Nonlinear spectral mixing in desert vegetation

ELSEVIER

Nonlinear Spectral Mixing in Desert Vegetation Terrill W. Ray* and Bruce C. Murray* L i n e a r mixing models are widely used in terrestrial remote sensing, with the errors in these models being often attributed to "nonlinear" mixing. Nonlinear mixing refers to the interaction of light with multiple target materials. Reflectance data from creosote bush in the Manix Basin of the Mojave Desert is used to show the existence and importance of nonlinear mixing in arid region vegetation. It shows that the difference in the reflectance spectrum of plants against a soil background and the spectrum of the plant against a dark background is well represented by light that has interacted with both the soil and the plant.

INTRODUCTION Land degradation in arid and semiarid regions is an important problem around the globe as the ever-growing world population forces the expansion of agriculture and human habitation into these marginal areas (Mainguet, 1994; Dregne, 1983). An important facet of the land degradation problem is the vegetation cover, which can serve as both a measure of present and recent degradation, as well as an indicator of areas most susceptible to future degradation (Ray, 1995; Mainguet, 1994; Dregne, 1983). The sparseness of the vegetation in such regions means that soil is a dominant factor in remotely sensed measurements in such regions. This means the measurement of plant cover in arid and semiarid areas is complicated by variability in the soil reflectance as well as spectral interactions between the sparse plant canopies and the soil. A large amount of work in terrestrial remote sensing has focused on the use of linear spectral mixing (e.g.,

*Division of Geological and Planetary Sciences, California Institute of Technology, Pasadena Address correspondence to Terrill W. Ray, Cahech, Mail Code 170-25, Pasadena, CA 91125. Received 31 October 1994; revised 8 July 1995. REMOTE SENS. ENVIRON. 55:59-64 (1996) @Elsevier Science Inc., 1996 655 Avenue of the Americas, New York, NY 10010

Adams et al., 1989; Smith et al., 1990). The general assumption made in the linear mixing models is that a "checkerboard" mixture of materials has a spectrum that can be expressed as a linear combination of the spectra of the individual materials (Smith et al., 1990). It is further supposed that the coefficient applied to the spectrum of each material is approximately equal to the amount of area covered by the corresponding material. The phenomenon of linear mixing is exemplified by the common use of dithering algorithms in the display of digital images or in the display of photographs in newspapers. Dithering produces colors or shades of gray that the display device (or printer) is unable to display directly by producing a pattern of displayable colors which the eye "averages" into the intended undisplayable color. The light coming from any patch has only interacted with the one patch, and no "mixing" occurs until the light enters the eye. However, it is known that an intimate mixture of minerals produce a spectral reflectance that is not simply a linear combination of the spectra of the materials being mixed (Nash and Conel, 1974). This nonlinear mixing arises as the materials are so closely associated that light is most likely to interact with multiple materials as it is scattered by the mixture. Boardman (1994) has aptly pointed out that linear mixing occurs in the instrument, but nonlinear mixing occurs primarily in the material being observed by the instrument. The phenomenon of nonlinear mixing between soil and vegetation spectra has been observed by numerous authors (e.g., Borel and Gerstl, 1994; Roberts et al., 1993; Roberts, 1991; Roberts et al., 1990; Smith et al., 1990; Huete, 1986; Huete et al., 1985). Usually these observations have involved attributing the residual between linear mixing results and the original spectrum being unmixed. Huete et al. (1985) found that significant changes in the observed "greenness" of a cotton canopy occurred as the underlying soil background was changed, and concluded that this was due to nonlinear spectral 0034-4257 / 96 / $15.00 SSDI 0034-4257(95)00171-V

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Ray and Murray

mixing. Huete (1986) used a principle-components-like factor analysis approach to characterize the multiple scattered component in plant-soil mixtures. Borel and Gerstl (1994) treat plant canopies as a layer of horizontal, nonoverlapping Lambertian disks and arrive at a BRDF (bidirectional reflectance distribution function), which includes powers of the product of leaf and soil reflectance. Boardman (1994) has suggested that these mismatches may be due to an incorrect choice of endmembers, while Smith et al. (1994) is of the opinion that this may be due to a "fuzziness" in the endmembers. This suggests that these residuals may be more due to an inadequacy in the linear model than due to nonlinear mixing caused by light interacting with multiple endmembers.

.

