geometric-optical modeling of a conifer forest...

IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. GE-23, NO. 5, SEPTEMBER 1985

Geometric-Optical Modeling of a Conifer Forest CanopyXIAOWEN LI AND ALAN H. STRAHLER

Abstract-A geometric-optical forest canopy model that treats coni-fers as cones casting shadows on a contrasting background can explainthe major portion of the variance in a remotely sensed image of a foreststand. The model is driven by interpixel variance generated from threesources: 1) the number of crowns in the pixel; 2) the size of individualcrowns; and 3) overlapping of crowns and shadows. The model usesparallel-ray geometry to describe the illumination of a three-dimen-sional cone and the shadow it casts. Cones are assumed to be randomlyplaced and may overlap freely. Cone size (height) is distributed lognor-mally, and cone form, described by the apex angle of the cone, is- fixedin the model but allowed to vary in its application.

The model can also be inverted to provide estimates of the size, shape,and spacing of the conifers as cones using remote imagery and a min-imum of ground measurements. Field tests using both 10- and 80-mmultispectral imagery of two test conifer stands in northeastern Cali-fornia produced reasonable estimates for these parameters. The modelappears to be sufficiently general and robust for application to othergeometric shapes and mixtures of simple shapes. Thus it has wide po-tential use not only in remote sensing of vegetation, but also in otherremote sensing situations in which discrete objects are imaged at res-olutions sufficiently coarje that they canot be resolved individually.

Keywords-Plant canopy model, remote sensing, forest reflectancemodeling, geometric probability

I. INTRODUCTIONMSATHEMATICAL modeling of plant canopies is a re-

search field that has been highly active in recentyears. With the Duntley equations as a basis, many modelshave been developed for optical wavelengths. They areparameterized by such variables as p and r (reflectanceand transmittance of the leaf, respectively), leaf area, andthe leaf angle distribution. Most of these models are one-dimensional, in that the canopies vary only with heightabove the soil surface.During the past three years, the authors have pursued

the development of a plant canopy model that adopts adifferent perspective, treating the plant canopy as an as-semblage of large solid three-dimensional objects. Utiliz-ing optical principles and parallel-ray geometry, we havemodeled a forest as a collection of randomly located conesthat are illuminated at an angle and cast shadows on abackground. We have also approached the modeling prob-lem with the idea that the canopy will be imaged by aremote-sensing device, and, further, that the resolution ofsuch a device will be sufficiently fine that the geometry ofthe pixel will interact with the size and placement of the

Manuscript received December 20, 1984; revised May 1, 1985. This workwas supported by NASA under Grant NAG 5-273.

X. Li was with the Department of Geography, University of California,Santa Barbara, CA 93106. He is now with the Department of Geology andGeography, Hunter College, City University of New York, NY 10021.

A. Strahler is with the Department of Geology and Geography, HunterCollege, City University of New Xork, NY 10021.

cones. (In other words, a forest imaged by multispectralscanner in which the pixel size is several times greaterthan average size of the trees.) This approach thereforeimplies that a reflectance model will have to predict notonly the average reflectance of the canopy, but also thevariance in reflectance from pixel to pixel.

Another concern has been to try and maintain inverti-bility in our model, either directly through the use of ap-propriate side constraints or indirectly through iterativeparametric estimation techniques. In this way, we have de-veloped procedures to estimate the parameters of size,shape, and spacing which drive the model from the reflec-tance values that are observed. Although invertible modelsare usually more difficult to formulate, the developmentof image-based models has allowed us to exploit interpixelvariance as an information source for inversion. -Our em-phasis on inversion, which is not a characteristic of mostexisting canopy models, is also somewhat unique.

It is important to understand that our model is basedmore strongly on geometric probability than the physicallaws of radiative transfer. As input, the model requiresspecifying the size, shape, and density of the cones, thesize of the pixel, the angle of illumination, and the relativebrightness of the cones and their background under con-ditions of both shadowing and direct illumination. Theoutput is the average brightness of a pixel and its variance.In inversion, the inputs are the pixel brightness values,along with parameters describing the angle of illuminationand the reference brightness values of the illuminated andshadowed cones and background. The solution estimatesthe mean height, apex angle, and density of the cones.

This paper fully describes our geometric-optical canopymodel, including as well some general derivations of geo-metric probability that are required along the way. It alsosummarizes our experience in applying the model to realimages of forests and testing its invertibility in two forestsstands for which the driving parameters of size, shape,and spacing are known. It concludes by analyzing the im-pact on the inversion procedure of some of the assump-tions that are made in constructing the model and com-menting on the applicability of our methods to dense forestcanopies and fine-resolution imagery.

II. PREVIOUS WORKA. Mathematical Plant Canopy Modeling

In the past decade, many mathematical plant canopymodels have been devised. Space does not permit a fullreview of these models here; instead, the reader is re-ferred to Smith and Ransom [1], Smith [2], and Strahler

0196-2892/85/0900-0705$01.00 © 1985 IEEE

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et al. [3], as well as other papers in this issue. A numberof models, however, have been developed that utilize ageometric-optical approach, and are thus relevant to thepresent discussion. Examples are the geometric-opticalmodels developed by Richardson et al. [4] and Jacksonet al. [5] for row crops. In addition, several models com-bine both radiative transfer and geometric optics ap-proaches. These include the extended versions of the one-dimensional Suits model [6] for row crops developed byVerhoef and Bunnik [7] as well as Suits [8], and the three-dimensional reflectance model of Norman and Welles [9]based on a rectangular array of ellipsoids simulating cropplants.

1) Three-Dimensional Models: The geometric-opticalmodel for a conifer forest canopy presented here relies onthe three-dimensional nature of the forest canopy. Only afew models have been developed for the reflectance ofthree-dimensional surfaces. Kimes and Kirchner [10] de-veloped a general modeling framework for a heteroge-neous scene that utilizes a three-dimensional radiativetransfer model. Colwell's three-dimensional model ofdesert vegetation [11] utilized bare soil, erect stems, andtheir shadows as components to determine the coverageof plants by observing Landsat reflectances at two differ-ent sun angles. Kimes [12] developed a thernmal model forrow crops as repeating extended rectangular solids withthree geometric parameters-width, height, and spacingbetween rows. By measuring thermal emission at differentviewing angles, Kimes inverted his model to obtain cropgeometry.The modeling effort that is perhaps most directly rele-

vant to our three-dimensional canopy model is that of Eg-bert [13], [14], who modeled optical bidirectional reflec-tance from shadowing parameters of surface projectionsor perturbations. More recently, Otterman [15], [16] hasdeveloped a model for a surface covered with vertical cy-lindrical perturbations. Like Egbert's model, Otterman'smodel assumes that the protrusions are small and numer-ous. Neither model is directly applicable to open forestcanopies, in which the perturbations (trees) can occur inlow densities and/or are relatively large with respect to thesize of the resolution cell.

2) Scene Component Models: Another feature of ourgeometric-optical canopy model is that it treats total re-flectance as a linear composite of scene components asweighted by their relative areas within the scene. Heimesand Smith [17] used this approach to investigate the var-iability in reflectance of two forest sites in mountainousterrain. By explicitly modeling the contribution of ashadow component, their study recognized the importanceof shadows in determining the spectral response of a three-dimensional scene.

B. Height and Spacing Functions for Conifer Stands1) Spacing Functions: The conifer forest canopy model

presented here requires specifying the height and spacingfunctions for trees within the stand. The spatial pattern ofplants has been a topic of theoretical interest to ecologists

and botanists for many years [18], [19]. Nearly all suchstudies have relied in a Poisson model-that all plant lo-cations are equilikely and independent. Although our pre-vious work showed that a two-parameter Neyman type Amodel [20] better fit counts of ponderosa/Jeifrey pines in80-m quadrats, our field work in the specific target standsused for field verification has shown a reasonable fit to thePoisson model for intermediate and dense stands at pixel(quadrat) sizes of 10-30 m.

2) Height Distribution: A normal distribution is per-haps the simplest choice to parameterize a random vari-able. The simple normal model, however, is not adequateto explain the variation of tree heights in forest stands.Instead, relatively simple but effective model, the lognor-mal distribution, has been widely applied [21], [22]. Ourfield work has also shown a good fit to a lognormal modelin most of the test stands we have examined. Accordingly,a lognormal height distribution is used in our model de-velopment.

