Thermal Microwave Radiation of Disperse Systems on the Sea Surface

Following from: Radiative transfer problems in nature and engineering

The theoretical models for thermal radiation of various disperse systems considered in the book are general and can be employed in both the visible and infrared spectral ranges. The rigorous electromagnetic scattering theory for single particles is universal and the Mie solution for spherical particles is used in the radio-wave range, too (see the articles, Radiative properties of metal particles in microwave spectral range and Water droplets and bubbles in microwave spectral range). The radiation transfer theory also does not have any wavelength restrictions. At the same time, application of the radiation transfer theory in the radio-wave range is accompanied by the following principal difficulties:

  • As a rule, the collective effects due to coherent scattering by closely positioned particles are more pronounced.

  • A considerable polarization, which is not taken into account in the scalar transfer theory, is found in the radio wave more often than that in the optical range.

Nevertheless, the traditional approach based on the scalar transfer theory and the assumption of independent scattering appears to be applicable to some problems of microwave thermal radiation.

With the development of remote sensing in the microwave, mainly from artificial satellites, more attention was given to the physical explanation of radio-brightness temperature (in terms of this book--spectral emissivity) data obtained for the ocean surface. This problem has been considered in some detail in monographs by Basharinov et al. (1974), Cherny and Raizer (1998), and Sharkov (2003, 2007). Early experimental investigations of the thermal radio emission of the ocean showed that a foam on the water surface leads to the extremely high emissivity in the millimeter spectral range (Stogryn, 1972; Raizer et al., 1976; Bordonskii et al., 1978; Vorsin et al., 1982). Some unsuccessful attempts for the physical explanation of this phenomenon were made at that time. The suggested theoretical models treated foam like a number of alternating layers of water and air or as a transitional layer with a smooth variation of parameters. These models could not give the observed high emissivity of foam on the water surface. In connection with this, it was reasonable to assume that the unusual microwave properties of the foam are due to some special features of the microwave radiation absorption and scattering by thin-walled water shells. A new theoretical model for thermal microwave radiation of foam on the water surface was developed by Dombrovsky (1979). This model includes the Mie solution for hollow spherical particles and analytical solutions for radiative transfer in DP0 and DP1 approximations. This approach is based on the assumption of negligible collective (dependent scattering) effects while single particles absorb and scatter the radiation independently from each other. This situation takes place for a “stretched” disperse system of separate water shells (bubbles) at sufficiently large distances between the randomly positioned bubbles, as well as between the bubbles and water surface. A comparison of theoretical predictions with the experimental data will give us an estimate of the collective effects contribution.

A rigorous solution for microwave radiation transfer can be obtained only by use of the vector radiative transfer equation (RTE) taking into account the polarization. As was suggested by Dombrovsky (1979), the scalar RTE was considered as a simplified approach; i.e., the random polarization of the radiation was assumed. The problem of thermal radio emission of a foam structure was formulated for a homogeneous isothermal plane-parallel foam layer of thickness h over the smooth water surface with emissivity εw. The spectral emissivity of this system can be obtained by use of the DP1 analytical solution:

(1a)

(1b)

(1c)

(1d)

(1e)

(1f)

Note that solution (1) is more general than solution (15) from the article, Radiation of isothermal plane-parallel layer, which corresponds to the special case of εw = 0. In the DP0 approximation, we have:

(2)

If the transport approximation is employed, the value of ξ in Eq. (2) is replaced by ξtr = ξ/ (1 - μpω). An estimate of the DP0 approximation error for a similar problem was given in the article, Radiation of isothermal plane-parallel layer.

The characteristics of a disperse system and water in the above equations are as follows: the equivalent thickness of foam layer kh (where k < 1 is the packing coefficient); the foam radiative properties β/ k, ξ, and μp; and the water surface emissivity εw. It is seen from Table 1 that the normal and hemispherical emissivities of a smooth water surface calculated by use of Fresnel’s equations for random polarization differ insignificantly from each other. Therefore, one can use either value as εw.

