Predicted Distribution of Visible and Near-Infrared Radiant Flux Above and Below a Transmittant Leaf
Dar A. Roberts, John B. Adams, and Milton O. Smith Department of Geological Sciences, University of Washington, Seattle
T h e effects of background reflectance, leaf size, and leaf height above the background on upward and downward radiant flux (dpu and ~b a) from a leaf were investigated using a computer model of a horizontal, isotropicaUy scattering leaf. This re- search was conducted to determine how these vari- ables influence the light environment above, below and adjacent to a leaf Leaf spectral properties for big-leaf maple (Acer macrophyllum) were mea- sured in the laboratory and used in the model. Model results were reported as relative radiant flux (qbr), defined as a percentage of the light entered into the model. The model showed that upward relative radiant flux ~)ur from a leaf was highly dependent on the reflectance, of the background and the wavelength of light. The greatest variation in ~bur was observed in the near infrared (NIR). The 49,r also varied depending upon the height of the leaf above the background and the size of the leaf. Leaves were brightest when placed the far- thest distance above the background. Small leaves reached maximum brightness at lower heights than larger leaves. Finally, ~bur varied spatially. Leaf edges reflected more light than the leaf center except for leaves positioned very close to the back-
Address correspondence to Dar A. Roberts, Department of Geological Sciences, AJ-20, University of Washington, Seattle, WA 98195.
Received 10 January 1990; revised 7 September 1990.
ground. Additional studies using the model showed that the intensity of light within a leaf shadow varied spatially, with the greatest downward rela- tive radiant flux 4) dr, occurring directly below the center of the leaf. Furthermore, ~dr within the shadow cast by the leaf decreased as the height of the leaf above the background increased. The rate of decrease depended upon the size of the leaf. The smaller the leaf, the greater was the change in (~ dr with change in leaf height. These results imply that NIR canopy reflectance, due to leaf transmittance, may be highly dependent upon the reflectance of its background. Furthermore, architecturally different canopies may show different degrees of dependence upon background reflectance in the NIR. These results extend to closed canopies, in which leaf size and spacing may vary the reflectance of the canopy. Finally, these results suggest that the amount of light scattered to the side by leaves increases the amount of NIR light measured from adjacent, un- shadowed backgrounds.
INTRODUCTION
A basic understanding of light interactions be- tween foliage and the underlying substrate is cru- cial to the interpretation of reflected visible and near-infrared (NIR) reflectance spectra of vegeta- tion collected in the field or from remote sensing
2 Roberts et al.
platforms. Models that do not take into account effects of substrate reflectance on canopy re- flectance may result in erroneous interpretations. As an example, Huete et al. (1985) and Huete and Jackson (1987) found that all three vegetation in- dices, the NIR//red ratio (VI), the normalized difference vegetation index (NDVI), and the per- pendicular vegetation index (PVI) were influenced by the reflectance of the substrate underlying a canopy. As a result, the same amount of vegetation on the ground may result in a different quantity as estimated from remote sensing data, if substrate reflectance varies across an image.
Interactions between canopy and substrate reflectance have been investigated previously through the use of canopy reflectance models [see Bunnik (1978) and Goel (1988) for reviews of canopy reflectance models], laboratory measure- ments of single leaves placed over light and dark backgrounds (Carlson and Yarger, 1971; Lillesaeter, 1982), and canopy reflectance mea- surements (Bunnik 1978; Huete et al., 1985; Bauer et al., 1986; Huete, 1987). Although several empir- ical and theoretical studies have been conducted, questions remain concerning light interactions within canopies. Most of the empirical measure- ments have been made at the scale of whole canopies, not individual canopy components. Em- pirical measurements of single leaves have been collected over bright and dark substates by Lillesaeter (1982); however, these measurements were made of leaves placed directly upon the background. The effect of canopy architectural variables, such as the size of a leaf, or height of a leaf above a background, have not been investi- gated at the scale of a single leaf.
The purpose of this research was to study the upwelling (reflected) and downwelling (trans- mitted) components of light intercepted by a leaf positioned above a uniform, reflecting background. This research was initiated, in part, to provide more information concerning the potential light environment within a canopy at the scale of a leaf. A second objective was to use a single leaf model as a preliminary step towards developing more complex models that incorporate multiple leaf lev- els, variable leaf orientation, canopy morphology, and shadowing.
To characterize light scattering by individual leaves within a canopy, multispectral images were synthesized of a single, horizontal leaf at resolu-
tions of 1 cm. Using this simple model, the effect of: 1) leaf height (the distance between the leaf and the background--leaf heights ranging from 0 cm to 25 cm were modeled), 2) leaf size (ranging from 1 cm to 10 cm), 3) background reflectance (ranging from 0% to 100%), and 4) leaf transmit- tance and reflectance (at 450 nm, 675 nm, and 900 nm for both upper and lower leaf surfaces) on radiant flux, ~b, from a leaf were investigated.
