Wavelength dependence of the bidirectional reflectance distribution function (BRDF) of beach sands

Katarina Z. Doctor; Charles M. Bachmann; Deric J. Gray; Marcos J. Montes; Robert A. Fusina

doi:10.1364/AO.54.00F243

1. INTRODUCTION

The dominant directional reflective properties of surfaces have been studied for decades in both laboratory and natural settings. Results of these studies have numerous applications, such as for non-nadir and multiangle space and airborne imagery systems, vegetation density analysis, trafficability studies, robotic sensors, unmanned vehicles, virtual autonomous navigation, and in virtual reality models. Most theories and models of natural surfaces are based on laboratory studies and involve simplifications of natural settings. Empirical studies of dominant directional reflective properties of natural surfaces in natural settings are important to further improve the theoretical models and applications.

A. Background

A common method to identify or model the dominant directional reflective properties of a surface is the bidirectional reflectance distribution function (BRDF) [1]. Bidirectional means that both the incoming light and the observed light are directional and measured in an infinitesimally small solid angle. In practice, the infinitesimally small solid angle does not include measurable amounts of radiant flux [2–4]. The theoretical treatments of a radiative transfer function, on the other hand, deal almost exclusively with the directional version of the reflectance distribution function, so all measurable quantities of reflectance are performed in the conical or hemispherical domain of geometrical considerations [2,3,5–8]. Sometimes the terminology bidirectional is used to refer to biconical measurements where the solid angles involved are relatively small [3,7]. By definition, BRDF relates the incident irradiance from one given direction to its contribution to the reflected radiance in another direction [2] according to the equation

f_{DD} (θ_{i}, ϕ_{i}; θ_{r}, ϕ_{r}) = \frac{d L_{r} (θ_{i}, ϕ_{i}; θ_{r}, ϕ_{r})}{d E_{i} (θ_{i}, ϕ_{i})},

where

f_{DD}

is the BRDF,

L

is the radiance,

E

is the irradiance, the

i

and

r

denote the incident and the reflected light, and

θ

and

ϕ

marks the zenith and the azimuth angles, respectively, as shown in Fig. 1. Table 1 shows the symbols and notations used in this paper. The units of BRDF are [1/sr].

BRDF describes the angular behavior by which light interacts with surfaces. To derive BRDF, one would measure the radiance and then calculate the bidirectional reflectance factor (BRF). BRF is the ratio of the directionally reflected radiance from the surface target and the nadir radiance of a Lambertian-scattering reference target [2,9–13]. A BRF derivation is demonstrated in the next section of this paper.

Fig. 1. Concept of incident and reflected angles in spherical coordinate system.

Download Full Size | PDF

Table 1. Symbols and Notations

View Table | View all tables in this article

Remote sensing technology has advanced to the stage where hyperspectral sensors, with hundreds of separate wavelength bands in the range of 350–2500 nm, are fairly common. There is interest in examining BRDF in the hyperspectral regime, which implies examining the directional reflective properties of hundreds of narrowly spaced wavelength bands. In this study, we attempted to determine whether there is meaningful variance in BRDF as a function of wavelength for the beach sands being studied, and to identify whether a subset of wavelengths is sufficient to characterize the dominant directional reflection for beach sands.

One method used in previous studies to examine BRDF wavelength dependency is the anisotropy factor (ANIF), which is the ratio of the BRF at a specific view direction to the nadir BRF, and is defined as

ANIF = \frac{R (θ_{i}, ϕ_{i}, 2 π; θ_{r}, ϕ_{r}, ω_{r}; λ)}{R_{0} (θ_{i}, ϕ_{i}, 2 π; θ_{r} = 0, ϕ_{r}, ω_{r}; λ)},

where

R_{0} (θ_{i}, ϕ_{i}, 2 π; θ_{r} = 0, ϕ_{r}, ω_{r}; λ)

denotes the reflectance factor acquired from the nadir direction.

The ANIF enables the separation of spectral BRDF effects from the spectral signature of the sample, and it is used to analyze the spectral variability of nadir reflectance with the viewing geometry [9,14–16]. Therefore, the ANIF shows the overall variation of the nadir reflectance factor with the viewing geometry. However, it does not show the wavelength variation of the dominant direction. Studies have shown that the BRDF of soil has minimal wavelength dependence using the ANIF method, especially in comparison to vegetation, which has high wavelength dependence [9,10,15,16].

In laboratory studies, Wang and co-workers [17,18] concluded that changes in the solar principal plane [the plane containing the sun and the target (Fig. 1)] of a soil BRF did not depend on wavelength for the four wavelengths they examined (460, 560, 670, and 860 nm). They pointed out that the rough surface had greater BRDF variation than the smooth soil surface. They also noticed differences in the range of values of BRDF at near-infrared (860 nm) and of the three other wavelengths, which implies wavelength dependence. However, they looked at sandy clay loam soil with large clods (average smallest clod dimension of soil that they called smooth was $2 cm \times 2 cm \times 2 cm$ ) simulating plowed soil, and a relatively narrow wavelength band. The type of soil they examined is rougher and a much coarser scale than the beach sands we studied for this paper.

A common method to show wavelength dependence in the field of remote sensing is to plot the BRDF in the solar principal plane (SPP) and color-code or mark the results differently for different wavelengths. Some previous studies assumed azimuthal symmetry, and only the SPP (or, in a laboratory setting, the light principal plane) was analyzed [10,16,19], while others have highlighted azimuthal asymmetry [14,17,18,20]. Since azimuthal asymmetry is generally expected in the field conditions, we also analyzed areas out of the solar principal plane. This can be time-consuming work if done manually and is computationally costly. Details on this and explanatory figures are in Section 3.B of this paper.

A more efficient method to analyze wavelength dependence is through principal component analysis (PCA). PCA is used to reduce the dimensionality of data by extracting the linearly independent, most informative subsample of a dataset to produce uncorrelated output bands and to segregate noise components [12,21–24]. PCA can be calculated from covariance or its standardized version, the correlation matrix, giving the same distribution result with a different scale factor. The elements of the correlation matrix are calculated from covariance matrix elements, such as $c_{i j} = \frac{v_{i j}}{\sqrt{v_{i i} v_{j j}}}$ , where $c_{i j}$ are the elements of the correlation matrix, $v_{i j}$ are elements of the covariance matrix, and $v_{i i}$ and $v_{j j}$ are the variances of the $i$ th and $j$ th wavelengths of the dataset [23].

In this study, we used PCA to analyze the wavelength dependency of the dominant directional reflective properties of beach sand. We use the resulting correlation matrix to identify the correlation between each pair of wavelengths and to identify clusters of correlated wavelengths. We also compared this method with two frequently used methods showing BRDF wavelength dependence, the ANIF and plotting BRDF in planar view.

