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Comparative study between a newnonlinear model and common linearmodel for analysing laboratorysimulated‐forest hyperspectral dataWenyi Fan a , Baoxin Hu b , John Miller b & Mingze Li aa Forestry College, Northeast Forestry University, Harbin, Chinab Department of Earth and Space Science and Engineering, YorkUniversity, Toronto, CanadaPublished online: 22 Jun 2009.

To cite this article: Wenyi Fan , Baoxin Hu , John Miller & Mingze Li (2009) Comparativestudy between a new nonlinear model and common linear model for analysing laboratorysimulated‐forest hyperspectral data, International Journal of Remote Sensing, 30:11, 2951-2962,DOI: 10.1080/01431160802558659

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Comparative study between a new nonlinear model and common linearmodel for analysing laboratory simulated-forest hyperspectral data

WENYI FAN*{, BAOXIN HU{, JOHN MILLER{ and MINGZE LI{

{Forestry College, Northeast Forestry University, Harbin, China

{Department of Earth and Space Science and Engineering, York University, Toronto,

Canada

The spectral unmixing of mixed pixels is a key factor in remote sensing images,

especially for hyperspectral imagery. A commonly used approach to spectral

unmixing has been linear unmixing. However, the question of whether linear or

nonlinear processes dominate spectral signatures of mixed pixels is still an

unresolved matter. In this study, we put forward a new nonlinear model for

inferring end-member fractions within hyperspectral scenes. This study focuses

on comparing the nonlinear model with a linear model. A detail comparative

analysis of the fractions ‘sunlit crown’, ‘sunlit background’ and ‘shadow’ between

the two methods was carried out through visualization, and comparing with

supervised classification using a database of laboratory simulated-forest scenes.

Our results show that the nonlinear model of spectral unmixing outperforms the

linear model, especially in the scenes with translucent crown on a white

background. A nonlinear mixture model is needed to account for the multiple

scattering between tree crowns and background.

1. Introduction

Spectral mixing at the sub-pixel scale is an inherent feature of remote sensing imagery

over forest landscapes amongst others, because natural scenes tend to be composed of

mixed materials at any scale (Mustard and Sunshine 1999). For example, a single pixel

may contain sunlit crown, sunlit background and shadow in a forest scene.

Quantitative studies of the mixing problem in remote sensing have been undertaken

for the last two decades (e.g. Adams et al. 1986, Li and Strahler 1986, Mustard and

Pieters 1987, Smith et al. 1990, Boardman and Goetz 1991, Sabol et al. 1992, Hall et al.

1995). In quantitative mixing analysis (Mustard and Sunshine 1999), the scene of

interest is assumed to be dominated by a small number of materials called end-

members. The spectral variation between pixels in the scene can be modelled by a

mixture of these components. If detected photons interact mainly with a single

component on the scene before they reach the sensor, the reflectance of a mixed pixel is

a linear combination of the reflectance of the end-members weighted by the area

coverage of each end-member in the pixel. In a linear spectral mixture model, the end-

members are assumed to have a relatively constant reflectance across the whole scene,and their fractions are calculated by solving constrained linear equations. This process

is also called linear spectral unmixing. If photons interact with multiple components

within the instantaneous field of view, such as multiple scattering between the

vegetation canopy and soil, the mixture analysis has the potential of becoming

significantly nonlinear (Johnson et al. 1983). Currently, linear spectral unmixing is

*Corresponding author. Email: fanwy@163.com

International Journal of Remote Sensing

Vol. 30, No. 11, 10 June 2009, 2951–2962

International Journal of Remote SensingISSN 0143-1161 print/ISSN 1366-5901 online # 2009 Taylor & Francis

http://www.tandf.co.uk/journalsDOI: 10.1080/01431160802558659

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receiving considerable attention in remote sensing image interpretation, due to the

development of hyperspectral imagers and the availability of commercial software.

Whether a linear or nonlinear process dominates the spectral signatures of mixed

pixels depends on a number of factors, mainly the nature of the scene (Boardman 1994).