Table 1. Emissivity of the smooth water surface: εwn, in the normal direction; εwh, in the hemisphere; εw(1), in the direction of 35° to the normal (at horizontal polarization)

λ (mm) εw(1) εwh εwn
2.6 0.6134 0.6046 0.54
8.6 0.4500 0.4665 0.39
20.8 0.3846 0.4095 0.33
80 0.3650 0.3922 0.31
180 0.3626 0.3900 0.31

Consider the first computational results for monodisperse system emissivity presented in Fig. 1. If scattering is small (see the article, Water droplets and bubbles in microwave spectral range), the thick disperse layer is near to being absolutely black. The maximum emissivity can be estimated by use of DP0 and the transport approximation:

(3)

Figure 1. Spectral emissivity of the water surface covered by a monodisperse foam layer of thickness kh = 1 mm (a) and 10 mm (b) at δ1 = 10 μm (I) and 50 μm (II): 1 - λ = 2.6 mm, 2 - 8.6 mm, 3 - 20.8 mm, 4 - 80 mm, 5 - 180 mm.

The value ελmax is reached at various layer thicknesses for different extinction coefficients. Therefore, despite ελmax increasing with the wavelength, the emissivity at fixed kh ≤ 10 mm is considerably greater in the short-wave range (see Fig. 1). It is interesting that ελ at λ > 20.8 mm and a > 0.3 mm does not practically depend on bubble radius a and bubble wall thickness δ1. This fact is very important and can be used in the analysis of experimental data. Note that approximate constancy of the emissivity in the range mentioned is a consequence of almost linear dependences Qa(a).

The spectral emissivity of a polydisperse system at the typical size distribution of water bubbles from the paper by Dombrovsky (1979) is shown in Fig. 2. The value of ελ increases monotonically with kh, more rapidly in the short-wave range. At wavelength 2.6 mm, the foam layer of millimeter thickness (as a matter of fact, monolayer) is almost thick-limiting. A similar critical value kh* at the wavelength 8.6 mm is about 3 mm, and we have kh*≈ 10 mm at λ = 20.8 mm. At wavelengths 80 and 180 mm, even the foam layer of centimeter thickness does not essentially influence water surface emissivity.

Figure 2. Spectral emissivity of the water surface covered by a polydisperse foam layer at δ1 = 10 μm (a) and 50 μm (b): I - Mie theory, II - Rayleigh theory; 1 - λ = 2.6 mm, 2 - 8.6 mm, 3 - 20.8 mm, 4 - 80 mm, 5 - 180 mm.

A comparison of Fig. 2 with Fig. 1 shows that the calculations using the monodisperse model with equivalent particle radius a* = a32 give practically the same results as the exact ones. One can see in Fig. 2 that the Rayleigh approximation gives quite accurate results in the long-wave range λ ≥ 8.6 mm. Thus, we can use Eq. (7) from the article, Water droplets and bubbles in microwave spectral range, instead of the Mie theory calculations and the monodisperse model with Sauter radius a32. By using DP0 instead of DP1, we obtain the following equation in the limit of relatively small scattering:

(4)

or

(5)

where

(6)

It is interesting that foam structure coefficient K at λ ≥ 8.6 mm differs insignificantly for variants of δ1 = 10 and 50 μm. Simple Eqs. (5) and (6) can be used for fairly accurate determination of the emissivity of foam on the water surface in the range of the theoretical model applicability. An error of the calculations is less than 5% over the considered range of the problem parameters. Equation (5) can also be used for experimental verification of the model: according to the model, the dependences of ln/ on foam layer thickness h are expected to be near to linear at all wavelengths. On the other hand, Eq. (5) (after the experimental confirmation) can be used as a basis of semi-empirical methods of foam structure characterization. Similar equations hold true at various polarizations (at an appropriate choice of εw). Indeed, the linear polarized radiation penetrates through a medium with small albedo without essential variation of the polarization. Mathematically, it corresponds to small components of the scattering matrix, when the vector RTE is reduced to a system of scalar equations weakly related to each other.