B. Deriving BRDF from the Measured Radiance Data

In practice, BRDF is not measured directly, but is calculated through some intermediate steps, including BRF [2,12], as shown in Fig. 2(a). Several papers [2,6,7,9–14] describe the reflectance factor as the ratio of radiance measurements of a target surface and a diffuse, Lambertian standard surface of white Spectralon. BRF is defined as a ratio of the radiant flux $d Φ_{r}$ reflected by a sample surface in a particular direction to the reflected radiant flux $d Φ_{r}^{id}$ , from an ideal, perfectly diffuse, Lambertian standard surface, and can be written as

BRF (θ_{i}, ϕ_{i}, ω_{i} \to 0; θ_{r}, ϕ_{r}, ω_{r}) = \frac{d Φ_{r} (θ_{i}, ϕ_{i}, θ_{r}, ϕ_{r})}{d Φ_{i}^{id} (θ_{i}, ϕ_{i})} = \frac{\cos θ_{r} \sin θ_{r} d L_{r} (θ_{i}, ϕ_{i}; θ_{r}, ϕ_{r}) d θ_{r} d ϕ_{r} d A}{\cos θ_{r} \sin θ_{r} d L_{i}^{id} (θ_{i}, ϕ_{i}) d θ_{r} d ϕ_{r} d A} = \frac{f_{DD} (θ_{i}, ϕ_{i}; θ_{r}, ϕ_{r})}{f^{id} (θ_{i}, ϕ_{i})} = π f_{DD} (θ_{i}, ϕ_{i}; θ_{r}, ϕ_{r}) .

Fig. 2. Geometrical aspects of reflectance factor nomenclature used in this study: the bidirectional (BRF) (a), the conical–conical (CCRF) (b), the hemispherical–conical (HCRF) (c), and the hemispherical–directional (HDRF) (d).

Download Full Size | PDF

Since the ideal diffuse surface reflects the same radiance in all view directions, the BRDF of an ideal Lambertian surface is $f^{id} (θ_{i}, ϕ_{i}) = 1 / π$ [2,11]. Therefore, the BRF of any surface is equal to its BRDF multiplied by $π$ .

As illustrated in Fig. 2(a), BRF specifies an infinitesimal beam for both the incoming and reflected light. In reality, the closest sensors can come to this ideal is to measure the incoming light field in conical beams, as shown in Fig. 2(b). In practice, an incoming narrow beam is often not measured directly; instead one measures the radiance returned from a Lambertian surface (e.g., a white Spectralon plaque) that is received from all downward directions. The result produces the hemispherical–conical reflectance factor (HCRF), which is shown in Fig. 2(c). From Nicodemus et al. [2] the HCRF equation is

HCRF (θ_{i}, ϕ_{i}, 2 π; θ_{r}, ϕ_{r}, ω_{r}) = \frac{1}{Ω_{r}} \int_{2 π} \int_{ω_{r}} f_{HC} (θ_{r}, ϕ_{r}) d Ω_{r} d Ω_{i} .

Further, the incident light under ambient sky conditions consists of direct and diffuse parts [5,7] and so HCRF can be written as

HCRF (θ_{i}, ϕ_{i}, 2 π; θ_{r}, ϕ_{r}, ω_{r}) = d R (θ_{i}, ϕ_{i}, ω_{i} \to 0; θ_{r}, ϕ_{r}, ω_{r} \to 0) + (1 - d) R (θ_{i}, ϕ_{i}, 2 π; θ_{r}, ϕ_{r}, ω_{r}),

where

R

is reflectance factor,

d

corresponds to the fractional amount of direct radiant flux, and

d \in [0,1]

[5]. In outdoor illumination conditions when the sky is perfectly clear with little diffuse scattering (low aerosol optical depth and no clouds), the diffuse component of the incident light is often neglected [7,10]. In this study, we also neglected the diffuse component.

For a sufficiently narrow field of view, the measured reflectance can be considered directional [3,7]. This yields the hemispherical–directional reflectance factor (HDRF) as shown in Fig. 2(d), which is from Nicodemus et al. [2] and is written as

HDRF (θ_{i}, ϕ_{i}, 2 π; θ_{r}, ϕ_{r}, ω_{r} \to 0) = \int_{2 π} f_{HD} (θ_{r}, ϕ_{r}) d Ω_{i} = \int_{0}^{2 π} \int_{0}^{\frac{π}{2}} f_{HD} (θ_{r}, ϕ_{r}) \cos θ_{i} \sin θ_{i} d θ_{i} d ϕ_{i} = \int_{0}^{2 π} \frac{1}{2} f_{HD} (θ_{r}, ϕ_{r}) d ϕ_{i} = π f_{HD} (θ_{r}, ϕ_{r}) .

In this study we calculated HDRF using the dual spectrometer approach described by Bachmann et al. [25]. This approach uses two spectrometers. The main spectrometer measuring the target is placed on a goniometer, which enables measurements to be taken at precise angular positions over a full hemisphere. The second spectrometer is at the base station near the goniometer, and it continuously measures the radiance of a white plaque Lambertian surface. HDRF using the dual spectrometer approach equation is

HDRF (θ_{i}, ϕ_{i}, 2 π; θ_{r}, ϕ_{r}, ω_{r} \to 0, λ) = C_{id, gon} (λ) \frac{1}{n m} \sum_{t^{'} = 1}^{n} \frac{L_{r}^{id, base} (t^{'}, λ)}{L_{r}^{id, gon} (t^{'}, λ)} \sum_{t = 1}^{m} \frac{L_{r}^{gon} (t, λ)}{L_{r}^{id, base} (t, λ)},

where “

gon

” and “

base

” subscripts denote the goniometer and the base station white Lambertian surfaces. Time

t^{'}

is the time when the spectrometer on the goniometer is measuring its white plaque at nadir,

t

is when it is measuring a target radiance,

n

is the number of nadir measurements of the white plaque at the goniometer, and

m

is the number of the target surface measurements made at a particular view geometry with the same spectrometer. Typically for the observations presented in this paper

m = 1

.

C_{id, gon}

is the known reflectance of the white Spectralon reference panel used at the goniometer, which serves to correct the deviation from the ideal Lambertian reflector. The approximated BRDF of our measurements, therefore, is

f_{DD} (θ_{i}, ϕ_{i}; θ_{r}, ϕ_{r}; λ) \approx f_{HD} (θ_{r}, ϕ_{r}; λ) = \frac{HDRF (θ_{i}, ϕ_{i}, 2 π; θ_{r}, ϕ_{r}, ω_{r} \to 0; λ)}{π} = \frac{1}{π} C_{id, gon} (λ) \frac{1}{n m} \sum_{t^{'} = 1}^{n} \frac{L_{r}^{id, base} (t^{'}, λ)}{L_{r}^{id, gon} (t^{'}, λ)} \sum_{t = 1}^{m} \frac{L_{r}^{gon} (t, λ)}{L_{r}^{id, base} (t, λ)},

where

λ

at

f_{DD} (θ_{i}, ϕ_{i}; θ_{r}, ϕ_{r}; λ)

represents spectral BRDF.