III. VARIABLES AND ASSUMPTIONS USED IN THIS PAPER

A. ContextUnlike many of the canopy methods described earlier,

our model was developed specifically within the contextof the remotely sensed digital image. In this context, themeasurements of exiting radiation available to calibrate,verify, or invert a model are obtained from a digital imagesuch as that produced by Landsat's multispectral scanner(MSS) or thematic mapper (TM) instruments. The indi-vidual pixels of the image from an area of homogeneousforest canopy can thus be taken as replicate measurementsof reflectance. The digital-image context also implies thatthe scene is illuminated at a constant angle that is knowna priori; a further assumption is that the sensor will benadir or nearly nadir looking. Atmospheric effects are alsoneglected, in that the signatures needed to characterizethe components within the scene are assumed to be sep-arable, distinct, and constant in the face of differential ab-sorption, backscattering, etc. In addition, our modelingeffort will be restricted to flat sites. This restriction meansthat we can ignore changes in geometry that occur withslope, since trees grow upright no matter what the incli-nation of the ground surface. Some of these assumptionsor restrictions will be relaxed later; however, they providea context that conforms well to Earth applications of re-mote sensing.

B. AssumptionsThe fundamental assumption underlying our geometric-

optical model is that conifer forest stands can be modeledgeometrically as arrays of cones casting shadows on a con-trasting background. The cones are of uniform shape, pa-rameterized by a single constant-the apex angle of thecone-although this assumption will be relaxed somewhatlater. The heights of the cones are lognormally distrib-uted. Although the mean height is not known, we willassume that the coefficient of variation (ratio of standard

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LI AND STRAHLER: GEOMETRIC-OPTICAL MODELING OF A FOREST CANOPY

deviation to mean) for the log height values has been de-termined. We will further assume that trees (cones) arelocated randomly within and between pixels-that is, thecounts of trees from pixel to pixel varies as a Poisson func-tion, depending on the size of the pixel and density of thetrees. In general, overlapping of crowns and shadows ispermitted. A final assumption, discussed in more detailearlier, is that the reflectance of a pixel can be modeled asa sum of the reflectances of its individual scene compo-nents as weighted by their respective areas within thepixel. Taken together, these assumptions imply that pixel-to-pixel variance in reflectance will arise from threesources: 1) variation in the number of trees (cones) frompixel to pixel; 2) variation in the size (height) of trees bothwithin and between pixels; and 3) chance variation inoverlapping of crowns and shadows within the pixel.

C. Variables and Notation1) Variables Associated with Tree Crowns:

a One-half of the crown apex angle. Assumed con-stant within the stand. This assumption is relaxedlater.

r Radius of crown as cone at the base. Lognormallydistributed.

h Crown height. Lognormally distributed. Since a isconstant, h = r cot a.

Ch Coefficient of variation (ratio of standard deviationto mean) for heights.

Cr Coefficient of variation of radius. If a is fixed, thenCh = Cr since r is a linear function of h.

2) Variables Associated with a Pixel:

A Pixel size. Usually taken as having a unit area.n Number of trees (cones) in a pixel. Distributed as

a Poisson; independent of other variables.R2 Average of squared radii within the pixel. That is

In

R2=- Erin i=i

i=1, * , n.

m Ratio of sum of squared cone radii to area ofpixel. This is

nR2 n

M = A~ ZnEr/A.iA

Dimensionless.

A problem arises with the boundary of the pixel-whathappens to a portion of the crown or its shadow that passesout of the pixel? Here we assume that the pixel will bereplicated in all directions (i.e., will be surrounded withitself), and thus any excluded area on one side of the pixelwill be included on the opposite side. This assumptionwill lead to a model that underestimates within-pixel var-iance slightly, but unless the pixel size is close to the treesize, the effect should not be large.

3) Variables Associated with the Timber Stand:

N Mean of n for all pixels. For the fully randommodel, this is the value of the Poisson param-eter.

Cd Dispersion coefficient (variance-to-mean ratio) ofn. That is, Cd= V(n)IN. If n is distributed asa Poisson function, Cd = 1. If not, Cd willdepend on the pixel size A. For the clumped orpatchy distributions that characterize largequadrats in natural forests, Cd will increasewith A.

H Population mean of h.E(r) Population mean of r.V(r) Population variance of r. V(r) = C2 (E(r))2.E(r2) Population mean of r2.V(r2) Population variance of r2.

If r is lognormally distributed, then r2 is also lognormallydistributed. We can then show from the definitions of Eand V that

E(r2) =( + C2) E(r)2and

V(r2) = w[E(r )]where

w = (1 + r)4 - .

R2RV(R2)

Mean value of R2 for all pixels. That is, E(R2).The square root of R2, i.e., E(R2).Variance of R8.

If n is a constant and r is randomly distributed in the spa-tial domain, then R2 E(r2). Also, R2 is a sample mean,and thus V(R2) = V(r2)/n.M Mean of m for all pixels in the stand.V(m) Variance of m.

IV. NONOVERLAPPING VARIANCE-DEPENDENT MODELAs the first stage of our modeling efforts, we developed

a simple mathematical model that applies to sparsely for-ested stands when overlapping effects between coniferouscrowns and/or shadows can be neglected [231, [20]. Thismodel is invertible, thus allowing the direct calculation ofheight and spacing of trees from remotely sensed reflec-tance values. Inversion of the model, however, requirescalculation of interpixel variance over a timber stand.Since the model requires interpixel variance to be known,we refer to it as the "variance-dependent" model.

A. Geometry of ModelFig. 1 shows the geometry of a cone illuminated at a

zenith angle 0. The apex angle of the cone is 2 a. If 0 >a, a shadow will be cast. The right-hand diagram in thefigure shows the projection of the cone and shadow ontothe horizontal plane (i.e., the cone is "flattened"). Theangle -y identifies the portion of the cone illuminated be-yond the cone half. It is relatively easy to show that

707


\ I f

I0\

h

Side View

I,

Top View

Fig. 1. Geometry of an illuminated cone and its shadow.

-= sin-' (tan a/tan 0).

The projection of the cone and shadow consists of threecomponents: illuminated crown, shadowed crown, andshadowed background. Some simple geometric analy-sis reveals that the illuminated area of the "flat cone"(i.e., the area of green canopy projected to the sensor) is(ir/2 + -y)r2; the shadowed area of the flat cone is(wr/2 - y) r2; and the shadow on the ground has area (coty- 7r/2 + -y) r2. The summation of these areas is (cot y +

ay + r/2) r2. We shall denote the quantity (cot y + y +7r/2) as r; it can be thought of as a dimensionless geo-metric form parameter associated with the cone and itsshadow that is dependent on the illumination angle and theapex angle of the cone.

B. Reflectance of an Individual PixelAs stated earlier, we model the reflectance of the pixel

as an area-weighted sum of the reflectances of the fourspectral scene components. New terms will need to bedefined.

1) Reflectance Vectors: Multispectral reflectance vec-tors. These can also be thought of as points in multidi-mensional feature space.

G Reflectance vector for a unit area of illuminatedbackground (constant).

C Reflectance of a unit area of illuminated crown(constant).

Z Reflectance of a unit area of shadowed background(constant).

T Reflectance of a unit area of shadowed crown (con-stant).

S Reflectance of a pixel. Variable; depends on numberand size of cones in pixel.

2) Areas and Proportions: Variables describing areasor proportions for scene components.Ag Area of illuminated background within the pixel.

Quantity (A - Ag) is termed "covered area" intext later.

AC Area of illuminated crown within a pixel.Az Area of shadowed background within a pixel.A, Area of shadowed crown within a pixel.Kg = Ag/A. Proportion of pixel not covered by crown

or shadow.Kc = A,/(A - Ag). Proportion of area covered by

crown and shadow that is in illuminated crown.K, = A,/(A - Ag). Proportion of covered area in shad-

owed crown.Kz = Az/(A - Ag). Proportion of covered area in shad-

owed background.

3) Geometric Relationships: From the geometry of thecone model, we have the following simple relations:

(A - A) E r72- Amr

(A - Ag) = A + At + AzKg = 1 - mr

Kc - (r/2 + y)/]PKz (r - r)/r

Kt, (=@2 - y)Ir1 = Kc + K, + K,.

4) Modeling theof a pixel can thenfour components.

Reflectance: The average reflectancebe written as a linear combination of

S - (Ag ' G + A, - C + Az * Z + At, T)IA.

