Analyzing Hyperspectral BRDF Data of a Grass Lawn and Watercress Surface Using an Empirical Model

This paper presents results from analyzing hyperspectral BRDF data of grass lawn and watercress. The intensity of the hotspot as a function of wavelength is determined from fitting an empirical (or rather phenomenological) model to the data. The model consists of a lambertian, bowlshape, hotspot and forward scattering component. An analytical, theoretically derived relation between the hotspot intensity and the lambertian component is given. The intensity of the multiple scattered radiation obtained from this relation can be explained qualitatively with the surface structures of the samples. INTRODUCTION The Bidirectional Reflectance Distribution Function (BRDF) describes the dependence of surface reflectance on incidence and viewing angles as a function of wavelength. With hyperspectral BRDF data, the dependence of the BRDF on the ’brightness’ of the surface at the respective wavelength can be studied effectively on a vegetation canopy, since the spectral albedo of vegetation varies considerably (e. g. from 0.026 to 0.43 between 500 nm and 1000 nm for a grass sample). The spectral signature affects the shape of the BRDF, because multiple scattering increases dramatically at high albedos. This paper focuses on the decrease of the hotspot relative to the albedo as multiple scattering increases. DESCRIPTION OF DATA SETS Data from an erectophile grass (Lolium perenne) and a planophile watercress (Lepidium sativum) surface were acquired under controlled laboratory conditions at the European Goniometric Facility (EGO) of the Joint Research Center in Ispra/Italy [1]. The angular resolution of the BRDF data is 5 and 15 in zenith and azimuth direction, respectively. The data is available from our website. In addition, a field data set of the same grass species is analyzed, measured with the FIGOS field goniometer with a resolution of 15 and 30 in zenith and azimuth, respectively [2]. All data were taken under a source zenith angle of 35 with zenith angles of reflection r ranging from 0 to 75 using a GER-3700 spectroradiometer with 323 channels between 500 nm and 1000 nm. EMPIRICAL BRDF MODEL Four basic BRDF components are identified and fitted to the data with an empirical function: 1) hotspot 2) forward scattering 3) bowlshape and 4) lambertian component. The forward scattering component is adopted from a specular term successfully used in similar studies [3] (see [4] for a detailed explanation of the specular term). The bowl shape is modeled as a simple linear function of the zenith angles. To model the hotspot, we chose a simple exponential function e b2 g which has a shape very similar to a function proposed by B. Hapke [5] (g is the relative angle between viewing and illumination direction). Our function has the advantage of approaching zero in the forward scattering direction much faster than Hapke’s function, which makes it easier to separate the hotspot from the forward scattering term in the inversion step. Since the decrease of reflectance from the hotspot towards larger viewing zenith angles is smaller than the decrease towards nadir, we multiplied the hotspot term with an exponential function eb1 ( i r)2 also used in the forward scattering component. The resulting BRF model has the following form: BRF = BRDF = a0 + a1 ( i + r) + a2 eb1 ( i r)2 e b2 g + a3 eb3 ( i r)2 eb4 2 (1) where is the relative angle to the specular direction, see [4]. The coefficients ai depend on wavelength, the coefficients bi are fixed for each sample. The underlying assumption is that the shape of the hotspot and the forward scattering term are primarily determined by the geometry of the canopy, whereas the intensity of these two terms can vary with wavelength due to the change of reflectance with wavelength. The coefficients have the following physical meaning: a0 = lambertian component (diffuse scattering) a1 = intensity of the bowlshape a2 = intensity of the hotspot a3 = intensity of the specular peak 1This research was supported by the German Research Foundation (DFG) and the Swiss National Science Foundations (NF)2. b2 = width of the hotspot For an explanation of b1; b3 and b4 see [4]. Based on the assumption that the surface consists of surface patches of one reflectance only (so the background does not influence the BRDF), we derived a physical (not empirical) relation between a0 and a2: a2= e b1 4 i 2 k ( K1 +qK2 1 4 kK0 ) with (2) K0=a0(x c) + a20 k(1 c) ; K1= a0(2 k c) + x where x [0; 1] is the areal proportion of shadowed surface patches at angles where the BRDF equals a0 (diffuse limit), k [0; 1] is the probability that scattered light will hit another surface patch, and c is a geometry factor (close to 1). x; k and c are wavelength independent quantities. The derivation of (2) and the