Nondimensionalization of the reflectance factor allowed us to compare the dominant directional reflective properties of different surfaces and also allowed us to compare the wavelength dependence. The nondimensionalized BRDF and nondimensionalized reflectance factor are calculated as

f_{HD, nonD} (θ_{r}, ϕ_{r}; λ) = \frac{R (θ_{r}, ϕ_{r}; λ_{x}) - R_{\min} (λ)}{R_{\max} (λ) - R_{\min} (λ)} 100 [%] = R_{nonD} (λ),

where

R (θ_{r}, ϕ_{r}, λ_{x})

represents the reflectance factor of view direction

(θ_{r}, ϕ_{r})

at

x

-th wavelength of one scan cycle of the sample, and

R_{\min}

and

R_{\max}

are the minimum and the maximum values of all reflectance factors at the

x

-th wavelength. All the comparisons of the dominant directional reflective properties of sand in this paper are in a nondimensionalized reflectance factor expressed in percent. This gives the same distribution results as BRDF in percent, since the BRDF is the reflectance factor divided by

π

[Eq. (8)]. The nondimensionalized BRDF is equal to the nondimensionalized reflectance factor.

C. Study Sites

The study sites for this work were three sandy beaches in eastern Queensland, Australia: Freshwater Beach, Port Clinton Pocket Beach, and Sabina Point Beach, as shown in Fig. 3 [26].

Fig. 3. Queensland, Australia: Freshwater Beach, Port Clinton Pocket Beach, and Sabina Point Beach.

Download Full Size | PDF

The tidal range is very high at these beaches, measuring approximately four to six meters. The shore tidal zone at Freshwater Beach and Port Clinton Pocket Beach is 200 m and at Sabina Point it is up to 500 m. Fresh water seeps from the ground at low tide, forming small dendritic flow patterns in the sand. Also at low tide, sand bubbler crabs (Dotilla fenestrate) emerge from their burrows in the ground to feed on organic matter stranded in the sand as the water retreats. These crabs scrape sand into their mouths and roll it around, forming a pellet, until the organic matter is taken from the sand. Then the crab drops the sand ball and repeats the process. The crab activity and the patterns formed by the sand balls on the beach affect the overall surface roughness of the sand and its optical properties, as shown in Fig. 4.

Fig. 4. Sand bubbler crab (Dotilla fenestrate) creates sand balls at low tide (a), and the pattern created from the sand balls on the smooth sand (b).

Download Full Size | PDF

The sand at these beaches is mostly fine and medium grained, and some sampling areas contained fine gravel. The sand is mainly composed of the mineral quartz (Fig. 4), and in some areas, black, very fine grained sand contains heavy minerals, mainly ilmenite [Fig. 4(a)]. Photos of sampling stations mentioned in this paper are shown together with the results in Section 3.C.

2. METHODS

A. Measurement Setup

Measurements of the radiance of beach sand at locations with various sand properties were taken with the goniometer for outdoor portable hyperspectral earth reflectance (GOPHER) instrument shown in Fig. 5(a) [26–28]. The instrument consists of a spectrometer mounted on a rotating arc, allowing the measurement of surface radiance between 0° and 360° azimuth and 0° to nearly 75° from nadir. The scan cycle starts at the GOPHER’s mast, and the arc moves in a counter-clockwise azimuthal direction. On the GOPHER frame, a Spectra Vista Corporation (SVC) HR-1024 full-range spectrometer collects spectra, in the 350–2500 nm spectral range, with sampling intervals of 1.5–3.9 nm, and a FWHM of 3.5–9.5 nm. The field of view is 1.5° in the zenith direction and 4.9° perpendicular to the arc and creates a pixel that is a rounded-edge rectangle of about $12 \times 4.5 cm$ at the nadir when the height of the spectrometer is set to about 2 m. The size and shape of the ground incident field of view vary with each measurement and depend on the height of the spectrometer, its zenith angle, the sand surface slope, the surface aspect, and view azimuth. One example of the field-of-view pattern during the whole scan cycle is shown in Fig. 5(b). While the individual pixels in the different view angles do not exactly overlap, we purposely selected areas that are homogeneous in the entire area viewed by the hemispherical scan.

Fig. 5. GOPHER instrument. (a) A schematic drawing of the GOPHER as viewed from above. (b) An example scanning grid pattern, where green dots represent the positions of the spectrometer. (c) The scan cycle starts at the nadir (0° zenith and 0° azimuth angles) and the red arrows show the first few positions of the spectrometer.

Download Full Size | PDF

The angular grid and pattern of the spectral measurement cycle is programmable. One of the patterns we used is shown in Fig. 5(c), where the grid represents the movement area of the spectrometer on the GOPHER’s arc and the green dots show only the angular position of the spectrometer when collecting data, not the field of view. The frequency and pattern of sampling was programmed to be at every 20° of azimuth and every 20° of zenith from 0° to 60° (or some cases, at 70°); with the addition of a 10° zenith angle scan, as shown in Fig. 5(c). Note that in this particular scanning pattern every ninth measurement is at the nadir, producing a set of ten nadir points for the scheme shown in Fig. 5. The sampling stations were carefully chosen to be as homogeneous as possible. After the scanning cycle was finished, we measured the slope and aspect of the sampled surface. Photographs of the sampling surface and the surrounding area were taken during and after the scan cycle, which captured qualitative information about the surface and light conditions. Sand samples were also collected and taken to the lab for grain size analysis.

For quality control, we discarded measurements for which the light conditions changed significantly during the scan cycle such that it affected the quality of the data. Our final dataset contains measurements from 36 sampling stations. Wavelength regions that are affected by atmospheric absorption were excluded from further analysis.

B. Correction of Changing Illumination Conditions Using a Dual Spectrometer Approach

Each hemispherical scan required between 10 and 40 min, depending on the scan pattern. During the scan time, the light conditions can change due to clouds or haze, and the motion of the sun. In our experiments, we made every effort to take measurements when there were constant light conditions during a whole scan; however, in nature there are no perfect, constant conditions.