The above presented model of microwave thermal radiation of foam on a water surface is based on the Mie theory and radiation transfer theory. The key assumption is as follows: the disperse system emissivity does not depend on the volume fraction of the water bubbles and coincides with that of a rarefied cloud of the bubbles in air. Applicability of the Mie theory and radiation transfer theory for a rarefied cloud of spherical particles is beyond any doubt. For this reason, a comparison of the theoretical predictions with experimental data gives information on the relative part of collective effects due to the touching of neighboring bubbles in the disperse system. Aside from that mentioned above, we use the following assumptions:

  • The lower layer of the water bubbles is at some distance over the smooth water surface.

  • Polarization effects are small and can be neglected.

  • All of the bubbles are spherical.

  • Bubble size distribution is the same across the foam layer.

  • The wall thickness of the water bubbles is the same for bubbles of various sizes.

The first assumption is of principal importance. It is a logical part of the general assumption of negligible collective effects. In the limits of the model, one can do without the second assumption using the vector RTE, but this assumption is acceptable in the problem parameter range under consideration. The rest of the assumptions are used to simplify the mathematics. The assumption of the same size distribution of bubbles across the foam layer is rather crude (Militskii et al., 1976), but it can be accepted due to the weak influence of the bubble radius in the range from 0.3 to 1 mm on the disperse system emissivity (Dombrovsky, 1996). The last assumption corresponds to the experimental data for the emulsion foam layer, but it should be verified for the foam of the cellular structure.

The emissivity calculations have been performed at parameters corresponding to the experimental conditions from Militskii et al. (1976) and Bordonskii et al. (1978). In addition, one should take into account that the calculations by Dombrovsky (1979) give hemispherical emissivity, whereas only the directional emissivity was measured in the experiments. Approximate relations for the directional emissivity can be obtained by use of RTE for a nonscattering medium. The spectral emissivity of the disperse medium in direction μ is given by

(7)

where εw(1) is the emissivity of the smooth water surface in direction μ. One can correct Eq. (7) for the case of weakly scattering medium by using Eq. (4) and substituting the ratio ελ(1)/ ελmax instead of εw(1). Finally, we obtain the following relation between directional and hemispherical emissivities of the disperse system:

(8)

A comparison of theoretical predictions and experimental data is shown in Fig. 3. Experimental points correspond to the above mentioned measurements by Bordonskii et al. (1978) at horizontal polarization at an angle of 35°. The values of εw(1) from Table 1 were used in the calculations.

Figure 3. Comparison of calculated emissivity (dashed lines) with experimental data by Bordonskii et al. (1978): (a) - emulsion foam layer of thickness 1 mm, (b) - cellular foam of thickness 10 mm.

In the case of an emulsion foam layer, good agreement between calculated and measured values takes place over the whole spectrum. In the case of a cellular foam structure, one can see a somewhat larger difference. Note that the last two of the above formulated assumptions are not correct for cellular structure foam. The radiation scattering by large and, probably, more thick-walled water bubbles of the cellular foam is expected to be greater. It corresponds to lower emissivity of the foam than for the above calculated values. For a more correct comparison, one should have experimental data both for horizontal and vertical polarizations at wavelength λ > 8.6 mm as well as measurements at various thicknesses of the disperse layer. Generally, there is satisfactory agreement between the theoretical predictions and experimental data of Bordonskii et al. (1978) both for the thin emulsion layer and for the thick foam layer of the cellular structure. It is interesting that the collective effects due to touching the neighboring water bubbles are not dominating and the model developed is applicable. High emissivity of foam on a water surface in the millimeter range is not a special effect of a densely packed disperse system, but it is a result of relatively small scattering of radiation by single thin-walled water bubbles.