METHODS
A leaf was modeled as a Lambertian scattering material. This model was used to simplify calcula- tions, and because photometric measurements ,of nonwaxy leaves have shown that leaves can ap- proximate a Lambertian scattering surface at nor- mal and near-normal incidence (Breece and Holmes, 1971; Norman et al., 1985; Walter-Shea et al., 1989). A computer program was developed to generate synthetic images of a horizontal leaf positioned a fixed distance above a uniform Lambertian background material. A square array consisting of 1 cm pixels was created and each pixel was classified as either leaf or as background (Fig. la). Direct incident light was modeled as entering from an infinitely distant point source positioned directly above the leaf. Combined di- rect and diffuse light were calculated at a position directly below and directly above the modeled leaf. Both the "light source" and the "detector" were modeled as being far enough away from the targets to minimize any measurable differences in flux due to proximity (Fig. lb). Non-Lambertian scattering of off-nadir multiply scattered light was not taken into account.
Diffuse skylight and penumbral effects due to the fact that the sun is not a point source (Miller and Norman, 1971) were not included in this model. Skylight would have its greatest effect at the shortest wavelengths, resulting in a measur- able increase at 450 nm and only a slight increase at 900 nm. The potential effects of a penumbra were calculated using an equation formulated by Miller and Norman (1971). For most of the cases presented, the maximum effect of a penumbra was to decrease the amount of direct incident light striking unshadowed pixels adjacent to the leaf by less than 0.8% and to add up to 0.8% of direct
Radiant Flux Distribution of Leaves 3
a)
b) Source Detector
r
[ ] Leaf
[ ] Background
) 1 cm Pixels
I = 0 E= 0
Leaf Level
cm height 2
Ground Level
Figure 1. Map view and side view showing a modeled 5 cm leaf positioned 2 cm above a uniform background. To model reflected radiation, the "source" and "detector" were placed an infinite distance above the modeled leaf. To model downward flux, the "detector" was placed within the plane of the background (I = angle of incidence, E = angle of exitance).
incident light to the shadowed region beneath the leaf. For the most extreme case of a 1 cm leaf positioned 5 cm above the background, the result- ing penumbra accounted for a 1.8% decrease adja- cent to the leaf and a 2% increase directly below the leaf. This proved only to be of significance for downward flux within the shadow cast by a small leaf at large height, proving to be of greatest significance at 450 nm and 675 nm.
Hemispherical leaf reflectance and transmit- tance spectra for both surfaces of a big-leaf maple (Acer macrophyllum) were entered into the model (Table 1, Fig. 2). These spectra were measured in the laboratory using a modified Beckman DK2A spectrophotometer attached to a halon-coated inte- grating sphere. M1 measurements were standard- ized to halon. Leaf reflectance was measured from both surfaces using a black background to elimi- nate multiple scattering.
Table i. Hemispherical Reflectance and Transmittance of the Upper and Lower Surfaces of a Big-Leaf Maple Leaf
Modeled materials having 0%, 25%, and 100% reflectance at all wavelengths were used as back- grounds. The black (0%) and white (100%) back- grounds were selected as lower and upper limits defining the extremes that may occur in the natu- ral world. A 25% reflectant background was se- lected as more representative of soils that occur in the field. In addition, a background having 5% reflectance at 450 nm, 7% reflectance at 675 nm, and 41% reflectance at 900 nm was used to simu-
4 Roberts et al.
0 ,r--
(,~
d 15 (9
m. Q
I--
o f , , , 0 n 400
~,50 675 900 Reflectance
800 1200 1600 2000 2400 Wavelength(rim)
Figure 2. Hemispherical transmittance and reflectance spec- tra of Acer macrophyllum. Reflectance spectra for the upper (--) and lower (---) surfaces of the leaf are plotted near the top. Transmittance spectra in the downward (--) and upward (---) directions are graphed below reflectance. The wave- lengths modeled (450 nm, 675 nm, and 900 nm) are shown as vertical lines on the graph.
late a single leaf positioned over a background consisting of leaves. This model, however, under- estimates the effects of a leaf background because it does not take into account transmittance of the lower leaf layer, which would be influenced by the reflectance properties of whatever surface was be- neath this layer.
Upwelling and downwelling radiant fluxes (~b u and ~b d) at each pixel in the synthetic image were calculated as the sum of a direct component and an indirect component emanating from all of the scattering sources. Scattered light sources in- cluded: 1) light transmitted through the leaf, "2) light reflected off of the background, and 3) up- welling light scattered off of the undersurface of the leaf. The light was allowed to scatter multiple times until additional scattering contributed less than 0.1% to the total flux. The largest number of scattering events calculated in this study was seven, for the case of NIR radiation passing through a highly reflectant and transmittant leaf positioned over a 100% reflectant background. At 450 nm and 675 nm, where leaves are relatively opaque, at most three scattering events were needed.
