1. Introduction

The use of Infrared polarization is becoming more common in remote sensing, especially for water, which is perhaps the most strongly polarized thermal infrared source in nature [1]. Infrared polarimetry has been proposed as a method for remote sensing of water surface roughness [1–3] and for identification of man-made objects in military surveillance [4,5]. Even non-polarimetric water remote sensing applications require consideration of polarization when the sensor response varies with polarization [6,7] Whenever water is considered in these analyses, a polarized emissivity is required for the water surface.

Emissivity is a number that varies from zero to one, which describes how efficiently something radiates thermal radiation. Thermal emission from any object can be calculated as the product of the object’s emissivity and the Planck function evaluated at the object’s physical temperature and the wavelength of interest. For example, a perfect blackbody has an emissivity of one, which implies that it radiates the full amount of energy that is predicted by the Planck function alone; conversely, a perfect reflector has an emissivity of zero, implying that it neither absorbs nor emits any energy at all.

Computing polarized emissivity for a smooth, specular water surface requires only simple application of the Fresnel equations [1,2]. However, a more realistic wind-roughened surface requires more complicated calculations [8–10]. Searching the literature for calculated emissivity values currently yields only unpolarized results [8,9]. This electronic paper addresses the need for tabulated values of polarized water emissivity by providing user-interactive tables from which data can be copied for a variety of wavelengths, viewing angles, and wind speeds. The intent is to provide reliable numbers for realistic analysis in a format that is easier and more reliable to use than printed tables, without the user needing to program rough-surface integrals.

We consider the thermal infrared spectral range of 3–15 µm in 0.1 µm steps, viewing angles of 0–70° with respect to the nadir, and wind speeds of 0, 1, 3, 5, 10, and 15 m s^-1. We do not present results for larger angles where the present algorithm does not adequately treat emission and multiple reflection from neighboring wave facets. The rough surface is modeled with the simple isotropic Gaussian term from the Cox-Munk slope-probability model [11]. A more detailed model could also include higher-order slope-statistics terms, as well as the variation with azimuthal roughness [11] and air-sea temperature difference [12].

2. Rough-surface emissivity model

Each polarization component of infrared emissivity for a smooth, specular water surface can be calculated as one minus the corresponding reflectivity,

where the reflectivity is given for each polarization state by the Fresnel equations:

and

In these equations, the superscripts s and p indicate the polarization state, with s polarization perpendicular to the plane of incidence and p polarization parallel to the plane of incidence. Thus, for a flat, horizontal water surface, s indicates horizontal and p indicates vertical. The other variables in eqns. (1) and (2) are wavelength λ, viewing angle in air θ, angle of refraction in the water θ_r, and complex refractive index n. We use the complex refractive index values given by Hale and Querry [13] for pure water because the the sea-water values change the emissivity by only about 0.1%. For a given viewing angle in air, the angle of refraction can be calculated from Snell’s Law:

Smooth-surface calculations are relatively simple, but a rough surface requires integration over the randomly oriented specular facets. The derivation and details of this procedure are described elsewhere [8–10] so we give only a brief overview here. We compute an effective emissivity for the rough-surface, which is found by multiplying the Fresnel emissivity for each wave slope by the slope probability density function (pdf), and integrating over the full range of possible slopes, weighted by the projected wave-facet area. For simplicity, we use the isotropic Gaussian term from the Cox-Munk model for wave-slope probability density function [11]:

This pdf describes the probability of a wave facet having slope θ_n with respect to the zenith for any wave-slope variance (mean-square slope) σ². In the Cox-Munk model, the mean-square slope depends linearly on wind speed w according to

Shaw and Churnside recently demonstrated that the mean-square slope also depends on the air-sea temperature difference [12] so the results shown here (as with all results from the Cox-Munk model) are valid strictly for neutral stability (equal temperatures of the water surface and immediately overlying air). Also, higher-order terms in the slope pdf are required to simulate a more realistic rough surface [11,12], but their relatively small contribution to the slope integral allows these terms to be dropped for the sake of simplicity.

The formulation used here for the effective emissivity of a wind-roughened surface is the same as that derived by previous authors [8–10], so the derivation is not repeated. Our adaptation of this formulation uses the following notation: µ_e=cos(θ) is the cosine of the emission viewing angle with respect to nadir; µ_n=cos(θ_n) is the cosine of the angle of the wave-facet normal with respect to zenith; ϕ is the azimuth angle of emission with respect to the wave-facet normal, and χ is the emission angle with respect to the wave-facet normal, calculated from

The effective emissivity is calculated as an expectation average for each polarization state as

with

and the normalization factor,

Because the wave-slope probability density function must be conditioned by both the wave-facet projected area [cos(χ)] and the Jacobian, the factor given by eq. (9) is required to normalize the density function in the numerator of eq. (7). Also, even with the isotropic Cox-Munk wave-slope probability, we computed the integrals in eqs. 6–9 over the indicated azimuth limits.

