2.1.

Microfacet Model for the mmBRDF

When light is incident upon a surface, the light–matter interaction can be quantified as a change to the Stokes vector. Linear effects are conventionally described by a $4\times 4$ matrix, which relates an incident Stokes vector to a reflected Stokes vector

Microfacets are perfect Fresnel reflecting elements that are small compared to the imaging system resolution.¹⁹ The magnitude of the total reflected polarized light is proportional to the quantity of microfacets that are oriented toward specular reflection at a given acquisition geometry. At a given acquisition geometry, the illumination vector and view vector are bisected by a surface normal $\widehat{\mathbf{n}}$, which reflects specularly. The angle $\beta$ is between $\widehat{\mathbf{n}}$ and the vector normal to the surface of the object ${\widehat{\mathbf{r}}}_s$. If at a given acquisition geometry and surface orientation, there are many microfacets in the $\beta$ orientation, then the specular reflection will be strong. Conversely, if at that value of $\beta$ the surface does not have many microfacets the specular reflection is weak. The microfacet structure of a surface is described by a scattering density function $pr(\beta ,\mathrm{\Psi })$, which parameterizes the density of microfacets at a polar and azimuth angle; denoted $\beta$ and $\mathrm{\Psi }$, respectively. The orientation of microfacets is usually assumed to be azimuthally uniform so that $pr(\beta ,\mathrm{\Psi })=pr(\beta )$. The assumption of azimuthal symmetry imposes an invariance in the polarized light scattering when rotating the source and the view direction together about the surface normal. For top of atmosphere measurements, the surface normal to the land surface can be reasonably approximated by the zenith $\widehat{\mathbf{z}}$. The surface normal of each object in a GroundMSPI image is not known a prior. In this work, two approaches to defining $\beta$ are compared: (1) with respect to the zenith $\widehat{\mathbf{z}}$ (as in Fig. 3) and (2) with respect to the object’s surface normal vector ${\widehat{\mathbf{r}}}_s$, which is not known a prior and is estimated from the measurements (as in Figs. 4 and 5). The variable ${\widehat{\mathbf{r}}}_s$ is inserted into Eq. (5) using the substitutions

Fig. 4

Microfacet mmBRDF fits when the surface orientation is estimated and true surface orientation is known. The mean value of the linear Stokes parameters within the yellow box in the RGB image is used for an LS fit to the model. The double $y$-axis plots show intensity in red (left $y$-axis) and $Q^s$ and $U^s$ in blue (right $y$-axis). The Stokes measurements of $U^s$ are triangle markers and $Q^s$ are square markers. The lines show the LS fit of the models without a shadowing function (solid line) and with a shadowing function (dashed line). The thicker solid and dashed lines indicate a Gaussian scattering density function and the thinner lines a Bréon function. Here, the surface normal is estimated from the measurements so the Bréon model contains three parameters and the Gaussian model contains four parameters. Two example measurements are shown in (a) and (c) and the surface orientation estimates are the green planes in (b) and (d). The fan of yellow vectors in (b) and (d) is the illumination angles over the time period of data collection. The blue vector in (b) and (d) is the direction of photon travel toward the camera.

Fig. 5

Microfacet mmBRDF fits when the surface orientation is estimated and true surface orientation is unknown. The mean value of the linear Stokes parameters within the yellow box in the RGB image is used for an LS fit to the model. The double $y$-axis plots show intensity in red (left $y$-axis) and $Q^s$ and $U^s$ in blue (right $y$-axis). The Stokes measurements of $U^s$ are triangle markers and $Q^s$ are square markers. The lines show the LS fit of the models without a shadowing function (solid line) and with a shadowing function (dashed line). The thicker solid and dashed lines indicate a Gaussian scattering density function and the thinner lines a Bréon function. Here, the surface normal is estimated from the measurements so the Bréon model contains three parameters and the Gaussian model contains four parameters. Two example measurements are shown in (a) and (c) and the surface orientation estimates are the green planes in (b) and (d). The fan of yellow vectors in (b) and (d) is the illumination angles over the time period of data collection. The blue vector in (b) and (d) is the direction of photon travel toward the camera.

The first term in Eq. (5) models the unpolarized portion of the reflected field. Here, $\mathbf{D}$ is a $4\times 4$ matrix that is zero except for the (1,1) element. Without the matrix $\mathbf{D}$, the first term in Eq. (5) reduces to a scalar-valued BRDF model referred to as a modified Rahman–Pinty–Verstraete (mRPV) that is currently used in aerosol retrievals by NASA’s MISR instrument.²⁰ The mRPV model uses three parameters to describe a nonpolarimetric intensity measurement: $a_{\lambda }$ to characterize the amount of reflectance at each wavelength, $k$ to characterize the anisotropy of reflectance, and $b$ to characterize the forward/backscattering contribution.^21,22