R~,x RI~/

(a) Background: Rb

RI, x Rb/

ANALYSIS The light detected by a spectrometer observing a desert plant can take many paths from the light source (usually the sun) to the instrument. The four most basic paths the light can take are: I) source ~ plant ~ instrument, 2) source ~ background ~ instrument, 3)source =* plant ~ background ~ instrument, and 4)source background =} plant = instrument. Let us begin by assuming that both the plant and the soil reflect light isotropically. We define the reflectivity of the plant to be Rp, which includes all the multiple reflections of light that occur between the various leaves and twig, which comprise the canopy as well as light directly scattered by the leaves and twigs. The reflectivity of the background is R~. We can now defined the effective reflectivity seen be light interacting with both the background and the plant, scattering once off each, to be Rp × R~. The reflectivity observed by the instrument is now given by R = aRp + ~Rb + yRpRb,

Leaf: R

~

/

Background: RI ~ Figure 1. The simplest paths for light reflecting from both leaves, with reflectance Rp and background, with reflectance Rb for opaque leaves, a) Horizontal leaves. This is the model of Borel and Gerstl (1994) for leaves with zero transmittance. The simplest reflectance term for interaction with both leaves and soil is Rb× Rp× Rb (or Rp× R~). b) The case for nonhorizontal leaves (or nonflat leaves). The simplest reflectance term in this ease is Rp× R~.

(1)

where a, p, and )' are weighting coefficients. Linear mixing assumes that the cross product is small or unimportant for the endmembers being considered. Borel and Gerstl (1994) predict a series of powers of Rp x R~ as part of their radiosity functions. The Borel and Gerstl model includes light transmitted through the leaves, but most desert vegetation have opaque leaves which transmit no light (Gates et al., 1965). Figure la shows the Borel and Gerstl model in the case of leaves that do not transmit light. The simplest path for light interacting with both soil and leaf requires light to pass between leaves, reflect from the soil, reflect from the underside of the leaves, and reflect from the soil again. The simplest multiple reflectance term in the Borel and Gerstl model when opaque leaves are considered is therefore Rb × Rp × R/~ (or np × R~). However, when the leaves are nonhorizontal (or nonflat), there is a simpler path for light, as shown in Figure lb, which results in

Rp × Rb as the simplest multiple reflectance term. The model with nonhorizontal leaves is more appropriate for plants in arid regions which tend to have nearly vertical leaves. In both the horizontal and nonhorizontal cases, there is an infinite series of multiple reflectance terms, but this analysis takes only the first multiple reflectance term. Consider the case when the spectrum of a desert plant has been acquired with two different backgrounds. If we take the difference between these two spectra, we get Rd = R' - R = aRp - aRp + ~R~ - ~Rb

+vR~n~-vnbnp,

nd -- # ( a ~ - as) + y (R~ap - abnp),

(2) (3)

where we have eliminated the direct contribution of the plant. If linear mixing is assumed and we ignore the

Nonlinear Spectral Mixing in Desert Vegetation 61

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Wavelength (/xm) Figure 2. Plot showing the reflectance spectra of soil, a creosote bush against the soil background (Creosotes), and the same creosote bush against a dark nonreflective target (Creosote~r).

cross terms, we expect the difference spectrum Re to be equal to some multiple of the difference between the two background spectra.

OBSERVATIONAL COMPARISONS Field spectra of plants in the Manix Basin area of the Mojave Desert (34 o56'N latitude, 116 °35'W longitude) were collected as part of a study of land degradation related to center-pivot irrigation agriculture. These spectra were collected using an Analytic Spectral Devices Personal Spectrometer 2 that covers the spectral range from 0.338/~m to 1.064/~m in 512 channels. In many cases the spectrum of the plant was collected twice, once with the natural background and once with a background composed of posterboard coated with KryIon ultra flat black paint. The dark target has a reflectance of less than 3% in the wavelength covered by the spectrometer, and it is less reflective in the near-infrared than in the visible. Figure 2 shows the spectrum of the soil along with two spectra of a creosote bush. The brighter of the creosote bush spectra is that with the soil background. These spectra were collected using the standard 25 ° FOV of the PS-II, and they were collected with the angle between the sun and the target as near to zero as possible while avoiding casting the instruments shadow on the target. For the case used here, five spectra were collected with the black background and five without. A Spectralon panel, which is supplied with the instrument, was used as the reference target. The total time elapsed during the collection of the three Spectralon reference spectra and all 10 spectra of the target plants was less than 5 min, so that motion of the Sun in the sky was insignificant.

0.6 0.8 Wavelength (#m)

1.0

Figure 3. Plot of the difference between the soil spectrum and the dark target spectrum (R~) and of the difference between the spectrum of the creosote bush against the soil and the creosote bush against the dark target (Rd). Clearly these two spectra are not multiples of each other.

