Remote Sensing of Environment 130 (2013) 188–204

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Remote Sensing of Environment

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Revisiting the vegetation hot spot modeling: Case of Poisson/Binomialleaf distributions

Abdelaziz Kallel a,⁎, Tiit Nilson b

a Institut Supperieur d'Electronique et de Communication de Sfax, 3000 Sfax BP 868, Tunisiab Tartu Observatory, 61602, Travere, Estonia

⁎ Corresponding author.E-mail addresses: abdelaziz_kallel@yahoo.fr (A. Kall

0034-4257/$ – see front matter © 2012 Elsevier Inc. Allhttp://dx.doi.org/10.1016/j.rse.2012.11.018

a b s t r a c t

a r t i c l e i n f o

Article history:Received 1 January 2012Received in revised form 15 November 2012Accepted 17 November 2012Available online xxxx

Keywords:Hot spot effectBidirectional gap distributionPoisson/Binomial distributionsHomogeneous discrete vegetation

The accuracy of spaceborne/airborne sensor measurements in the solar domain keeps increasing over time. Highresolution, multi-directional and hyperspectral image acquisitions start to be abundant. With regard to themulti-angular remote sensing data, the hot spot, i.e. the exact backscattering direction of direct sunlight togetherwith its neighboring directions, is of special interest. Accurate hot spot models have to be used to adequatelysimulate the hot spot signature and to allow reliable inversion of multi-angular data. In this paper, we proposea physical hot spot model (Leaf Spatial Distribution based Model, LSDM) assuming that for a given point insidethe vegetation to be sunlit (respectively, observed) it should be located within a cylinder free from leaf centers.The cylinder is oriented to the sun (respectively, sensor) direction. Assuming a leaf random, regular or clumpedspatial distributions, the gap probabilities in the sun and sensor directions are expressed as a function of thesecylinder volumes. Based on the same hypothesis, the bidirectional gap probability is estimated as a function ofthe total volume of the two cylinders. The evaluation of the needed common volume of two cylinders havingdifferent radii is reduced to calculation of some elliptic integrals. Finally, the hot spot signature is estimatedbased on the bidirectional gap probability distribution. Different model versions with different leaf spatial distri-bution functions are compared. Particularly, it is shown that compared to the random distribution, the regular(the clumped, respectively) distribution increases (decreases, respectively) the reflectance due to single scat-tering contribution from foliage. The proposedmodel is validated using the ROMCweb-based tool and its betterperformance compared to the Semi-Discrete Model and Kuusk's model is confirmed.

© 2012 Elsevier Inc. All rights reserved.

1. Introduction

The results of spaceborne multi-angular remote sensing mea-surements have been available at least during the last 15 years.New sensors such as CHRIS, which is an imaging spectrometer car-ried on board the space platform PROBA, allow high resolutionmulti-angular and hyperspectral acquisitions. In terms of multi-angular observations, this sensor can be pointed off-nadir in bothalong-track and across-track directions. The sophistication of suchinstruments keeps increasing over time, particularly the sensor agil-ity is a hot topic. It allows to rapidly acquire off-nadir targets, in orderto sequence images of the same area in different observation anglesleading to sampling of the directional reflectance factor of the canopy[e.g., Pleiades-HR constellations (Lebegue et al., 2010)]. Althoughthe hot spot region, corresponding to the bright area close to theexact backscattering direction, has been recognized as a potentiallyinformative angular region, a majority of existing instruments withmulti-angular capability do not currently measure in the hot spot di-rection. However, a considerable amount of images showing the hot

el), nilson@aai.ee (T. Nilson).

rights reserved.

spot region as well as its angular signature have been recorded in thelast two decades. For instance, the airborne version of the Multi-angle Imaging SpectroRadiometer (AirMISR) (Gerstl et al., 1999),and the spaceborne Polarization and Directionality of the Earth's Re-flectances (POLDER) (Grant et al., 2003) instruments provided theBidirectional Reflectance Distribution Function (BRDF) signaturesthat included the hot spot region. Recent studies have shown that anumber of biophysical features can be retrieved from a sampledBRDF (just two or three parameters per inversion). For instance,one can cite the canopy architecture (Schlerf & Atzberger, 2006)(i.e., the tree spatial distribution, canopy cover, leaf area index), thetree macro structure (Mõttus et al., 2006) (e.g., tree height, the sizeand shape of the crowns and leaves), the understory reflectance(Canisius & Chen, 2007) and the clumping index (He et al., 2012).As the main aim of canopy remote sensing is to derive canopy prop-erties (Combal et al., 2002) from the BRDF, it is important to ade-quately model it as a function of canopy features and scenegeometry. For that a lot of theoretical works are trying to increasethe accuracy of the BRDF modeling (Widlowski et al., 2006b), there-fore the proposed approach complexities keep increasing and themodels are becoming time consuming, particularly those basedon Monte Carlo (MC) ray-tracing (for which the inversion is not

189A. Kallel, T. Nilson / Remote Sensing of Environment 130 (2013) 188–204

practical) (Widlowski et al., 2006b). Especially the hot spot, whichneeds simulating light paths in a canopy layer considering its 3D ar-chitecture, is among the BRDF features for which MC models providemore reliable results than 1D models, which approximate the archi-tecture by considering vertical variations only. In particular, thetraditional 1D models assume a random leaf distribution (Poissondistribution) neglecting any scales of leaf spatial organization(Widlowski et al., 2005). So, increasing the performance of 1Dmodeling of the hot spot signature can be a step towards better re-trieval of canopy features (Liang et al., 2000).

Before being studied in the vegetation canopy, the hot spot effectwas known one century ago. In fact, in 1887, Seeliger (1887) pro-posed the first hot spot model to interpret the phase curve of theSaturn rings showing a maximum brightness at the opposition direc-tion and the phenomenon was known as the opposition effect.Seeliger assumed that the rings' particles have a spherical shape. Tobe sunlit and observed, such particles should be contained in cylin-ders free from other particles having the same radius as one particlewith cylinders' axis respectively in sun and observation directions(cf. Fig. 1, in the particular case rs= ro). Thus, the hot spot problemwas reduced to the estimation of the common volume of the two cyl-inders. After that a lot of studies were carried out to enhance thismodel (for more details, the reader is referred to Kuusk (1991)). Inhis study of the moon surface in 1963 and 1986, Hapke (1963,1986) proposed a different method to model the hot spot effect. Heassumed that the moon surface contained cylindrical pores. After-ward, the developed model was applied to investigate the BRDF ofterrestrial bare soil surfaces (Pinty et al., 1989). However, in their studyof the coherent baskscattering calculation, Liang and Mishchenko(1997) and Mishchenko (1992) showed that this theory is not valid forsuch surfaces composed of fine particles that do not have well-definedshadows.

Canopy components (e.g., leaves or needles) can be assumed tobe considerably larger than the wavelength and almost opaque, sothey cast shadows on neighboring components. This way, the hotspot theory can be applied. As far as we know, Suits (1972) was thefirst to include such a phenomenon in his study of the canopy BRDFin 1972. He simply inserted an empirical correction term to the bidi-rectional gap probability expression. Then, a lot of theoretical workstrying to model the physical phenomenon have been proposed. For

M

A B

s

z

t

0 rs ro

Fig. 1. M on depth z, A and B both on depth t are located within the vegetation.M is sunlitand observed from the directions Ωs and Ωo, respectively. A and B are belonging to thelines passing from M with director vectors Ωs and Ωo, respectively. Two cylinders withhorizontal base of radii rs and ro, respectively and of axis passing throughM and in direc-tions Ωs and Ωo, respectively, are located from the depth z to the top of the layer. Theintersections between the cylinders and the horizontal plane at depth t are the disksSs(t) and So(t), respectively. The gray surface corresponds to the disks' intersection.

example, Nilson (1977, 1980) has analytically described the bidirec-tional gap probabilities and the part of the hot spot effect caused bysunlit crowns and ground. In 1985, Kuusk proposed the first theoret-ical hot spot model adapted to a homogeneous vegetation canopy.He divided the vegetation layer into thin superposed sublayers, as-sumed independent in terms of leaf positions, and computed the bi-directional gap correlation in each sublayer. Then, by integrationover the vegetation depth, the bidirectional gap probability was de-rived. Based on the Kuusk theory, Jupp and Strahler (1991) and Qin(1993) proposed to replace the joint gap correlation by a Booleanmodel and an overlap function, respectively. These canopy hot spotmodels and others are summarized and compared in Qin and Goel(1995). Such approaches suffer from the assumption of indepen-dence between sublayers. In fact, it is true that a horizontal leafmay belong only to one thin sublayer. However, the more inclinedis the leaf, the larger number of successive sublayers it passesthrough and the assumption is violated. To overcome such a prob-lem, Kallel (2010a, 2010b, 2012) proposed in his model, FDM/FDM2, the calibration of the hot spot factor (which is originally de-fined as the ratio between the leaf radius and the vegetation height)for each leaf distribution (e.g., for the planophile and the erectophiledistributions, the original parameter is multiplied by 1.5 and 3,respectively).