It is important that the radiative properties of an elementary volume of the disperse system can be estimated on the basis of the Mie theory. As to emissivity calculations, the use of radiation transfer theory for a rarefied “stretched” disperse system is not principal. An alternative approach can be employed. This approach is based on the description of the foam layer as a nonscattering homogeneous medium with a complex index of refraction m* obtained through the complex amplitude of scattering. A corresponding model for microwave thermal radiation of foam, based on the Lorentz-Lorenz equation in the Rayleigh region, was suggested by Raizer and Sharkov (1981). The spectral emissivity of the foam-water system was then determined by use of the macroscopic theory of electromagnetic wave reflection from the plane-parallel dielectric layer.

The electromagnetic interaction between absorbing and scattering water bubbles effects the applicability of the radiation transfer theory or electrodynamic model of the homogeneous medium. For instance, at wavelength 2.6 mm, the conditions of medium homogeneity are not satisfied and scattering takes place, whereas the scattering cannot be correctly described by the electrodynamic model. In the centimeter range, the conditions of applicability of the radiation transfer theory are obviously not satisfied. Some intermediate situation takes place over the spectral range where the foam layer affects considerably the water surface emissivity; i.e., both the transfer theory and Lorentz-Lorenz equation are not quite correct. It is surprising that both descriptions give satisfactory qualitative and even quantitative results when the radiative properties and the electric dipole moments of single bubbles are determined by use of the Mie theory.

When foam in the ocean is produced by sea breaker collapse, the structure of the disperse system appears to be very complex and one should consider both foam and water spray simultaneously (Bortkovsky and Timanovsky, 1982; Bortkovsky, 1983; Bezzabotnov et al., 1986; Cherny and Sharkov, 1988). Formation of such a disperse system over the water surface leads to great variations of the radiative characteristics of the ocean in the microwave. A combined radiophysical model of this disperse system based on radiation transfer theory for the droplet spray and macroscopic electrodynamics of the foam layer has been developed by Dombrovsky and Raizer (1992). The model considers a stratified medium that consists of three layers: water, foam, and spray. The spray layer is treated as a disperse system of spherical water droplets, but the foam layer is treated as a homogeneous medium. Hemispherical emissivity of a disperse system is determined by use of the solution of the scalar RTE in the transport approximation. Note that one should solve the vector RTE to determine the polarization characteristics. Only thermal microwave radiation in the nadir direction was analyzed by Dombrovsky and Raizer (1992). In this case, the polarization of radiation of the foam on the water surface can be ignored both in the millimeter and near-centimeter spectral ranges (Raizer and Sharkov, 1981). The radiation transfer in the spray layer was calculated by Dombrovsky and Raizer (1992) using the DP1 approximation. The spectral emissivity of a sub-layer (foam layer on water surface) was determined by use of electrodynamic model of a homogeneous medium with a complex index of refraction calculated by the Lorentz-Lorenz equation. For simplicity, the universal type of size distribution (10) from the article, Radiative properties of polydisperse systems of independent particles, was chosen for water droplets and bubbles.

We will not reproduce here the results reported by Dombrovsky and Raizer (1992). Nevertheless, it is interesting to remember some physical conclusions, which were formulated for the important value of radio-brightness contrast due to spray:

(9)

where T0 is the temperature of the system, ελ is the spectral hemispherical emissivity of the disperse system as a whole, ελf,w is the corresponding emissivity of the sub-layer (f, for the foam on water surface; w, for the smooth water surface). More correct values of radio-brightness contrast ΔTb can be determined after averaging of ΔTb0 for the known distribution of the foam layer thickness. The computational analysis showed that the brightness contrast is always positive when there is no foam on the water surface. With the foam layer, the contrast may be both positive and negative [this physical effect was first discussed by Shifrin (1953)]. The spectral dependence of the brightness contrast due to spray ΔTb(λ) appears not to be trivial, as in the earlier suggested models of transitional dielectric layers with smoothly varying parameters. This dependence appears to be sign-alternating: the contrast may be both positive and negative. It is important that the theoretical model of Dombrovsky and Raizer (1992) takes into account the main special features of real foam-spray structures and enables us to find preferable spectral ranges for microwave remote sensing of the ocean surface. This approximate model is also used at present in the analysis of the remote sensing data (Camps et al., 2005; Raizer, 2005, 2006, 2007). At the same time, one should refer to applications of the general dense media radiation transfer theory for microwave remote sensing of foam covered ocean (Guo et al., 2001; Rose et al., 2002; Chen et al., 2003).