Substituting expressions from above into this equationS =Kg 'G + (1 Kg)

' (Kc - C + Kz - Z + Kt, T). (1)Since K, K, and K, sum to one, the expression (Kc -

C + K4 * Z + K, - T) represents a point in multispectralfeature space lying within a triangle with vertices at C, Z,and T. Its position is dependent on K, Kz, and K, whichin turn are simple functions of the scene's geometry (apexangle a and illumination angle 0). We will refer to thispoint as X; it can be thought of as the average reflectanceof a cone and its associated shadow. The only variable inthe right side of (1) is thus Kg, which is a linear functionof m. When m varies, S will vary along a straight lineconnecting points G and X.

Substituting the geometric expressions earlier for K, K,and K, into (1) yields

S = G-GmF + Xmr.

Rearranging, we have

mr(G - X) = (G -S). (2)In the last expression, G - S and G - X are vector dif-ferences; however, G - S lies on the line G - X andtherefore the equation is actually scalar. Using the nota-tion GS to indicate the length of the vector connecting Gand S, we have

IGSrFIGX (3)

Although there is thus in theory a unique solution for m,noise will always be present in S, ae, and the component

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signatures that determine X. Therefore, the position of Swill deviate from the line segment GX. Under these con-ditions, (2) is no longer a scalar relationship but a set oflinear equations in which m is overdetermined.One solution to this problem is to use a maximum like-

lihood approach for the best fit of m to the reflectancesignatures [24]. The approach we use here, however, se-lects a projection in feature space that maximizes the sig-nal-to-noise ratio. In the absence of noise, (1) becomes ascalar equation in that projection and thus possesses anexact solution. Note by taking the partial derivative of (2)for S, G, and C that the error in m will be proportional to1/j GX. Thus a projection for which the line segment GXhas the greatest length will have the least sensitivity toerror. In other words, we choose the band combination inwhich the background is spectrally most different from theaverage reflectance of the cone and its shadow.

C. Inverting the Model using the Variance ofmAssume that a timber stand consists of K pixels, i =

1, * , K. From (2), we can obtain a value of m for eachpixel. Then, the values of m will have a mean and variancewithin the timber stand

I K

M = - E niR2;

IK_VW(=)-K E (niP2-M)2Let us now assume that height (and thus r) is independentof density. Thus expressions for the mean and variance ofindependent products will apply

M = E(nR2) = E(n) - E(R2) = NR2

Thus given sample estimates of the mean and varianceof M determined from the reflectances of pixels in thestand, we can solve for R2, and then for N, yielding theaverage size and density of trees in the stand.The assumption underlying the use of the sample vari-

ance of r2 as V(R2) is that each pixel is an independentsample of values of r2.Other approximations can be alsoapplied to (5). For example, if the interpixel variation ofr is more significant than intrapixel variation, we mayuse V(R2) directly as an approximation of V(r2). Then (8)becomes

V(m) = (1 + w) MR2 + wM2

and we obtain

R2 V(m) wM2(1 + w) M

Also, if the dispersion coefficient of n is significantly dif-ferent from 1, we may use V(n) = NCd. Then (9) becomes

2 ((Cd + W)2 M2 + 4V(m) wCd)"2-(Cd + w) M2wvCd

The choices basically depend upon what a priori infor-mation we have.No matter which approximation is used, however, the

underlying expressions (4) and (5) will always yield rela-tionships such that for a given M, the larger the V(m), thelarger is R2. To make this point clearer, let us further applythe approximation formula I + x 1 + x/2 to (9). Weobtain

R2 V(m)(1 + w) M(4)

and

V(m) = V(nR2) = (R2)2 V(n) + N2V(R2) + V(n) V(R2).(5)

Since n is a Poisson function

V(n) = N. (6)

Further

V(R2)= V(r2)n V(r2)IN= w(E(r2))21N. (7)

Substituting (6) and (7) into (5), we finally obtain

V(m) (N + wN + w)(R2)2=(M + wM + wR2) R2. (8)

In order to derive (8), R2 and V(R2), which are parametricterms, are approximated using the sample mean and var-iance of r2. Small errors are introduced by these approx-imations, but they may be ignored for our purposes. Solv-ing (8) for R2, we obtain

((1 + w)2 M2 + 4V(m) w)12 -(1 + w) M2w

(10)

Although (10) may not be sufficiently accurate to applywhen V(m) is fairly small, the expression shows that for agiven m, the larger is R2, the higher will be the varianceof m-and since S is a linear function of m, the "rougher"the surface will appear. This is obviously true for bothmanual interpretation of aerial photographs and digitalprocessing of remotely sensed imagery.Although the variance-dependent model has performed

well in earlier work [20], it is only applicable to cases oflow and moderate stocking. Thus we turned to a fullerconsideration of how the reflectance of pixels is influencedby the variation in overlapping of objects within them.

V. OVERLAPPING MODELThe problem of two-dimensional objects overlapping in

a spatial field has been examined by both geographers andmathematicians. There are two ways to deal with thisproblem. The first is to assume that the centers of objectsare randomly distributed at an average density over an in-finitely large region. In 1971, Getis and Jackson [25] ap-plied a Poisson model for the mean area polluted by ran-domly scattered pollution sources. In their model, a pointis not polluted if it is not contained within the area asso-ciated with a pollution source. Thus the mean area not

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polluted by any of N sources is

Ag Ae N(a/A) (11)

where A is the overall area, Ag is the unpolluted area, anda is the area polluted by one source. More recently, Serra[26] proves a similar relation and a formula for the prob-ability that neither of two points is covered by the objectsfor any multidimensional Poisson process locating pointsor volumes. Although this model predicts the mean, it doesnot account for the pixel-to-pixel variance.Another approach is to assume a specific number of ob-

jects are distributed within a specific area, and then toassume this number varies as a Poisson or other function.In 1947, Garwood [27] proved that the mean proportionof undamaged area in a building or factory complex of sizeA is

E(Ag/A) - (1 - a/B)k (12)

where B is a finite target area which contains A; k is thenumber of bombs which fall randomly within B; and a isthe area damaged by a single bomb. He also obtained thesecond order moment of Ag/A

2C (B B (,Y k

E((AgIA)2) = 2a+ q(x, Y)) dx dylA (13)A

where q(x, y) is the common area which is damaged bytwo bombs falling at (0, 0) and (x, y). Garwood's resultis directly relevant to the problem of cones with shadowsfalling randomly in a square or rectangular pixel.

In Garwood's formulation, B is required to eliminatedifficulties at the boundary. An alternative method is toassume, as stated previously, that the pixel is replicated inall directions. This assumption also allows the use of Fou-rier transforms and theorems. We have obtained expres-sions for the first- and second-order moments of As/A [28]that are quite similar to the results of Garwood

E(Ag/A) = (1 - a/A)k

E((Ag/A)2) =

A

A- 2a4+ q(x,y)V A dxb dy/A

to yield

V(Kg) =

A

A- 2 a + q(x, Y)VA

dx dylA - (1 - a/A)2k.

(14)

(15)

Thus (14) and (15) present relatively simple formulas forthe mean and variance of Ag/A. These expressions con-

form with Ailam's more general result for the moments ofcoverage and coverage spaces of randomly distributed ob-jects [29]. The formulas are also validated by the goodagreement observed between calculated values and resultssimulated by Monte Carlo modeling, as discussed in a latersection. The formula for the mean agrees with the Getis-Jackson model (11) since e Na/A is a very good approxi-

mation of (1 - a/A)N when N is large and a/A is small.(This can be seen easily by expanding both expressionsinto a Taylor series.) When alA is somewhat larger, oursimulations show better agreement with (1 - a/A)N bute-Na/A will still be adequate for the purposes of this paper.Our general expression for the variance (15) also conformswith the specific result of Solomon [30], who employed atheorem of Robbins (1944, cited in [30]) to derive a re-cursive integral equation for calculating the mean and var-iance of coverage of random caps on a sphere. Other au-thors [31] have also applied Robbins' theorem to differentspecific coverage problems with similar results.We should emphasize that (15) is a very general result.