As first proposed by Seeliger, Verstraete et al. (1990) developed ahot spot model for semi-infinite canopies assuming that to be sunlitand observed, a given point should belong to cylinders free fromleaves with horizontal circular bases in sun and observation direc-tions (cf. Fig. 1). In this case, the radii of the two cylinders were as-sumed equal to the radius of an average equivalent horizontal sunfleck [the distribution of sunlit holes along a horizontal plane belong-ing to the vegetation layer (Miller & Norman, 1971)]. To extend it tothe finite canopy case, Gobron et al. (1997) proposed somemodifica-tions to the approach. In particular, they derived a new formulationof the size of holes between leaves that are assumed disk-shaped.

The traditional hot spot model assumes leaves are randomly dis-tributed (Poisson law) within a considered vegetation layer withoutany constraint about their spatial organization. However, differentlevels of foliage clustering could occur as already shown in a numberof works dealing with gap frequencies within forest canopies(Nilson, 1999; Pisek et al., 2011; Wang et al., 2007). According toNilson (1999), a stand is composed by a number of trees coveringthe area, the tree distribution could be Poisson as well as Binomial(i.e., regular or clumped). Moreover, inside a given crown, twokinds of foliage clustering could be considered: shoot-level andstructures larger than shoot-level, e.g. crown level (Wang et al.,2007). In a number of works (e.g. Pisek et al. (2011)), all these structur-ations are taken into account in the derivation of the gap frequency, andallow inversion of LAI based on these frequency measures (Nilson,1999). According to Pisek et al. (2011), such foliage organization has ahuge impact on the gap fraction, which implies an important varia-tion of the radiative regime and therefore the canopy reflectance incomparison with the Poisson case. The radiative transfer problemcorresponding to media composed by big structures (e.g. stand orcrown levels) is complex and can be solved using Monte Carlo 3-Dmodels as they allow the simulation of heterogeneous medium re-flectances (Widlowski et al., 2006b). However, when the clustersare too small (e.g. shoot level), the considered medium can be as-sumed homogeneous and radiative regime could be approximatedusing just 1-D models as it will be shown in this paper.

Our proposed approach is as follows. The assumption that an ob-served or sunlit point is belonging to a cylinder free from leaves isadopted. However, compared to Verstraete et al. (1990) or Gobronet al, (1997), where cylinder bases are equal to the sun fleck (themean sunlit surface within the vegetation), the cylinders in ourcase are the smallest volumes free from leaf centers to ensure thata given point within the vegetation is sunlit or observed. Therefore,

190 A. Kallel, T. Nilson / Remote Sensing of Environment 130 (2013) 188–204

the probabilities that the point is sunlit or observed (i.e., the gap frac-tion) are none other than the probabilities that the cylinders are freefrom leaf centers. Such probabilities depend on the cylinder volumesand the leaf spatial distribution. The latter could be totally random(Poisson) as assumed in traditional hot spot model (Gobron et al.,1997; Kuusk, 1985; Verstraete et al., 1990) or extended to the Bino-mial distribution (regular or clumped), as our model offers the pos-sibility to deal with any spatial distribution. Following the samereasoning, it is possible to estimate the bidirectional gap probabilityas it corresponds to the probability that both cylinders (in sun andobservation directions) are free from leaf centers. It is noted herethat cylinders share a common volume which explains why theprobability to be sunlit and observed is dependent. It is noted herethat this link between gap and bidirectional gap probabilities wasnot done in the models of Verstraete et al. (1990) and Gobron et al.(1997). Indeed, they only propose a correction of the bidirectionalgap probability computed assuming the independence betweensun and sensor paths based on some normalization by the ratio be-tween the two cylinder total volume and the sum of their volumes.

This paper is structured as follows. First, we present the theoret-ical background of our model (Section 2). Then, we provide modelexperimental results (Section 3). Finally, we present our main con-clusions and future perspectives (Section 4).

2. Theoretical background

In this section, wewill present the theoretical basis for estimationof the gap fraction, the bidirectional gap distributions and the reflec-tance in the homogeneous vegetation canopy.

2.1. Estimation of gap probability

All along this section, a discrete vegetation layer located betweenthe depth −h and 0 with a given leaf area index (LAI) is considered.The layer is assumed homogeneous, so the spatial distribution of leafcenters, leaf angular and area distributions are assumed the same foreach point within the vegetation layer. Leaves are assumed to be flatdisks with a given size distribution. However, in this work, leaf sizedistribution is ignored and all leaves are supposed to be disks ofidentical size, the radius of the leaf of mean area is denoted by rl.

The leaf spatial distribution is described by the leaf center posi-tions. The mean number of leaf centers per m3 or the leaf number vol-ume density, nl, is given by

nl ¼LAIhπr2l

: ð1Þ

The spatial leaf center distribution within the vegetation canopycould be simulated by a Poisson process, negative or positive Binomialdistributions, as suggested by Nilson (1971).

The Poisson distribution assumes an ‘entirely’ random distribu-tion of leaf centers (i.e., no assumption about any systematic patternin the leaf positions). For a given spatial region V of volume v, theprobability to find m leaf centers is given by

ppoi;nl m; vð Þ ¼ nlvð Þm exp −nlvð Þm!

: ð2Þ

Conversely, the Binomial distributions could be interpreted ashaving a vertical stratification of the foliage into independent spatialregions (Δv), but dependent leaf position inside each one. Two kindsof dependency can be assumed: (i) positive Binomial distribution(p+) assuming regular foliage dispersion; and (ii) negative Binomial

distribution (p−) describing a clumped dispersion of foliage. Theyare modeled by Nilson (1971)

pþ m; vð Þ ¼ CmN nlΔvð Þm 1−nlΔvð ÞN−m; m ¼ 0;…;N;

p− m; vð Þ ¼ CmNþm−1 nlΔvð Þm 1þ nlΔvð Þ−N−m; m ¼ 0;…;N;…

ð3Þ

where

N ¼ vΔv

: ð4Þ

One could notice that, for N real, Eq. (3) would not be valid. How-ever, as we are interested in the probability of gap that it will beshown that it is none other than the probability that a given volumeis empty and therefore can be estimated using Eq. (3). Form=0, thevalidity problem will not be encountered since for any value of N,CN0=CN+0−1

0 =1, otherwise an extension to the real number casemust be proposed.

Now, let M be a point inside the vegetation at depth z∈ [−h, 0]receiving sunlight from a direction Ωs=(θs,φs). Let V (of volume v) bea cylinder located from the depth z to the top of the layer. The cylinderhas its axis oriented atΩs and horizontal base in a form of a disk, S, withcenter M, area s and radius r≥0. Then, according to Eqs. (2) and (3),the probabilities that V is free of leaf centers, pc,…(s,z) (… ∈ {poi,+,−}corresponding to the Poisson, positive and negative Binomial distribu-tions), are given by

pc;poi s; zð Þ ¼ exp −nlvð Þ;

pc;þ s; zð Þ ¼ 1−nlΔvð ÞvΔv;

pc;− s; zð Þ ¼ 1þ nlΔvð Þ−vΔv:

ð5Þ

The volume v of V is a function of r and |z|. It is given by

v ¼ π r2 zj j: ð6Þ

Note that, pc,… is different from the probability of a gap coveringthe whole area of S (i.e. the probability that the entire S be sunlit),pgap,…(s,z). Although no leaf centers are in the cylinder V, it remainspossible that parts of leaves with their center outside of V can blocksome of the radiation inside V. Thus, obviously

pgap;…b pc;…: ð7Þ

In the particular case, when the disk S is reduced to only one pointlocated at altitude z, pc …(0,z)=1, whereas pgap …(0,z) could not beequal to one, otherwise all points inside the vegetation would be sunlit.In the following, we will present a method to express pgap …(0,z).