The scattered component (transmitted by or reflected off leaf surfaces or reflected off the back- ground) was determined using a multiple step process. This procedure is shown schematically in
Figure 3. The first step was to calculate a Lamber- tian scattering function for each leaf height mod- eled. This function describes the intensity of reflected or transmitted light emanating from a perfectly reflectant or transmittant 1 cm square Lambertian surface centered above a much larger target array consisting of a grid of 1 cm elements (Fig. 3a).
The second step was to calculate the down- ward radiant flux ~b a across a grid centered below a 1 cm leaf element. This was accomplished by assigning the corresponding ~b~ value for a Lambertian source (determined in part 3a) to each background element based upon its position rela- tive to the source, and then multiplying these values by hemispherical downward leaf transmit- tance measured using the spectrophotometer (Fig. 3b).
This procedure was repeated for each leaf element and the corresponding ~b d value stored for each background element as the sum of the ~b d from all of the leaf elements. Finally, direct ~b d was added to diffuse ~b d for unshadowed back- ground elements (Fig. 3c).
The fourth step was to calculate the intensity of ~b u. The procedure was similar to the one used to calculate the ~b d. In this instance, however, the source elements were the background elements, and the target elements were the leaf elements. The intensity of light upwelling from each back- ground element was calculated as the product of the total incident ~b d and the hemispherical re- flectance of the background. The intensity of ~b u striking a specific leaf element from a specific background element was calculated using the cor- responding ~b u from a perfect Lambertian source and multiplying this value by the source intensity at ground level. This procedure was repeated for each background element. The total ~b u at a spe- cific leaf element was calculated as the sum of the ~b~ emanating from every background element be- low the leaf (Fig. 3d).
The last step was to calculate the upward radiant flux ~b u above the leaf. This value was calculated by multiplying the ~b~ below the leaf by the hemispherical upward leaf transmittance and adding this to the light reflected from the upper leaf surface determined in the absence of any multiple scattering (Fig. 3e).
Multiple reflections were incorporated by first calculating the intensity of the ~b u reflected off the
Radiant Flux Distribution of Leaves 5
lower surface of the leaf at each leaf pixel, and then adding this to the original ~b a (Fig. 3e). This provided a new, augmented value for the ~b,l, which was used to replace the original value of downward hemispherical transmittance used in the second step of the model. New values for the ~b d at ground level, the ~b u at ground level, the ~b, at leaf level, and the ~b, above the leaf were calcu- lated using the augmented downward hemispheri- cal transmittance. This procedure, of augmenting the ~b~l at the leaf level with reflected ~b u, was continued until the change in the ~b~ resulted in less than a 0.1% change in the ~b,.
Limitations in computer time restricted the size of the synthetic image generated. The largest array used in this study for a Lambertian scatter- ing function was 65 cm of a side. This placed upper limits on the height of the leaf and size of leaf that could be modeled. Based upon a compari-
son between model calculations for ~b, from a complete canopy and theoretical values for a leaf stack, it was determined that the upper limit for a 5 cm square leaf and a 65 cm x 65 cm Lambertian scattering function was a height of 5 cm. Above that height, too much light remained outside of the 65 cm area modeled to calculate accurately the total scattered light. To obtain average q~, values for larger leaves and greater heights than 5 cm, the size of the elements and the heights were modified proportionally. Thus, a 10 cm leaf at a height of 10 cm could be calculated by running a model of a 1 cm leaf at 1 cm.
The results of the model were output from each pixel along a transect starting left of the modeled leaf and progressing to the center of the leaf. The modeled leaf could be viewed from two geometries, either directly above (looking down- wards, showing the ~b,) or directly below (looking
a) Calculate Lambertian filter function
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Figure 3. Schematic diagram showing the procedure for calculating diffuse downward and upward radiant flux from a Lambertian leaf positioned over a Lambertian background. The grids consist of 1 cm elements. Grey levels correspond to different intensities of radiant flux. Black arrows show the direction of diffuse and direct flux. White arrows indicate that the background array is larger than the area shown on the figures, a) Calculation of the Lambertian scattering function for one height. The high- est intensity is shown as light grey and is positioned directly below (or above) the 1 cm source, b) Calculation of the downward diffuse light component from a single leaf element. Downward flux is highest at the center of the grid and lowest at the corners, c) Calculation of the total downward diffuse light emanating from all leaf elements summed across the background. A direct compo- nent is added to the unshadowed background elements making them the brightest elements in the image, d) Calculation of upwelling flux using the background as the source and the undersurface of the leaf as the target, e) Calculation of upward flux above the leaf. Multiple reflections are incorporated by reflecting light off the undersurface of the leaf.