Because (1 - a/A)2 can be expressed in terms of q(x, y),the variance in Ag is a simple function of the autocorrela-tion function q(x, y) and the number of objects. Although(15) refers to variance in the uncovered area, it also de-scribes the variance of the covered area as well since thetwo areas are complementary. Thus it will apply to an ob-ject of any shape, as long as we can describe its autocor-relation function. When the sizes and shapes of the objectswithin the pixel are not the same, (14) and (15) become

N

E(K,) = *I1 (1 - ai/A)I=

and

V(Kg) i A, (A- 2a + qi(x Y))

N. dx dy/A - Y1 (1 -aiA)2.

i =1(17)

In the first expression, e- ai/A can be used as a good ap-

proximation of E(Kg) if the ai are small with respect to A.Thus E(Kg) can easily be calculated for mixtures of simpleshapes. In the second expression, the variance is a func-tion of the products of the individual autocorrelation func-tions for each shape. Note that the cross correlation func-tion for one shape with another need not be determined.In practice, this means that such scenes as mixtures ofconical conifers and hemispherical deciduous trees can beexplicitly modeled. In fact, it should be relatively easy tomodel any scene that can be described as a mixture ofrandomly distributed simple shapes. Thus our results havewide implications beyond the modeling of forest canopyreflectance.

VI. MONTE CARLO SIMULATIONIn order to confirm the mathematical proofs of the pre-

ceding section, as well as to explore various other aspectsof geometric modeling of forests, a Monte Carlo computermodel was prepared to simulate pixels composed of cone-

shaped tree crowns. In the simulatioin, the pixel is mod-eled as a two-dimensional array of many small subpixels,each of which is assigned a cover type (e.g., bright crown,

shadowed crown, bright background, shadowed back-ground). Much of our work used Landsat-sized pixels,

(16)

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Fig. 2. Simulated image of cones on a plane.

each 80 m by 80 m, with 1-m subpixels. SPOT-sized pixelswere also modeled; these were 10 m by 10 m, with a sub-pixel size of approximately m.The computer program carrying out the simulation is

exercised in several steps. First, tree counts for as manypixels as are to be simulated are selected from either ofthree distributions: uniform (i.e., constant), Poisson, orNeyman Type A. Given that the number of trees to belocated within the pixel has thus been determined, the nextstep is to specify the position of each. To locate each tree,a subpixel is chosen at random as its center. If a pure Pois-son model has been chosen, trees may overlap fully andno restrictions are placed on the location of a center. How-ever, we have also implemented a "hard-core" model inwhich the center of a tree may not be located within thecrown of another tree. In this case, the crowns may touchand intersect only to a limited degree. The purpose of thisoption is to simulate situations in which intertree compe-tition may be important in the location process.

Next, the program selects the height of each tree froma lognormal distribution using a mean and variance thatare specified as input parameters. Since the apex angle ofthe cone and the illumination angle relative to the zenithare also specified, the program can then compute the ge-

ometry of the cone and its shadow and assign subpixels tosunlit and shadowed crown and background. In addition,the program also calculates the height of the center of eachsubpixel. This information is used to assign the appropri-ate cover type to each subpixel when shadows fall oncrowns and crowns intersect. The height matrix is alsoused when a nonnadir viewing angle is simulated. In thiscase, the program calculates whether or not each subpixelis obscured. When the specified number of trees have beenplaced in the pixel, the program counts the subpixels ineach type of cover and then goes on to the next simulatedpixel. Statistics are accumulated as the program runs andsummaries and tabulations are output at the end.

Fig. 2 presents an image of a simulated pixel generatedby the Monte Carlo modeling program. The image is pro-duced by assuming Lambertian reflectance from thecurved surface of each cone. This 80 m by 80 m pixel uses

100 20° 40060°800Fig. 3. Bidirectional reflectance of a simulated pixel. Polar plot; angle 09

indicates azimuth angle between illumination source and viewing plane;length p indicates reflectance. The solar zenith angle is 300. Curves arefor viewing angles in 10° increments.

a very fine subpixel mesh of approximately 0.3 m to yielda high-quality continuous-tone image. For this simulation,the solar zenith angle (0) is set at 30°, the apex angle ofthe cones (2 ai) is 200, the trees are uniformly 20 m high,and the pixel contains 80 cones. Viewing is from thezenith.

A. Modeling Bidirectional ReflectanceThe bidirectional reflectance distribution function

(BRDF) for this pixel was also calculated using the ge-ometry of the scene and is shown in Fig. 3. Both the coneand background are assumed to be Lambertian surfaceswith albedos for sunlit background, sunlit crown, andshadow of 0.7, 0.3, and 0.0, respectively. Fig. 3 presentsthe bidirectional reflectance plotted in a polar coordinatesystem in which s° indicates the difference in azimuth an-gle betwen sun and sensor, and p indicates the correspond-ing reflectance. The family of curves shows different sen-sor zenith angles, indicated in units of 100.The influence of the three-dimensional geometry on re-

flectance is clearly visible in the figure. At small azimuthdifferences, sun and sensor lie in nearly the same plane.The shadows can barely be seen and thus the scene ap-pears brightest. As the viewing angle comes around andthe sensor looks into the sun, shadowing becomes muchmore important and the reflectance decreases. The geo-metric effects also cause reflectance to decrease as theviewing angle becomes increasingly oblique. At such"low" viewing angles, the light background becomes in-creasingly obscured and thus the scene appears darker. Theminor irregularities that produce spikes on the curves ap-pear to be due to the chance placement of cones in thisparticular pixel (Fig. 2). For example, the spike at s =1700 is obviously explained by a gap in that direction,which reveals more sunlit background and produces agreater reflectance.The shape of this reflectance function generally resem-

bles that of Kriebel [32], and also exhibits features similarto these observed and modeled by Kimes et al. [40] forsparse crop canopies. These results emphasize the impor-tance of the three-dimensional geometry of the canopy indetermining its BRDF.

B. Simulations ofLandsat-sized PixelsThe Monte Carlo model was first exercised to simulate

80-m pixels. The driving parameters for this simulation

711


1.1.8

I.6

0

04-

.2

0 .5 1 1.5 ;Coverage Index

Fig. 4. Calculated and simulated values for mean of 1 Kg (area covered

by cones and shadows) 80-rn pixel.

C.'

~0x

0

002)

0

~ ~ ~ ~ ~ smuae~ ~ ~ cac lae

0 .5 1.0 1.5 2.0Coverage Index

Fig. 5. Calculated and simulated values for the variance of 1 Kg 80-mpixel.

were the same as those used in the BRDF simulation ear-

lier, except that a coarser 1-m subpixel size was used. Thecount of trees per pixel was varied in steps of 10 from 1

to 200; with cones of 20-m height, this range spans can-

opy coverages from about 5 to 70 percent.Fig. 4 plots 1 - Kg, the area covered by crowns and

shadows, as a function of mr, which is the proportion ofarea covered by cones and shadows without overlapping.Also plotted are values calculated using e-mP; the

curves are nearly coincident. The simulated and calcu-lated variance of Kg (which is equal to the variance of1 -Kg) is presented in Fig. 5. The two curves show rea-

sonable agreement, although the simulated values seem

slightly higher than the calculated values and depart froma smooth function due to random variation. The formereffect may well be due to the fact that the same randomnumber seed was used for each choice of mr, which wouldcause some serial correlation in the results.

Fig. 6 presents the means of Kc, Kz, and K, as a functionof mr. As the tree count increases, Kz, the proportion ofthe covered area in shadowed background, decreases. Thiseffect arises because progressively more of the shadowsfall on surrounding trees as the density of treesincreases. For the same reason, K, the proportion of shad-owed crown, increases. With K, decreasing and K, in-creasing, Kc, the proportion of the covered area remainingin sunlit crown, remains nearly constant. Fig. 7 presents

Fig. 6. Simulated means of Kz, K, and K, 80-m pixel.

2 .15 -

0

,_ VlkC

a

a(kc)

0 .5 1.0 1.5 2.0Coverage index

Fig. 7. Simulated variances of K, KC, and K1 80-m pixel.

the variance of these three quantities as observed by sim-ulation.