REFERENCES

Basharinov, A. E., Gurvich, A. S., and Egorov, S. T., Radioemission of the Earth as a Planet, Moscow: Nauka (in Russian), 1974.

Bezzabotnov, V. S., Bortkovsky, R. S., and Timanovsky, D. F., On the structure of two-phase medium generated at wind-wave breaking, Izv., Atmos. Oceanic Phys., vol. 22, no. 11, pp. 1186-1193, 1986.

Bordonskii, G. S., Vasil’kova, I. B., Veselov, V. M., Vorsin, N. N., Militskii, Yu. A., Mirovskii, V. G., Nikitin, V. V., Raizer, V. Yu., Khapin, Yu. B., Sharkov, E. A., and Etkin, V. S., Spectral characteristics of the microwave emissivity of foam structures, Izv., Atmos. Oceanic Phys., vol. 14, no. 6, pp. 464-469, 1978.

Bortkovsky, R. S. and Timanovsky, D. F., On the microstructure of wind-waves breaking crests, Izv., Atmos. Oceanic Phys., vol. 18, no. 3, pp. 327-329, 1982.

Bortkovsky, R. S., Heat- and Moisture Transfer between Atmosphere and Ocean at Storm Conditions, Leningrad: Hydrometeoizdat (in Russian), 1983.

Camps, A., Vall-Ilossera, M., Villarino, R., Reul, N., Chapron, B., Corbella, I., Duffo, N., Torres, F., Miranda, J., Sabia, R., Monerris, A., and Rodrigues, R., The emissivity of foam-covered water surface at L-band: Theoretical modeling and experimental results from the FROG 2003 field experiment, IEEE Trans. Geosci. Remote Sens., vol. 43, no. 5, pp. 925-937, 2005.

Chen, D., Tsang, L., Zhou, L., Reising, S. C., Asher, W. E., Rose, L. A., Ding, K.-H., and Chen, C.-T., Microwave emission and scattering of foam based on Monte Carlo simulations of dense media, IEEE Trans. Geosci. Remote Sens., vol. 41, no. 4, pp. 782-790, 2003.

Cherny, I. V. and Sharkov, E. A., Remote radiometry of the sea wave breaking cycle, Earth Explor. Space, vol. 2, pp. 17-28 (in Russian), 1988.

Cherny, I. V. and Raizer, V. Y., Passive Microwave Remote Sensing of Oceans, New York: Wiley, 1998.

Dombrovsky, L. A., Calculation of the thermal radiation emission of foam on the sea surface, Izv., Atmos. Oceanic Phys., vol. 15, no. 3, pp. 193-198, 1979.

Dombrovsky, L. A., Radiation Heat Transfer in Disperse Systems, New York: Begell House, 1996.

Dombrovsky, L. A. and Raizer, V. Yu., Microwave model of a two-phase medium at the ocean surface, Izv., Atmos. Oceanic Phys., vol. 28, no. 8, pp. 650-656, 1992.

Guo, J., Tsang, L., Asher, W., Ding, K.-H., and Chen, C.-T., Applications of dense media radiative transfer theory for passive microwave remote sensing of foam covered ocean, IEEE Trans. Geosci. Remote Sens., vol. 39, no. 5, pp. 1019-1027, 2001.

Militskii, Yu. A., Raizer, V. Yu., Sharkov, E. A., and Etkin, V. S., On scattering of UHF-radiation by foamy structures, J. Tech. Phys. Lett., vol. 2, no. 18, pp. 851-855 (in Russian), 1976.

Raizer, V., A combined foam-spray model for ocean microwave radiometry, Proc. of IGARSS’2005, Seoul, Korea, July 25-29, 2005.