For example, an intimate mixture of minerals produces a spectral reflectance that is not

simply a linear combination of the reflectance of these materials. However, if the

reflectance is transferred to the single scattering albedo, the intimate mixture is a linear

problem (Hapke 1993). Mustard and Sunshine (1999) suggested that the error in the

fractions may be as much as 30% absolute for mineral ensembles, if a linear mixture

model is applied to a nonlinear system. Effects of the nonlinear mixture between

vegetation and soil on linear unmixing have been reported for the vegetated landscape as

well (e.g. Huete et al. 1985, Smith et al. 1990, Roberts et al. 1993). However, these

observations are based on the residual between the original reflectance and the

reflectance predicted by a linear mixture model. Borel and Gerstl (1994) used a radiosity-

based model to demonstrate that nonlinear mixture effects occur when multiple

scattering effects are considered and, in the near-infrared range, the surface reflectance

obtained by their model was almost twice as high as with the linear mixture model.

Boardman (1994) used a nonlinear unmixing approach with mineral surfaces to

Airborne Visible and Infrared Imaging Spectrometer (AVIRIS) data. Roberts et al. (1993)

derived end-member fractions from a nonlinear spectral mixture of green vegetation and

shade. Canopy shade was calculated by rearranging the standard linear mixing equation

and solving for shade. In this study, considering vegetation is the main cover of the Earth’s

surface and is more complex than mineral materials due to transmission and multiple

scattering, we put forward a new nonlinear model to depict the vegetation scattering

properties for inferring end-member fractions within hyperspectral scenes. A comparative

analysis of this mixture estimation method with the commonly used linear mixture model

was carried out using a database of laboratory simulated-forest scenes.

2. Data and data preprocessing

A few imagery scenes were designed to simulate the natural forest landscape in the

laboratory. Two kinds of objects, opaque and translucent, were mounted on stems to

simulate forest crowns on trunks. Their dimensions are listed in table 1. These ‘trees’

were randomly (Soffer 1995) placed on a mounting board, covered, in turn, with tree

backgrounds: dark, green and white. Opaque and translucent trees were separate and

each mounting board occupied an area of 40 cm640 cm on the same mounting board.

The four edges of each scene (with opaque or translucent trees) were clearly marked

on the mounting board to identify the scenes from remote sensing imagery. Canopies

of both opaque and translucent trees were designed with two tree densities: sparse and

dense. For the sparse canopies, 40 trees were planted in an area of 40 cm640 cm,

while 100 trees were used for the dense canopies. The scene illumination was generated

using a 1000 W tungsten lamp at the focal point of a 120 cm focal length Fresnel lens,

which provided incident collimated radiation (Soffer 1995). For these experiments an

illumination angle of 40u was used. The light source was fixed with respect to the scene

and with respect to the sensor. As described by Soffer (1995), the illumination

intensity is not uniform across the illumination field because of irregularity in the

Fresnel lens. As a result, a white uniform panel was placed on the mounting board, as

a reference to correct the non-uniformities of the illumination field. Hyperspectral

images of these simulated forest scenes were acquired by the Compact Airborne

Spectrographic Imager (CASI), a push-broom imager, by moving the entire scene

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perpendicularly at a constant rate with respect to the CASI field of view. The

hyperspectral CASI images were acquired in 72 spectral channels, covering the

spectral region 414–914 nm, at a nominal spectral resolution of 7.5 nm. With the target

scene observed within ¡15.3u from the CASI nadir, nadir viewing was assumed for all

of the subsequent image analysis.

CASI imagery was also collected over a white Spectralon panel under the same

imaging geometry. The radiance of the white Spectralon panel was used as a

reference to calculate the reflectance of the scenes. During the data pre-processing

process, the raw CASI imagery over the simulated scenes and the white Spectralon

panel was first converted to radiance data using calibration coefficients determined

in the laboratory by the Center for Research in Earth and Technology. The

illumination non-uniformity was then corrected by scaling the radiance of each pixel

by the relative response of the white uniform panel in the same location in the

across-track direction, as described by equation (1).