C. Simulations of SPOT-sized PixelsBecause both Landsat data and an aircraft multispec-

tral-scanner simulation of SPOT satellite data (discussedin a following section) were available for testing the model,SPOT-sized pixels were also simulated. For these runs, a

1-ft square subpixel size was used in a pixel measuring 33ft by 33 ft-thus approximatng a 10-m pixel. Althoughthis subpixel grid, composed of 1089 cells, is coarser thanthe 6400-cell grid used in the Landsat pixel simulation,note that the objects are proportionately much larger. Thusthese results should be comparable.To match the characteristics of our SPOT simulator data

and ground observations (described in more detail in a

later section), we used a solar zenith angle 0 - 260 anda cone apex half-angle of a = 8.480. Simulated and cal-

culated values of means and variances for 1 - Kg areshown in Figs. 8 and 9. Again, the simulated and calcu-lated curves show good agreement. The simulated meansof Kc, Kz, and K, are shown in Fig. 10. They show thesame trends as Fig. 6. Note that they are presented as a

function of m -r (rather than mr ), which is the ratio of thetree base area to the area of the pixel. The simulated var-

iances for Kg K, K1, and K, are shown in Fig. 11.Because K, the proportion of sunlit crown within the

total covered area, remains relatively constant, its vari-ance is much smaller than that of Kg. Since K, + K, =

1 -K, the variance of K, + K, is also small. Both K: and

= simulateD- -calculated

r - x712


.84,0

(0@ .6

X .40

.2r

4._h

0CL0.

C .1

00

C)

(U

0 2 4 6 -8Coverage Index

Fig. 8. Calculated and simulated values for mean of I - Kg 10-m pixel.

0)4,01(c

00

S

cI

Fig. 9. Calculated and simulated values for the variance of 1 -Kg 10-mpixel.

.6

0

°.4 kc

0

0

e.2-i

0 2 4

Crown base area index

Fig. 10. Simulated means of Kz, Kc, and K, 10-m pixel.

K, are shadowed cover types and will have similar reflec-tances; thus we will be able to assume in later modelingthat the overlap variances of K, K,and K, can be ignored.

VII. MODELING SPECTRAL REFLECTANCEOF THE CANOPY

Given the spectral reflectances of the four scene com-ponents, it is possible to model the spectral reflectance ofthe whole pixel as a function of the height, density, andshape of the cones it contains. For this effort, we will con-sider the tree crown to be a medium-bright green reflector.

0 2 - 4Crown base area index

Fig. 11. Simulated variances of Kz, Kc, and K, 10-m pixel.

(00

cC4)a)C,

Brightness

Fig. 12. Idealized plot of brightness-greenness spectral space with com-ponent signatures and diagrammatic coverage trajectory.

To provide a strong contrast, we will assume that thebackground is snow-a bright uniform spectral reflectorin the visible and near infrared. Shadowed crown andshadowed background will be darker, but spectrally sim-ilar to their directly illuminated counterparts.

For convenience, we will work in a two-dimensionalspectral space in which the reflectances are orthogonal anduncorrelated, corresponding approximately to a "green-ness-brightness" transform [33]. This linear transformwas derived from analysis of many Landsat images of ag-ricultural scenes, which showed that the four spectralbands of Landsat data could be decomposed into two or-thogonal axes of variation-one related to the density anddepth of the plant canopy (greenness) and the other relatedto the brightness -of the soil. From our viewpoint, thegreenness, brightness transform represents the best twoorthogonal axes separating tree crowns from background.

A. Coverage and Apex TrajectoriesFig. 12 presents an idealized plot of the four spectral

components in a greenness-brightness feature space. Thesnow background (G) has the greatest brightness value,but as a fairly uniform spectral reflector it has low green-ness. Illuminated crown (C) has the most intense green-ness, and is also somewhat bright. Shadowed crown (T)is the darkest component of all, but it still exhibits somegreenness. Shadowed background snow (Z) is lighter thanshadowed crown, but rates a very low greenness. A pixel

*1 -I--

713


without trees will be pure background, and thus will havereflectance G. As trees are added to the pixel, the reflec-ance of the pixel S will move along a line toward the pointX0, which represents the average reflectance of a singletree with its three spectral components. Overlapping,however, will occur, reducing the relative proportion ofshadowed background K, and increasing the relative pro-portion of shadowed crown K,. As a result, the path of thepixel as coverage increases will diverge away from X0.As tree density goes to the limit, all the background willbe obscured and K, will become zero. The limiting reflec-tance XO, will then lie somewhere along the line TC, withthe exact position depending on the apex angle of the coneand the angles of illumination.To model this situation mathematically, we first recon-

sider (1) for the proportions expressed as means

S (K&C + KzZ + KtT)(l - Kg) + KgG. (18)As before, the point X KCC + KgZ + Kt T will representthe average reflectance of a cone and its associatedshadow. Our analysis from the preceding sections, how-ever, has yielded expressions for the average proportions.First, we know that

Kg =e-MrThen, since KC is approximately a constant, and the meanarea occupied by the bases of the cones is A(l -e'),we can show that

Kz e-r(r- or)Iand

Kt1 [(ir /2 - y) + (r - 7)(1 - e-nvlr.By substituting these relationships into (18), the point Xbecomes a function of m and a. At m = 0

Xo = [(X12 + 7) C + (P12 - y) T + (r - -r)z/Prwhich is the average reflectance of a crown and shadowwith no overlapping. At m =o-

Xoo = T + (r/2 + 'y)(C - T)

which is the average reflectance of a shadowed crown withcomplete overlapping. Thus the 4"coverage trajectory" ofS (Fig. 12) with increasing m will be determined by theapex angle. If m is fixed and at is allowed to vary, an "apextrajectory" will result.

Fig. 13 presents families of coverage and apex trajec-tories for a reasonable range of values of m and a. (Thevalues of C, Z, T, and G on greenness-brightness axes arechosen from analysis of SPOT simulation data, discussedin a following section.) The trajectories are calculated ex-plicitly from (18). The results of our simulation trails mayalso be plotted as coverage trajectories to verify the cal-culations.

B. Relevance to Tasseled Cap TransformIt has long been known that the spectral reflectance pat-

tern of crop canopies in Landsat MSS bands is basically

Brightness Units

Fig. 13. Apex and coverage trajectories for m and a.

two-dimensional. The pattern can be represented by theso-called "tasseled cap" shape, in which the crop canopybecomes increasingly darker and greener as it progres-sively obscures the lighter soil beneath it. As the cropcanopy reaches maturity, its reflectance trajectory changesdirection, turning to become brighter, before losinggreenness as the crop scenesces. This same pattern hasalso been recognized for forests [34],and has been attrib-uted to the overlapping of shadows onto tree canopies [11].The geometric-optical model we derived earlier presentsa deterministic explicit description of this effect, at leastfor forests. It arises from the progressive obscuring of thesoil or understory background coupled with a reduction inshadowed area as high coverages are reached. Althoughthey are not explicitly modeled here, similar geometric ef-fects may well explain the "tasseled cap" shape for crops.

C. Numerical Solution of (18)Our expression for the greenness and brightness of a

pixel with overlapping (18) is essentially a set of two non-linear simultaneous equations with two unknowns, m anda. Although it is difficult to obtain a closed-form solutionfor these equations due to the nonlinear terms e-m and F,values of m and a can be found numerically without muchdifficulty. Because these values are estimated from themodel, we will refer to them as m' and ae'.

Since the signature of each pixel is a modeled as a linearfunction of the signatures of the four components it con-tains, the solution of (18) exists for any S inside thepolygon ZCTG (Fig. 12). When S falls outside of this pol-ygon due to noise or other error, this situation can be rec-ognized and the value can be omitted from further pro-cessing. There may also be some pixels with little or nocanopy cover. The reflectance of these pixels should thenbe G, in the ideal case. However, as a result of topographicvariation, shadowing from other pixels, variation in sur-face composition, etc., these pixels will present a rangeof reflectance patterns analogous to the "soil axis" ofKauth and Thomas [33]. Some of these values will fallinto the polygon ZTCG and (18) will give nonzero vales ofm' and a' as solutions for them. If such values are not

714


close to Z, the resulting m' will be very small, i.e., closeto a correct solution. If the value lies near the segmentZT, the resulting m' value may be large; however, the valueat will be very small. These types of errors can be easilyrecognized and such pixels can be discarded.When S values lie on or near the segment TC, the so-

lution of (18) will be unstable. In this case, both e-m ande-mr are near zero, and the sensitivity of m'to changes inS will be proportional to em. For this reason, the solutionof (18) will be increasingly subject to error as m becomeslarge. Thus there will probably be a practical upper boundon stand densities for which m' can be derived reliably.

Without noise or overlap variance, we could simply ap-

ply (3)-(8) to the m' values derived from (18) and invertthe model directly. Noise and overlap variance, however,exist; further, the solution of (18) is sensitive to noise atlarge m values. Under these conditions, the mean and es-

pecially the variance of m' will depart from modeled val-ues. In spite of these problems, it is still possible to invertthe model and obtain estimates of size and spacing for theobjects in the scene. To show this, let us describe the prob-lem as follows.

