different cases. They are differentiated by large loss tangent and small loss tangent where loss tangent is and σ is the conductivity of the medium. For large loss tangent, it represents the case that conductive current is much larger than displacement current, while the opposite is true for the small loss tangent case. It has been shown by extensive simulations [Tsang et al., 1992] that QCA is valid for the small loss tangent case. The QCA theory has been successful in dry snow [Tsang et al., 2000a]. However, the QCA theory is less successful when conductive current dominates [Chew et al., 1990]. The conductive current can go through several connected particles. On the other hand, QCA is limited to pair distribution functions. Thus the applicability of QCA is also dependent on frequency through the dependence on . In this paper, we see that the results of QCA are in better agreement with those of Monte Carlo simulations at high frequency of 36.5 GHz, and less successful at low frequency of 10.8 GHz. The experiment in Guo et al.  was at 19.0 GHz, and QCA was reasonably successful. Since Monte Carlo simulations provide the exact solution of Maxwell equations, this model will be applied to calculate the four Stokes parameters in the following studies.
Table 1. Parameters of Foam Model for the Simulated Results in Figures 2 and 3
Table 2. Quantities of Foam Based on QCA
Absorption rate κa, cm−1
Scattering rate κs, cm−1
Extinction rate κe, cm−1
Effective permittivity εeff
1.4573 + i0.1245
1.2944 + i0.231
Table 3. Quantities of Foam Based on Monte Carlo Simulations
Absorption rate κa, cm−1
Scattering rate κs, cm−1
Extinction rate κe, cm−1
Effective permittivity εeff
1.4900 + i0.1700
1.1600 + i0.220
5. Numerical Illustration of Four Stokes Parameters
 The foam model based on Monte Carlo simulations is used to calculate the extinction, absorption and scattering coefficients. The parameters of foam model are listed in Table 4, and the quantities of foam layer are described in Table 5. To describe the wind-induced wave on the ocean surface, the empirical surface spectrum W proposed by Durden and Vesecky  was used. The choice of this spectrum is because much of the observed dependence of the radar cross section on frequency, polarization, incidence angle, and wind velocity was predicted by using this spectrum function. There were some typograghical errors in Durden and Vesecky . The expressions of the spectrum were corrected in Yueh et al.  and listed in Appendix A. In the model, a0 is the absolute magnitude of the spectrum. Since a parametric analysis has been carried out showing that a0 = 0.008 gives the best agreement with experiment data in Yueh et al. , a0 of 0.008 is used for calculations in this paper. The wave-number cutoff kd is a key parameter in the spectrum. The quantity, kd, can be selected to be sufficiently large to satisfy the small perturbation method condition [Yueh et al., 1994; Yueh, 1997]. In the following studies, we choose kd equal to 100 rad/m for 10.8 GHz, 120 rad/m for 19 GHz and 230 rad/m for 36.5 GHz, with kh equal to 0.25, 0.38 and 0.41, respectively. The parameter h is the rms height of small-scale surface. Note that the values of kh are all less than 0.42. As investigated in Yueh et al. , the SPM should be applicable to these cases. The permittivity of seawater is calculated from the model of Klein and Swift  with the water temperature of 284 K and salinity of 20 per thousand. The permittivities of the seawater at 10.8 GHz, 19 GHz and 36.5 GHz are 49.1493 + i40.1053, 28.9541 + i36.8340 and 13.4480 + i24.7844, respectively.
Table 4. Parameters for Foam Model
Table 5. Quantities of Foam Using Monte Carlo Simulations
Extinction Rate κe
Effective Permittivity εeff
1.791 × 10−4
1.4971 + i0.1475
1.763 × 10−3
1.446 + i0.1623
1.642 × 10−2
1.3717 + i0.1504
 To demonstrate the effects of the thickness of the foam layer, we plot the brightness temperature of four Stokes as a function of thickness of foam layer in Figure 4. As the thickness of the foam layer increases, the mean values of Tv and Th will increase correspondingly and then saturate at particular thickness of the foam layer. On the other hand, U and V components will decrease to zero. For horizontal and vertical polarizations, saturation occurs around 7 cm, 3 cm and 2 cm of foam thickness at 10.8 GHz, 19 GHz and 36.5 GHz, respectively. For U and V components, saturation occurs at a larger layer thickness of 14 cm, 7 cm and 4 cm, respectively.
Figure 5 illustrates the Stokes parameters as a function of the zenith angle θ with a fixed azimuth angle of ϕ = 45°. We also compare the results of ocean rough surface calculated by SPM. Note that at those frequencies, the mean values of first two Stokes parameters are increased with the presence of foam, and the third and fourth parameters, on the contrary, are reduced. Figure 6 compares the azimuthal variations of four Stokes parameters over a 360° circle for viewing with θ = 30°. It is shown that the Stokes parameters have a cos2ϕ variation in azimuth for Tv and Th, and sin2ϕ for U and V. That is, Tv and Th have an even symmetry with respect to the wind direction, while U and V have an odd symmetry. Comparing with the results of SPM, the magnitudes of azimuthal variations become decreased. The level of decrease is bigger as the frequency becomes bigger. Tv and Th of azimuthal variations are decreased by 0.1 K and 0.085 K at 10.8 GHz; 0.7 K and 0.44 K at 19 GHz; 1.45 K and 1.4 K at 36.5 GHz, respectively. The decrease is caused by attenuation through foam layer. For prediction of values of azimuthal variations, some other effects should be included (such as, tilting effects of large-scale surfaces) which are not considered in this paper.
 To demonstrate the contribution of ocean roughness, we compare the results with those of foam-covered smooth ocean surfaces in Figure 7. It can be seen that the brightness temperatures of Tv and Th are approximately increased by 10K at nadir viewing, after considering the effects of rough ocean surfaces.
 The four Stokes parameters of brightness temperature are sensitive to wind speeds. In Figures 8–10, the results for foam-covered rough ocean surfaces at wind speed U19.5 = 10 m/s are presented and compared with those at U19.5 = 20 m/s. U19.5 is the wind speed at elevation of 19.5 m. At lower wind speed, all four Stokes parameters are reduced. The magnitudes of azimuthal variations at 10 m/s are approximately half of those at 20 m/s.
 For partially foam-covered surfaces the emissivity may be written as [Ulaby et al., 1982]
where F is the fractional foam coverage, esw is the emissivity of the wind-roughened ocean surface, and ef is the emissivity of the foam. The Stokes brightness temperatures can be calculated for a given F and using the results of this paper.
 Four Stokes parameters of brightness temperature for foam-covered rough ocean are theoretically analyzed. Important features are shown by the results at 10.8 GHz, 19 GHz and 36.5 GHz for the four Stokes parameters. These are important to determine how the foam affects the brightness temperatures and the retrieval of the ocean surface wind vector. The first two Stokes parameters are increased with the presence of foam, and the third and fourth parameters are reduced. The azimuthal variations of polarimetric brightness temperature are also illustrated. The first two Stokes parameters are even function of ϕ, while the last two parameters are odd functions. Emissions with various wind speeds and foam layer thickness are also studied. The four Stokes parameters of brightness temperature are dependent on wind speeds and foam thickness.