L’s, i, j lð Þ~Ls, i, j lð Þ L�

p lð ÞL�

p, i lð Þ ð1Þ

where L’s, i, j lð Þ and Ls,i,j(l) are the corrected and uncorrected radiances of pixel

(i,j) in a simulated scene for a given channel (l); L�

p(l) and L�

p,i(l) are the average

radiance for the entire white uniform panel (on the same mounting board as the

simulated scene) and the average panel radiance for the given pixel’s column i. The

corrected spectral radiance was then converted to the reflectance Rs,i,j(l) using

equation (2). The corrected spectral radiance is divided by the average corrected

spectral radiance L’r, i, j lð Þ of the white Spectralon panel, L�

r(l).

Rs, i, j lð Þ~ L’s, i, j lð ÞL�

r lð Þ ð2Þ

The scenes designed for this study are summarized in table 1.

3. Methods

3.1 End-members

The most crucial step in spectral mixture analysis is the selection of spectral end-

members. Four approaches are commonly used (Baoxin et al. 2004). (1) Obtain the

Table 1. Basic information concerning the designed scenes.

Data file Background Density (trees per 100 cm2) Lines Columns Bands

F03_opaque dark Dense (6.25) 200 148 72F03_translucent dark Dense (6.25) 213 148 72F04_opaque green Sparse (2.5) 216 148 72F04_translucent green Sparse (2.5) 185 148 72F05_opaque white Sparse (2.5) 204 148 72F05_translucent white Sparse (2.5) 184 148 72F06_opaque green Dense (6.25) 192 148 72F06_translucent green Dense (6.25) 203 148 72F07_opaque white Dense (6.25) 186 149 72F07_translucent white Dense (6.25) 207 148 72F08_opaque dark Sparse (2.5) 200 148 72F08_translucent dark Sparse (2.5) 178 148 72

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end-member spectra from a spectrum library. The successful use of this approach is

strongly dependent on the calibration of the image data used. It is complicated

further when differences between inherent and apparent reflectance of surface

features are considered as a result of the 3D structure of the canopy. (2) Derive the

end-member spectra from the pure pixels of the image itself. This approach can be

used only in cases where there are some pure pixels in the image data examined. (3)

Automatically obtain end-member spectra using factor analysis. (4) Automatically

construct end-member spectra using convex geometry. In this study, it is clear that

the laboratory-simulated scenes with opaque, translucent tree crowns on a green

background, a dark background and a white background are composed mainly of

sunlit crown, sunlit background, shaded crown and shaded background. The

reflectance of each end-member was calculated by method (2). Because the

reflectance spectrum of shaded crown is very close to that of the shaded background

– the average relative error between shaded crown and shaded background from

414–914 nm is 1.1032% – we mixed the shaded crown and shaded background

together to produce the end-member shadow. This method was also used by Hall et

al. (1995) and Baoxin et al. (2004). The end-members and their spectra are presented

in figure 1. The sunlit crown and green background have typical vegetation spectra,

shadow has a low reflectance and white background has a strong reflectance.

3.2 Linear and nonlinear unmixing model

The linear spectral mixture model describes the surface reflectance spectrum as a

linear combination of a finite number of spectra corresponding to the pure

components, called end-members. With an ideal choice of end-members, the

coefficients representing the component fractional areas in the linear model are non-

negative and sum to 1.

r lið Þ~Pm

j~1

Fjrj lið Þze lið Þ

Pm

j~1

Fj~1 Fj§0

ð3Þ

where r(li) is the reflectance of a pixel and is the function of wavelength l, i51, 2, ...,

n represents band number; rj(li) is the spectral reflectance of the end-members, j51,

2, ..., m represents end-members; Fj is the area fractions of the end-members; e(li) is

the error. The matrix form of equation (3) can be written as:

r~Pfze ð4Þ

where r is an n61 reflectance vector, P is a n6m matrix where each column

represents the end-member vector, f is a m61 fraction of the end-member vector

and e is a n61 error vector.