Using a geometric-optical approach, we have modeledhow the distributions of r2 and n produce a distribution ofS values within an image. Even though the procedure issomewhat complicated, the only unknown parameters are

E(r2) and N for the modeled procedure. We also have a

sufficient number of observations of S. The problem is thenwhether we can estimate the unknown parameters fromobservations of S. The problem is a typical one ofparametric point estimation, and many techniques, suchas maximum-likelihood estimation, minimum x2, leastsquares, etc., are available for its solution.The analysis of (18) in the preceding section showed

that noise in S values may prevent accurate inversion.Since some of the techniques cited earlier do not requirethat the observations cover the whole range of the distri-bution, some observations can be discarded if we knowthey are likely to contain error. In this application, theseare most likely to be values that produce large values fori'. An appropriate strategy is then to make the inversionof the model rely on only a limited range of the m distri-bution. This amounts to fitting a curve in an area of thecurve for which the data are most accurate.

VIII. FIELD TESTSA. Remotely Sensed Data and Field MeasurementsTo test the model on real data, we used 10-m simulated

SPOT imagery, produced by a modified Daedalus scanner

on a Lear jet aircraft as part of the 1983 SPOT simulationcampaign. The image was acquired on June 25, 1983.Fourbands of data were supplied: a 10-m panchromatic band(B4, 0.51-0.73 ,um nominal), and three spectral bands inthe green (B1, 0.50-0.59 ,um), red (B2, 0.61-0.68 ,um),and infrared (B3, 0.79-0.89 Mm) at 20-m resolution. Forour tests of model inversion, the multispectral bands were

expanded to 10-m resolution by replicating pixels. Alsoused was a Landsat-4 MSS image from June 11, 1983. The

original pixel values corresponding to a resolution cell sizeof 56 by 79 m were used. These data were processed usingprincipal components into transformations that resembledbrightness and greenness. The transformation for theSPOT data is

Brightness = 0.28B1 + 0.43B2 + 0.75B3 + 0.41B4

Greenness = 0.65B3 - 0.40B1 - 0.40B2 - 0.51B4.

The transformation for the Landsat data is

Brightness = 0.50B1 + 0.51B2 + 0.54B3 + 0.45B4.

Greenness = 0.36B3 + 0.64B4 - 0.50B1 - 0.46B2.

For coordinataed ground study, two test areas were se-lected in the Goosenest Ranger District of the KlamathNational Forest (northeastern California): a dense maturehigh-elevation red fir stand, and a more open mixed con-ifer stand, largely of ponderosa/Jeffrey pine with somewhite fir, at a lower elevation. Field measurements in eachplot were taken in July, 1983; they are described more fullyin [28].

Analysis of the field data showed the mixed conifer standto be composed of ponderosa and/or Jeffrey pine (78 per-cent) and white fir (21 percent), with a canopy coverageof about 30 percent. For this plot, the background under-story consisted of a mixture of grass, bare soil, and a fewshrubs.The red fir site was dominated by red fir (69 percent)

with white fir (26 percent) and pine (< 5 percent) com-prising the remaining components. Canopy coverage wasabout 80 percent. The understory was largely a litter layerof cones, needles, and branches at the time of the fieldwork, but was covered by snow at the time the imagerywas collected.

Further analysis of the data [28] showed that the treecounts within the pixels were fit well by a Poisson distri-bution at the small (10-m) pixel size. The dispersion coef-ficient Cd, however, increased with pixel size, requiringspecific correction at Landsat pixel size. Tree height wasobserved to fit a lognormal model well. A fuller descrip-tion and analysis of tree data are planned for a futurepublication.

B. Inversion ProcedureTo test the geometric-optical canopy model in the red

fir and mixed conifer stands, we used the following pro-cedure.

1 Using air photos, identify and delineate the two standson the imagery.

2 Pooling pixels from the two sites, carry out a prin-cipal components axis rotation and project the data ontothe first two axes. Figs. 14 and 15 plot the rotated SPOTsimulator data for the red fir and mixed conifer stands,respectively.

3 Determine the rotated signatures of the four spectralcomponents from the imagery by careful examination ofvalues for individual pixels. The apex and coverage tra-jectories shown in Fig. 13 are plotted using these spectral

715


100 200Brightness Units

30

TABLE IRESULTS OF MODEL INVERSION, 10-m SPOT SIMULATION

Red Fir Site Mixed Conifer SiteParameter Observed Calculated Observed Calculated

E(r) 5.7 ft 4.2 ft 5.2 ft 6.9 ftN 15.5 14.0 3.9 3.0H 60.5 ft 37.4 ft 36.3 ft 40.8 fta 8.5 6.4 9.3 9.6

TABLE 1IRESULTS OF MODEL INVERSION, 80-m MSS SIMULATION

Red Fir Site Mixed Conifer SiteParameter Observed Calculated Observed Calculated

E(r) 5.7 ft 4.4 ft 5.2 ft 7.0 ftN 725 4 93 181 192H 60.5 ft 44.5 ft 36.3 ft 48.0 fta 8.5 5.6 9.3 8.3D

Fig. 14. Plots of red fir pixels in brightness-greenness space (SPOT sim-ulation).

0 1

0 100 200 300Brightness Units

Fig. 15. Plots of mixed conifer pixels in brightness-greenness space (SPOTsimulation).

component signatures and are at the same scale as Figs.14 and 15. A direct comparison of Figs. 14 and 15 with 13shows that the red fir site has more overlapping and a

smaller mean apex angle than the mixed conifer site. Thisobservation conforms well with the field data collected.4 Use (18) to determine m' and a' for each pixel. (At

present, this is implemented through a table look-up pro-

cedure.) Thus we transform each pair of greenness andbrightness values to a pair of m' and ao' values. By pro-jecting these pairs to two one-dimensional distributions,we obtain a "projected m distribution" and a "'projecteda distribution." From the latter, we estimate the mean

apex angle a.

5 Begin an iterative procedure to find values of N andR2 that will generate an m distribution matching that ob-served. Select starting values for N and R2 to generate an

m distribution. To reduce computing time, use a normaldistribution with the mean and variance derived from (4)and (8). From the normal, generate a probability distri-bution p(m) for values of m.

6 For every value of mi generate a normal distributionp(Kg m) using values for the mean and variance derived

from (14) and (15). Also calculate values for Kz, Kt, andK; as discussed earlier, their variances can be ignored.

7 Now find the distribution of Kg as

p(Kg) = P(m) p(KgIm) dm.

From this, use (18) to generate an S distribution that there-fore characterizes the values chosen for N and R2.

8 Apply the lookup table based on (18) to the generatedS distribution, obtaining a "generated m distribution"which contains overlap variance. Calculate the meansquare error at a limited range of m' between the gener-ated and projected m distributions.

9 Repeat the four preceding steps using different pairsof N and R2. Use the pair of N and R2 values which yieldsthe least mean square error as the result.

10 Last, apply the relations E(r) = RI (1 + C2) andH = E(r) cot (a) to obtain mean height.

Since 80-m Landsat MSS data were also available forthe two sites, the procedure was repeated using the Land-sat imagery as well.

C. ResultsTable I presents the results for 10-m SPOT simulator

data and compares them to values as measured in the field.In inversion of the model, Cr is required; the value ob-tained by field measurement is used.A close examination reveals that the calculated results

do not reproduce the field measurements exactly, but differfrom them in some respects. For the red fir site, the cal-culated average radius and apex angle are somewhatsmaller than observed, and the height is significantlysmaller. The discrepancy in average height may be ex-plained in part by the field sampling procedures, whichoverestimated the mean height [28]. In spite of these dif-ferences, the contrast between the two sites is evident.The red fir site is much more densely covered than themixed conifer site, and includes trees that have narrowercrowns (i.e., red firs as compared to ponderosa/Jeffreypines). Thus the inverted model clearly reveals the majorstructural differences between the two sites.

Table II presents the results of model inversion for

0

"I

co

LOco

0la

0

, II o

Oj.

