1. Introduction
In remote sensing, on which applications from weather forecasting to environmental studies rely, understanding the interaction between light and different materials offers useful information about the underlying properties of each material. Central to this understanding is the concept of polarization, since it offers potentially significant information useful for a number of applications, for example by providing a means of directly estimating the index of refraction [
1]. In the context of hyperspectral imaging, there are a relatively limited number of past studies that have treated polarization in remote sensing analyses. Most of these have focused on remote sensing of the water column, atmospheric correction, or snow [
2,
3,
4,
5]. There have been only a few quantitative radiative transfer models used in the analysis of polarimetric hyperspectral data [
6]. For remote sensing of granular materials (sediments, planetary regoliths), attempts to create a quantitative model for polarization have had only limited success [
7,
8,
9] and by and large, these analyses have typically been applied to multi-spectral data.
Studies of the interactions of polarized light with surfaces or interfaces have a long history dating back to the early work of Fresnel [
10]. In astronomy, historical roots can be traced back to planetary observations of the moon and planets by Arago (1811) and Lyot [
7,
11]. From a practical perspective, polarization ratios provide both useful information and the added advantage that absolute detector calibration is not required. In this work, we focus on the linear polarization ratio in the context of a hyperspectral imaging system and explore the effects of particulate grain size on the response of the linear polarization ratio as a function of phase angle. Our emphasis is on particulate media where the grain size relative to the wavelength of incident light can be considered to be in the geometric optics regime where radiative transfer models are appropriate. Models for smaller particle sizes have been considered from the perspective of Mie theory [
12].
Between phase angles of 0 and 20 degrees, in some particulate media, the linear polarization ratio is negative, a region of the linear polarization curve referred to as the negative branch, and the much larger positive branch extends from 20 to 180 degrees. Due to the complexity of modeling complex scattering interactions in terms of polarization, theoretical studies of polarization have dealt with the positive and negative branches separately [
7,
8]. The linear polarization ratio is defined in terms of a radiance ratio of the difference and sum of two orthogonal polarization states as defined in Equation (
1):
where
and
represent the perpendicular and parallel components of the radiance.
When a negative polarization branch exists, the typical minimum value in the negative branch is relatively small, ∼−0.01, at a phase angle below
and the maximum value on the positive side is around 0.1 at approximately 100–110
[
7]. Influences on polarization come from varying forms of scattering, as explained in
Figure 1.
As
Figure 1 illustrates, in a particulate medium, these observations stem from contributions due to surface scattering from particles (positively polarized light), volume scattering within particles (negatively polarized light), and multiple scattering between particles (negatively polarized light at small angles due to coherent backscatter effects [
7,
13], but otherwise unpolarized at larger angles). The coherent backscatter opposition effect (CBOE) has been well studied as one mechanism for the increased brightness at small phase angles observed in scattering from granular media, specifically at very small phase angles (typically
), and analysis and comparison with experiment suggest that low-order multiple scattering contributions (≤4 scattering events) typically contribute most at these small phase angles [
7,
13]. For light originating from a larger number of scattering events, the trend is toward randomization of the polarization, which in the aggregate leads to an unpolarized contribution.
Figure 1 also illustrates that, as was observed previously, in a close-packed medium, diffraction effects can generally be ignored in the far field because interference in the near field is blocked by the presence of nearby particles [
7].
Hapke developed a polarization model based on their Isotropic Multiple Scattering Approximation (IMSA) radiative transfer model [
7], which will be discussed further in the next section. However, his model only focuses on the positive branch of polarization and does not agree with observational data at larger phase angles. This may be due in part to the assumptions made in model development, such as the neglect of negative polarization stemming from light transmitted through particles, the equal division of multiple scattering between the parallel and perpendicular radiance components, and the chosen isotropic form of multiple scattering used [
7].
Prior to 1994, Shkuratov conceived several studies to model the negative branch in terms of different scattering phenomena. Shkuratov eventually summarized these models dividing them into 4 categories [
8]. The first three are multiple reflection, refraction, and diffraction which were based on Lyot’s three hypotheses [
8]. The final and most promising category used models of coherent backscatter.
This study examines the effect of grain size distribution on observed polarimetric hyperspectral reflectance data. The observed link between these provides insight that may help to improve models of polarization overall through analysis of physical parameters which impact polarization, allowing for better inversion and retrieval of these parameters. In our study, we focus on granular materials which may have both surface and volume scattering. In these materials, as noted above, the scattering mechanisms exhibited by granular materials, which appear in
Figure 1, include multiple effects that contribute to the polarization. This contrasts with materials that exhibit primarily strong surface scattering and are better described by the Fresnel equations. While the impact of grain size on polarimetric data has been studied before by those mentioned above, here this relationship is studied using hyperspectral polarimetric data, giving much more insight into the wavelength dependence of polarimetric imagery of particulate surfaces. We also derive specific relationships between grain size and observed changes in an approximately linear region of the positive polarization branch.