The linear spectral mixture model describes the surface reflectance spectrum as a

linear combination of a finite number of spectra corresponding to the pure components,

end-members. Actually, the reflectance of materials within a pixel correlates with each

other. Spectral mixtures of tree crowns have the potential of being nonlinear due to

crown transmission. Linear mixing occurs when light interacts only with one material.

Transmission by crown leads to multiple scattering and nonlinear mixing (Roberts et al.

1991). In this case, the reflectance of the mixed pixel is not a linear combination of the

reflectance of these materials and it should exhibit nonlinear behaviour.

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Assume the nonlinear model is:

r lð Þ~f Fi, ri lð Þð Þze ð5Þ

where f(Fi, ri(l)) is a nonlinear function. Usually, it is hard to know the f(Fi, ri(l))

exactly. The nonlinear model that we put forward is not a radiosity model. We

assume the reflectance of the mixed pixel is a nonlinear model and it can be

approximated by a polynomial function. When the Taylor series expansions of the

function are made and only the first-order items retained, the model will be very

simple, and only the second scatterings between end-member j (j51, 2, … m) andend-member s (s51, 2, … m) are considered. Fj or Fs are the area fractions of end-

members j and s. rj(li) or rs(li) is the spectral reflectance of end-members j and s. So

Figure 1. The reflectance spectra of the end-members: (a) opaque crowns on darkbackground; (b) translucent crowns on dark back ground; (c) opaque crowns on greenbackground; (d) translucent crowns on green background; (e) opaque crowns on whitebackground; (f) translucent crowns on white background.

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the factor Fjrj(li)Fsrs(li) would represent the fraction of light that reflects off

surface j to surface s and then to the observer, and also the j would scatter to other

materials. This intercrossing item represents the second scattering between end-

member j and end-member s. Then the nonlinear model (equation (5)) is

approximated by a polynomial and rewritten as below:

r lið Þ~Xm

j~1

Fjrj lið ÞzXm

j, s~1jvs

FjFsrj lið Þrs lið Þze lið Þ ð6Þ

where r(li) is the reflectance of a pixel and is the function of the wavelength l.

Notice that, in the right of equation (6), the first item is just the linear model and it

would represent only the single scattering among the end-members; the second item

might represent the second scattering between tree crowns and background causedby crown transmittance. In this study, for simplicity and ease of solving, only the

second scattering was considered.

Let:

A~

r1 l1ð Þ r2 l1ð Þ � � � rm l1ð Þ r1 l1ð Þr2 l1ð Þ r1 l1ð Þr3 l1ð Þ � � � r1 l1ð Þrm l1ð Þ r2 l1ð Þr3 l1ð Þ � � � rm{1 l1ð Þrm l1ð Þr1 l2ð Þ r2 l2ð Þ � � � rm l2ð Þ r1 l2ð Þr2 l2ð Þ r1 l2ð Þr3 l2ð Þ � � � r1 l2ð Þrm l2ð Þ r2 l2ð Þr3 l2ð Þ � � � rm{1 l2ð Þrm l2ð Þ

..

. ... ..

. ... ..

. ... ..

. ... ..

. ... ..

.

r1 lnð Þ r2 lnð Þ � � � rm lnð Þ r1 lnð Þr2 lnð Þ r1 lnð Þr3 lnð Þ � � � r1 lnð Þrm lnð Þ r2 lnð Þr3 lnð Þ � � � rm{1 lnð Þrm lnð Þ

2

66664

3

77775ð7Þ

g~ F1 F2 � � � Fm F1F2 F1F3 � � � F1Fm F2F3 � � � Fm{1Fm½ �T ð8Þ

where A is a n6k matrix and k is the item number of equation (6); g is a k61 vector.

r~

r l1ð Þr l2ð...

r lnð Þ

2

66664

3

77775

. . . e~

e l1ð Þe l2ð Þ...

e lnð Þ

2

66664

3

77775

ð9Þ

So the matrix form of equation (6) can be written as:

r~Agze

Xm

j~1

Fj~1ð10Þ

The meaning of the symbols is the same as in the linear model. For the algorithm

of solving equation (10), see Appendix A.