6 Roberts et al.
d) Calculate upward diffuse light below an array of leaf elements
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upwards, showing the ~ba). All of the values are reported as relative radiant flux (~bur and ~ba~), defined as a percentage of the quantity of light entered into the model. As a result, it is possible to generate values greater than 100% in areas where direct light is augmented by scattered light.
RESULTS
In the first example, the ~i)ur and the ~/)dr w e r e
calculated for a 5 cm leaf placed a distance of 2 cm above a 100% reflectant background. Along a tran- sect beginning to the left of the leaf and progress- ing to the center of the leaf, ~bdr at 900 nm increased from 100% (direct light only) to over 120% (direct+diffuse) adjacent to the leaf (fig. 4a). Upon entering the shadow of the leaf, 900 nm q~dr abruptly decreased to ~ 50% and then in- creased again to over 65% in the center of the leaf. Similar trends also occurred at 450 nm and 675 nm; however, the magnitude of change was much smaller.
The manner in which light is distributed along the transect, showing its greatest intensity in the area adjacent to, but not shadowed by the leaf, and its lowest intensity along the inner margin of the leaf, is a result of a gradient in the intensity of diffuse light. The intensity of diffuse light scat-
Figure 3. (Continued)
tered by the leaf decreases from the center of the leaf outward. At the edge of the leaf, in the unshaded portion of the background, combined diffuse and direct light result in the highest inten- sity of downwelling radiation. In the shadow of the leaf only diffuse light remains, resulting in the lowest intensity of downwelling radiation occur- ring in those areas located farthest from the leaf center, along the inner margin of the leaf.
The intensity of 900 nm ~bur above this model ranged from 120% adjacent to the modeled leaf to 78% at the center of the modeled leaf (Fig. 4b). Like the ~bdr, the ~ur varied across the leaf, however, the trend was reversed; the margins of the leaf were brighter than the center. The re- versed trend in the ~ur is due to the presence of unshadowed background pixels adjacent to the leaf which supply a greater amount of upwelling light to the edge of the leaf than to the center.
The f~ur and ~)dr a t 900 nm were calculated for the same modeled leaf placed over three more backgrounds, one 0% reflectant, one 25% reflec- tant, and one 41% reflectant (Fig. 5). The ~)dr a t
the center of the leaf increased from less than 45% to 65% from the 0% to 100% reflectant background (Fig. 5a). The difference between these two ex- tremes is due to the absence of multiple reflection in the case involving the 0% reflectant back- ground.
Radiant Flux Distribution of Leaves 7
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450 nm
675 nm
900 nm
o I t o 5 10
Distance from Leaf Center (cm)
Figure 4. a) Downward relative radiant flux ~bdr for a 5 cm modeled leaf positioned 2 cm above the background. The intensity of light at three wavelengths (450 nm ( - - - ) ; 675 nm (- . . . . ); 900 nm ( - - ) ) is shown along a transect that starts 7 cm left of the modeled leaf and extends to the center of the modeled leaf. 4) Upward relative radiant flux ~b.r.
The 900 nm ~b,r above this modeled leaf decreased from 80% along the inner margin of the leaf positioned over the 100% reflectant back- ground to a uniform value of 41% over the leaf positioned over the 0% reflectant background (Fig. 5b). The presence of a highly reflectant back- ground resulted in a near doubling of leaf re- flectance due to multiple scattering and an up- wardly transmittant leaf.
To investigate the effect of changes in leaf height over the background on the radiant flux, synthetic images of a 5 cm square leaf were gener- ated at heights of 0 cm, 0.2 cm, 0.5 cm, 1 cm, 3 cm and 5 cm (a - f in Fig 6). According to the model, the ~b,t r changes as a function of leaf height (left halves of a-f). The ~bar at 900 nm was highest in the leaf shadow when the leaf lay fiat upon the background (at 82%; Fig. 6a). This decreased to 20% at a height of 5 cm. Furthermore, the differ- ence between the dpd r along the margin of the
leaf, and the center of the leaf, changed with leaf height. The greatest discrepancy occurred at a leaf height of 1 cm (Fig. 6d). To demonstrate the relative importance of transmission and light scat- tering by leaves at the different leaf heights, ~bdr below and adjacent to an opaque leaf with a 0% reflectant lower surface are shown for comparison (dashed lines on Fig. 6).
The ~ur at 900 nm also changed with increas- ing leaf height (right halves of a - f in Fig. 6). It decreased initially from a value of 85% for a leaf placed directly on the 100% reflectant background (Fig. 6a) to an average of 80% at 1 cm (Fig. 6d) and then increased to 88% for a leaf at height of 5 cm (Fig. 60. There was also a discrepancy be- tween upwelling flux at the edge of the leaf and flux at the center. At heights of 0.2 cm and 0.5 cm, the ~bur was slightly greater at the center of the leaf than it was at the edge (Fig. 6b and 6c). At heights of 1 cm or higher, the trend was reversed
8 Roberts et al.
i ° m
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a) DOWNWARD FLUX (~r) b) UPWARD FLUX (~r)
5 cm leaf 2 cm above 0%, 25%, 41% & 100% Reflectant Backgrounds
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100%
41%
~.. . . . .~ 25%
~ . ~ 10 5 0
Distance from Leaf Center (cm)
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Figure 5. a) The ~)dr at 900 nm for a 5 cm modeled leaf positioned 2 cm above a white (100%) and a dark background (0% reflectance), b) The ~)ur at 900 nm.
