### Abstract

- Top of page
- Abstract
- 1. Introduction
- 2. Formulation
- 3. FOAM Model Parameters
- 4. Comparison With QCA
- 5. Numerical Illustration of Four Stokes Parameters
- 6. Conclusions
- Appendix A: Empirical Sea Surface Spectrum
- Acknowledgments
- References
- Supporting Information

[1] 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).

### 1. Introduction

- Top of page
- Abstract
- 1. Introduction
- 2. Formulation
- 3. FOAM Model Parameters
- 4. Comparison With QCA
- 5. Numerical Illustration of Four Stokes Parameters
- 6. Conclusions
- Appendix A: Empirical Sea Surface Spectrum
- Acknowledgments
- References
- Supporting Information

[2] 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.* [2001] used the Quasi-Crystalline Approximation (QCA) model for foam, while *Chen et al.* [2003] used the Monte Carlo Simulations. Controlled experiments of radiometric measurements of foam microwave emissions have also been made [*Rose et al.*, 2003].

[3] 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.

### 3. FOAM Model Parameters

- Top of page
- Abstract
- 1. Introduction
- 2. Formulation
- 3. FOAM Model Parameters
- 4. Comparison With QCA
- 5. Numerical Illustration of Four Stokes Parameters
- 6. Conclusions
- Appendix A: Empirical Sea Surface Spectrum
- Acknowledgments
- References
- Supporting Information

[13] We assume that there are *N*_{s} 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; *a*_{j}, outer radius of coated air bubbles of the *j*th species; *b*_{j}, inner radius of coated air bubbles of the *j*th species; *f*_{j}, fractional volume of coated air bubbles of the *j*th species

where *n*_{sj} is the number of the *j*th species per unit volume. The parameter *f* is the total fractional volume of air bubbles

and parameter *f*_{w} is the fractional volume of seawater in foam

[14] 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 *a*_{j} and *b*_{j}, *f*_{v} 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 μ*m* to 5000 μ*m* with median between 900 μ*m* and 1000 μ*m*. An average foam layer is 2.8 *cm*.

### 4. Comparison With QCA

- Top of page
- Abstract
- 1. Introduction
- 2. Formulation
- 3. FOAM Model Parameters
- 4. Comparison With QCA
- 5. Numerical Illustration of Four Stokes Parameters
- 6. Conclusions
- Appendix A: Empirical Sea Surface Spectrum
- Acknowledgments
- References
- Supporting Information

[15] 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.

[16] 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 *b*_{1} and *b*_{2}. 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.* [2001] 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 3Parameter | Value |
---|

*d* | 3 *cm* |

2*a* | 2000 μ*m* |

2*b*_{1} | 1220 μ*m* |

2*b*_{2} | 1995.9 μ*m* |

*f* | 74% |

*f*_{w} | 10.3% |

*f*_{1} | 12.9% |

*f*_{2} | 61.1% |

Table 2. Quantities of Foam Based on QCAParameter | 10.8 *GHz* | 36.5 *GHz* |
---|

Absorption rate κ_{a}, *cm*^{−1} | 0.1763 | 0.8154 |

Scattering rate κ_{s}, *cm*^{−1} | 0.0567 | 0.7299 |

Extinction rate κ_{e}, *cm*^{−1} | 0.2330 | 1.5453 |

Albedo | 0.2435 | 0.4724 |

Effective permittivity ε_{eff} | 1.4573 + *i*0.1245 | 1.2944 + *i*0.231 |

Table 3. Quantities of Foam Based on Monte Carlo SimulationsParameter | 10.8 *GHz* | 36.5 *GHz* |
---|

Absorption rate κ_{a}, *cm*^{−1} | 0.2990 | 0.9400 |

Scattering rate κ_{s}, *cm*^{−1} | 0.0165 | 0.6183 |

Extinction rate κ_{e}, *cm*^{−1} | 0.3155 | 1.5583 |

Albedo | 0.0552 | 0.6578 |

Effective permittivity ε_{eff} | 1.4900 + *i*0.1700 | 1.1600 + *i*0.220 |

### 5. Numerical Illustration of Four Stokes Parameters

- Top of page
- Abstract
- 1. Introduction
- 2. Formulation
- 3. FOAM Model Parameters
- 4. Comparison With QCA
- 5. Numerical Illustration of Four Stokes Parameters
- 6. Conclusions
- Appendix A: Empirical Sea Surface Spectrum
- Acknowledgments
- References
- Supporting Information

[17] 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* [1985] 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* [1985]. The expressions of the spectrum were corrected in *Yueh et al.* [1994] and listed in Appendix A. In the model, *a*_{0} is the absolute magnitude of the spectrum. Since a parametric analysis has been carried out showing that *a*_{0} = 0.008 gives the best agreement with experiment data in *Yueh et al.* [1994], *a*_{0} of 0.008 is used for calculations in this paper. The wave-number cutoff *k*_{d} is a key parameter in the spectrum. The quantity, *k*_{d}, 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 *k*_{d} 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.* [1994], the SPM should be applicable to these cases. The permittivity of seawater is calculated from the model of *Klein and Swift* [1977] 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 + *i*40.1053, 28.9541 + *i*36.8340 and 13.4480 + *i*24.7844, respectively.

Table 4. Parameters for Foam ModelParameter | Value |
---|

*d* | 1 *cm* |

2*a* | 500 μ*m* |

2*b*_{1} | 260 μ*m* |

2*b*_{2} | 498.9 μ*m* |

*f* | 74% |

*f*_{w} | 10.2% |

*f*_{1} | 11.4% |

*f*_{2} | 62.6% |

Table 5. Quantities of Foam Using Monte Carlo SimulationsParameter | Albedo | Extinction Rate κ_{e} | Effective Permittivity ε_{eff} |
---|

10.8 *GHz* | 1.791 × 10^{−4} | 0.2723 *cm*^{−1} | 1.4971 + *i*0.1475 |

19.0 *GHz* | 1.763 × 10^{−3} | 0.5363 *cm*^{−1} | 1.446 + *i*0.1623 |

36.5 *GHz* | 1.642 × 10^{−2} | 0.9802 *cm*^{−1} | 1.3717 + *i*0.1504 |

[18] 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 *T*_{v} and *T*_{h} 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.

[19] 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 *T*_{v} and *T*_{h}, and sin2ϕ for *U* and *V*. That is, *T*_{v} and *T*_{h} 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. *T*_{v} and *T*_{h} 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.

[20] 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 *T*_{v} and *T*_{h} are approximately increased by 10*K* at nadir viewing, after considering the effects of rough ocean surfaces.

[21] 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 *U*_{19.5} = 10 *m*/*s* are presented and compared with those at *U*_{19.5} = 20 *m*/*s*. *U*_{19.5} 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*.

[22] For partially foam-covered surfaces the emissivity may be written as [*Ulaby et al.*, 1982]

where *F* is the fractional foam coverage, *e*_{sw} is the emissivity of the wind-roughened ocean surface, and *e*_{f} is the emissivity of the foam. The Stokes brightness temperatures can be calculated for a given *F* and using the results of this paper.