3.3 Comparing the linear model with the nonlinear model

To validate the results obtained using the nonlinear spectral unmixing approach, the

fractions of sunlit crown, sunlit background, and shadow were calculated using the

linear and nonlinear unmixing models. For this purpose, all scenes were classifiedinto three types, sunlit crown, sunlit background and shadow, based on many

training pixels using the Spectral Angel Mapper (SAM) to perform the supervised

classification on the CASI datasets, although the classification maps are based on

ð7Þ

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per-pixel classifiers, where a single cover class is mapped to each pixel. We

calculated the fractions of sunlit crown, sunlit background and shadow for all scenes

in 72 spectral channels, covering spectral region 414–914 nm, and compared the

modelling results with the classified results (types of sunlit crown, sunlit background

and shadow) in the 72 bands. We defined the absolute error in fraction of each end-

member as the absolute difference between the fraction obtained by the linear or

nonlinear unmixing model and that by the classification. The relative error is defined

as the ratio of the absolute error and the fraction obtained by classification. The

absolute and relative rms. errors (RMSE) were calculated using the formula as

follows:

RMSE %ð Þ~1N

PN

K~1

fck{fmkj j2� �1=2

f�

c

|100% ð11Þ

RMSE~1

N

XN

K~1

fck{fmkj j2 !1=2

ð12Þ

where fck is the classification results and fmk is the model result; f�

c is the average

of the classification results.

4. Results and analysis

4.1 Visualization

We selected Matrix Laboratory (MATLAB) as the platform to program for the

linear unmixing model and nonlinear model, which is described in section 3.2. To

validate our programming, the results of the linear unmixing model solved by the

software of Environment for Visualizing Images (ENVI) and by our MATLAB

program were compared with the same scenes. Fortunately, the results were the

same. So we proceeded with confidence to perform the programmed modules, linear

and nonlinear, for all scenes to obtain the fraction of sunlit crown, sunlit

background and shadow. Here, only the results of the dense translucent tree crowns

with a white background are presented (figure 2). From figure 2, it is clear that the

sunlit crown is distinguished more easily from others by the nonlinear model than by

the linear model.

4.2 Comparing the modelling results with classification

The purpose of our study is to compare the accuracy of linear spectral unmixing

with the nonlinear approach. We used the supervised classification results as a

benchmark for comparing with the fractions of sunlit crown, sunlit background and

shadow obtained by nonlinear and those by linear unmixing model. All scenes were

classified into three types – sunlit crown, sunlit background and shadow – and then

statistics were used to count how many pixels were classified into each type.

Subsequently, the fractions of each type were calculated: for example, for the scene

with opaque crowns and sparse tree density on a white background, 2605 pixels were

classified into sunlit crown, 23 057 pixels into sunlit background and 4530 pixels into

shadow. Thus, the fractions of sunlit crown, sunlit background and shadow were

0.0863, 0.7637 and 0.15, respectively. Similarly, we calculated the fractions of sunlit

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crown, sunlit background and shadow for other scenes, and compared the

classification results with the fractions of the end-members (sunlit crown, sunlit

background and shadow) obtained by linear and nonlinear models. The detailed

results are presented in table 2 and figure 3.

From table 2 and figure 3, the following phenomena can be observed. (1) The

errors in the fractions of sunlit crown, sunlit background and shadow obtained by

the linear model are larger than those by the nonlinear model for all scenes. (2) For

the scene with a white background, the errors in the fractions are larger than those

for the scenes with green backgrounds and dark backgrounds. These observations

make sense. For the scenes with white background, the multiple scattering between

tree crowns and the white background, and multiple scattering between crowns, is

large because the white background has a very high reflectance (see figures 1(e) and

(f)). The reflectance of the scene with a white background tends to be a nonlinear

mixture of the reflectance of sunlit crown, sunlit background and shadow. The

Figure 2. The spectral unmixing results for the scene with translucent crowns and dense treedensity on the white background. (a) Linear unmixing module solved by the software ENVI.(b) Linear model solved by our linear model program. (c) Nonlinear model solved by ournonlinear module program. Output from MATLAB.