At the beginning of each hemispherical scan cycle radiance from a white Spectralon plaque was measured with the GOPHER spectrometer, as shown in Fig. 6(c). This measurement was used to calculate the reflectance factor from the measured radiance of the sand surface. We plotted the resulting reflectance factors for the whole hemispherical scan cycle in Fig. 7(a). The gray shows spectra measured from off-nadir directions, whereas the purple indicates spectra measured from the nadir direction. As shown in Fig. 5(c), nadir measurements are taken at the beginning, end, and periodically in the middle of the cycle when the sensor goes back to the nadir position on the zenith arc. In a full hemispherical scan, 10–37 nadir and 87–135 off-nadir radiance measurements are taken, depending on the scan pattern.

Fig. 6. Dual spectrometer approach. A spectrometer measuring the radiance of the reference white Spectralon plaque (a) continuously at the base station while the goniospectrometer (b) is taking the hemispherical scan at the GOPHER’s station. The two spectrometers are located close to each other, so the light conditions are similar. Prior to each spectral measurement cycle, radiance from a reference white plaque of standardized material Spectralon (c) was measured from the nadir position, and was used to calculate the reflectance from the measured radiance as a part of the dual-spectrometer algorithm [25].

Download Full Size | PDF

Fig. 7. Example of improved spectra using the dual-spectrometer method. These are plots of spectra from a full hemispherical scan cycle. The gray shows the spectra measured at off-nadir directions, whereas the purple and green indicate the spectra measured while observing at the nadir (perpendicular to the leveled horizontal plane). Reflectance spectra processed using only the GOPHER’s white Spectralon reference obtained prior to the hemispherical ground scan (a) and reflectance spectra that are radiometrically corrected using the dual spectrometer approach (b).

Download Full Size | PDF

We used a dual-spectrometer approach [25] to reduce the effects of varying light conditions over the course of the measurement cycle. We did this by constantly measuring the down-welling irradiance with one spectrometer at a base station, as shown in Fig. 6, while the other spectrometer (on GOPHER) was obtaining the ground scan, thus ensuring that one spectrometer was always monitoring changes in illumination conditions. From the measurements of both Spectralon panels and the sand sample, the radiometrically corrected reflectance factor shown in Fig. 7(b) was calculated using Eq. (7).

Comparing Figs. 7(b) to 7(a) demonstrates the improvement of spectra from one selected scan cycle by applying the dual spectrometer approach. A camera mounted on the GOPHER’s frame constantly recorded sky conditions and it showed some thin cloud layer movement during the scan cycle, which contributed to the large variations in the reflectance factor [Fig. 7(a)], removed by the dual-spectrometer approach, as shown in Fig. 7(b).

C. Visual Representation of the BRDF Data

Processed reflectance factor measurements are visualized using a stereographic polar grid system. As shown in Fig. 8, the grid marked in the BRDF rendering figures is the scanning grid of the GOPHER’s spectrometer, where 0° azimuth is the starting position of the scan cycle, at which time the arc that holds the spectrometer is in the GOPHER’s reference plane [Figs. 5(b) and 5(c)]. The grid on the rendering is rotated so the sun’s location is at the top of the figure, and the SPP is a vertical imaginary line that goes through the middle of the figure. The magenta ellipse in each plot indicates the sun’s position (Fig. 8). The horizontal axis of the magenta ellipse represents the sun’s movement in azimuth during the full hemispherical scan and its vertical axis represents its movement in zenith direction. If the sun’s zenith angle is more than the maximum measured view zenith angle (in this particular case, 60°), then the sun position is drawn just inside the grid, so that on the illustration the sun is never plotted outside of the outer grid circle.

Fig. 8. Elements of the GOPHER spectra BRDF rendering. (a) The grid is the goniospectrometer’s hemispherical sampling pattern of the scanning cycle. The solar zenith angle throughout this scan is between 50.3° and 52.5° and the solar azimuth angle is between 319.6° and 316.3° from north in clockwise direction. The range of the sun’s movement during the scan and its position is plotted in magenta. The letter “N” indicates the geographical north direction in relation to goniospectrometer’s grid. (b) The mechanisms responsible when the dominant reflection is in each of the labeled areas.

Download Full Size | PDF

The angle between the reference plane of the GOPHER and the SPP at the start of the scan cycle is indicated on the grid azimuth angle at the bottom of the figure (Fig. 8). Geographical north is marked with the letter “N” on each rendering.

To illustrate BRDF on the positioned grid, the nondimensionalized reflectance factor is rendered using a stereographic polar projection and the inverse distance interpolation method. The interpolated data is on a grid system that is not marked on the renderings; however, the grid will be further referred to in this paper. This interpolated data’s grid system is shown in Fig. 8(a) in blue. The grid system starts at the bottom of the figure and increases in a counter-clockwise direction, so the sun is always at 180° in the interpolated data grid system and 0° is on the bottom of the figure. The mechanism responsible when the dominant reflection is in each of the labeled areas is illustrated in Fig. 8(b).

The BRDF renderings in Figs. 9 and 10 illustrate the variations of dominant directional reflective properties. The dominant directional reflective properties are seen in red, and are referred collectively as the “peak(s).” The BRDF renderings in Figs. 9 and 10(a) show various degrees of forward scattering, while Fig. 10 shows backscattering.

Fig. 9. Beach sand sampling station FWB-1-2. (a) and (c)–(f) show the renderings of nondimensionalized reflectance factor [%] at 450.6, 702.9, 1159.7, 1542, and 2212.8 nm, respectively. (b) Plot of the spectra of the full hemispherical scan cycle. Spectra in gray were measured from off-nadir direction, and teal indicates the spectra measured from nadir direction. The peaks in the BRDF renderings are similar at each selected wavelength.

Download Full Size | PDF

Fig. 10. Beach sand sampling station FWB-6-2. (b): plot of the spectra of the full hemispherical scan cycle. Spectra in gray were measured from off-nadir direction, and teal indicates the spectra measured from nadir direction. (a) and (c)–(f) show the renderings of nondimensionalized reflectance factor [in percent] at 450.6, 702.9, 1159.7, 1542, and 2212.8 nm, respectively. The peaks of renderings (d) and (e) (702.9, 1159.7 nm) are similar. This is also the case for the renderings in (c) and (f) (1542, 2212.8 nm), whereas the rendering at (a) (450.6 nm) also has a prominent forward-scattered peak.

Download Full Size | PDF

To get measurements of the opposition effect, we set up the GOPHER so the SPP is nearly aligned with the GOPHER’s reference plane. This nearly perfect alignment produced strong self-shadowing. To correct this effect, we identified the affected region and replaced it with values interpolated using the average of nearest neighbors, which inherently results in underestimating the true value. In other cases, when GOPHER was offset a few degrees as shown on the grid in Fig. 8(a), self-shadowing was not present in the data.