5. Results and Discussion
Figure 6 shows the linear polarization ratio for a representative wavelength (633 nm) for each sample measured, averaged along each line of the sample in the image data as described earlier, with one mean value in the plot for every row of the image after masking out the background. As shown, the olivine from Washington Mills displays a much more distinct separation and trend as a function of grain size than the other geometric-optics regime samples, including the AGSCO olivine. However, the AGSCO olivine does have narrower steps between grain size categories than the Washington Mills olivine, so the behavior might have been more similar if the samples could have been binned the same way, and we note that the separation in polarization between the largest and smallest grains is actually similar between the two olivine samples. The AGSCO olivine, however, does have more separation than the nepheline or silica. This may be due to a combination of differences in material properties and the specific grain size categories chosen. The Washington Mills olivine shows a much greater increase in the polarization maximum with increasing grain size than the other samples, for which the maximum increases more gradually. The reason for this will require further study. However, when we examine the best-fit polynomials to each curve in
Figure 7, every sample among the geometric optics regime-sized samples shows some separation between curves for the different grain sizes at phase angles of 45 deg or higher.
The final plots in
Figure 6 and
Figure 7 show the nepheline sample with resonance regime-sized particles. Here, the particle size distribution is better described by Mie Scattering due to the relative size of the particles compared to the wavelength. Because of its distinct behavior, in
Figure 6, a version of these data is shown with a zoomed-in scale to provide greater detail. A point of interest is that the curve for this sample is significantly flatter beyond a phase angle of 30 deg than for the other samples. None of the samples showed a negative polarization branch, except for the resonance regime-sized sample below a phase angle of
. This is in line with Shkuratov’s results for resonance-sized samples. We return to this observation in our discussion of the results later in this section.
Figure 7 shows a 5th-order polynomial fit to each of the datasets appearing in
Figure 6. This offers an easier analysis of the behavior of the data in the macro scale, i.e., across the full range of phase angles. The general trend for the Washington Mills olivine remains consistent for the averaged data across the different grain size distributions; however, the slope increases significantly as grain size increases. In contrast, the other geometric-optics regime samples all have grain size-designated curves that intersect the other curves more than once. On the other hand, consistent with the Washington Mills olivine sample, the other geometric optics regime samples each have regions in which there is increasing polarization with grain size. The resonance sized sample is entirely distinct, even having a negative polarization branch and multiple significant slope changes.
Figure 7 also shows the RMSE of fifth-order polynomial fits to the data sets. The fits to linear polarization with phase angle did not exhibit significant wavelength dependence and had comparable RMSE provided wavelengths were sufficiently far from the edges of the spectral range of the imaging system, avoiding lower signal-to-noise (SNR) regions of the imaging system. In our study, we found satisfactory data quality between 400 and 900 nm. Additionally, the necessary order of the model to achieve sufficient accuracy did not vary significantly with different grain size ranges within a sample. First, second and third-order fits were also considered, but had higher RMSE, while fifth-order fits captured the data most accurately. Fifth-order polynomial fits may not be sufficient to model samples displaying measurements taken at higher phase angles than the maximum range measured in this study (120 deg). Orders higher than 5 were tested and found to be unnecessary for the range of phase angles measured in our study, as these higher orders did not noticeably decrease the RMSE.
In addition to the trends already discussed, our measurements indicated a region of the polarization curve which was linear between phase angles of 20 and 80 deg for all of the samples considered. A linear fit at a wavelength of 633 nm for each sample appears in
Figure 8, and for 425 and 850 nm in
Figure 9, demonstrating the consistency of this trend across the wavelength range.
Figure 8 shows these fits to be as accurate as the fifth order fits to the full-range data in
Figure 6, and additional analysis not shown in this paper proved that the RMSE is not significantly changed between a linear fit and the fifth order curve fit within the 20–80 deg region, suggesting that this region is basically linear.
Figure 9 demonstrates that the RMSE of linear fits of data is still sufficient at 850 nm. For some of the samples, there is an increase in RMSE at 425 nm compared to the other two wavelengths shown, possibly because the 425 nm wavelength is too close to the lower end of the spectral range where SNR is lower. Nevertheless, a linear region was still observed at even those wavelengths at the extreme end of the spectral range (i.e., the RMSE even at 425 nm was not significantly worse for a linear vs. 5th order curve fit).
We analyzed the impact of grain size, wavelength, and material type on the slope of the linear region. The results are shown for each respective material type in
Table 1. The Table shows that, to some degree, this slope varies with all three of the aforementioned properties. The one material which has a different trend, compared to the others, is the nepheline sample. As we discuss later, this material appears to be more strongly surface scattering than the other materials, which may have led to some of the apparent differences. We analyze these differences further below.
At each wavelength, we extracted a single maximum value for the average polarization along with the corresponding phase angle at each wavelength. The results appear in
Figure 10. For the silica and nepheline samples, the lowest value of the polarization maximum occurs near 810 nm. Physically this wavelength corresponds to a local minimum of water absorption [
30]. Although all samples were dried in an oven prior to measurement, hygroscopic absorption could still play a role, and this feature near 810 nm may indicate that hygroscopic moisture might be present in the sample data. However, this lowest value of the polarization maximum is in a broad minimum in the silica and nepheline plots, and we also observe that the lowest value of the polarization maximum for the olivine samples is closer to 700 nm, which also casts doubt on the presence of hygroscopic moisture.