Table 2. RMSE (absolute and relative RMSE%) for scenes with different crown transparencyand tree density on dark, green and white background, using linear and nonlinear models

comparing with the results of superclassification.

Background DiaphaneityDensity (treesper 100 cm2) Abscissa

AbsoluteRMSE

Relative RMSE(%)

Linearmodel

Nonlinearmodel

Linearmodel

Nonlinearmodel

Dark Opaque Dense (6.25) 1 0.0489 0.0229 22.1130 15.1323Sparse (2.5) 2 0.0226 0.0137 15.0218 11.7085

Translucent Dense (6.25) 3 0.0988 0.0539 31.4297 23.2268Sparse (2.5) 4 0.1551 0.0573 39.3882 23.9386

White Opaque Dense (6.25) 5 0.1881 0.0326 43.3660 18.0630Sparse (2.5) 6 0.2111 0.0498 45.9427 22.3240

Translucent Dense (6.25) 7 0.2331 0.0053 48.2839 7.2903Sparse (2.5) 8 0.2250 0.0110 47.4355 10.5027

Green Opaque Dense (6.25) 9 0.0829 0.0315 28.7952 17.7609Sparse (2.5) 10 0.1216 0.0295 34.8783 17.1877

Translucent Dense (6.25) 11 0.1002 0.0234 31.6612 15.2914Sparse (2.5) 12 0.0982 0.0344 31.3390 18.5594

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linear unmixing model leads to large errors in the fraction of sunlit crown, sunlit

background and shadow. For example, the errors in the fractions of sunlit crown,

sunlit background and shadow obtained by the linear model for the scene with

opaque crown, sparse tree density on a white background are 0.2111 (absolute error)

and 45.9427 (relative error %), and the errors decrease to 0.0498 and 22.3240 with

the nonlinear model. Similarly, the errors decrease from 0.1881 and 43.366 to 0.0326

and 18.0630 for the scene with dense opaque crown on a white background. The

errors decrease from 0.2250 and 47.4355 to 0.0110 and 10.5027 for the scene with

sparse translucent crown on a white background. The errors decrease from 0.2331

and 48.2839 to 0.0053 and 7.2903 for the scene with dense translucent crown on a

white background. For details on the errors, see table 2 and figure 3. In addition, we

can see that the scene with translucent crown on a white background has the largest

errors. The translucent crown transmittance and the non-uniformity of the tree

crown surface irradiance can cause variation in the observed apparent reflectance.

The mixture analysis has the potential of becoming significantly nonlinear. From the

above analysis, it is easy to draw a conclusion that our nonlinear model outperforms

the commonly used linear model.

5. Conclusion

In this study, a few imagery scenes were designed. Two kinds of objects, opaque and

translucent, were mounted on stems to simulate forest crowns on trunks. These ‘trees’

were placed randomly on a mounting board covered, in turn, with one of three

backgrounds: dark, green and white. A new nonlinear model was proposed for inferring

end-member fractions within hyperspectral scenes. A detailed comparative analysis of

these nonlinear mixture estimation methods with the commonly used linear mixture

model was carried out through visualization and comparison with the supervised

classification using the database of laboratory simulated-forest scenes. We used the

supervised classification results as a benchmark for comparing with the fractions of

sunlit crown, sunlit background and shadow obtained by nonlinear and linear

unmixing models. The results show that the errors in the fractions of sunlit crown, sunlit

background and shadow obtained by the linear model are larger than those obtained by

the nonlinear model for all scenes, especially for the scene with translucent crown on a

white background. Our research results show that nonlinear mixture models are needed

to account for the multiple scattering between tree crowns and background, and the

nonlinear spectral unmixing model outperforms the linear model.

Figure 3. The chart of RMSE (%) corresponding with table 2. The abscissa 1–4 represent adark background, 5–8 represent a white background, and 9–12 represent a green background.

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Acknowledgements

This work is sponsored by ‘863’ program (2006AA12Z104) of China. The authors

would like to thank the Natural Science and Engineering Research Council of

Canada (NSERC) for financial support and Mr Jinnian Wang for help with the

experiments.