Raizer, V., Macroscopic foam-spray models for ocean microwave radiometry, Proc. of IGARSS’2006, Denver, July 31-Aug. 04, 2006.

Raizer, V., Macroscopic foam-spray models for ocean microwave radiometry, IEEE Trans. Geosci. Remote Sens., vol. 45, no. 10, pp. 3138-3144, 2007.

Raizer, V. Yu. and Sharkov, E. A., Electrodynamic description of densely packed disperse media, Radiophys. Quantum Electron., vol. 24, no. 7, pp. 553-560, 1981.

Raizer, V. Yu., Sharkov, E. A., and Etkin, V. S., Sea foam: Physical and chemical properties, emission and reflection characteristics, Space Research Institute of the USSR Academy of Sciences, Preprint No. 306 (in Russian), 1976.

Rose, L. A., Asher, W. E., Reising, S. C., Gaiser, P. W., St. Germain, K. M., Dowgiallo, D. J., Horgan, K. A., Farquharson, G., and Knapp, E. J., Radiometric measurements of the microwave emissivity of foam, IEEE Trans. Geosci. Remote Sens., vol. 40, no. 12, pp. 2619-2625, 2002.

Sharkov, E. A., Passive Microwave Remote Sensing of the Earth: Physical Foundations, Chichester, UK: Praxis, 2003.

Sharkov, E. A., Breaking Ocean Waves: Geometry, Structure, and Remote Sensing, Chichester, UK: Praxis, 2007.

Shifrin, K. S., On the albedo theory, Trans. GGO, vol. 39, pp. 244-257 (in Russian), 1953.

Stogryn, A., The emissivity of sea foam at microwave frequencies, J. Geophys. Res., vol. 77, no. 9, pp. 1658-1666, 1972.

Vorsin, N. N., Glotov, A. A., Mirovskii, V. G., Raizer, V. Yu., Troika, I. A., Sharkov, E. A., and Etkin, V. S., Natural radio emissive measurements of sea foam structures, Sov. J. Remote Sens., vol. 2, no. 3, pp. 520-525, 1982.