Let us consider a disk-shaped leaf inside the canopy of surfacesl=πrl2 (rl the leaf radius) and with normal orientation Ωl(θl,φl).The leaf is illuminated from the direction Ωs. Its projection on thehorizontal plane has the area sl

p, such that

spl ¼ Ωs:Ωl

μssl; ð8Þ

with

μs ¼ cos θsð Þ: ð9Þ

Thus, the average of slp over all leaf normal orientations, SHS, whichdefines the mean projection area of a leaf on the horizontal plane, isgiven by Nilson (1971)

sHS ¼Gs

μssl; ð10Þ

191A. Kallel, T. Nilson / Remote Sensing of Environment 130 (2013) 188–204

with Gs the Ross–Nilson geometry function (G-function) (Nilson,1971). It is clear that for each leaf orientation, Ωl, the leaf projectionon the horizontal plane, SHS(Ωl), is generally elliptically-shaped andtaking into account all the possible leaf orientations, the mean sur-face, SHS, is a complex geometrical form. However for the sake of sim-plicity, we assume that SHS are disks, SHS, having all the same area(SHS). The corresponding disk radius rHS is given by

rHS ¼ffiffiffiffiffiGs

μs

srl: ð11Þ

Now, let us derive pgap …(0,z) for a pointM inside the vegetation ataltitude z. To ensure that M is observed from the direction Ωs, noleaves should cross the line D of direction Ωs linking M to the top ofthe canopy. Therefore, it is necessary that for any altitude t∈ [z,0],the distance between the line D and the centers of leaves be largerthan rHS. Then, to make sure that M is in the gap when viewed fromthe direction Ωs, M should be inside a cylinder free from leaf centersof axis D and radius rHS. The cylinder volume (vgap) satisfies

vgap ¼ πr2HS zj j: ð12Þ

So, the probabilities that M is in a gap are given by

pgap;poi 0; zð Þ ¼ exp −nlvgaph i

;

pgap;þ 0; zð Þ ¼ 1−nlΔvð Þvgap

Δv ¼ exp −nl;þ Δvð Þvgaph i

;

pgap;− 0; zð Þ ¼ 1þ nlΔvð Þ−vgap

Δv ¼ exp −nl;− Δvð Þvgaph i

:

ð13Þ

with nl,+ and nl,− the equivalent number of leaf centers per unit volumefor the positive and negative Binomial distributions, respectively,

nl;þ Δvð Þ ¼ − log 1−nlΔvð ÞΔv

;

nl;− Δvð Þ ¼ log 1þ nlΔvð ÞΔv

:

ð14Þ

Considering the Poisson distribution and combining Eqs. (11)–(13)give

pgap;poi 0; zð Þ ¼ exp nlGs

μsslz

� �;

¼ expGs

μsLAI

zh

� �;

¼ exp kzð Þ;

ð15Þ

with k, the extinction coefficient, as defined by Verhoef (1984)

k ¼ Gs

μs

LAIh

: ð16Þ

The same gap probability expression was already found out byNilson (1971). Note that, although this probability is provided inthe discrete case, it does not depend on the leaf size (i.e., radius rl).However, one must notice that when we consider non-null leafsize, leaf center positions cannot be totally random but they mustbe distributed distant respecting the size and orientation of eachleaf. Therefore, for an actual canopy which is a discrete medium,one must consider Binomial leaf distributions rather than Poissonones.

Similarly to the Poisson distribution, for the Binomial ones, the gapprobabilities for a point are given by

pgap;þ 0; zð Þ ¼ exp kþz� �

;pgap;− 0; zð Þ ¼ exp k−zð Þ: ð17Þ

with k+ and k− are respectively the extinction in the positive andnegative Binomial distribution cases and given by

kþ ¼ nl;þ Δvð Þnl

k;

k− ¼ nl;− Δvð Þnl

k:ð18Þ

As we can see from Eqs. (15) to (17), the shape does not have anyinfluence on the estimation of the gap probability. However, in thenext section, the computation of the bidirectional gap probabilityneeds the estimation of a common volume between two cylinders.

It is noted finally that the gap probability for a given point (M) isjust its probability to be sunlit. We showed here that it correspondsto the probability that M belongs to a given cylinder free from leafcenters. Now, it is true that if M is sunlit it belongs to an arbitraryshaped sun fleck of a certain area, however the estimation of theprobability of gap for M is not directly related to the sun fleck fea-tures. Particularly, the base area of the considered cylinder freefrom leaf centers is independent of the corresponding sun fleckarea as it is claimed by Verstraete et al. (1990) and Gobron et al.(1997).

2.2. Bidirectional gap distribution

Fig. 1 shows a point M(x,y,z) inside a vegetation layer sunlit andobserved from the directions Ωs and Ωo=(θo,φo), respectively. Twopoints A and B having the same z-coordinate (t, t>z), such that Aand B are belonging to the lines passing fromMwith director vectorsΩs and Ωo, respectively. According to Kuusk (1991), the distance AB,denoted by dz(t), can be written as

dz tð Þ ¼ t−zð ÞΔ Ωs;Ωoð Þ; ð19Þ

with

Δ Ωs;Ωoð Þ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffitan2 θsð Þ þ tan2 θoð Þ−2cos φoð Þtan θsð Þtan θoð Þ

q: ð20Þ

To be sunlit and observed, respectively, from the directionsΩs and

Ωo, the point M must belong to the cylinders Vs and Vo of radii rs ¼ffiffiffiffiGs

μs

rrl and ro ¼

ffiffiffiffiffiGo

μo

rrl (with Go the G-function in the observation direc-

tion and μo=cos(θo)) around the axis passing throughM in directionsΩs and Ωo, respectively (cf. Fig. 1). The corresponding volumes [vgapas defined by Eq. (6)] of the cylinders free from leaf centers of Vs

and Vo are denoted by vs and vo, respectively. Thus, to ensure that Mis sunlit and observed, the spatial region Vso(z)=Vs∪Vo must befree from leaf centers. Its volume, vso(z), is given by

vso zð Þ ¼ ∫0

zsSs tð Þ∪So tð Þdt;

¼ ∫0

zss tð Þ þ so tð Þ−sSs tð Þ∩So tð Þdt;

¼ vs þ vo−∫0

zsSs tð Þ∩So tð Þdt;

¼ πr2s zj j þ πr2o zj j−∫0

zsSs tð Þ∩So tð Þdt;

ð21Þ

where so(t) and ss(t) are the area of the disks So(t) and Ss(t) at depth twhich correspond to the intersection between the horizontal plane

1 The multiplication by π was introduced by Suits (1972) to define Eo as an equiva-lent irradiance corresponding to the scattered radiance in the direction Ωo assumingthat it is created by a horizontal Lambertian surface.

192 A. Kallel, T. Nilson / Remote Sensing of Environment 130 (2013) 188–204

(z= t) and the cylinders Vs and Vo, respectively (cf. Fig. 1). sSs tð Þ∪So tð Þand sSs tð Þ∩So tð Þ are the areas of Ss(t)∪So(t) and Ss(t)∩So(t), respectively.

To ensure that Ss(t)∩So(t) is not an empty set, the distance be-tween the disk centers (t−z)Δ(Ωs,Ωo) should be smaller than rs+ro.Therefore, the altitude t must meet

t≤ rs þ roΔ Ωs;Ωoð Þ þ z ¼ zmax: ð22Þ

Moreover, it is possible tofind Ss(t)⊂Ss(t)∩So(t) or Ss(t)⊂Ss(t)∩So(t)when rs≤ro or rs≤ro, respectively. The distance dz(t) meets the conditionin this case

dz tð Þ≤ rs−roj j⇒t≤zþ rs−rj joΔ Ωs;Ωoð Þ ¼ zmin: ð23Þ

In this case, the area of intersection between the disks is the areaof smallest disk between ss(t) and so(t) (i.e., πmin(rs,ro)2). When,rs=ro the last case does not happen only for t=z (i.e., zmin=z). Theintegration of sSs tð Þ∩So tð Þ in Eq. (21) is therefore decomposed into twoterms (i.e., whether tbzmin or t>zmin).