ReferencesADAMS, J.B., SMITH, M.O. and JOHNSON, P.E., 1986, Spectral mixture modeling: a new

analysis of rock and soil types at the Viking Lander 1 Site. Journal of Geophysics, 91,

pp. 8098–8112.

BAOXIN, H., MILLER, J.R., CHEN, J.M. and HOLLINGER, A., 2004, Retrieval of the canopy leaf

area index in the BOREAS flux tower sites using linear spectral mixture analysis.

Remote Sensing Environment, 89, pp. 176–188.

BOARDMAN, J.W., 1994, Geometric mixture analysis of imaging spectrometry data. In

Proceedings of IEEE International Geosciences and Remote Sensing Symposium ’94,

(New York: IGARSS), pp. 2369–2371.

BOARDMAN, J.W. and GOETZ, A.F.H., 1991, Sedimentary facies analysis using AVIRIS data:

a geophysical inverse problem. In Proceeding of the 3rd Airborne Visible/infrared

Imaging Spectrometer (AVIRIS) Workshop, JPL Publication 91–28.

BOREL, C.C. and GERSTL, S.A.W., 1994, Nonlinear spectral mixing models for vegetative and

soil surfaces. Remote Sensing of Environment, 47, pp. 403–416.

HALL, F.G., SHIMABUKURO, Y.E. and HUEMMRICH, K.F., 1995, Remote sensing of forest

biophysical structure using mixture decomposition and geometric reflectance model.

Ecology Application, 5, pp. 993–1013.

HAPKE, B., 1993, Theory of Reflectance and Emittance Spectroscopy (New York: Cambridge

University Press), pp. 110–112.

HUETE, A.R., JACKSON, R.D. and POST, D.F., 1985, Spectral response of a plant canopy with

different soil backgrounds. Remote Sensing of Environment, 17, pp. 37–53.

JOHNSON, P.E., SMITH, M.O., TAYLOR-GEORGE, S. and ADAMS, J.B., 1983, A semi-empirical

method for analysis of the reflectance spectra of binary mineral mixtures. Journal of

Geophysics, 88, pp. 3557–3561.

LI, X. and STRAHLER, A.H., 1986, geometrical-optical modeling of a conifer forest canopy.

IEEE Transactions on Geoscience and Remote Sensing, 23, pp. 705–721.

MUSTARD, J.F. and PIETERS, C.M., 1987, Abundance and distribution of ultramafic

microbreccia in Moses Rock Dike: quantitative application of mapping spectrometer

data. Journal of Geophysics, 94, pp. 13619–13634.

MUSTARD, J.F. and SUNSHINE, J.M., 1999, Spectral analysis for earth science investigation. In

Remote Sensing for the Earth Sciences, pp. 509–564 (New York: John Wiley & Sons

Inc.).

ROBERTS, D.A., SMITH, M.O. and ADAMS, J.B., 1991, Leaf spectral types, residual and

canopy shade in an AVIRIS image. Proceedings of the 3rd Airborne Visible/Infrared

Imaging Spectrometer (AVIRIS) Workshop, R.O. Green (Ed.), Pasadena, CA, pp.

43–50.

ROBERTS, D.A., SMITH, M.O. and ADAMS, J.B., 1993, Green vegetation, nonphotosynthetic

vegetation, and soils in AVIRIS data. Remote Sensing of Environment, 44, pp.

117–126.

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 Sensing of

Environment, 31, pp. 1–26.

SABOL, D.E., ADAMS, J.B. and SMITH, M.O., 1992, Quantitative sub-pixel spectral detection

of targets in multispectral images. Journal of Geophysics, 97, pp. 2659–2672.

SOFFER, R., 1995, Bidirectional reflectance factors of an open tree canopy by laboratory

simulation. Master Thesis, York University, Toronto, Canada.