Les références

  1. Basharinov, A. E., Gurvich, A. S., and Egorov, S. T., Radioemission of the Earth as a Planet, Moscow: Nauka (in Russian), 1974.
  2. Bezzabotnov, V. S., Bortkovsky, R. S., and Timanovsky, D. F., On the structure of two-phase medium generated at wind-wave breaking, Izv., Atmos. Oceanic Phys., vol. 22, no. 11, pp. 1186-1193, 1986.
  3. Bordonskii, G. S., Vasil’kova, I. B., Veselov, V. M., Vorsin, N. N., Militskii, Yu. A., Mirovskii, V. G., Nikitin, V. V., Raizer, V. Yu., Khapin, Yu. B., Sharkov, E. A., and Etkin, V. S., Spectral characteristics of the microwave emissivity of foam structures, Izv., Atmos. Oceanic Phys., vol. 14, no. 6, pp. 464-469, 1978.
  4. Bortkovsky, R. S. and Timanovsky, D. F., On the microstructure of wind-waves breaking crests, Izv., Atmos. Oceanic Phys., vol. 18, no. 3, pp. 327-329, 1982.
  5. Bortkovsky, R. S., Heat- and Moisture Transfer between Atmosphere and Ocean at Storm Conditions, Leningrad: Hydrometeoizdat (in Russian), 1983.
  6. Camps, A., Vall-Ilossera, M., Villarino, R., Reul, N., Chapron, B., Corbella, I., Duffo, N., Torres, F., Miranda, J., Sabia, R., Monerris, A., and Rodrigues, R., The emissivity of foam-covered water surface at L-band: Theoretical modeling and experimental results from the FROG 2003 field experiment, IEEE Trans. Geosci. Remote Sens., vol. 43, no. 5, pp. 925-937, 2005.
  7. Chen, D., Tsang, L., Zhou, L., Reising, S. C., Asher, W. E., Rose, L. A., Ding, K.-H., and Chen, C.-T., Microwave emission and scattering of foam based on Monte Carlo simulations of dense media, IEEE Trans. Geosci. Remote Sens., vol. 41, no. 4, pp. 782-790, 2003.
  8. Cherny, I. V. and Sharkov, E. A., Remote radiometry of the sea wave breaking cycle, Earth Explor. Space, vol. 2, pp. 17-28 (in Russian), 1988.
  9. Cherny, I. V. and Raizer, V. Y., Passive Microwave Remote Sensing of Oceans, New York: Wiley, 1998.
  10. Dombrovsky, L. A., Calculation of the thermal radiation emission of foam on the sea surface, Izv., Atmos. Oceanic Phys., vol. 15, no. 3, pp. 193-198, 1979.
  11. Dombrovsky, L. A., Radiation Heat Transfer in Disperse Systems, New York: Begell House, 1996.
  12. Dombrovsky, L. A. and Raizer, V. Yu., Microwave model of a two-phase medium at the ocean surface, Izv., Atmos. Oceanic Phys., vol. 28, no. 8, pp. 650-656, 1992.
  13. Guo, J., Tsang, L., Asher, W., Ding, K.-H., and Chen, C.-T., Applications of dense media radiative transfer theory for passive microwave remote sensing of foam covered ocean, IEEE Trans. Geosci. Remote Sens., vol. 39, no. 5, pp. 1019-1027, 2001.
  14. Militskii, Yu. A., Raizer, V. Yu., Sharkov, E. A., and Etkin, V. S., On scattering of UHF-radiation by foamy structures, J. Tech. Phys. Lett., vol. 2, no. 18, pp. 851-855 (in Russian), 1976.
  15. Raizer, V., A combined foam-spray model for ocean microwave radiometry, Proc. of IGARSS’2005, Seoul, Korea, July 25-29, 2005.
  16. Raizer, V., Macroscopic foam-spray models for ocean microwave radiometry, Proc. of IGARSS’2006, Denver, July 31-Aug. 04, 2006.
  17. Raizer, V., Macroscopic foam-spray models for ocean microwave radiometry, IEEE Trans. Geosci. Remote Sens., vol. 45, no. 10, pp. 3138-3144, 2007.
  18. Raizer, V. Yu. and Sharkov, E. A., Electrodynamic description of densely packed disperse media, Radiophys. Quantum Electron., vol. 24, no. 7, pp. 553-560, 1981.
  19. Raizer, V. Yu., Sharkov, E. A., and Etkin, V. S., Sea foam: Physical and chemical properties, emission and reflection characteristics, Space Research Institute of the USSR Academy of Sciences, Preprint No. 306 (in Russian), 1976.
  20. Rose, L. A., Asher, W. E., Reising, S. C., Gaiser, P. W., St. Germain, K. M., Dowgiallo, D. J., Horgan, K. A., Farquharson, G., and Knapp, E. J., Radiometric measurements of the microwave emissivity of foam, IEEE Trans. Geosci. Remote Sens., vol. 40, no. 12, pp. 2619-2625, 2002.
  21. Sharkov, E. A., Passive Microwave Remote Sensing of the Earth: Physical Foundations, Chichester, UK: Praxis, 2003.
  22. Sharkov, E. A., Breaking Ocean Waves: Geometry, Structure, and Remote Sensing, Chichester, UK: Praxis, 2007.
  23. Shifrin, K. S., On the albedo theory, Trans. GGO, vol. 39, pp. 244-257 (in Russian), 1953.
  24. Stogryn, A., The emissivity of sea foam at microwave frequencies, J. Geophys. Res., vol. 77, no. 9, pp. 1658-1666, 1972.
  25. Vorsin, N. N., Glotov, A. A., Mirovskii, V. G., Raizer, V. Yu., Troika, I. A., Sharkov, E. A., and Etkin, V. S., Natural radio emissive measurements of sea foam structures, Sov. J. Remote Sens., vol. 2, no. 3, pp. 520-525, 1982.
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