Ce

4f)._

e

coc 0

C,44)

c:

0

716

eq 1

0tDL_


TABLE IIISENSITIVITY ANALYSIS, RED FIR SITE

Signature E(r) N a

GCrZ 4.2 ft 14.0 37.4 ft 6.4G'=l.lG 4.18 ft 12.5 32.6 ft 7.3C=-0.8(C-T)+T 4.35 ft 12.0 43.8 ft 5.7T'=0.8(T-Z)+Z 4.75 ft 10.0 41.9 ft 6.5Observed 5.7 ft 1 15.5 60.5 ft 8.5

TABLE IVSENSITIVITY ANALYSIS, MIXED CONIFER SITESignature E(r) N H a

GCTZ 7.0 ft 3.0 40.8 ft 9.7G'=0.95G 6.8 ft 3.0 32.5 ft 11.9G'=1.05G 7.2 ft 3.0 48.0 ft 8.5Observed 5.2 ft 3.9 36.3 ft 9.3

80-m MSS data. Again, there are some differences be-tween observed and calculated results-the spacing pa-rameter for the red fir stand is significantly underesti-mated, for example-but the contrasts between the twosites emerge well.

Inversion of the model for 80-m data required an addi-tional piece of information-Cd. Since earlier studies ofspacing in similar conifer stands showed that this valuediffers from one at larger pixel sizes [35], we estimatedthis parameter from field data by regression of Cd valueswith pixel sizes in the range of 14.2 ft square to 56.7 ftsquare. Obviously, the 80-m Landsat pixel is larger, andthus our estimates may be subject to significant error fromthis source.

IX. DISCUSSION

A. Sensitivity to Compound Signatures

An assumption that underlies our inversion procedure isthat the signatures of the four spectral components areknown accurately. Without field radiometric measure-ments, however, we are forced to estimate the signaturesfrom the image itself Thus it is reasonable to examine thesensitivity of the inversion procedure to error in determi-nation of the component signatures.An analysis of (18) reveals that error in a component

signature will influence the signature of the pixel in directproportion to the area within the pixel having that signa-ture and in inverse proportion to the distances between thesignature in error and the others. The first source of errorwill depend on the cover and illumination geometry forthe shape; the second will depend on the spectral separ-

ability of the four components.As a practical test of sensitivity, we modified the sig-

natures for the SPOT data and reinverted the model for thered fir and mixed conifer sites. Due to limits in computertime, it was not possible to carry out a systematic sensi-tivity test of each parameter. The results are summarizedin Tables III and IV. In general, the results of the inversionare more sensitive for the mixed conifer site than for thered fir site. This effect arises because the shrub-coveredbackground at the mixed conifer site is less separable fromtrees and shadows than in the snow-covered backgroundof the red fir site. The average background proportion (0.7)

for the mixed conifer site is also larger, multiplying thiseffect.The tables show the effects of changing the separability

between some of the signatures by as much as 20 percent.The model responds with varying estimates of size, spac-ing, or apex angle by somewhat lower percentages. Yet inspite of such changes, the primary differences betweenthe two sites remain obvious. Thus we conclude that themodel is not overly sensitive to error in the determinationof component signatures.

B. Field CalibrationThe field calibration required to invert the canopy model

includes determining Cr for SPOT data, and in addition,Cd for MSS. The inversion method used here could alsobe applied to the solution of a four-parameter model thatincludes Cr and Cd in addition to E(r) and N as unknowns(e.g., methods used by Goel [36]-[39]). This approach ispossible because the m distribution is governed by thesefour parameters and it should be possible to estimate themfrom the observed m values. This, however, may be verydifficult due to the existence of noise as well as the limi-tations on the number of pixels that can be sampled froma homogeneous stand. On the other hand, it may be pos-sible to estimate Cd from multiresolution imagery, sinceCd is proportional to the size of the pixel [35]. This ap-proach remains to be developed. Cr, however, will prob-ably have to be established through field calibration, al-though it may be that sufficient forestry data will exist toestimate Cr at many locations without ground measure-ments.

C. ResolutionFor inversion of the canopy reflectance model, the

10-m resolution of the simulated SPOT data is superior tothe 80-m resolution of Landsat MSS. Since V(m) is in-versely proportional to the pixel size, V(m) will be -largerat smaller pixel sizes for a given m value. With V(m) larger,the signal-to-noise ratio will be enhanced, leading to moreaccurate inversion. Fine resolution will also provide morepixels, thus further increasing accuracies by increasingsample size. Accuracy should be further enhanced by thegreater level of quantization of the SPOT data. As the re-sult, the least-square error of fitting projected and gener-ated m distributions to SPOT data will be much smallerthan that of MSS data, and hence the estimated parame-ters of size and density will have smaller associated con-fidence intervals. Yet another advantage of the 10-m res-olution is that we may safely assume the Poisson spacingmodel [35], and therefore Cd will be unity.

In light of these considerations, it was surprising to notethat using the 80-m MSS data in inversion yielded esti-mates of size, height, spacing, and apex angle that werefairly close to the values obtained by field measurement(Table II). We view this result with caution, however, sinceit relies on a value for Cd that is obtained by projecting alinear regression well beyond the points used to fit the line.

717


D. Covariance Between Size and Spacing

In our earlier work with the nonoverlapping variance-dependent model [25], we noted that the assumption ofindependence between height and spacing did not hold forMSS pixels-rather, stands of large trees tended to be lessdense than stands of smaller trees. Thus we used an em-

pirical correction for this covariance in our inversion pro-

cedure that was calibrated by measurements of tree sizeand spacing in MSS-sized pixels using air photos. It ispossible to go beyond a simple empirical correction andmodel the covariance explicitly under the assumption thattree counts in pixels are distributed as either normal or

Poisson functions. Although space limitations precludepresenting a full derivation here, it is possible to show that

cov (n, RJ2) = -CdE(r2) (19)

under the assumption that the area of space nearest to any

tree is directly proportional to its size (as R2). Althoughthe field data we collected in the red fir and mixed conifersites which could be used to verify this relationship are

limited, they suggest that the covariance is much smallerthan (19) indicates for small pixel sizes (12-m or less).Thus we have ignored the effects of this covariance in thederivation of our model. At larger pixel sizes, and forsparse forest stands, it may be important to correct for thiseffect explicitly.

E. Apex Angle

The mathematical derivations of the geometric-opticalmodel assume that the apex angle of the cones is a con-

stant. Field data, however, show that measured apex an-

gles may have a coefficient of variation as large as 0.585.When we invert the model through the least squares fittingprocedure, we partly relax this assumption by treating a

as a variable, projecting the two-dimensional tasseled cap

pattern into a one-dimensional m distribution and obtain-ing reasonably accurate estimates of N and R2. At the sametime we obtain a projected a distribution; if this distri-bution agrees well with the field measurements, then thetasseled cap brightness pattern of the coniferous forestforest is quantitatively and accurately explained by our

geometric-optical model.The variation in apex angle, however, raises the problem

of the meaning of the average apex angle as an

aggregation variable for the pixel or timber stand. If a isindependent of r, we can take the average of a as para-

meterizing the form of the trees. The strong negative cor-

relation between a and r, however, means that the simpleaverage of a has little physical meaning. Under our as-

sumptions, even if the n cones within a pixel have differenta and r2 values, the reflectance of the pixel is still mod-eled as if it can be approximated by n cones that have thesame mean size R2 and the same apex angle az. Therefore,an obvious meaningful definition of the "average" a fora pixel is an equivalent angle such that the fixed apex anglemodel approximately holds. It is difficult to achieve a very

accurate equivalence because the variation in form will

affect all four signature components and influence theoverlapping effect as well. Some approximations have tobe made to simplify the problem. Since all the area termsof the reflectance components are of the form r2 multi-plied by a function of oa, the average apex angle we obtainfrom the projected a distribution will resemble a weightedaverage E(r2a)/R2 for the pixel. This means that the largercones will have greater weight in determining a for thepixel. The projected a distribution, we believe, should beunderstood in this sense.

If a similar weighted mean is calculated from the fielddata, the mean of the projected a distribution overesti-mates this value by about 13 percent. If we change thesignatures as discussed in the section on sensitivity, themean of the projected a distribution continues to overes-timate the observed weighted mean by 2-45 percent. Ifwe compare the mean of the projected a distribution tothe simple average of a, the simple average is underesti-mated by 25 percent, or 39-4 percent in sensitivity anal-ysis. In short, our model produces an estimate of the meanapex angle at, which lies between a simple average and anaverage weighted by r2.

In our field work, we also noted that the coefficient ofvariation of the projected a distribution is about one-thirdof the coefficient of variation of measured crown apex an-gles. This underestimation results because the projecteda values are averages over whole pixels. Our present fielddata, however, are insufficient to evaluate this aggrega-tion effect quantitatively, since only four apex angles weremeasured at each subplot location.