According to Eqs. (22) and (23), Eq. (21) becomes

vso zð Þ ¼ πr2s zj j þ πr2o zj j−πmin rs; roð Þ2 min 0; zminð Þ−zð Þ

−∫min zmax ;0ð Þmin zmin ;0ð Þ sSs tð Þ∩So tð Þdt:

ð24Þ

When t∈]min(zmin,0),min(zmax,0)[, the intersection between disks(Ss(t)∩So(t)) is a non-zero surface included both in Ss(t) and So(t) (cf.the gray area in Fig. 1). Therefore based on the geometric propertiesof the disks and the distance between their centers, it is possible toshow that

sSs tð Þ∩So tð Þ ¼ r2s arccos12rs

r2s−r2odz tð Þ þ dz tð Þ

!" #þ r2oarccos

12ro

r2o−r2sdz tð Þ þ dz tð Þ

!" #

−dz tð Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2o−

14

r2o−r2sdz tð Þ þ dz tð Þ

!2vuut :

ð25Þ

Eq. (24) becomes

vso zð Þ ¼ πr2s zj j þ πr2o zj j−πmin rs; roð Þ2 min 0; zminð Þ−zð Þ

−∫min zmax ;0ð Þmin zmin ;0ð Þ r

2s arccos

12rs

r2s−r2ot−zð ÞΔ Ωs;Ωoð Þ þ t−zð ÞΔ Ωs;Ωoð Þ

!" #dt

−∫min zmax ;0ð Þmin zmin ;0ð Þ r

2oarccos

12ro

r2o−r2st−zð ÞΔ Ωs;Ωoð Þ þ t−zð ÞΔ Ωs;Ωoð Þ

!" #dt

þ∫min zmax ;0ð Þmin zmin ;0ð Þ t−zð ÞΔ Ωs;Ωoð Þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2s−

14

r2s−r2ot−zð ÞΔ Ωs;Ωoð Þ þ t−zð ÞΔ Ωs;Ωoð Þ

!2vuut dt:

ð26Þ

By substitution, u=(t−z)Δ(Ωs,Ωo),

vso zð Þ ¼ πr2s zj j þ πr2o zj j−πmin rs; roð Þ2 min 0; zminð Þ−zð Þ

− r2sΔ Ωs;Ωoð Þ∫

min rsþro ;−zΔ Ωs ;Ωoð Þð Þmin rs−roj j;−zΔ Ωs ;Ωoð Þð Þ arccos

12rs

r2s−r2ou

þ u

!" #du

− r2oΔ Ωs;Ωoð Þ∫

min rsþro ;−zΔ Ωs ;Ωoð Þð Þmin rs−roj j;−zΔ Ωs ;Ωoð Þð Þ arccos

12ro

r2o−r2su

þ u

!" #du

þ rsΔ Ωs;Ωoð Þ∫

min rsþro ;−zΔ Ωs ;Ωoð Þð Þmin rs−roj j;−zΔ Ωs ;Ωoð Þð Þ u

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1− 1

2rs

r2s−r2ou

þ u

!" #2vuut du:

ð27Þ

The estimation of the integrals in Eq. (27) is given in Appendix A.1.Finally, similar to the gap probability Eqs. (17) and (15), the bidi-

rectional gap probabilities Pgap,… are given by

Pgap;poi Ωs;Ωo; zð Þ ¼ exp −nlvso zð Þð Þ;Pgap;þ Ωs;Ωo; zð Þ ¼ exp −nl;þvso zð Þ

� �;

Pgap;− Ωs;Ωo; zð Þ ¼ exp −nl;−vso zð Þ� �

:

ð28Þ

Note that actually the base of the projection of a leaf of orientation,Ωl, on a horizontal plane for the direction of sun and observation are el-lipses (SHS,s(Ωl) and SHS,o(Ωl), respectively) (cf. Section 1). Thereforeto be sunlit and observed, the point M must belong to elliptic-cylinders (EVs(Ωl) and EVo(Ωl), respectively) free from leaf centersof base SHS,s(Ωl) and SHS,s(Ωl), respectively. The bidirectional gapprobability is therefore the probability that both elliptic-cylindersare free from leaf centers. Therefore, the estimation of the totalelliptic-cylinders volume is needed (EVs(Ωl)∪EVo(Ωl)), then whenintegrating over all leaf orientations, Ωl, an averaged total volumeis obtained. The result replaces vso(z) estimated in Eq. (27). Thedrawback of this method is the estimation of the common area be-tween two ellipses of both different centers and axes. A (semi)ana-lytical expression of the global volume cannot be obtained in thiscase as proposed in this work. However a numerical solution couldbe addressed, but such a solution suffers from long running time asclaimed by Verhoef (1998).

2.3. Reflectances due to single scattering

The reflectances due to the single scattering (of the solar irradi-ance) contribution from the foliage and the soil background are stud-ied in this section.

In this section, first we present the derivation of the reflectancesdue to single scattering contribution in the general case, and secondwe present our model formulation.

2.3.1. OverviewThis section deals only with the case of vegetation layer such that

the leaf center spatial distribution is described by a Poisson distribu-tion. Indeed, traditionally reflectance is calculated for a totally ran-dom leaf distribution without any condition about leaf spatialrepartition which corresponds to the Poisson case (Ross, 1981).

When radiation strikes a leaf, the light is either absorbed orscattered (reflected or transmitted). The amount of each part is afunction of the size, the orientation, the refractive index and the me-sophyll structure of the leaf (Jacquemoud & Baret, 1990) as well asthe wavelength and the incidence angle of the incident radiation.Considering the scattering phenomenon, each leaf having differentinclination and azimuth angle has a different angular diagram ofscattering. The scattering diagram averaged over all leaf orientationsis called the bi-directional area scattering coefficient,w (Suits, 1972).It defines the infinitesimal radiance times π, dEo,1 in the view direc-tion Ωo(θo,φo) caused by scattering of direct transmitted irradiance,Es, from the directionΩs(θs,φs), within an infinitesimal layer of thick-ness dz,

dEodz

¼ wEs: ð29Þ

In general, leaves are assumed bi-Lambertian with hemisphericalreflectance ρ and transmittance τ. Let us consider a leaf inside thevegetation with a normal orientation Ωl(θl,φl) receiving the directtransmitted radiation flux from an incident direction Ωs(θs,φs) and

193A. Kallel, T. Nilson / Remote Sensing of Environment 130 (2013) 188–204

observed from direction Ωo(θo,φo). The bi-directional area scatteringcoefficient,wl, for this leaf normal orientation is derived as follows. IfEs is the flux received by a unit area of the horizontal layer dz,the amount of intercepted flux by foliage of orientation Ωv is givenby Es×(Ωs.Ωl/μs) multiplied by the total area of leaves per unit areain the infinitesimal layer, LAI/h×d z. As the leaf is assumedbi-Lambertian, the amount of scattered radiance is given by ρ or τdepending on the leaf normal orientation relative to the incidentand view directions. Finally the equivalent amount of “radiance”(“.” are added to point out that we deal with radiance times π) creat-ed by a horizontal layer is givenmultiplying the obtained radiance byΩo.Ωl/μv. It follows finally that

wl Ωs;Ωoð Þ ¼ LAIh

Ωs:Ωlð Þ: Ωl:Ωoð Þμsμo

� ρ; if Ωs:Ωlð Þ: Ωl:Ωoð Þ≥0;τ; otherwise:

ð30Þ

Assuming that the leaf orientation density function is given by afunction g(θl,φl) satisfying the normalizing condition

∫2π0 ∫1

0 g θl;φlð Þdμ ldφl ¼ 1; ð31Þ

with

μ l ¼ cos θlð Þ: ð32Þ

The global bi-directional area scattering coefficient, w, taking intoaccount the LIDF is therefore given by

w Ωs;Ωoð Þ ¼ ∫2π0 ∫1

0wl Ωs;Ωoð Þg θl;φlð Þdμ ldφl: ð33Þ

Next we will present the turbid case corresponding to a mediumwith null-size particles and for which the hot spot effect does notoccur followed by the discrete case. Particularly, we present the mod-ifications to be introduced when the finite-size leaves are considered.

2.3.1.1. Turbid case. In the following, the vegetation layer is assumedto be a turbid medium located between altitudes −h and 0. Let usconsider a thin vegetation layer of depth dz at altitude z∈ [−h, 0] re-ceiving solar radiation from direction Ωs and observed from direc-tion Ωo. The “radiance” exiting the thin layer as a function of theincident radiation is therefore given by w(Ωs,Ωo)d z. Moreover,when traveling from the top of the canopy until reaching the consid-ered thin layer, the incident radiation intensity decreases. Similarly,the created “radiance” decreases when traveling from the thin layerto the top. Both decreases of radiation/radiance in downward andupward directions are governed by the gap probabilities. They are re-spectively given by [cf. Eq. (15)]

pgap Ωs; zð Þ ¼ exp kzð Þ;pgap Ωo; zð Þ ¼ exp Kzð Þ; ð34Þ

where k and K are the extinction coefficients per unit length in thesun and sensor directions, respectively.