3. RESULTS

A. Anisotropy Factor

The BRDF renderings of beach sand from some sample stations showed some wavelength dependencies while others did not. Figure 9 shows a case in which the peaks in the BRDF are highly similar in all selected wavelengths (sampling station FWB-1-2), while Fig. 10 shows a case in which the structure of peaks show some variation (sampling station FWB-6-2). The spectral plot (b) in both figures shows the reflectance factor of the full hemispherical scan cycle in gray and from the nadir direction only in teal. The BRDF renderings showing the dominant directional reflective properties are the nondimensionalized reflectance factor in percent [(Eq. (9)], so they can be easily compared.

Figure 11(a) shows the ANIF of FWB-1-2 (corresponding to Fig. 9) and Fig. 11(b) shows the ANIF of FWB-6-2 (corresponding to Fig. 10). Figures 11(a) and 11(b), which represent the beach sand in this study, show very little wavelength dependence, regardless of whether the BRDF renderings show variation in dominant directional reflectance with wavelength. The measurements from small zenith angles have lower ANIF than measurements from higher view zenith angles. The ANIF shows the overall variation of the nadir reflectance factor with viewing geometry, but does not show the variation in the dominant direction.

Fig. 11. (a) ANIF of FWB-1-2and (b) the ANIF of FWB-6-2, where the colors represent the view zenith angles in degrees. The nondimensionalized reflectance factor in the solar principal plane of these same stations are shown in (c) FWB-1-2 and (d) FWB-6-2, where the colors denote the different wavelengths, of which every tenth is plotted for figure clarity reasons. The 180° represents the backscattering and 0° the forward-scattering direction.

Download Full Size | PDF

B. Wavelength Plot in a Planar View of BRDF

An alternative way to show wavelength dependence is to plot the nondimensionalized reflectance factor in the SPP [Figs. 11(c) and 11(d)], where the colors indicate different wavelengths. FWB-1-2 shows very minimal wavelength dependence, whereas FWB-6-2 clearly demonstrates wavelength dependence, especially in the forward-scattering (0°) direction. The peak in forward scattering (0°) is only visible in shorter wavelengths below 500 nm [Figs. 10(a) and 11(d)].

Some previous works have assumed azimuthal symmetry for different reasons, and only the solar principal plane (or in laboratory setting, the light principal plane) was analyzed [10,16,19], while others have highlighted the azimuthal asymmetry [14,20]. In the renderings from this beach sand study the peaks are not always in the SPP. Moreover, in some cases there was more than one peak and in more than one azimuthal plane. At sampling station SPB-1-1, there was no peak in the SPP, but only in the plane of 30°–210° (Fig. 12). One must be careful when using the solar principal plane plot because it does not contain all the information about the dominant directional reflective properties.

Fig. 12. Sampling station SPB-1-1: (a) rendering of nondimensionalized reflectance factor at 1159.7 nm, (b) solar principal plane, and (c) plane of 30 and 210° plot of every tenth of wavelength.

Download Full Size | PDF

C. PCA of BRDF Renderings

As mentioned earlier, PCA is a more efficient method to analyze wavelength dependence. Performing PCA on beach sand BRDF data shows high correlation between the wavelengths at all sample stations. The majority of sample stations contained above 90% of the variance in the data in the first principal component (PC1) and less than 4% variance in the second principal component (PC2). Figure 13 shows the first two principal components of sample stations FWB-1-2 and FWB-6-2, where the second components are rendered using the same scale as the first components to be able to compare them. As shown in Fig. 9, the peaks in renderings at sample station FWB-1-2 are very similar in each selected wavelength, which is reflected in the principal components: PC1 [Fig. 13(a)] contains 99.29% of the variance from all wavelengths and PC2 contains only 0.48% of additional variance [Fig. 13(b)]. As shown in Fig. 10, the renderings at sample station FWB-6-2 show significant differences in peaks in selected wavelengths, which is also reflected in the principal components: PC1 [Fig. 13(c)] contains 94.78% variance from all wavelengths and PC2 contains only 2.57% of additional variance [Fig. 13(d)]. Even though PC1 at FWB-6-2 is less than at FWB-1-2, it is still a high percentage of the total information. Since we are interested in the dominant directional reflective properties, the first principal component provides sufficient information.

Fig. 13. First and second principal components (PC1 and PC2) of (a and b) FWB-1-2 and (c and d): FWB-6-2 sand sampling stations. The scale of PC2 renderings is the same as the corresponding PC1 for comparison. PC1 of FWB-1-2: (a) contains 99.29% of variance from all wavelengths and PC2, and (b) contains 0.48% additional variance, which is 99.77 cumulative % variance. PC1 of FWB-6-2 (c) contains 94.76% of variance from all wavelengths, and PC2 (d) contains 2.67% additional variance, which is 99.77 cumulative % variance.

Download Full Size | PDF

The correlation matrix from the PCA results is an excellent way to visualize the wavelength dependence of the BRDF. A correlation matrix is a two-dimensional (2D) diagonal matrix giving the correlation between all pairs of wavelengths used. Both axes of the matrix are the wavelengths used and the values in the matrix express the correlation between each pair of wavelengths. The correlation values are from 0 (no correlation) to 1 (identical: along the diagonal). Since the correlation at all the sampling stations is very high, the scale on each correlation figure is limited to between 0.8 and 1.

The axes show the wavelengths, which are not equally spaced because the SVC-HR-1024 spectrometer’s sampling interval is between 0.9 and 3.9 and different for each wavelength. The nominal spectral range of the spectrometer is between 350 and 2500 nm, and after removing the bands that are affected by atmospheric absorption, there are 865 bands remaining, with wavelengths ranging from 347.4 to 2401.9 nm (Fig. 14). The breaks at 1000 nm in the correlation matrix are because of the transition between sensors.

Fig. 14. Correlation matrix from: (a) FWB-1-2 and (b) FWB-6-2 sampling stations. The axes show the wavelengths, which are not equally spaced. The white area represents no data, which is the region of the removed bands due to unacceptable atmospheric absorption (bad bands). The highest correlation is 1 (along the diagonal) and the lowest is 0. Since the correlation at all the sampling stations is very high, the scale on each correlation figure is limited to between 0.8 and 1, so they can be compared.

Download Full Size | PDF

Figure 14 shows the correlation matrices of sample stations FWB-1-2 and FWB-6-2. The correlation of wavelengths is above 0.95 in all wavelengths, for sampling station FWB-1-2 reflecting the low wavelength dependence seen previously. At sampling station FWB-6-2, three clusters are noticeable with wavelength ranges of 350–500 nm, 500–1350 nm, and 1450–2400 nm, excluding the atmospheric water vapor absorption bands [Fig. 14(b)]. Therefore, the dominant directional reflective properties of beach sand at sampling station FWB-1-2 can be shown with one BRDF rendering at any wavelength. On the other hand, at sampling station FWB-6-2, three BRDF renderings chosen from the aforementioned three groups would be sufficient to represent all 865 wavelengths.