(Fig. 6d); the ~b,r was greatest at the margins of the leaf. The greatest disparity between the ~bur at the center and the ~ur at the edge of the leaf occurred at a height of 3 cm (Fig. 6e). Above that height, the ~ur became more uniform (Fig. 6f). The quantity of light contributed by multiple scat- tering and transmittance at each leaf height is readily observed by comparing the ~b,r above a transmittant leaf to ~bur above an opaque leaf (dashed lines in the figure).
Changes in ~dr and ~b,~ with leaf height and spatial variability in fluxes across the leaf result from changes in the relative importance of light loss from leaf to background elements along the leaf margins, and the competing process of light recovery from unshaded background. Differences in the proximity of leaf elements relative to the background (i.e., center versus edge) result in preferential light loss and light recovery along the margins, accounting for the spatial variability. This reached its greatest extreme at heights of 1-3 cm (d and e in Fig. 6). Light loss outweighed light
recovery as leaf height increased from 0 cm to 0.5 cm, accounting for the net decrease in ~b,r at these heights (a-c in Fig. 6). Above these heights, light recovery outweighed light loss, resulting in a net increase in ~b,r (d - f in Fig. 6).
Light loss outweighed light recovery at the lowest heights due to a loss of multiply scattered light. When the leaf was placed directly on the background, no light was lost to, or recovered from, unshaded regions of the background. With an increase in leaf height, light was no longer trapped between the leaf and the background. A small, but significant quantity of light leaked along the margins. This light loss was compensated by light recovered from directly illuminated, unshad- owed background, resulting in no net change. However, during subsequent scattering events, multiply scattered light also was lost along the margins. Little of this light was recovered, result- ing in a net decrease in ~bur.
Above a height of 0.5 cm, light recovery out- weighed light loss. This was due to a decrease in
Radiant Flux Distribution of Leaves 9
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Distance from Leaf Center Figure 6. The ~bdr and ~bur at 900 nm for a 5 cm modeled leaf lying directly on a 100% reflectant background (height = 0 cm) and positioned at heights of 0.2 cm, 0.5 cm, 1 cm, 3 cm, and 5 cm above the background ( - - ) . Radiant flux for the case of an opaque leaf ( - - - ) .
]0 Roberts et al.
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Figure 7. Average intensity of ¢~dr at 900 nm within the shadow of I cm, 3 cm, 5 cm, and 10 cm modeled leaves plotted against leaf height above 100% (--) and 41% (---) reflectant backgrounds. Numbers to the right of each plot correspond to leaf size. The case of 100% leaf cover for both backgrounds is shown as vertical lines at 82% and 59.5%.
the importance of multiple scattering coupled with an increase in the importance of light scattered from unshadowed background. At these heights, light recovered from adjacent, unshadowed back- ground was high enough to outweigh light lost along the margins and compensate for light lost during subsequent scattering events.
The effect of combined changes in leaf size and leaf height were investigated for a 100% re- flectant background. The average intensity of ~b,t r at 900 nm within the shadow of the leaf decreased rapidly with increased leaf height (Fig. 7). The rate of decrease varied with leaf size. For example, within the shadow of a 10 cm leaf, ~bar decreased from 82% at 0 cm to 33.2% at 5 cm. Over the same height range, a 1 cm leaf showed a decrease from 82% to 1.1%. The ~)ur above the leaf showed a different trend initially decreasing from 85% to 80% followed by an increase to 95% at the greatest leaf height (Fig. 8). The case of an opaque leaf is
shown in the left side of the figure. Because this leaf does not transmit light, it shows no change in qbur with height. A 0% reflectant background would produce the same result (not shown in this figure).
The case of 100% leaf cover (i.e., the canopy is a monolayer and no gaps or overlapping leaves are present in the leaf layer) is shown on both Figures 7 and 8. These figures demonstrate that, where leaf coverage is complete and there are no unshad- owed edges, leaf size and leaf height have no effect upon the ~)dr and f~ur" In this case, the leaf layer would behave as a leaf stack (Allen and Richardson, 1968).