It follows that the elementary “radiance”, d Eo(0), created at thetop of the vegetation layer by scattering of the direct transmittedflux in the elementary layer, dz, is given by

dEo 0ð Þ ¼ pgap Ωo; zð Þw Ωs;Ωoð ÞdzEs zð Þ¼ pgap Ωo; zð Þw Ωs;Ωoð Þdzpgap Ωs; zð ÞEs 0ð Þ: ð35Þ

By integration over the altitude z, one can find the single scatter-ing “radiance” for the whole vegetation layer

Eo 0ð Þ ¼ Es 0ð Þw Ωs;Ωoð Þ∫0z¼−h pgap Ωo; zð Þpgap Ωs; zð Þdz

¼ Es 0ð Þw Ωs;Ωoð Þ1−exp −h kþ Kð Þ½ �kþ K

:ð36Þ

The reflectance due to single scattering contribution from foliage(ρso1 ) defined as the ratio between the top layer exiting radiance (with-out multiplication by π) and the radiance of a white Lambertian surfacein the same illumination and observation conditions (π−1Es(0)), istherefore given by

ρ1so ¼ w Ωs;Ωoð Þ1−exp −h kþ Kð Þ½ �

kþ K: ð37Þ

The last reflectance expression corresponds to the reflectance dueto single scattering contribution from foliage in the turbid case asshown in works of Suits (1972), Ross (1981) and Verhoef (1984), val-idated by Widlowski et al. (2006b) or Kallel et al. (2008).

The reflectance due to single scattering contribution from soil back-ground, ρso0 , corresponds to the ratio between the radiance created byradiation scattering by the soil and the radiance created by a whiteLambertian soil for the incident irradiance and the same observation di-rection. In this case, the sun radiation is transmitted through the vege-tation layer from the top to the bottom, scattered by the soil andtransmitted to the top without any interaction with canopy elements.The soil scattering is described by the soil bidirectional reflectance,rsoil, whereas the probabilities of upward and downward transmittanceare given by the gap probabilities pgap(Ωs, −h) and pgap(Ωv, −h),respectively. Therefore,

ρ0so ¼ rsoil Ωs→Ωoð Þpgap Ωo;−hð Þpgap Ωs;−hð Þ

¼ rsoil Ωs→Ωoð Þexp −h kþ Kð Þ½ �: ð38Þ

2.3.1.2. Discrete case. When the medium is assumed discrete, the re-flectances should be increased since the downward and upwardtransmittance probabilities are partly correlated as already explainedin Section 2. The bidirectional gap probability (Pgap(Ωs,Ωo,z)) shouldreplace the product pgap(Ωo,z)pgap(Ωs,z) in reflectance expressions[cf. Eqs. (38) and (39)] (Kuusk, 1985). It yields to

ρ1so ¼ w Ωs;Ωoð Þ∫0

−H Pgap Ωs;Ωo; zð Þdz;ρ0so ¼ rsoil Ωs→Ωoð ÞPgap Ωs;Ωo;−hð Þ:

ð39Þ

To model the bidirectional gap probability, Kuusk (1985) proposesthe decomposition of the vegetation layer into thin sub-layers assum-ing that gaps are independent from a sub-layer to another. Correla-tion between gap probabilities in sun and observation directionsinside a thin layer is modeled as a negative exponential function ofthe distance between downward and upward paths. In this case, thebidirectional gap probability (Pgap,ku) is given by

Pgap;ku Ωs;Ωo; zð Þ ¼ pgap Ωo; zð Þpgap Ωs; zð ÞCHS Ωs;Ωo; zð Þ; ð40Þ

with CHS a correction factor,

CHS Ωs;Ωo; zð Þ ¼ exp

ffiffiffiffiffiffikK

p

b1−exp bzð Þð Þ

" #; ð41Þ

where b is a function of the vegetation features, the different solid an-gles and the hot spot factor d defined as the ratio between the leaf di-ameter and the layer height (Kuusk, 1991).

In his dissertation, Verhoef (1998) shows that the termffiffiffiffiffiffikK

pcan

be interpreted as a sort of overlap between leaf projections on a hor-izontal plane for the direction of sun and observation. He proposes toestimate numerically the exact overlap parameter. Then compared tothe original Kuusk's model, a difference of about 2% was found formoderate canopy parameters. Although this method allows a theoret-ical derivation of the overlap function, it does not overcome the as-sumption of vertical independency between foliage which leads toreflectance underestimation.

194 A. Kallel, T. Nilson / Remote Sensing of Environment 130 (2013) 188–204

Verstraete et al. (1990) and Gobron et al. (1997) propose a differ-ent hot spot model. Their model is currently called the semi-discreteand it is based on cylinder intersection. In contrast to our approach,the cylinder volume is not used directly to derive the gap probability,but a corrective term is added to the turbid bidirectional gap proba-bility. The obtained bidirectional gap probability Pgap,Discrete(Ωs,Ωv,z)is given by

Pgap;Discrete Ωs;Ωo; zð Þ ¼ exp λ zð Þ kþ Kð Þz½ �; ð42Þ

where λ is the correction term

λ zð Þ ¼ 1−ϑs zð Þ∩ϑo zð Þϑo zð Þ ; ð43Þ

where ϑs(z) and ϑo(z) are cylinders free of scatterers in the directionsof sun and observation, respectively. These cylinders have the sameradius which is the mean sun fleck radius. The bidirectional gap prob-ability presented in Eq. (43) could be the same as the one in ourmodel, (cf. Eq. (28), Poisson case) if ϑs(z)=Vs(z) and ϑo(z)=Vo(z).However, the cylinder definitions in these two approaches arecompletely different. In our case, the cylinders are defined as the min-imal volumes not containing leaf centers and allowing the given pointto be sunlit and observed. Therefore the probability that both cylin-ders are free from leaf centers is no other than the bidirectional gapprobability. Conversely, as defined by Verstraete et al. (1990) andGobron et al. (1997), the cylinder bases correspond to the mean sunfleck area within the vegetation (average over depth and orientation).In terms of complexity, their cylinder bases do not vary as a functionof the sun or sensor directions, therefore ϑs(z) and ϑo(z) have thesame radii and their intersection computation is easier than in ourcase. Moreover in physical point of view, sun fleck size and probabil-ity of gap do not have the same meaning and the expression given byVerstraete et al. (1990) and Gobron et al. (1997) can be just viewed asan approximation of the bidirectional gap probability.

The two hot spot models (i.e., Kuusk's model and the Semi-Discrete model) presented in this section will be compared to ourmodel by numerical simulations in Section 3. For an extensive largeoverview about the different vegetation hot spot models, the readersare referred to Qin and Goel (1995).

2.3.2. Our model reflectanceIn the Poisson case, our model reflectance fits with the expres-

sions in Eq. (40). Therefore, the reflectances due to single scatteringcontribution from foliage and soil background, denoted respectivelyby ρso,poi

1 and ρso,poi0 , are given by [cf. Eq. (28)]

ρ1so;poi ¼ w Ωs;Ωoð Þ∫0

−h Pgap;poi Ωs;Ωo; zð Þdz¼ w Ωs;Ωoð Þ∫0

−h exp −nlvso zð Þð Þdz;ρ0so;poi ¼ rsoil Ωs→Ωoð ÞPgap;poi Ωs;Ωo;−hð Þ

¼ rsoil Ωs→Ωoð Þexp −nlvso −hð Þð Þ:

ð44Þ

Let us derive the reflectance for the case of the Binomial distribu-tions. According to Eq. (18), for Δv>0, k−bkbk+ which means thatthe probabilities of radiation interception and therefore scatteringfor positive (respectively negative) Binomial distribution are higher(respectively lower) than these probabilities in the Poisson case. Itimplies that the positive (respectively negative) Binomial distributioncreates more (respectively, less) diffuse flux than the Poisson one. Itfollows that w−bwbw+, where w+ and w− are the bi-directionalarea scattering coefficient relative to the positive and negative Bino-mial distributions, respectively. Moreover, as the scattering intensityis linearly linked to the intercepted radiation (Kallel et al., 2008),the bi-directional area scattering coefficient is proportional to the ex-tinction coefficient. It follows that the increase (respectively, de-crease) of the extinction in the regular (respectively, clumped) case

by the factor nl;þnl

(respectively, nl;−nl) [cf. Eq. (18)], leads to an increase

(respectively, decrease) of the bi-directional scattering coefficientby the same factor, therefore

wþ ¼ nl;þ Δvð Þnl

w;

w− ¼ nl;− Δvð Þnl

w:

ð45Þ

The reflectances due to single scattering contribution from foliagein the Binomial cases can be therefore deduced from the Poisson oneEq. (28)

ρ1so;þ ¼ wþ Ωs;Ωoð Þ∫0

z¼−h Pgap;þ Ωs;Ωo; zð Þdz;ρ1so;− ¼ w− Ωs;Ωoð Þ∫0

z¼−h Pgap;− Ωs;Ωo; zð Þdz: ð46Þ

The reflectances due to single scattering contribution from soilbackground in the Binomial cases are given by

ρ0so;þ ¼ rsoil Ωs;Ωoð ÞPgap;þ Ωs;Ωo;−hð Þ;

ρ0so;− ¼ rsoil Ωs;Ωoð ÞPgap;− Ωs;Ωo;−hð Þ: ð47Þ

To estimate the integrals in the single scattering from foliage reflec-tance expression [cf. Eqs. (45) and (47)], we adopt the same numericalsolution as proposed by Verhoef (1998); details are given in AppendixA.2. Such processing is complex since it includes an iterative algorithmas it divides the vegetation layer into 60 sublayers to approximate theintegration over the depth [−h, 0]. In addition, in each iteration a nu-merical estimation of the volume vso [cf. Eq. (27)] based on elliptic inte-gral codes (Abramowitz & Stegun, 1964) is needed. However, despitethese complex computations, the running time is very short: few milli-seconds (Intel Centrino Laptop).

Note that when the leaf radius (rl) tends to zero, all the estimatedreflectance tends to the turbid case as proved in Appendix B.

Note finally, as for regular (respectively, clumped) distributionkþ ¼ nl;þ

nlk andwþ ¼ nl;þ

nlw (respectively, k− ¼ nl;−

nlk andw− ¼ nl;−

nlw), for Bi-

nomial leaf spatial distribution, the estimation of the reflectance dueto single scattering contribution boils down to the estimation of thereflectance in the Poisson case considering an apparent LAI value,LAI+ and LAI− for regular and clumped distribution, respectively,they are given by,

LAIþ ¼ nl;þ Δvð Þnl

LAI;

LAI− ¼ nl;− Δvð Þnl

LAI:ð48Þ

3. Results of numerical experiments

Below our model is denoted by LSDM (acronym of Leaf Spatial Dis-tribution based Model). In this section, the canopy reflectances andalbedo simulated by different versions of the LSDM, correspondingto the different leaf spatial distributions are inter-compared andthe impacts of regularity and clumping of the leaf distribution arestudied. Also, the LSDM simulation results are compared to those pro-vided by the Kuusk and Semi-Discrete models. Then the validation ofthe model is done using the ROMC web-based tool (Widlowski et al.,2008).

In our experiments, the leaf orientation density, results of numer-ical experiments, is assumed uniformly distributed on azimuthangle. For the sake of simplicity, results of numerical experimentsare integrated over azimuth angle and replaced by the leaf inclina-tion distribution function (LIDF), noted results of numerical experi-ments (Verhoef, 1984). The Bunnik parametrization is used to

195A. Kallel, T. Nilson / Remote Sensing of Environment 130 (2013) 188–204

model the =LIDF. So, the distribution (fl) as a function of the leafnormal's zenith angle (θl) is given by

f l θlð Þ ¼ 2π

al þ blcos 2clθlð Þð Þ þ dlsin θlð Þ; ð49Þ

with al=1, bl=1, cl=1 and dl=0 for a planophile distribution, al=1, bl=−1, cl=1 and dl=0 for an erectophile distribution, and al=0,bl=0, cl=0 and dl=1 for a uniform distribution. It is noted that theterm dlsin(θl) was not introduced by Bunnik (1978) but added re-cently by researchers from the Earth Observation Science team ofthe Joint Research Center (JRC) institute to model the uniform distri-bution (i.e., fl(θl)= sin(θl)). However, such distribution was namedspherical by a lot of other researchers in the community [e.g., Verhoef(1985); Wang et al. (2007)] as the frequency of leaf inclination is thesame as for surface elements of a sphere. Moreover, the uniform distri-bution as reported by Verhoef (1984), is given by f l θlð Þ ¼ 2

π, because inthis case all the leaf inclinations have the same frequency. Anyway aswe will use the JRC ROMC database (Widlowski et al., 2008) for valida-tion, we will adopt their notation.

3.1. Model comparison

In addition to be compared to the Kuusk model and NADIM code(i.e. Gobron et al.'s semi-discrete code), the simulations by ourmodel with different leaf spatial distributions are inter-compared.

(c) Principle plane, erectophile LIDF

9060300-30-60-90

0.4

0.3

0.2

LSDMNADIMKuusk

(a) Principle plane, planophile LIDF

θo

θo

ρ1 soρ1 so

9060300-30-60-90

0.4

0.3

0.2

Fig. 2. Single scattering from foliage reflectance as a function of the view zenith angle in thLAI=5, θs=20° and rl=0.05 m.

3.1.1. Comparison with Kuusk and Semi-DiscreteFig. 2 shows the results of simulation of the reflectance due to

scattering contribution from foliage. Its angular variations in theprincipal and cross planes are shown for planophile and erectophileLIDF, using the LSDM, Kuusk as well as NADIM models. The consid-ered LSDM leaf distribution in this case is the Poisson one becauseboth the Kuusk and NADIM models assume totally random foliagedistribution.

The comparison between the two LIDF shows that the reflectanceis higher in the planophile case than in the erectophile one. Indeed,the more foliage is horizontal, the more it scatters light when thesun is high above the horizon. The hot spot peak is shown in the prin-cipal plane for θo values close to θs. Moreover, the hot spot peak islarger in the planophile case than the erectophile one since the mu-tual shadowing between leaves is higher for more horizontal leaves(Kallel, 2010b; Kallel et al., 2008).

The Kuusk model reflectances are lower than the LSDM ones. Thesimulation in the principal plane shows that the hot spot peak is thenarrowest for the Kuusk model mainly in the erectophile case. In-deed, the Kuusk model takes into account the light path (downwardbefore scattering and upward after scattering) correlation only inhorizontal thin sublayers. This assumption is adequate only whenleaves are horizontal. However, leaves are not necessarily horizontal.Vertical leaves cause path dependency in non-horizontal directions,too, which means that Kuusk model underestimates the path corre-lation and so the single scattering from foliage reflectance. This prob-lem is more pronounced in the erectophile case since foliage normal

(d) Cross plane, erectophile LIDF

9060300-30-60-90

0.4

0.3

0.2

(b) Cross plane, planophile LIDF

9060300-30-60-90

0.4

0.3

0.2

θo

θo

ρ1 soρ1 so

e principal and cross planes. Both leaf reflectance and transmittance are equal to 0.5,

196 A. Kallel, T. Nilson / Remote Sensing of Environment 130 (2013) 188–204

orientations are closer to the horizontal (i.e. more vertical leaves)than in the planophile case. NADIM-simulated curves are differentfrom the LSDM curves, too. In particular, NADIM-simulated curvesare the highest, particularly in the planophile case, and are veryclose to the LSDM with regular distribution with Δv=5×10−4. Infact, NADIM uses a different foliage parametrization, namely the cyl-inder radius is taken equal to the mean sun fleck size (Gobron et al.,1997). In general, the sun fleck radius is larger than the radii of thecylinders in direction of sun and observation [rs and ro, respectively,cf. Eq. (11)] as considered by the LSDM model, since for this modelthe radiuses are defined as the minimal distances between a givenpoint inside the vegetation and leaf centers allowing to this pointto be sunlit and observed, respectively. The relative contribution ofthe cylinders' common volume considered by the NADIM code istherefore larger than in the LSDM one which implies that NADIM bi-directional gap probability is higher than the LSDM one.