2960 W. Fan et al.

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Appendix A: Algorithm for solving equation (10)

For equation (6), differentiate r with respect to F as follows:

Lr

LF1~r1 lið ÞzF2r1 lið Þr2 lið ÞzF3r1 lið Þr3 lið Þz � � �zFmr1 lið Þrm lið Þ

Lr

LF2~r2 lið ÞzF1r1 lið Þr2 lið ÞzF3r2 lið Þr3 lið Þz � � �zFmr2 lið Þrm lið Þ

Lr

LF3~r3 lið ÞzF1r1 lið Þr3 lið ÞzF2r2 lið Þr3 lið Þz � � �zFmr3 lið Þrm lið Þ

..

. ...

Lr

LFm

~rm lið ÞzF1r1 lið Þrm lið ÞzF2r2 lið Þrm lið Þz � � �zFm{1rm{1 lið Þrm lið Þ

Then:

dr~Lr

LF1

Lr

LF2

Lr

LF3� � � Lr

LFm

� �

dF1

dF2

dF3

..

.

dFm

2

66666664

3

77777775

ðA1Þ

For equation (6), Taylor series expansions can be obtained at point (F10, F20, ... Fm0),

and only the first-order items are retained:

r lið Þ&r0 lið Þz CizBif0ð ÞT dFze lið Þ ðA2Þ

where (F10, F20, ... Fm0) is the result of the linear model; substitute it into equation (6)

to get r0(li).

LrLF1

LrLF2

LrLF3

� � � LrLFm

h i~ CizBif0ð ÞT

dF~ F1{F10, F2{F20, � � �Fm{Fm0ð ÞT~ dF1, dF2, � � � dFmð ÞT

dFj~Fj{Fj0

Ci~ r1 lið Þ, r2 lið Þ, � � � rm lið Þð ÞT

Bi~

0 r1 lið Þr2 lið Þ r1 lið Þr3 lið Þ � � � r1 lið Þrm lið Þr2 lið Þr1 lið Þ 0 r2 lið Þr3 lið Þ � � � r2 lið Þrm lið Þ� � � � � � � � � � � � � � �

rm lið Þr1 lið Þ rm lið Þr2 lið Þ � � � rm lið Þrm{1 lið Þ 0

2

6664

3

7775

f0~ F10, F20, � � �Fm0ð ÞT

Let Pi5Ci + Bif0

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Then equation (A2) is written as:

r lið Þ~r0 lið ÞzPTi dFze lið Þ ðA3Þ

The matrix form can be written as:

r~

r l1ð Þr l2ð Þ� � �

r lnð Þ

2

6664

3

7775

r0~

r0 l1ð Þr0 l2ð Þ� � �

r0 lnð Þ

2

6664

3

7775

P~

PT1

PT2

� � �PT

n

2

6664

3

7775

e~

e l1ð Þe l2ð Þ� � �

e lnð Þ

2

6664

3

7775

f~

F1

F2

� � �Fm

2

6664

3

7775ðA4Þ

r~r0zPdFze ðA5Þ

dF~ PTP� �{1

PT r{r0ð Þ ðA6Þ

Fj~Fj0zdFj ðA7Þ

In order to calculate the Fj, an iteration loop would be needed. The algorithm of the

iteration loop is described as follows:

(1) Obtain the matrix of the end-member spectral reflectance: the matrix name

is defined as EndSpec (bands,end-members).

(2) Calculate total items K of nonlinear equation (6).

(3) Construct matrix A from the matrix EndSpec(bands,end-members). A isdefined in equation (7).

(4) Open the hyperspectral image files with 72 bands.

(5) Read out a pixel vector r, which is a pixel with 72 bands value.

(6) Calculate the initial value f05(F10, F20, ... Fm0)T, which is a m61 fraction of

the end-member vector.

(7) Began the iteration loop.

(8) Calculate g, which is a k61 vector.

(9) Calculate r05Ag.(10) Calculate Pi5Ci + Bif0.

(11) Calculate the P matrix of the nonlinear model:

P~

PT1

PT2

� � �PT

n

2

6664

3

7775

(12) Calculate dF5(PTP)21PT(r2r0) and Fj5Fj0 + dFj.

(13) Repeat the iteration loop until the calculated result is less than a tolerance.

(14) Save the result.

(15) Read out the next pixel vector r and repeat the steps (6) to (14) until the end

of the image file

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