The correlation matrices of all the 36 beach sand sampling stations from this project in general show high correlation. In cases where the sand surface has uniform and fine grain size and the surface expression is smooth, there is just one wavelength cluster in the correlation matrix (Fig. 15). When the sand had a visually rough surface, there are three clusters of wavelengths (Fig. 16). Overall, higher surface visual roughness leads to greater wavelength dependency.

Fig. 15. Examples of sand samples measured at GOPHER stations exhibiting smooth surfaces. The station names are indicated on the left side of the photos. The photos were taken with a portable digital microscope (second column), and the scale bar shown on them is a millimeter scale. The wooden ruler on the photos in the first column is in metric scale. The length of the ruler is 50 cm. The correlation matrices related to the sand samples are in the third column.

Download Full Size | PDF

Fig. 16. Examples of sand samples measured at GOPHER stations exhibiting visually rough surfaces. The station names are indicated on the left side of the photos. The photos in the second column, (a) and (c)–(e), were taken with a portable digital microscope, and the scale bar shown on them is a millimeter scale. The wooden ruler on the photos in the first column is in metric scale. The length of the ruler is 50 cm. The correlation matrices are in the third column.

Download Full Size | PDF

There are several factors that may cause the surface expression to be visually rough: larger grain size, sand balls from crabs, sand waves, and tiny dendritic patterns from freshwater seepages. Figure 16 demonstrates several examples where the correlation matrix shows wavelength clusters. Table 2 lists the grain size distribution of the sand samples from Figs. 15 and 16. The smallest grain size in the beach sand samples from this study was the very fine sand, from 63 to 125 μm, which is much larger than the wavelengths we were observing (0.35–2.5 μm). Therefore, the scale of surface roughness was larger than the wavelength in all cases.

Table 2. Grain Size Distribution

View Table | View all tables in this article

Even though a sand surface expression seems smooth and the sand grains are small, surface visual roughness can be increased by water-saturated sand where the water creates tiny dendritic shaped patterns as it seeps from the ground. The grooves of these tiny seepages increases the surface visual roughness, resulting in higher wavelength dependence, as seen at FWB-3-3 in Fig. 16(e). Visually rough surfaces can also result from sand waves, as shown in examples of fine, dry sand at FWB-7-1 [Fig. 16(b)] and pore water-saturated medium sand with some gravel at SPB-2-4 [Fig. 16(f)]. FWB-6-1 [Fig. 16(d)] is very fine sand; however, the sparsely distributed shells have a much bigger surface than the sand surrounding them and together they form a visually rough surface and show wavelength dependence. Although the shells are not technically part of the sand, they are a natural part of the beach environment. Some heterogeneity is introduced with the shells and on the scale of the observation represents mixed pixels. FWB-6-2 [Fig. 16(a)] is fine sand and the sand balls made by crabs make the sand surface visually rough and wavelength dependent.

Despite the variety of causes of visual surface roughness, only three wavelengths are needed to represent the dominant directional reflective properties of these beach sands. The spectral range of the groups varies slightly from case to case. However, there are three clear wavelength ranges that represent these beach sands: 350–450 nm, 700–1350 nm, and 1450–2400 nm, excluding the atmospheric water vapor absorption bands.

Therefore, only a single wavelength is needed to represent the directional dependence of a visually smooth sand surface. For a visually rough beach sand surface, however, we need three wavelengths (from the appropriate clusters) to represent the directional dependence for the entire wavelength range.

4. CONCLUSIONS AND DISCUSSION

Principal component analysis and the related correlation matrix of wavelengths on BRDF provided a thorough, straightforward method to identify BRDF wavelength dependence. In our analysis, the first principal component shows, on average, greater than 95% of the variance of the information, and the correlation coefficient between the BRDF renderings at all wavelength pairs is above 0.67 at all sampling stations. While examination of individual BRDF renderings is highly recommended to appreciate the complexity and variation in a collection of soil samples, the PCA method is more efficient and overcomes the weakness of the planar plots that only show one plane at a time.

For the beach sands in this study the BRDF has weak wavelength dependence that can be captured in three broad wavelength clusters. These clusters are associated with higher visual roughness of sand surfaces. The BRDF of visually smoother sands show virtually no wavelength dependence. Thus, the weak wavelength dependence is more evident in surface expressions of greater visually rough than in visually smooth surfaces. The weak wavelength dependence of the BRDF’s of beach sand can be captured with three broad wavelength regions instead of hundreds of individual wavelengths. The spectral range of the groups varies slightly from case to case; however, all are represented by three clear wavelength ranges: 350–450 nm, 700–1350 nm, and 1450–2400 nm, excluding the atmospheric water vapor absorption bands.

This study’s results should represent beach sands with characteristics similar to those in the Australian beaches we examined. Further studies of other beaches and types of soils will be necessary to establish the applicability of these findings to other areas.

Funding

Office of Naval Research (ONR) (N0001413WX00008); Office of the Under Secretary of Defense for Acquisition, Technology and Logistics (OUSD (AT&L)) Coalition Warfare Program (DWAM20026).

Acknowledgment

We would like to thank Roy J. Hughes and Stephen Carr (Defence Science and Technology Organisation, Australia), Gordon Mattis (South Carolina Research Authority, USA), Reid Nichols (Marine Information Resources Corporation, USA), and Andrei Abelev, Rong-Rong Li, and Michael Vermillion (U.S. Naval Research Laboratory) who collaborated with us on data collection. We would like to thank William Philpot (Cornell University, USA) who also collaborated with us for data collection and, in addition, provided the photographs of the sand surface from a portable microscope. We also thank Brad Weymer (Texas A&M University, USA) for the grain size analysis of sand samples and Andrew Hammond (Central Queensland University, Australia) for help with the geologic interpretation of the coastal sands.

REFERENCES

1. F. E. Nicodemus, “Directional reflectance and emissivity of an opaque surface,” Appl. Opt. 4, 767 (1965). [CrossRef]

2. F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, and T. Limperis, “Geometrical Considerations and Nomenclature for Reflectance,” Tech. Rep. (U.S. Department of Commerce, National Bureau of Standards, 1977).