Fig. 3 shows the variation of the albedo due to scattering contribu-tion from foliage, ρsd1 , as a function of LAI for the compared models fortwo LIDF, erectophile and planophile, and for two hot spot sizes 0.05and 0.1. Remember that,

ρ1sd Ωsð Þ ¼ 1

π∫2πφo¼0∫

1μo¼0 ρ

1so Ωs→Ωoð Þcos θoð Þdμodφo: ð50Þ

Similarly to the bidirectional reflectance, the simulated albedo forthe erectophile leaves is smaller than the one for planophile leaves.LSDMwith Poisson distribution, Kuusk andNADIMmodel-simulated al-bedo increase as a function of LAI, since the increase in LAI is an increase

NADIMLSDMKuusk

(a) rl = 0.05 & planophile LIDF

(c) rl = 0.05 & erectophile LIDF

LAI

ρ1 sd

7654321

0.3

0.2

0.1

LAI

ρ1 sd

7654321

0.3

0.2

0.1

Fig. 3. Single scattering contribution from foliage directional–hemispherical reflectance as aLSDM with Poisson distribution models. Both leaf reflectance and transmittance are equal t

in the number of scatterers. For these models, the albedo increases as afunction of the leaf size, too. Indeed, remember that to be sunlit and ob-served, each pointM should belong to the cylinders in sun and observa-tion directions (cf. Fig. 1). Along with an increase in leaf size, thecylinder diameters and the percentage of the cylinder common volume[cf. Eq. (21)] increase. Now, the hot spot correction term in the bidirec-tional gap probability, λ(z) as defined by Eq. (43) (also valid for LSDM),is a decreasing function of the common volume, which means that thereflectance is an increasing function of the common volume and there-fore it is an increasing function of rl. Besides, as in Fig. 2, the comparisonbetween these models shows that the Kuusk model-simulated curvesare the lowest and those by NADIM are the highest. These differencesincrease together with increasing LAI and rl. Indeed, for low LAI values,the reflectance depends mainly on the well-exposed leaves for whichthe bidirectional gap probability is almost equal to one for all themodels, which explains the relatively close reflectance simulations inthis case. Conversely, at high LAI values an important proportion ofthe reflectance is provided from photon collisions within deep layers(i.e., collisions with leaves located under a large number of leaves),thus the differences in the simulation of the bidirectional gap probabil-ity become more evident, explaining the simulation divergence be-tween the models. Another reason of divergence between NADIM andLSDM for high LAI values could be the coarse discretization of thesemi-discrete equations using the NADIM code decreasing thereforeits accuracy for deep layers (LAI>5). Indeed, this code was developedat a time when computing constraints were an issue. In terms of varia-tion as a function of the hot spot size, plots show that for small rl valuesall the model reflectances converge to the turbid medium reflectance,

(b) rl = 0.1 & planophile LIDF

(d) rl = 0.1 & erectophile LIDF

LAI

ρ1 sd

7654321

0.3

0.2

0.1

LAI

ρ1 sd

7654321

0.3

0.2

0.1

function of LAI (varying between 0.1 and 7) for the Kuusk model, the NADIM code, theo 0.5 and θs=20°.

197A. Kallel, T. Nilson / Remote Sensing of Environment 130 (2013) 188–204

large rl values cause larger differences in the simulated bidirectional gapprobabilities and larger differences in simulated albedo.

3.1.2. Leaf distribution comparisonIn this section the impact of the regularity and the clumping on a

vegetation layer reflectance is studied. As in the last section, the BRDFand the directional–hemispherical are studied. Now, as the curveshapes remains the same when the Binomial distribution are consid-ered (just a reflectance increasing and decreasing for the regular andclumped distribution, respectively), we will focus here on the impactof regularity/clumping on reflectance. We propose to study a normal-ized difference parameter, named Δx…, with …∈{+,−} for positiveand negative Binomial distributions, respectively and x∈ρso1 , ρsd1

when the BRDF and the directional–hemispherical reflectances areconsidered, respectively. Δx is given by

Δx… ¼ x…−xpoixpoi

: ð51Þ

Fig. 4 shows the same simulation experiments as presented inFig. 4 except the leaf distribution that is assumed Binomial, Δρso1 isplotted in this case for the different leaf distribution parameter Δv.In comparison between different LSDM versions (Poisson, regularand clumped foliage distributions), Fig. 4 shows that the reflectanceis the lowest (Δρso,−1 b0) for the clumped distribution and the highestfor the regular one (Δρso,+1 >0). Moreover, the more regular the dis-tribution is (Δv increases), the higher the reflectance is. Conversely,the more clumped is the distribution (Δv increases), the lower the re-flectance is. This variation can be explained as follows. Compared tothe Poisson case, when a clumped (respectively, regular) leaf distribu-tion is considered, the sun radiation penetrates deeper (respectively,shallower) into the canopywhich implies that the probability of photonto be scattered back is less (respectively, higher).

Before interpreting the angular variation, let us remember that inthe turbid case and for positive and negative Binomial leaf distributions,

(c) Principle plane, erectophile LIDF

Cl Δ υ = 10−3Rg Δ υ = 10−3

Cl Δ υ = 5 . 10−4Rg Δ υ = 5 . 10−4

(a) Principle plane, planophile LIDF

θo

Δρ1 so

9060300-30-60-90

0.05

0

-0.05

θo

Δρ1 so

9060300-30-60-90

0.05

0

-0.05

Fig. 4. Δρso1 as a function of the view zenith angle in the principal and cross planes. Both leaf rm3.

the BRDF (ρso,+1 and ρso,−1 , respectively) are given by [cf. Eqs. (38) and(47)]

ρ1so;… ¼

w… Ωs;Ωo

� �k…þ K…ð Þh� 1− exp −h k…þ K…ð Þ½ �ð Þ

¼w Ωs;Ωo

� �kþ Kð Þh � 1− exp −h k…þ K…ð Þ½ �ð Þ:

ð52Þ

where … ∈ {+,−}.Note that, for Eq. (52) the first part of the formula does not depend

on the leaf distribution.Fig. 4 shows that for high sensor inclination angle (θ0≈90°), all

Δρso,…1 values are close to zero which means that the dependencyon leaf distribution is low. Remember that in this case the observa-tion direction is far from the sun direction, which means that it ispossible to approximate the BRDF using Eq. (52). Moreover, for oneside, it is clear that the difference between the different formula isgiven by (1−exp[−h(k … +K …)]), from the other side for suchhigh inclination μo approaches 0 and therefore K… approaches infin-ity [cf. Eq. (16)], therefore all the formula differences approach 1 thatexplains the convergence of the different reflectances to the samevalue and Δρso,…

1 to 0. After that, when the inclination decreases,the hot spot contribution becomes non-negligible and differentfrom a leaf distribution to another, i.e. wide peak for planophileand narrow peak for erectophile, and the corresponding reflectancediverge accordingly. Indeed, remember that for given directions ofsun and observation, the hot spot effect contribution depends onthe pourcentage of common volume between cylinders free fromleaf centers [cf. Eq. (28)]. Moreover, the wider the cylinders, themore important is the pourcentage of common volume. Now, asthe cylinder bases are linearly linked to the extinction [cf. Eq. (16)]and k−bkbk+, it implies narrow hot spot peak for the clumped dis-tribution (low reflectance) and wide hot spot peak for the regulardistribution (high reflectance) which explains such absolute value

(d) Cross plane, erectophile LIDF

(b) Cross plane, planophile LIDF

θo

Δρ1 so

9060300-30-60-90

0.05

0

-0.05

θo

Δρ1 so

9060300-30-60-90

0.05

0

-0.05

eflectance and transmittance are equal to 0.5, LAI=5, θs=20° and rl=0.05 m. Δv unit is

198 A. Kallel, T. Nilson / Remote Sensing of Environment 130 (2013) 188–204

of Δρso,…1 increasing. This norm is larger for planophile thanerectophile leaf distribution because beam interception (for highsun elevation) is more important for more horizontal leaves(planophile) which implies higher extinction for planophile distri-bution and therefore wider hot spot peak. Close to the direction ofbackscattering, Δρso …

1 decreases again mainly in the planophilecase. Again, this phenomenon could be explained by the width ofthe hot spot region, indeed close to the backscattering region the cyl-inders free from leaf centers in direction of sun and sensor becomeclose and the pourcentage of common volume increases accordinglyto reach 100% in the exact backscattering direction independentlyfrom the cylinder sizes. It implies that the reflectance difference

Clumped, Δ v = 10− 3Regular, Δ v = 10− 3

Clumped, Δ v = 5 . 10− 4Regular, Δ v = 5 . 10− 4

Clumped, Δ v = 3 . 10− 4Regular, Δ v = 3 . 10− 4

(a) rl = 0.05 & planophile LIDF & without HS

(e) rl = 0.1 & planophile LIDF & without HS

(c) rl = 0.05 & erectophile LIDF & without HS

(g) rl = 0.1 & erectophile LIDF & without HS

LAI

Δρ1 sd

7654321

0.1

0.05

0

-0.05

LAI

Δρ1 sd

7654321

0.1

0.05

0

-0.05

LAI

Δρ1 sd

7654321

0.1

0.05

0

-0.05

LAI

Δρ1 sd

7654321

0.1

0.05

0

-0.05

Fig. 5. Δρsd1 as a function of LAI (varying between 0.1 and 7) for the LSDM Poisson, positive anhot spot effect and the right side plots take it into account. Both leaf reflectance and transm

due to the hot spot contribution to the reflectance is reduced closeto the backscattering direction which explains the norm of Δρso …

1

decreasing. This effect is more pronounced in the planophile case be-cause the extinction is the largest and therefore the cylinders freefrom leaf centers are the widest in this case, therefore the commonvolume pourcentage is themost important in this case which impliesa fast convergence to 100%.