In Figure 3, we see the difference between the creosote bush against the soil background and the same creosote bush against the dark background, which we call Rd. The difference between the soil spectrum and the dark target spectrum, which we call R~, is plotted on the same scale. Clearly these spectra are not multiples of each other, which is reinforced by Figure 4, which shows the results of dividing Re by Rbd. Spectra of other creosote bushes and Russian thistle (tumbleweed) give similar results. Most spectra of white bursage show very little difference between the different backgrounds, probably because the bursage canopies have much denser tangles of twigs than do creosote bush or tumbleweed. This result demonstrates that nonlinear mixing is

Figure 4. Plot of Rd/R~a. These weighting coefficients are what each wavelength of Rbd would need to be multiplied by to get Ra. If linear mixing were occurring, all of these weighting coefficients would be equal. 0.40 [ ~

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62 Ray and Murray

Table 1. Comparison of Spectral Mixing Model Results a Results of Spectral Mixing Models for Creosote Bush Endmember Abundances Model Type

Creosote Bush

Bare Soil

Plant-Soil Interaction

Model 1: conventional linear (plant and soil) Model 2: plant-soil interaction (relative abundances)

79.6% 22.5%

20.3% 3.3%

74.1%

Model 1: conventional linear (plant and soil) Model 2: plant-soil interaction (weighting coefficients) Model 3: plant only (weighting coefficients)

0.796 0.536 1.553

0.203 0.080

1.768

--

-

The relative abundances for Models 1 and 2 assume that the total abundance in the pixel is 100%. Both Models 1 and 2 were two-component models. occurring, because the only thing that changed between the acquisition of the two creosote bush spectra was the background. This inference can be tested in a crude way by attempting to unmix the difference spectrum using the bare soil and the product of the soil and plant reflectance spectra. The creosote bush against the dark target can be taken as the "true" creosote bush reflectance, although it is probably slightly brighter than the true plant reflectance because there is some spectral mixing with the dark target. Creosote bushes have very open canopies, and any observation of a creosote bush canopy includes holes through which the background can be seen. Ideally, the background should return no photons, and the dark target is the best available approximation of such a target. The holes in the creosote bush canopy are as much a part of the creosote bush canopy as are the leaves and twigs. However, as shown in Figure 5, the spectrum produced by the mixing model is very similar to the difference spectrum that we were

trying to match. The mixing model suggests that the difference between the creosote bush spectrum against the soil and the creosote bush against the dark target is 4.3% bare soil and 95.6% soil-plant interaction. If we now return to the original spectrum of creosote bush against the soil background, we find that if we do a two-component linear unmixing using the creosote bush against the dark target as the first e n d m e m b e r and the composite cross-product plus soil spectrum as the second endmember, we get 22.5% creosote bush, 3.3% bare soil, and 74.1% plant-soil interaction. This compares to 79.6% creosote bush and 20.3% bare soil predicted by a conventional linear mixing between creosote bush against the dark target and bare soil. Figure 6 shows the errors in fit for each of the two unmixing models used above. Clearly the mixing model using the creosote bush against the dark background and the plant-soil cross term as endmembers has a much smaller error than the creosote bush against dark

Figure 5. Comparison of the difference between the spectrum of creosote bush against soil and creosote bush against the dark target (Ra) with the model fit of 4.3% bare soil and 95.6% plant-soil interaction. Notice that the model predicts too much red reflectance and slightly too little nearinfrared reflectance. The model is only valid in the range 0.4-0.9/~m.

Figure 6. Residuals for two mixing models. Model 1 was a conventional linear mixing model using creosote bush against the dark target and bare soil as endmembers. Model 2 used creosote bush against tlae dark target and the model fit from Figure 4 as endmembers. Both models overestimate visible reflectance and underestimate near-infrared reflectance, but Model 2 has smaller residuals.

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Nonlinear Spectral Mixing in Desert Vegetation

63

EVALUATION

ties, but the leaves of the creosote bush tend to have very low transmissitivies (Gates et al., 1965). The spectrum of the creosote bush against the dark target is almost certainly darker than the hypothetical true reflectance of the creosote bush not having any background, and since the reflectance of the dark target decreases in the near-infrared wavelengths, there is a slight suppression in the near-infrared reflectance. An examination of Figure 4 shows stronger visible-wavelength chlorophyll features which may be indicative of an enhancement in multiple scattering inside the canopy due to more light being reflected from the soil into the canopy. This particular creosote bush was re-observed in an identical manner with the ASD PSII spectrometer and also with the JPL GER (Geophysical Exploration and Research) spectrometer 7 months after the data presented here were acquired. The same nonlinear effect was observed each time, with the difference between NIR reflectance and red reflectance always enhanced by the removal of the dark target. This effect was observed in other creosote bushes as well. One creosote bush actually showed an inverse effect where the NIR and red reflectance difference decreased when the dark target was removed, and this occurred consistently across two observation dates with the ASD PSII and one observation date with the GER spectrometer. This indicates that shrub architecture plays an important role in the nonlinear mixing, which suggests that detailed models of the plant canopy using the reflectance characteristics of the constituent leaves and twigs is needed to fully analyze these effects. This illustration shows that nonlinear mixing is important in the observation of plants in arid environments. This example also shows that the error observed in the simple linear mixing model can be accounted for by the interaction of light with multiple endmember materials, which demonstrates that "nonlinear" mixing does not just mean the error in a linear mixing model. The enhancement in apparent vegetation through this nonlinear mixing is a double-edged sword in that it makes vegetation more detectable, which is useful, but it also makes accurate quantitative assessments of vegetation cover more difficult. Finally, since the strength of the plant-soil interaction can vary |~om vegetation type to vegetation type, nonlinear mixing effects may confound conclusions about the relative abundances of different plant types in communities observed by remote sensing.