At 450 nm and 675 nm, there was considerably less change in the magnitude of JPdr with in- creased leaf height above a 100% reflectant back- ground (Fig. 9). Furthermore, unlike at 900 nm, the ~bar increased slightly as the leaf was elevated, and then decreased. The ~bur at 450 nm and 675 nm also changed considerably less, increasing less
Radiant Flux Distribution of Leaves 11
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90Ohm Upward Relative Radiant Flux ((~ur) Figure 8. Average intensity of ~ur at 900 nm for 1 cm, 3 cm, 5 cm, and 10 cm modeled leaves plotted against leaf height above 100% ( - - ) and 41% ( - . - ) reflectant backgrounds. The case of 100% leaf cover for both backgrounds is shown as vertical lines at 85.1% and 54.2%.
tO
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Downward Relative Radiant Flux ((lDdr) Figure 9. The ~Par at 450 nm ( . . . . . ), 675 nm ( - . - ) , and 900 nm ( - - ) for a 1 cm leaf positioned above a 100% reflectant background.
12 Roberts et al.
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Figure 10. The ~dr at 450 nm, 675 nm, and 900 nm for a 1 cm opaque ( - ' ) and transparent ( - - ) modeled leaf positioned above a 100% reflectant background.
than 10% with an increase in the leaf height above the background (Fig. 10). In addition, the trend differed from the changes modeled at 900 nm; the ~bur increased monotonically at 450 nm and 675 nm. At all wavelengths, the ~bur was greater than it would have been for an opaque leaf (shown as dashed lines in the figures), demonstrating the importance of multiple scattering and transmit- tance even at 450 nm and 675 nm. A 0% reflectant background would produce the same results as an opaque leaf.
The major differences between ~br modeled at 450 nm and 675 nm and those modeled at 900 nm are due to low reflectance and transmittance prop- erties of leaves at the shorter wavelengths. Be- cause both leaf transmittance and reflectance were low, multiple scattering was considerably less im- portant. As a result, light recovered from adjacent, unshadowed background overcompensated for light lost along the margins at all leaf heights resulting in a monotonic increase in ~bur. This was further demonstrated by the change in the ~bd~ at 450 nm with increased leaf height. Because leaf
reflectance on the lower surface at 450 nm was four times greater than the downward transmit- tance (Table 1), light recovered from unshaded pixels and reflected off of the lower leaf surface actually outweighed the loss of transmitted light along the leaf margins, resulting in an increase in ~dr up to a height of 0.25 cm.
DISCUSSION
Validity of the Model
A test of the validity of this model is whether it converges upon upper and lower limits. The lower limit would be the case where the leaf is placed directly on the background. The upper limit would be the case where the leaf was placed an "infinite" distance above the background.
Theoretically, the lower limit of a leaf that reflects and transmits light equally in all direc- tions, placed over a Lambertian reflecting surface
Radiant Flux Distribution of Leaves 13
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P u ~ Leaf
S u b s t r a t e Ps
Figure 11. a) Schematic diagram showing the pathway of light as it passes through a leaf and undergoes a series of multiple reflections off the background and the lower surface of the leaf. The leaf is positioned so closely to the back- ground that light reflecting off unshaded, adja- cent background pixels does not contribute sig- nificantly to light striking the lower surface of the leaf. The leaf height is small relative to the dimensions of the leaf. b) Schematic diagram showing the same leaf positioned a greater distance above the background. In this case, the proportion of shaded background is very small relative to unshaded background. As a result, the only significant light source below the leaf is unshaded background. Tile leaf height is large relative to the dimensions of the leaf. In both diagrams Pu, Pt, %, and r t equal leaf reflectance and leaf transmittance from the upper and lower surfaces of the leaf, respectively.
is oo
qb.~ = p. + ~o ~'*P'~+'*P~'*~'I dn, (1)
where Pu equals the reflectance of the upper surface of the leaf in the absence of multiple scattering, r , equals the transmitted light in the downward direction, Ps equals the reflectance of the background, Pl equals the reflectance of the lower surface of the leaf, r 1 equals the transmit- tance of a leaf in the upward direction, and n equals the number of times a packet of light reflects off the lower surface of the leaf. A graphi- cal representation of this equation is shown in Figure l la . The equation is derived in the Ap- pendix.
Based upon Eq. (1), the theoretical lower lim- its for the ~)ur above a leaf that has the spectral
properties used in the model (Table 1) is 5% at 450 nm, 7.9% at 675 nm, and 85.1% at 900 nm for a 100% reflectant background. Modeled values converged upon these values as the height of the leaf above the background was reduced (Table 2).
Calculation of the theoretical upper limit is simpler. At an infinite distance, the area shadowed by the leaf becomes insignificant in proportion to the unshadowed area (Fig. l lb). In addition, be- cause the leaf and background are far apart, multi- ple reflection becomes negligible. As a result, Eq. (1) simplifies to
6u =pu+p* u. (2)
The theoretical upper limits for a leaf with the spectral properties entered into the model, posi-
Table 2. Convergence of Reflectance to Lower and Upper Limits
tioned an infinite distance above a 100% reflectant background, are 7%, 16%, and 95% at 450 nm, 675 nm, and 900 nm, respectively. From heights of 5-20 cm, the 5 cm modeled leaf converged upon these values (Table 2).