To use these equations for calculating the polarized rough-surface emissivity, the polarized smooth-surface emissivity components for each facet are calculated from eqns. (1) and (2) to generate each ∊^s,p(λ,χ) inside the integrals of eq. (8). The entire calculation procedure is repeated separately for the s- and p-polarized emissivity components.

The resulting degree of polarization D for emitted radiance is given as

where the superscripts s and p denote the corresponding polarization components of emissivity. Note that this is the degree of polarization for the emitted radiance alone; the total radiance observed in an actual polarimetric measurement would also include reflected background radiance and atmospheric absorption [1].

3. Results

We calculated polarized rough-surface emissivity for a similar range of wavelengths, viewing angles, and wind speeds as did Masuda et al. [8] for unpolarized emissivity, except that we used angles up to only θ=70° because at larger angles a more sophisticated scheme is required to include the effect of emission and multiple reflections from neighboring facets [9,14]. Our calculations are for wavelengths of 3 µm to 15 µm in 0.1 µm steps, viewing angles of 0°, 10°, 20°, 30°, 40°, 45°, 50°, 60°, and 70°, at wind speeds of 0, 1, 3, 5, 10, and 15 m s^-1.

Figure 1 shows the s- and p-polarized emissivity on the left and the resulting degree of polarization on the right, both versus wavelength (3–15 µm) for zero wind speed and 70° viewing angle. Clicking on this graph activates a Java applet that displays the spectral emissivity and degree of polarization for different angles and wind speeds. The slider at the bottom of the interactive graph selects different combinations of wind speed and angle.

 

Fig. 1. An example of how infrared emission from water becomes increasingly p-polarized as the viewing angle increases. The left-hand graph shows the polarized emissivity components and the right-hand graph shows the resulting degree of emission polarization. This example is for zero wind speed and 70° viewing angle. Click the figure to activate an interactive version. [Media 1]

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Table 1 provides values of s- and p-polarized emissivity components vs wavelength. Clicking the table activates an interactive version that loads the emissivity values for a user- selected wind speed and viewing angle. The emissivity and wavelength numbers can be copied and pasted electronically from this electronic table to your own application.

In the interactive table, use the arrow keys to move up and down in the data columns. The columns in the upper right-hand corner provide access to each individual polarized emissivity component, while the one at the bottom allows you simultaneously to scroll through the wavelength and emissivity columns. Clicking once selects the entire column, allowing you to copy and paste the full array of data.

 

Table 1. Click the table to obtain polarized emissivity values for the chosen wind speed and viewing angle. [Media 2]

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4. Discussion

In Figures 1 and 2, notice how the emissivity polarization components change with viewing angle as the angle passes the Brewster angle (~49°–56°). At nonzero angles, but below the Brewster angle, the p-polarized emissivity is larger than the zero-angle value and the s-polarized emissivity is smaller. The p-polarized component reaches a peak near unity at the Brewster angle and becomes steadily smaller as the angle is increased beyond the Brewster angle. Note that the behavior of emission polarization at the Brewster angle is fundamentally different from that of reflection polarization (see Fig. 2). Whereas light reflected from a perfect dielectric at the Brewster angle is 100% s-polarized (less than 100% for a lossy dielectric, but still maximum), emission polarization does not achieve 100% (or even maximum) polarization at the Brewster angle because neither of the two emissivity components is zero there. In fact, for pure emission polarization, the degree of polarization magnitude increases steadily as the viewing angle ranges from 0° to 90°.

 

Fig. 2. Degree of polarization versus viewing angle for reflection from water (top) and emission from water (bottom) at two thermal infrared wavelengths. Note that the significantly large imaginary component of the refractive index at 12 µm results in peak reflection polarization less than 100%.

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It is important to remember that the actual radiance observed from water in nature contains a combination of emission and reflection polarization [1]. Water observed in nature at thermal infrared wavelengths typically exhibits a net degree of p-polarization that peaks near 75°–80°, and Fig. 1 shows only the polarization of the radiation emitted by the water surface. Furthermore, strictly speaking, the background reflection must be included inside the rough-surface integral, making our results valid only for the case of a uniform clear-sky background. However, at least for the low wind speed conditions, the effective emissivities presented here can be used in Fresnel calculations to approximate the scene polarization for a given background.

Averaging the s and p emissivity values from our table for any one set of conditions reproduces the unpolarized emissivity values published by Masuda et al. [8] to within ±0.0001 (~0.01% difference) except for a few values at high wind speed and large angle where the numerical algorithm stability seems to become questionable. For example, the data for a wind speed of 15 m s-1 and angle of 70° are included in the table, but their agreement with the previously published average values varies from about 0.1 to 0.01 throughout the spectrum (approximately 1–10% error).

5. Conclusion

We have provided an electronic table from which the reader can copy columns of polarized emissivity versus wavelength for a rough water surface. An interactive graph also provides the opportunity to observe how the emissivity polarization components change with viewing angle, wind speed, and wavelength. We hope that this provides the interested reader with a chance to become more familiar with the concepts of emission polarization and to obtain readily usable polarized emissivity values for use in modeling and analysis of infrared polarimetry.