F. Pixel Boundary ProblemRecall that in both the Monte Carlo simulation and the

mathematical modeling of overlap between crowns andshadows, the pixel boundary problem was treated by rep-licating the portion of a crown or shadow that falls outsideof the pixel on the inside of the pixel at the oppositeboundary. In effect, the pixel overlaps onto itself Thissimplification should underestimate pixel-to-pixel vari-ance, and it is appropriate to examine its implications.

In the running of the Monte Carlo simulation, statisticswere kept that recorded the proportions of the areas of Ac,Az, and A, falling outside of the pixel boundary that werereplicated within the pixel. At the 80-m pixel size, only afew percent of each area was affected; thus the influenceon Kc, Kz, and Kt will be negligible at MSS resolution. At10-m resolution, however, these percentages increase toseveral tens, and thus the variances of the proportions in-crease significantly. And for large m values, this cross-boundary variance can even be larger than the overlappingvariance, since the latter is very small in this case (seeFig. 1). The field measurements also show that ourexpression for the variance with overlapping (15), whichassumes infinite replication of the pixel, does not hold wellat a pixel size smaller than about 12 m. This observationalso indicates that cross-boundary variation will be aproblem at this pixel size.

In order to evaluate the effects of this variation on ourmodel, let us assume that n trees are located within a pixel,

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and they produce components Ac, A, and A, as we previ-ously modeled. Because of cross-boundary effects, partsof these areas, A', A', and A', will fall outside of the pixel,while AJ,7 A", fand A" from trees outside the pixel will fallinside. Let us consider the following simplified cases.

1) At/A"g A;/A;' Az/AJ: In this case, the m'valueinverted from (18)will deviate from nR2 along the cover-age trajectory. This value, however, can be thought of asa correct solution because pixel size is arbitrary and noth-ing prevents us from defining n as a real number, ratherthan an integer. Thus the deviation along the coverage tra-jectory does not affect the accuracy of the model.

2) Al+Al + Al A' + At' + Ag; In this case, theinversion of (18) will have errors basically only in a, sinceAg does not change and it is the dominant factor deter-mining the location of S along the coverage trajectory. Un-less the value of a obtained is far from the true value, theaccuracy of the model will not be seriously affected, al-though the variance of ay will be inflated somewhat.The actual situation will be a combination of these two

cases, and will be more difficult to model exactly. How-ever, it appears that unless the resolution is so fine thatmany pixels contain pure Ac, At, Az, or only fractions ofcrowns, the accuracy of this model will not be signifi-cantly degraded. Further Monte Carlo simulations con-ducted at 10-m resolution and using trees of the size en-countered in the two study sites also show little errorcaused by the boundary problem. Thus our method of ac-commodating the boundary problem is probably reason-able for the types of forests and imagery we have encoun-tered thus far in this research.

G. Further Application of the Geometric/Optical ModelFrom the numerical viewpoint, our model is driven by

the variance encountered when the number of trees, theirsizes, and their overlap varies from one pixel to the next.If this variance is not large, or contains a lot of noise, theninversion of the model will be inaccurate. As (15) shows,this can happen in two ways. First, if the pixel size is large,then a, which is the ratio of the area of the object to thearea of the pixel, will become smaller, in turn reducingV(Ag). Second, if the number of objects N is large, thenthe kernel of (14) will become smaller since q is a positivefunction between 0 and 1. In other words, large pixels willhave low variance because of averaging, and densely cov-ered pixels will have low variance since nearly all thebackground is obscured.

In our work to date, we have applied our model to pixelswith sizes equal to or smaller than 80 m on a side and tocanopy coverages of M < 0.46, which gives a cover ofabout 75 percent (red fir site). Even in the case of canopycovers greater than 75 percent, there may still be a suffi-ciently ample number of pixels from the low end of the mdistribution to invert the model through the least-squaresprocedure (e.g., Fig. 16). Thus the procedure should workwith even denser stands provided that the trees are not toosmall.As to the problem of large pixel size, one helpful ap-

proach will be to estimate or calibrate the noise variance

a6)0cr

(U

S

0 2 4 6 8Crown base area index

Fig. 16. Projected m distribution for red fir site.

contained in V(m); since it will be independent and addi-tive, it can be separated from the true variance unlessV(m) is so small that it lies below the quantization level ofthe remotely sensed signal. This approach, however, isnot likely to be needed, since we have already demon-strated sufficient variance at 80 m with 75-percent canopycover. Given the abundance of Landsat MSS data at 80-mresolution, and the prospect of 7-band 30-m data from TM,it seems likely that the model will be widely applicable.Another advantage of our approach is that it can be used

to identify pixels that do not fit the model for the stand asa whole. If the individual value of a or M obtained for apixel departs greatly from modeled values,, it can be clas-sified as nonconifer and excluded from parametric esti-mation. In this way, the present requirement for predelin-eating a timber stand may be relaxed somewehat.

X. CONCLUSIONOur geometric-optical canopy model demonstrates that

regarding a remotely sensed scene as consisting of three-dimensional objects casting shadows on a background canbe highly productive. Our simulations show that the three-dimensional geometry alone can go a long way toward ex-plaining the bidirectional reflectance distribution functionof forest canopies, thus emphasizing the importance ofshape, form, and shadowing of objects in influencing im-ages of real scenes.

In addition, explicit modeling of shadowing geometryin the presence of overlapping shadows and objects hasallowed the development of an invertible model that yieldsthe average size and spacing of objects, even when theobjects are considerably smaller than the pixels. The in-version requires specifying and modeling the interpixelvariance within a uniform area, and in this sense our modelis probably the first canopy model to utilize the fact thatpixels from a uniform area may be regarded as differentrealizations of random processes containing informationabout the processes themselves. Our model is thereforespatial in the sense that the realizations are drawn from aspatially connected field, and requires an image if it is tobe inverted. Thus the geometric-optical model differs fromnearly all other plant canopy models in that it is designedspecifically to explain, utilize, and exploit the spatial vari-ation inherent in digital imagery.

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Although our geometric-optical model exploits spatialvariation, it does not truly utilize the spatial dimension inthat the absolute or relative position of each measurementis not required. It therefore does not make full use of allthe information inherent in a digital image. A plannedfuture thrust of our modeling activity is to use spatial po-sition directly in model formualtion. When resolution cellsize is small and the objects are large (as with SPOT orTM imagery), adjacent pixels will influence one anotherbecause objects are discrete entities that extend acrosspixel boundaries. In explicitly modeling this effect, thecross-boundary variance is therefore not a problem to beovercome, but a source of information that can be used toadvantage to determine the size and spacing of the objectsmore accurately. Such "third-generation" models willtruly exploit the full potential of remote sensing to revealthe spatial parameters and processes that shape the Earth'slandscape.

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[39] -, "Inversion of vegetation canopy reflectance models for estmat-ing agronomic variables. IV. Total inversion of the SAIL model," Re-mote Sensing Environ., vol. 15, pp. 237-253, 1984.

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Xiaowen Li graduated from the Chengdu Instituteof Radio Engineering, China, in 1968. He re-ceived the M.A. and Ph.D. degrees in geographyand the M.S. degree in electrical and computer en-

gineering from the University of California atSanta Barbara in 1981 and 1985, respectively.

Since 1979, he has been on leave from the In-stitute of Remote Sensing Application, ChineseAcademy of Science. He is currently a post-doc-toral fellow at Hunter College, City University ofNew York. His interests are in remote sensing and

diffraction tomography.

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LI AND STRAHLER: GEOMETRIC-OPTICAL MODELING OF A FOREST CANOPY 721

Alan H. Strahler was born in New York City, NY, research since 1978. He has been a Principal Investigator on numerouson April 27, 1943. He received the B.A. and Ph.D. NASA contracts and grants. His primary research interests are in spatialdegrees from the Johns Hopkins University in 1964 modeling and spatial statistics as they apply to remote sensing, and in geo-and 1969, respectively. metric-optical modeling of remotely sensed scenes.

He is currently Professor and Chair of the De-partment of Geology and Geography at HunterCollege of the City University of New York. Hehas held prior academic positions at the Universityof California, Santa Barbara, and at the Universityof Virginia. Originally trained as a biogeographer,he has been actively involved in remote-sensing

geometric-optical modeling of a conifer forest...

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