Model Limitations
Model limitations were as follows: 1) leaf orienta- tion was restricted to horizontal leaves, 2) the number of leaves within a layer was limited to one except in the case where 100% leaf cover was modeled, 3) only one leaf layer was modeled (ex- cept in the case where the background was mod- eled as consisting of leaves), 4) the look direction was limited to a nadir or zenith view, 5) the primary light source (equivalent of the sun) was restricted to the zenith, 6) all materials were treated as Lambertian scatterers, 7) penumbral effects were not modeled, and 8) diffuse skylight was not modeled.
Due to the limitations of the model, this model could not be used to create an accurate rendition of a real canopy. Leaf orientation is neither con- stant nor horizontal in most canopies. Further- more, most canopies consists of more than one leaf layer, and most canopy layers contain variable numbers of leaves. Limitations in the position of the source and the detector also restrict the model to a very limited acquisition time. Also, leaves are only approximately Lambertian in the NIR at nor- mal or near-normal incidence. At off-nadir geome- tries they may show markedly non-Lambertian behavior, particularly in visible light (Breeee and Holmes, 1971; Norman et al., 1985, Walter-Shea et al., 1989).
Real canopies are considerably more complex than the model; however, the phenomena ob- served in this study will still occur to some extent in the more complex cases. In a real canopy, the effects of leaf size, leaf height, and substrate re- flectance may not be easily separated from the effects of multiple leaf layers, leaf shadowing, and leaf orientation. Leaf orientation (Kimes, 1984; Jackson and Pinter, 1986), multiple leaf layers (Huete et al., 1985), and leaf shadowing (Kriebel, 1978) have all been shown to affect significantly canopy reflectance. Investigation of the effects of the canopy architectural variables which were modeled in this study on canopy reflectance may provide a useful direction for further research.
Penumbral effects, which were not modeled, will become increasingly more important as the ratio of the size of the leaf to its height above the background decreases. The greatest influence will occur in the intensity of ~b a within the leaf shadow. This is because of the low intensity of ~b d at 450 nm and 675 nm, and at all three wavelengths within the shadows of small leaves positioned a large height above the background. For example, based upon our calculations, a penumbra would contribute at most 2% of the direct incident light to the shadow cast by a 1 cm leaf at a 5 cm height. Although this is small, under these same condi- tions, the 450 nm and 675 nm ~b,~ calculated using our model was lower.
Implications of This Research
NIR leaf refectance can be highly dependent upon the reflectance of the background. To re- move the effects of the substrate from the re- flectance of the leaf/canopy, the transmitted com- ponent needs to be taken into account. Several vegetation indices [including the vegetation index (VI: Jordan, 1969), normalized difference vegeta- tion index (NDVI: Rouse et al., 1973; 1974), the perpendicular vegetation index (PVI: Richardson and Wiegand, 1977), and the greenness vegetation index (GVI: Kauth and Thomas, 1976)] have been applied to remote sensing data and used to assess such diverse vegetation properties as leaf area index (LAI), vegetation cover, and green biomass (see Tucker, 1979). All of these indices rely heav- ily upon spectral contrast between the NIR and the red. None of these techniques take into ac- count the effects of transmitted light. Until vegeta- tion indices or techniques are developed which incorporate leaf transmittance as a variable, the amount of green vegetation assessed using remote sensing will depend upon substrate reflectance. Huete (1987) developed one technique which does incorporate leaf transmittance. This technique, however, does not account for variation which may be due to canopy architecture. The technique de- veloped by Huete (1987) assumes that second order or higher scattering is negligible (light was allowed to pass through a canopy only once on its way downward and once on its way upward). In contrast, we found a significant change in ~bur after as many as seven scattering events of NIR light.
Radiant Flux Distribution of Leaves 15
Based upon this research, it can be concluded that canopy reflectance of architecturally different canopies will show different degrees of depen- dence on substrate reflectance. Consider two canopies consisting of leaves with the exact same spectral properties, leaf spacing, and mean leaf height above the substrate. The canopies differ only in that one canopy has much smaller leaves. The model results imply that the canopy having the smaller leaves will be more reflective, having been influenced by substrate reflectance to a greater extent.
The implications of this research extend to closed canopies where the underlying substrate is another layer of leaves. All variables except leaf size being equal, the canopy having the smaller leaves would reflect more NIR light and relatively similar amounts of red light. Consider a compari- son of Acer macrophyllum to Alnus rubra (red alder). Based upon laboratory measurements, the leaves of Acer macrophyllum and Alnus rubra are spectrally similar. In addition, both species tend to have approximately horizontally distributed leaves (Alnus more so than Acer). One major difference between the two species is thatAcer leaves aver- age between 10 cm and 20 cm in diameter, whereas Alnus leaves average 6-12 cm in length (Munz and Keck, 1973). The results of this study suggest that an Alnus canopy having the same LAI as an Acer canopy will have greater NIR reflectance if the leaves are placed in the same configuration. The only case where leaf size and leaf height did not prove to be of importance was the case of a closed canopy in which there were no gaps be- tween leaves within a canopy layer and the leaves were horizontal.