The second term in Eq. (5) models polarization. $\mathbf{P}(n,\gamma )$ is a $4\times 4$ matrix of Fresnel reflection coefficients of a material where $n$ is the index of refraction and $\gamma$ is the angle of incidence. A wavelength subscript on $\mathbf{P}(n,\gamma )$ is not used due to the assumption that the refractive index of the surface is spectrally neutral within the measured waveband. Since specular reflection from each microfacet is assumed the scattering angle is related to the angle of incidence by $\cos \mathrm{\Omega }=-\cos 2\gamma$. Details about the structure and elements of the matrix $\mathbf{P}(n,\gamma )$, for applications in GroundMSPI analysis, have been described in Ref. 1. This model assumes all reflected polarized radiance is generated by a single reflection of light from the top surface. The matrix $\mathbf{M}({\alpha }_0)$ rotates an incident Stokes vector from the solar meridian plane to the scattering plane where ${\alpha }_0$ is the angle between the two planes. If results are desired in meridian, rather than scattering, coordinates then another rotation matrix can be inserted on the left-hand side of $\mathbf{P}(n,\gamma )$. The second term in Eq. (5) models the polarimetric contribution to the microfacet mmBRDF using only a single linear parameter: $\xi$. The model describes the functional relationship between the linear Stokes parameters and the acquisition geometry and fitting $\xi$ to the polarimetric measurements simply scales this functional form. This simplification is useful since polarimetric measurements show a similar trend with respect to scattering angle for rough materials and the Earth’s natural landscapes.⁷

Shadowing effects manifest themselves both in photometric and polarimetric characteristics of scattered radiation.²³ Without a shadowing function, microfacet mmBRDF models assume that every element of the surface contributes to the reflected field. This assumption neglects the shadowing of points on the surface caused by neighboring points, an effect that is increasingly important at large angles of incidence. A shadowing theory described in Refs. 24 and 25 derives the probability that a backscattering specular point on the surface will not be shadowed for a Gaussian slope density function. Analytic results for a non-Gaussian example are presented in Ref. 26. The slope density function is related to the scattering density function $pr(\beta )$ from which a shadowing function $S({\mu }_i,{\mu }_r)$ can be calculated so that the shadow-modified scattering density is $pr(\beta )S({\mu }_i,{\mu }_r)$. Both the Bréon^27,28 and Gaussian scattering density functions are used in this work and reported in Table 2. Measurement comparisons to these models will be computed both with and without a shadowing function.

Table 2

Scattering density and shadowing functions.

Note that the Bréon shadowing function only depends on the acquisition geometry. The Gaussian shadowing function depends on both the acquisition geometry and the width of $pr(\beta )$, denoted as $\sigma$. This width parameter of the Gaussian model is an additional dependent parameter as compared to the Bréon model. For empirical analysis, an optimal model imposes the strictest constraints possible while characterizing a realistic population of measurements. In this application, mmBRDF constraints are valuable for untangling land surface and atmospheric polarized light scattering for the task of aerosol retrieval. Since the Bréon model imposes a more strict constraint on the surface polarized light scattering, it would be preferable to the Gaussian model in the case where equivalent RMSE values were observed.

A Gaussian scattering density function is also called the Cox-Munk distribution in reference to the first researchers to use this model to describe ocean roughness.²⁹

Where,

Derivations of Eqs. (9) and (11) can be found in Refs. 24, 26, and 28.

Recent work on microfacet modeling for polarized light scattering has demonstrated that shadowing functions introduce errors at large grazing angles, i.e., ${\mu }_i$, ${\mu }_r\to 0$.³⁰ In this paper, only GroundMSPI measurements that are directly solar illuminated are considered. Therefore, the range of reported acquisition geometries does not include grazing angles because of the long shadows physical objects cast on each other when the sun is near the horizon.

2.2.

Assumptions for Passive Outdoor Scenes

Only GroundMSPI measurements at 660 nm are included in this work given that the spectral invariance hypothesis is well supported for polarized light scattering from the Earth’s surface. GroundMSPI measurements have been analyzed with respect to the spectral invariance hypothesis in other work.^1,31 The spectral invariance hypothesis assumes that both the magnitude and the angular shape of the polarimetric term in the mmBRDF are spectrally neutral. Researchers have noted only small exceptions, such as wavelength dependencies in the shortwave infrared (SWIR) at large scattering angles, that could affect aerosol retrievals over land when the AOT is small.³² A conventional approach is to fit the parameters of a mmBRDF model for land surface reflectance in the SWIR where, in general, the atmospheric contribution is small. Since the polarized surface reflectance depends weakly on the wavelength, the fitted values for the parameters are used at other wavelengths to predict the contribution of polarized light scattering from the land surface.