In
Figure 10, we see that the phase angle of the maximum falls generally within the 100
–120
range predicted by theory, excluding the noise in some of the samples. For the typical indices of refraction of the materials used in this study, the Fresnel equations predict a polarization maximum in this range. We also observe that the phase angle of the linear polarization ratio maximum of these samples is only very weakly correlated with wavelength in some of the materials measured; instead, the phase angle of the maximum appears to be at certain discrete levels. The phase angle of the maximum has some weak correlation with grain size, but the degree to which this is true varies with the material.
For each sample, we also analyzed the impact of the average grain size on the slope of the linear region of the polarization curve for each sample. The results appear in
Figure 11. To illustrate the trends, we calculated slopes of the linear region of the polarization curves for four representative wavelengths at even intervals in the middle of the spectral range of the hyperspectral imaging system.
Figure 11 shows the slope of the linear region of the sample data for the average grain size of each sifted sample for each material. Plots appear in this Figure for each of the four example wavelengths in the spectral range of the hyperspectral imaging system. Comparing linear and quadratic models, we found that quadratic fits had the lowest RMSE and best
values for all samples, with the exception of the Washington Mills olivine for which a linear model was sufficient. In all cases, the
goodness of fit ranged between 0.85 and 1.0. However, one notable distinction is the difference in the concavity of the AGSCO nepheline quadratic fit compared to the quadratic fits obtained for the AGSCO olivine and AGSCO silica samples. Material properties or potentially even particle shape may play a role in this. Since nepheline is highly reflective over a broad range of wavelengths with grains that visually exhibit glint and sparkle, there is likely greater surface scattering and less volume scattering and absorption in this material compared to the others used in our study. Also, the impact of grain size on the slope of the linear region appears to be much stronger for olivine, though the order of the relationship was inconsistent between the two olivine samples.
To elaborate, as noted in
Figure 1, surface scatter will tend to increase polarization while light that has been transmitted through particles and emerges from the particle will tend to be negatively polarized. In particular, the amount of negatively polarized light emerging from particles will depend on the particle diameter because the mean ray path, or average transit length through a particle, should be directly proportional to particle diameter. The theoretical basis of this argument is an equivalent slab model for particle absorption and transmission discussed by Hapke [
7]. In his model, the mean ray path
is proportional to the average particle diameter
D:
where
represents the real part of the index of refraction of the material relative to the surrounding medium (in this case air). Thus, the contribution of negatively polarized light that has been transmitted should be expected to decrease as particles grow larger and extinction increases with the longer average path within the particle. This leads to an increasing polarization maximum as particles increase in size, which is the effect that we observe in
Figure 6. The increased peak translates also to a steeper slope in the approximately linear region of the polarization curves between 20
–80
, and thus the observed correlation and increase in the slope in this region as grain size increases; we observe this trend for all but the nepheline sample, which had significantly smaller particle sizes, and for which as noted earlier, the particles are likely dominated by surface scattering rather than volume scattering.
Returning to the question of why no negative branch of polarization was observed in the geometric optics regime samples, it seems likely that the absence of a negative branch may also be related to a similar mechanism. In backscattered light, especially close to the opposition direction with phase angles less than where the negative branch would be observed if present, light that has been reflected from the surface will be positively polarized, while contributions from light that refract into the particle and then are scattered backward within the volume or from somewhere in the interior surface eventually emerging from the particle, would be negatively polarized. However, in larger particles, this latter contribution from light scattering from the particle volume or interior particle surface and eventually emerging from the particle in a direction toward a sensor positioned at smaller phase angles (<20) will diminish with longer mean ray paths and therefore greater extinction. Thus, larger particles will have greater polarization and may not have a negative polarization branch at all, which is what we observed in our samples. In the case of the sample with smaller 1–5 m nepheline particles, that are in the resonance regime, this contribution may be as small as it is because the material has such strong surface scattering, which is positively polarized, leading to the observed very small scale of the negative branch of linear polarization observed below .
The particulate samples in our analysis were illuminated with a directional source in a controlled laboratory setting. Although this illumination geometry simulates direct sunlight, skylight also contributes to the illumination in outdoor field conditions. Skylight is highly polarized due to Rayleigh scattering, and the polarization of the sky is dependent on the local Sun angles, the distribution of aerosol scatterers, and other atmospheric parameters. For bluer wavelengths, we estimated diffuse illumination in past field experiments by measuring the radiance from a shaded Spectralon panel and found that this radiance is approximately 30% compared to that from an unshaded Spectralon panel. For longer wavelengths past 450 nm, this value drops to 3–5%; therefore, we could expect samples illuminated in outdoor environments to be more similar in longer wavelengths to our present laboratory analysis. The effect of skylight illumination, which is also polarized, on the polarimetry of the scene is a complex problem.