There are several sources of error in the approach used here. First, our assumption that the plant and the soil reflect isotropically is not correct, from the simple fact that the plant will not be reflecting as much light on its shadowed side as its sunlit side. Also, it has been noted that the upper and lower surfaces of the leaves of many plants have different reflectivities (Gates et al., 1965). The leaves of some plants have significant transmissivi-

The authors would like to thank Curtiss Davis at the Naval Research Laborato~! for providing the ASD PSII spectrometer used in this study. Christopher Elvidge at the Desert Research Institute and Jim Westphal at Caltech both provided useful advice on the experimental design. Caltech undergraduates Lisa Gaskell and Becky Zaske assisted in the collection of the field data. Substantial funding for this work was provided by NASA Graduate Student Fellowship NGT-5891.

target and bare soil model, and, since both models used an equal number of endmembers (two), the model including the reflectance cross-product is the better model. The creosote bush-soil model produces a much higher visible reflectance and much lower near-infrared reflectance than is observed. The mixing model that includes the nonlinear interaction between the soil and the plant does a significantly better job in both the visible and near-infrared. It is noteworthy that a slightly better fit than that using the soil-plant interaction can be arrived at by simply multiplying the creosote bush against dark target spectrum by 1.553, which represents a 50% inflation in the apparent amount of vegetation. This shows that the general result of nonlinear mixing is to increase the apparent amount of vegetation. While the residuals for the mixing between the creosote bush and the plant-soil interaction are encouragingly small, we must nevertheless still consider how well this corresponds to reality. The creosote percentages listed above were the percentage of the total abundance, and if we take the weights directly as the "abundances" we get 53.6% creosote bush against dark target, 8.0% bare soil, and 176.8% soil-plant interaction. It should be pointed out that since 100% creosote bush against dark target was used to find the difference spectra, "abundances" greater than 100 % are permissible in this case. An "abundance" exceeding 100% is also not unreasonable for the cross-product term since it is composed of two reversible paths. This is close to the 10% coverage by directly lit soil seen in pictures of creosote bushes, and 50% cover by directly lit creosote bush does not seem unreasonable based on photographs. The rest of the area is covered by shadows and dimly lit creosote bush, which is probably reflecting light reflected from the soil beneath the plant. The plant-soil interaction spectrum is less than half as bright in the visible as are fully lit leaves, which supports this conclusion. We see, therefore, that nonlinear spectral mixing between plant and soil can be the dominant factor in the remotely sensed spectra of desert regions and that they can significantly alter the apparent abundance of vegetation inferred from remotely sensed data. Although the error introduced over an image pixel may only be a few percent, the low abundance of plant cover in arid regions makes even an error of a few percent in vegetation cover serious.

64 Ray and Murray

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Mainguet, M. (1994), Desertification: Natural Background and Human Mismanagement, 2nd edition, Springer-Verlag, New York, 314 pp. Nash, D. B., and Conel, J. E. (1974), Spectral reflectance systematics for mixtures of powdered hypersthene, labradorite, and ilmenite, J. Geophys. Res. 79:1615-1621. Ray, T. W. (1995), Remote monitoring of land degradation in arid / semiarid regions, Ph.D. thesis, California Institute of Technology, 415 pp. Roberts, D. A. (1991), Separating spectral mixtures of vegetation and soil, Ph.D. dissertation, University of Washington, 180 pp. Roberts, D. A., Adams, J. B., and Smith, M. O. (1990), Transmission and scattering of light by leaves: the effect of spectral mixtures, in Proc. IEEE Int. Geosci. and Remote Sensing Syrup. "90, IEEE, New York, pp. 1381-1384. Roberts, D. A., Smith, M. O., and Adams, J. B. (1993), Green vegetation, nonphotosynthetic vegetation, and soils in AVIRIS data, Remote Sens. Environ. 44:117-126. Smith, M., Roberts, D., Hill, J., et al. (1994), A new approach to quantifying abundance of materials in multispectral images, in Proc. IEEE Int. Geosci. and Remote Sensing Symp. "94, IEEE, New York, pp. 2372-2374. Smith, M. O., Ustin, S. L., Adams, J. B., and Gillespie, A. R. (1990), Vegetation in deserts: I. A regional measure of abundance from multispectral images, Remote Sens. Environ. 31:1-26.