We will restrict the modeling to unpolarized solar illumination [i.e., ${\mathbf{s}}_0=[\mathrm{1,0},\mathrm{0,0}]$ in Eq. (4)]; this precludes scattered skylight that is polarized. In the first GroundMSPI paper,¹ modeling solar illumination as partially polarized did not significantly improve the model and measurement agreement for data collected on a clear day. When the index of refraction is real-valued, the Fresnel reflection coefficients are also real-valued. In this work, $n=1.5$ is assumed. Putting together the two assumptions of a real index of refraction and unpolarized illumination leads to an important constraint on the model for the reflected Stokes vector: the third component is zero (i.e., $U^s=0$) and therefore ${\mathrm{AoLP}}^s=90\mathrm{deg}$ at all acquisition geometries. Other researchers also use this assumption.⁷ Others have also noted ${\mathrm{AoLP}}^s\ne 90\mathrm{deg}$ in their measurements of land surface scattering with errors up to 15%.³³

Following methods from Ref. 1, the polarimetric and nonpolarimetric parameters are estimated sequentially. This is because specular reflection from the microfacets contributes to the total radiance field, as well as the polarized field, but the polarized field is unaffected by the remaining parameters. First $\xi$ (and $\sigma$ if using a Gaussian facet distribution model) are estimated from a least-squares (LS) fit to $Q^s$ measured at a series of different scattering angles. These estimates parameterize an estimate of the first term in Eq. (5), which is then subtracted then from the measured radiance and an LS fit to the first term in Eq. (5) (i.e., $a_{\lambda }$, $k$, and $b$) is performed.

Although the surface orientation affects both the nonpolarimetric and the polarimetric term in this work, we choose to estimate ${\widehat{\mathbf{r}}}_s$ from an LS fit of the polarimetric term (i.e., simultaneous with estimating $\xi$) and then use that estimate when fitting the other parameters in the nonpolarimetric term.

3.1.

Assumed Surface Orientation

Figure 3 shows four GroundMSPI intensity images alongside double $y$-axis graphs of linear Stokes measurements and the predictions from four models: Bréon and Gaussian; each with and without shadowing [see Eq. (7)]. Here, the surface orientation is assumed to be horizontal and therefore the surface normal vector equals the zenith. The four RMSE values of the models for each scene are in Table 3.

Table 3

RMSE values between GroundMSPI measurements and models in Fig. 3.

In Fig. 3(a), a large region of gravel is selected and the average linear Stokes values in this region are fit to the microfacet mmBRDF models. For this example, the assumed horizontal surface orientation is correct. The Bréon models, both with and without shadowing, produce lower RMSE than the Gaussian models. The shadowed-Bréon model RMSE is 17%, and the unshadowed-Bréon model RMSE is 22%. In Fig. 3(b), a region on the wall of a brick building is selected. The true surface orientation is vertical yet the model assumes horizontal. Even with this modeling mismatch, all four microfacet mmBRDF models trend correctly with the measurement dependence on scattering angle. RMSE of unshadowed-Bréon and unshadowed-Gaussian models are both 12%, adding a shadowing function increases RMSE in both cases. The region used for analysis in Fig. 3(c) is the leafy part of a tree and RMSE of three of the models is $22\pm 1\%$; adding shadowing to the Bréon model reduces the RMSE to 9%. Measurements in Fig. 3(d) are of smooth cement and the Bréon model actually trends nearly opposite of the measurements; the Gaussian RMSE is more than four times lower. As shown in Table 3, shadowing does not improve the RMSE for the Gaussian distribution. In summary, the model with the lowest RMSE is the shadowed-Bréon for the tree and gravel measurements, the unshadowed-Gaussian for the smooth cement measurement, and a tie between these two models for the brick measurement.

3.2.

Estimated Surface Orientation

Here two angular parameters to describe the object’s surface normal are included in the LS parametric fit, see Figs. 4 and 5. This surface normal is ${\widehat{\mathbf{r}}}_s$ defined in Eq. (6). As reported in Table 4, all RMSE values are below 8% and most RMSE values are below 5% when the object’s surface normal is estimated as part of the microfacet mmBRDF model. These RMSE differences are barely noticeable in the Stokes plots in Figs. 4 and 5. The difference in the estimated surface normals is very noticeable in Figs. 4 and 5. The four GroundMSPI measurements are split into the two figures: one for which the true value of the surface normal is known in Fig. 4, and the other for which the surface normal is unknown or undefined in Fig. 5.

Table 4

RMSE values between GroundMSPI measurements and models in Figs. 4 and 5.

In Fig. 5(a), the true-value of the surface orientation is undefined since the selected pixels in the tree contain numerous leaves, shadows, and even the background seen through the tree branches. The purpose of including this example is to demonstrate the improvement in RMSE even when the surface orientation parameter is undefined. In Fig. 5(c), the smooth cement is on a south-facing vertical surface that is angled toward the sky. Both the shadowed-Gaussian and shadowed-Bréon are south-facing. The shadowed-Bréon is close to perfectly vertical. The shadowed-Gaussian is angled close to a 45 deg angle, which appears steeper than the true-value. The true-value of the surface orientation is somewhere between the shadowed-Gaussian and shadowed-Bréon surface orientation estimates.

As reported in Table 4, the addition of a shadowing function perturbs the RMSE values by a few percent, both increasing and decreasing depending upon the sample. The Gaussian and Bréon RMSE values are very similar and the rank-ordering of model RMSE is different for each of the four examples. For gravel shadowed-Bréon and shadowed-Gaussian are tied at 1%. For the brick and tree examples, the unshadowed-Bréon and unshadowed-Gaussian are tied at 3% and 7%, respectively. The Gaussian model RMSE is 2% for the smooth cement example, both shadowed and unshadowed.