3. R. McCluney, Introduction to Radiometry and Photometry (Artech House, 1994).

4. C. Walthall, J. L. Roujean, and J. Morisette, “Field and landscape BRDF optical wavelength measurements: experience, techniques and the future,” Remote Sens. Rev. 18, 503–531 (2000). [CrossRef]

5. G. Schaepman-Strub, M. Schaepman, T. Painter, S. Dangel, and J. Martonchik, “Reflectance quantities in optical remote sensing—definitions and case studies,” Remote Sens. Environ. 103, 27–42 (2006). [CrossRef]

6. E. J. Milton, “Review article principles of field spectroscopy,” Int. J. Remote Sens. 8, 1807–1827 (1987). [CrossRef]

7. C. S. Bourgeois, A. Ohmura, K. Schroff, H.-J. Frei, and P. Calanca, “IAC ETH goniospectrometer: a tool for hyperspectral HDRF measurements,” J. Atmos. Oceanic Technol. 23, 573–584 (2006). [CrossRef]

8. H. Croft and K. Anderson, “Directional reflectance factors for monitoring spatial changes in soil surface structure and soil organic matter erosion in agricultural systems,” in EGU General Assembly, Vienna, Austria (2012), vol. 14, p. 9089.

9. S. R. Sandmeier and K. I. Itten, “A field goniometer system (FIGOS) for acquisition of hyperspectral BRDF data,” IEEE Trans. Geosci. Remote Sens 37, 978–986 (1999). [CrossRef]

10. G. Strub, U. Beisl, M. Schaepman, D. Schläpfer, C. Dickerhof, and K. Itten, “Evaluation of diurnal hyperspectral HDRF data acquired with the RSL field goniometer during the DAISEX’99 campaign,” ISPRS J. Photogramm. Remote Sens. 57, 184–193 (2002). [CrossRef]

11. J. V. Martonchik, C. J. Bruegge, and A. H. Strahler, “A review of reflectance nomenclature used in remote sensing,” Remote Sens. Rev. 19, 9–20 (2000). [CrossRef]

12. J. R. Schott, Remote Sensing: The Image Chain Approach, 2nd Ed. (Oxford University, 2007).

13. M. Buchhorn, R. Petereit, and B. Heim, “A manual transportable instrument platform for ground-based spectro-directional observations (ManTIS) and the resultant hyperspectral field goniometer system,” Sensors 13, 16105–16128 (2013). [CrossRef]

14. S. Sandmeier, C. Müller, B. Hosgood, and G. Andreoli, “Physical mechanisms in hyperspectral BRDF data of grass and watercress,” Remote Sens. Environ. 66, 222–233 (1998). [CrossRef]

15. F. Camacho-de Coca, F. J. Garcia-Haro, and M. A. Gilbert, “Quantitative analysis of cropland’s BRDF anisotropy using airborne POLDER data,” in Proc. 1st International Symposium on Recent Advances in Quantitative Remote Sensing, Torrent (2002), pp. 1–7.

16. G. Schaepman-Strub, “Spectrodirectional reflectance analysis and definition for the estimation of vegetation variables,” Ph.D. thesis, (University of Zurich, 2004).

17. Z. Wang, C. A. Coburn, X. Ren, D. Mazumdar, S. Myshak, A. Mullin, and P. M. Teillet, “Assessment of soil surface BRDF using an imaging spectrometer,” Proc. SPIE 7830, 783010 (2010). [CrossRef]

18. Z. Wang, C. A. Coburn, X. Ren, and P. M. Teillet, “Effect of surface roughness, wavelength, illumination, and viewing zenith angles on soil surface BRDF using an imaging BRDF approach,” Int. J. Remote Sens. 35, 6894–6913 (2014). [CrossRef]

19. H. Zhang and K. J. Voss, “Bidirectional reflectance study on dry, wet, and submerged particulate layers: effects of pore liquid refractive index and translucent particle concentrations,” Appl. Opt. 45, 8753–8763 (2006). [CrossRef]

20. B. W. Hapke, Theory of Reflectance and Emittance Spectroscopy, 2nd ed. (Cambridge University, 2012), Vol. 47.

21. K. H. Esbensen, D. Guyot, F. Westad, and L. P. Houmoller, Multivariate Data Analysis–in Practice, 5th Ed. (CAMO Software AS, 2004).

22. M. T. Heath, Scientific Computing–An Introductory Survey (McGraw Hill, 2002).

23. J. A. Richards and X. Jia, Remote Sensing Digital Image Analysis: An Introduction, 4th Ed. (Springer-Verlag, 2006).

24. K. Varmuza and P. Filzmoser, Introduction to Multivariate Statistical Analysis in Chemometrics (CRC Press, 2009).

25. C. M. Bachmann, M. J. Montes, C. E. Parrish, R. A. Fusina, C. R. Nichols, R.-R. Li, E. Hallenborg, C. A. Jones, K. Lee, J. Sellars, S. A. White, and J. C. Fry, “A dual-spectrometer approach to reflectance measurements under sub-optimal sky conditions,” Opt. Express 20, 8959–8973 (2012). [CrossRef]

26. C. M. Bachmann, R. J. Hughes, A. Abelev, E. van Roggen, G. Mattis, R. A. Fusina, R.-R. Li, K. Z. Doctor, D. Gray, M. J. Montes, W. Philpot, S. Carr, S. Kharabash, A. Brady, G. Smith, J. Gardner, D. R. Korwan, W. D. Miller, C. Snow, M. Vermillion, R. Liang, D. Ball, R. Hagen, G. Frick, Z. Peterson, R. Winter, J. Vrbancich, J. Eggleton, L. Hamilton, T. White, A. Fairley, D. Bongiorno, K. Hong, J. Wilson, G. L. Mason, B. Q. Gates, A. Hammond, J. Junek, and G. Christie, “Tools for hyperspectral exploitation of multi-angular spectra (THEMAS): May 2012 and June 2013 experiments,” Technical Report (Naval Research Laboratory, 2015).

27. C. M. Bachmann, A. Abelev, W. Philpot, K. Z. Doctor, M. J. Montes, R. A. Fusina, R.-R. Li, and E. van Roggen, “Retrieval of sand density from hyperspectral BRDF,” Proc. SPIE 9088, 9 (2014).

28. C. M. Bachmann, D. Gray, A. Abelev, W. Philpot, M. J. Montes, R. Fusina, J. Musser, R.-R. Li, M. Vermillion, G. Smith, D. R. Korwan, C. Snow, W. D. Miller, J. Gardner, M. Sletten, G. Georgiev, B. Truitt, M. Killmon, J. Sellars, J. Woolard, C. Parrish, and A. Schwarzschild, “Linking goniometer measurements to hyperspectral and multisensor imagery for retrieval of beach properties and coastal characterization,” Proc. SPIE 8390, 1–9 (2012).