Although the model was restricted to a phase angle of zero, resulting in no shadows as "seen" by the detector above, the results for the dpd r imply that, in multispectral remote sensing data, the shadows cast by leaves will bear the spectral sig- nature of vegetation. This observation has implica- tions for the estimation of vegetation percent cover, where shadows cast by leaves may be misinter- preted as leaves, and thus result in overestimation of percent cover. The effects of leaf transmittance will be particularly significant when NIR data is used to estimate percent cover.
Finally, the results of this research imply that the amount of light scattered to the side by a plant may significantly alter the spectral quality of light
incident on adjacent, unshaded substrates. In the model, the background closest to the leaf received up to 20% more light than those elements farthest from the leaf. Furthermore, the magnitude of scat- tered light will depend upon proximity to the plant. Substrates closer to the plant will receive a larger proportion of their diffuse light from vegeta- tion than substrates at a greater distance. These findings suggest that it may be difficult to remove entirely or separate the vegetation component from the substrate component in reflected light mea- sured remotely over vegetated areas. However, a modeling approach, such as the one used in this study, may enable a better definition of the rela- tionship between substrate reflectance and canopy reflectance, thus facilitating better methods for separating vegetation and substrate spectral signatures.
SUMMARY
A leaf was modeled as an isotropic scattering medium that was positioned horizontally a fixed distance above a spectrally uniform background. The effects of variable background refectance, leaf size, and leaf height above the background were investigated using this model. Upward relative radiant flux above a leaf, ~bur, was highly depen- dent upon background reflectance; NIR qbur in- creased by a factor of 2 from the 0% reflectant background to the 100% reflectant background. Furthermore, leaf size and the height of the leaf above the background affected the dPur. In the NIR, the qbur showed an initial decrease with increasing leaf height, followed by an increase to a constant value (equal to the qbur emanating from a leaf positioned an infinite height above the back- ground). The size of the leaf controlled the height at which the maximum ~b~r was obtained. Small leaves attained a maximum dpu r value at a much lower leaf height than larger leaves.
Leaf size, leaf height, and background re- flectance also affected the intensity of down- welling radiation directly beneath and adjacent to the leaf. The ~b~ beneath a leaf decreased both as a function of increasing leaf height and decreasing leaf size. The ~bdr increased over 20% from a modeled leaf positioned over a 0% reflectant back- ground to one positioned over a 100% reflectant background. Lastly, light scattered to the side by
16 Roberts et al.
leaves increased NIR light incident on adjacent, unshadowed backgrounds by as much as 20%.
These model results suggest that transmittance and scattering need to be incorporated as variables when at tempting to assess vegetation indepen- dent ly from substrate reflectance, or when remov- ing the effects of vegetation from substrate spec- tra. In addition, leaf height and leaf size, which influenced the degree to which 0 , r from a leaf was affected by background reflectance, may need to be incorporated into canopy models and taken into consideration when assessing vegetation using vegetation indices.
A P P E N D I X
Variables:
Pu = leaf reflectance, upper surface, Pl = leaf reflectance, lower surface, ~u = downward leaf transmittance, r l --- upward leaf transmittance, p~ = substrate reflectance, n = number of reflections off of lower leaf sur-
face, 0 , r = total leaf reflectance.
Assumptions: All surfaces behave as Lambert ian materials. The leaf is placed directly upon the background.
Derivation:
Our = Pu, (A.1)
where Ps or r t equal zero. I f light is allowed to pass once through the leaf, strike the soil, and then pass once back upwards, Eq. (A.1) becomes
T*~n+ Our = Pu "~- u Ps 1,~/./,p~. (A.2) Because n = 0, this equation becomes
Our = P , + r*"*ru ~', t, (A.3)
which is identical to the equation developed by Lillesaeter (1982) and Huete (1987). I f light is allowed to reflect off the lower surface of the leaf once, n = 1 and Eq. (A.2) becomes
(A.4) Our = Pu + "u v's "t ~'l"
Total reflectance can be calculated by integrat- ing from n = 0 to n = 0% which give,;
oo
f0 , n+ Our ~-" Pu "~- Zu Ps 1,p~,../.i dn. (A.5)
We thank Dr. Elizabeth Van Volkenburgh and two anonymous reviewers for their helpful suggestions for improving the manuscript and Steven Willis for maintaining the computer systems and for invaluable assistance in programming. This research was supported by NASA Grant NAGW 1319 and by a grant from the W. M. Keck foundation for computer equip- ment.
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