assumptions made will be presented in a later publication. MODEL INVERSION We used a numerical routine from the software package ’PVWAVE’ based on the least-squares-deviation to fit the parameters of equation 1 to the BRDF data. Allowing all parameters to float freely did not yield reasonable results. So we adopted a step by step procedure. First, we fitted parameter a1 for the bowlshape component for each wavelength taking into account only BRDF measurements in the cross principle plane (relative azimuth ' = 90 ) where the effects of the hotspot and the forward scattering term are assumed to be smallest. Then, we estimated parameters bi (the shape-determining parameters of the hotspot and forward scattering term) from BRDF measurements in the principal plane (' = 0 =180 , respectively), taking into account the effect of parameter a1. For the forward scattering term, some freedom remains in the choice of b3 and b4. This term was designed to fit to a specular peak with a distinct maximum, but our data only shows a rise and no decline (if there is a maximum, it must be at r > 75 ). The parameters b1 and b2 could be determined well, because the hotspot peak (at i = r = 35 ) is well within the range of measured angles ( r = [0 :::75 ]). For the field measurement of grass, the same coefficients bi were used as for the laboratory measurement of grass because the angular resolution of the field measurement was much coarser. In the final step, we fitted the parameters a0; a2 and a3 to all data points. RESULTS AND DISCUSSION The agreement between measurements and fitted function (1) is satisfying, the mean relative deviation is only a few %, see table 1. The assumption that the shape of the hotspot does not change with wavelength was confirmed clearly for the grass laboratory sample. The coefficients bi are given in table 1. b1 equals 1.5 for all samples. b2 for cress is greater than b2 for grass, this means that cress has a sharper hotspot. The coefficients ai are shown in Fig. 1. All 3 samples show the rising edge characteristic for vegetation at about 700 nm, see plots A-C (neglecting BRDF effects, the coefficient a0 is approximately proportional to the albedo). The bowlshape component described by a1 for grass is very different from cress (plots D-F). a1 varies only by a factor of about 2 for cress (cress has almost no bowlshape at NIR), whereas for grass, a1 goes to zero for small a0. Thus, a1 for grass is roughly proportional to a0 for a0 < 0:3. For wavelengths larger than 800 nm, we suspect that the inversion routine did not distinguish well between a1 and a3 for cress, resulting in the negative values for a1. However, the impact of a1 relative to a0 is negligible for cress for a0 > 0:3. Coefficient a3 (plots J-L) tends to zero for small a0 and reaches a maximum at about a0 = 0:25 for grass. We don’t know whether this spectral behaviour is consistent with the assumption that a3 can be interpreted as a specular peak. If the Fresnel Reflectance of the scattering surface was constant (which is propably not the case here), a3 would be constant. a3 for cress is about a factor of 10 smaller than a3 for grass. If there was no multiple scattering (MS), we would expect the relation of a2 (the hotspot intensity) and a0 to be linear. Obviously, this relation is not linear (plots G-I). The reason is, that as a0 increases, the intensity of MS increases too. MS reduces the hotspot because the shadowed components of the canopy are less dark since they are illuminated by the MS. Thus, for large values of a0 (corresponding to high albedos) a2 rises much slower than for small values of a0. For small a0 and a2, (2) can be transformed to x a0 a0+a2 . x was estimated from a0 a0+a2 , an exact determination was not possible due to the ambiguous behaviour of a0 a0+a2 for small a0. k and c of (2) were fitted to all values of a0 and a2. As can be seen from the solid lines in plots G-I, (2) describes the relation of a0 and a2 very well. Note that (2) covers values of a0 from 0.014 to 0.306 for grass (lab.) and 0.019 to 0.561 for cress, which is a difference of more than a factor of 20. The coefficient k is proportional to the multiple scattered radiation. The difference of the coefficient k for grass and cress can be explained by the canopy structure: cress is a planophile canopy, not allowing much MS, whereas grass is erectophile with a lot of possibilities for MS. k for the laboratory measurement of grass is lower than k for the field measurement of grass. This is consistent with the above explanation, because the grass in the laboratory measurement has longer blades and is therefore not as erectophile as the field grass sample, and thus produces less MS. Coefficient c for the grass laboratory sample deviates significantly from 1, this may indicate problems with the application of (2) to this sample.