Symbols
$A$	Area $[m^{2}]$
$Φ$	Radiant flux $[W]$
$E$	Irradiance $[{Wm}^{- 2}]$
$L$	Radiance $[{Wm}^{- 2} {sr}^{- 1}]$
$ρ$	Reflectance $= d Φ_{r} / d Φ_{i}$ [dimensionless]
$R$	Reflectance factor $= d Φ_{r} / d Φ_{i}^{id}$ [dimensionless]
$R_{0}$	Reflectance factor measured from the nadir direction
$Θ$	Zenith angle, in spherical coordinate system [rad]
$ϕ$	Azimuth angle, in spherical coordinate system [rad]
$ω$	Solid angle $= \int d ω [sr]$
$Ω$	Projected solid angle $= \int \cos θ d ω [sr]$
$λ$	Wavelength [nm]
$f_{DD}$	Bidirectional reflectance distribution function [1/sr]
$f_{HD}$	Hemispherical–directional reflectance distribution function [1/sr]
$f_{HC}$	Hemispherical–conical reflectance distribution function [1/sr]
$f ()$	Distribution function
Sub- and Superscripts
$i$	Incident
$r$	Reflected
id	Ideal (lossless) and diffuse (Lambertian)
gon	At goniometer station
base	At base station
nonD	Nondimensionalized
0	Nadir direction
Terms
GOPHER	Goniometer for outdoor portable hyperspectral earth reflectance
SPP	Solar principal plane
BRDF	Bidirectional reflectance distribution function
BRF	Bidirectional reflectance factor
CCRF	Conical–conical reflectance factor
HCRF	Hemispherical–conical reflectance factor
HDRF	Hemispherical–directional reflectance factor
ANIF	Anisotropy factor
PCA	Principal component analysis

	Visually Smooth Surface						Visually Rough Surface
	FWB-1-2	FWB-3-2	FWB-6-4	FWB-7-3	PCB-1-3	FWB-5-4	FWB-6-2	FWB-7-1	SPB-2-3	FWB-6-1	FWB-3-3	SPB-2-4
Gravel[%]	0	0	0	0	0	0	0	0	2.6	0	0	0.6
Sand[%]	100	100	100	100	100	100	100	100	97.4	100	100	99.4
Mud[%]	0	0	0	0	0	0	0	0	0	0	0	0
Fine gravel[%]	0	0	0	0	0	0	0	0	1.2	0	0	0
Very fine gravel[%]	0	0	0	0	0	0	0	0	1.4	0	0	0.6
Very coarse sand[%]	0	0	0	0	0	0	0	0	5.6	0	0	2
Coarse sand[%]	0.5	0.2	0.3	0	0	0	0.1	0	40.4	0	0.1	16.9
Medium sand[%]	19.4	9.5	13.1	1.8	19.6	3.3	4.7	2.1	48.0	0	8.5	67.4
Fine sand[%]	76.9	89.1	82.6	94.3	72.4	94.4	89.8	93.5	3.2	36.4	90.5	12.5
Very fine sand[%]	3.2	1.2	3.7	4.0	7.9	2.3	5.4	4.4	0.1	63.6	0.9	0.7

Symbols
$A$	Area $[m^{2}]$
$Φ$	Radiant flux $[W]$
$E$	Irradiance $[{Wm}^{- 2}]$
$L$	Radiance $[{Wm}^{- 2} {sr}^{- 1}]$
$ρ$	Reflectance $= d Φ_{r} / d Φ_{i}$ [dimensionless]
$R$	Reflectance factor $= d Φ_{r} / d Φ_{i}^{id}$ [dimensionless]
$R_{0}$	Reflectance factor measured from the nadir direction
$Θ$	Zenith angle, in spherical coordinate system [rad]
$ϕ$	Azimuth angle, in spherical coordinate system [rad]
$ω$	Solid angle $= \int d ω [sr]$
$Ω$	Projected solid angle $= \int \cos θ d ω [sr]$
$λ$	Wavelength [nm]
$f_{DD}$	Bidirectional reflectance distribution function [1/sr]
$f_{HD}$	Hemispherical–directional reflectance distribution function [1/sr]
$f_{HC}$	Hemispherical–conical reflectance distribution function [1/sr]
$f ()$	Distribution function
Sub- and Superscripts
$i$	Incident
$r$	Reflected
id	Ideal (lossless) and diffuse (Lambertian)
gon	At goniometer station
base	At base station
nonD	Nondimensionalized
0	Nadir direction
Terms
GOPHER	Goniometer for outdoor portable hyperspectral earth reflectance
SPP	Solar principal plane
BRDF	Bidirectional reflectance distribution function
BRF	Bidirectional reflectance factor
CCRF	Conical–conical reflectance factor
HCRF	Hemispherical–conical reflectance factor
HDRF	Hemispherical–directional reflectance factor
ANIF	Anisotropy factor
PCA	Principal component analysis

	Visually Smooth Surface						Visually Rough Surface
	FWB-1-2	FWB-3-2	FWB-6-4	FWB-7-3	PCB-1-3	FWB-5-4	FWB-6-2	FWB-7-1	SPB-2-3	FWB-6-1	FWB-3-3	SPB-2-4
Gravel[%]	0	0	0	0	0	0	0	0	2.6	0	0	0.6
Sand[%]	100	100	100	100	100	100	100	100	97.4	100	100	99.4
Mud[%]	0	0	0	0	0	0	0	0	0	0	0	0
Fine gravel[%]	0	0	0	0	0	0	0	0	1.2	0	0	0
Very fine gravel[%]	0	0	0	0	0	0	0	0	1.4	0	0	0.6
Very coarse sand[%]	0	0	0	0	0	0	0	0	5.6	0	0	2
Coarse sand[%]	0.5	0.2	0.3	0	0	0	0.1	0	40.4	0	0.1	16.9
Medium sand[%]	19.4	9.5	13.1	1.8	19.6	3.3	4.7	2.1	48.0	0	8.5	67.4
Fine sand[%]	76.9	89.1	82.6	94.3	72.4	94.4	89.8	93.5	3.2	36.4	90.5	12.5
Very fine sand[%]	3.2	1.2	3.7	4.0	7.9	2.3	5.4	4.4	0.1	63.6	0.9	0.7

Wavelength dependence of the bidirectional reflectance distribution function (BRDF) of beach sands

Abstract

1. INTRODUCTION

A. Background

B. Deriving BRDF from the Measured Radiance Data

C. Study Sites

2. METHODS

A. Measurement Setup

B. Correction of Changing Illumination Conditions Using a Dual Spectrometer Approach

C. Visual Representation of the BRDF Data

3. RESULTS

A. Anisotropy Factor

B. Wavelength Plot in a Planar View of BRDF

C. PCA of BRDF Renderings

4. CONCLUSIONS AND DISCUSSION

Funding

Acknowledgment

REFERENCES

Cited By

Figures (16)

Tables (2)

Equations (9)

Applied Optics