Note that if the hot spot effect is not taken into account Δρso …1 re-

mains close to zero in both principle and cross planes.Fig. 5 plots the variation of Δρsd

1 as a function of LAI for differ-ent LIDF and leaf sizes. To separate the contribution of the leafspatial distribution from the hot spot effect, plots on the left

(b) rl = 0.05 & planophile LIDF

(f) rl = 0.1 & planophile LIDF

(d) rl = 0.05 & erectophile LIDF

(h) rl = 0.1 & erectophile LIDF

LAIΔ

ρ1 sd7654321

0.1

0.05

0

-0.05

LAI

Δρ1 sd

7654321

0.1

0.05

0

-0.05

LAI

Δρ1 sd

7654321

0.1

0.05

0

-0.05

LAI

Δρ1 sd

7654321

0.1

0.05

0

-0.05

d negative Binomial distribution models. The left side plots do not take into account theittance are equal to 0.5 and θs=20°. Δv unit is m3.

Table 1List of the ROMC experiments (the experiment HOM03 is divided up into HOM03a andHOM03b).

Experiment θs Vegetation LAI LIDF Wavelength

HOM01 0°, 30°, 60° TUR 1 ERE, PLA NR1HOM11 0°, 30°, 60° DIS 1 ERE, PLA NR1HOM02 0°, 30°, 60° TUR 2 ERE, PLA NR1HOM12 0°, 30°, 60° DIS 2 ERE, PLA NR1HOM03a 20°, 50° TUR 3 PLA, UNI NIR,REDHOM13 20°, 50° DIS 3 PLA, UNI NIR,REDHOM03b 20°, 50° DIS 3 ERE NIR,REDHOM05 0°, 30°, 60° TUR 5 ERE, PLA NR1HOM15 0°, 30°, 60° DIS 5 ERE, PLA NR1

(a) HOM11_DIS_PLA_NR1 (b) HOM11_DIS_ERE_NR1

(c) HOM12_DIS_PLA_NR1 (d) HOM12_DIS_ERE_NR1

(e) HOM15_DIS_PLA_NR1 (f) HOM15_DIS_ERE_NR1

Fig. 6. Single scattering from foliage reflectance variation in the principal plane (brfpp_co_sgl) for LSDM, NADIM and Kuusk models as well as the ROMC reference (ROMCREF). θs=30° and θo variation covers the range −75° to 75° with 2° intervals. The simulation names are explained in Table 1. Different gray levels show the 1%, 2.5% and 5% deviations fromthe ROMCREF mean reflectance.

199A. Kallel, T. Nilson / Remote Sensing of Environment 130 (2013) 188–204

size present the same experiments as those on the right side butwithout taking into account the hot spot effect. Thus, the simula-tions without hot spot effect are firstly studied and the conclu-sions about the impact of foliage clumping/regularity are given,then the hot spot effect is added and its impact on reflectanceis analyzed.

For low LAI values, all Δρsd1 values approach zero whichmeans thatthe reflectances are the same for the different leaf distributions. In-deed, for low LAI values, nl is low [cf. Eq. (1)], nl, + and nl, − approachnl [cf. Eq. (14)] and thus Pgap,+ and Pgap,− (respectively, w+ and w−)approach Pgap,poi [cf. Eq. (18)] (respectively, w [cf. Eq. (46)]). There-fore, ρso,+1 and ρso,−1 approach ρso1 .

(a) Turbid (b) Discrete

Fig. 7. Single scattering from foliage reflectance distribution in the principal plane as simulated by the LSDM, NADIM and Kuusk models compared with the ROMC reference(ROMCREF) reflectance. The considered experiments for sub-figure (a) [sub-figure (b), respectively] are the six turbid (discrete, respectively) simulations HOM01, HOM03 andHOM05 (HOM11, HOM13 and HOM15, respectively) for the two LIDF: planophile and erectophile and a unique sun zenith angle: θs=30°. For simulation names see Table 1.

200 A. Kallel, T. Nilson / Remote Sensing of Environment 130 (2013) 188–204

Then, when LAI increases the plots show that first scattering albedois the lowest for the clumped distribution (Δρsd,−1 b0) and the highestfor the regular one (Δρsd,+1 >0). Also, the albedo decreases (respectivelyincreases) along with an increase in Δv for the clumped (respectivelythe regular) distribution. The comparison between the leaf sizes showsthat for small leaf size, the norm of Δρsd1 is the highest, which meanthat for small leaves the reflectance corresponding to the different fo-liage spatial distributions is far from each other. In fact, on one hand,the regular (respectively clumped) distribution depends on the numberof leaves per m2 (nl), i.e., more leaves imply more regular (respectivelyclumped) spatial pattern. On the other hand, at a fixed LAI, the numberof leaves is inversely proportional to leaf size. So, the dependence ofLSDM-simulated albedo on Δv is higher for small leaves (rl=0.05)than for big leaves (rl=0.1). Indeed, the increase in LAI implies an in-crease in nl, particularly when nl approaches Δv−1, nl,+ approaches in-finity (nl,+ is not defined for nl≥Δv−1) and when nl approachesinfinity, nl,−/nl approaches zero. Therefore, the increase of LAI leads toan increase (respectively, decrease) of the apparent LAI+ (respectively,LAI−) [cf. Eq. (49)] which explains such a huge increase of the norm ofΔρsd1 for a low rl value. After this increase, the norm of Δρsd1 decreasesagain and approaches zero. In fact, when LAI increases, k… and K… ap-proach infinity and all the reflectance ρso,…1 converges towards w Ωs ;Ωoð Þ

kþKð Þh(that is independent from both leaf spatial distribution and LAI).

When the hot spot is taken into account, we see that the curvesare almost similar to those without hot spot for low LAI values(LAI≤2), but when LAI increases and when the hot spot effect istaken into account the norm of Δρsd1 increases again. Such effect issimilar to the shown phenomenon when the BRDF is studied at thebeginning of this subsection. In fact, this huge difference of reflec-tance simulations between the different spatial distributions isexplained by the increase of the hot spot effect contribution to reflec-tance and the difference between the peak width, i.e. huge (respec-tively, narrow) peak for regular (respectively, clumped) distributionwhich leads to an increase (respectively, decrease) of reflectance incomparison with the Poisson case.

3.2. Validation

The validation of the proposed model is performed using the RAMIOn-line Model Checker (ROMC) exercise (Widlowski et al., 2008)which is a web-based tool allowing the evaluation of the canopy RT

models. The evaluation consists of comparing the tested model reflec-tances to the results of simulation by reference models provided bymodel benchmarking within the frames of the third RAdiation transferModel Intercomparison (RAMI) (Widlowski et al., 2006b). The referenceis estimated based on the surrogate truth 3D Monte Carlo models suchas DART (Gastellu-Etchegorry et al., 1996), FLIGHT (North, 1996) andRayspread (Widlowski et al., 2006a) which show generally very closeresults to each other.

In the ROMC present version, three canopy structure cases arepresented: homogeneous (HOM), heterogeneous (HET) and a combina-tion between them (HETHOM). For each case, two vegetation kinds areassumed: turbid (TUR) and discrete (DIS). The leaf radius is assumed tobe zero for the turbid case and equal to 0.05 m for the discrete one. Twowavelengths (red andnear-infrared) and a purist corner cases are consid-ered, they are noted RED, NIR and NR1, respectively. For these cases, theleaf reflectance and transmittance (ρ, τ) values (0.0546, 0.0149),(0.4957, 0.4409) and (0.5, 0.5) are assumed, respectively. For thesethree cases, the soil reflectance is assumed to be equal to 0.127, 0.159and1, respectively. Three LIDFs are considered: planophile (PLA), uniform(UNI) and erectophile (ERE). Table 1 shows the nine related simulationexperiments tested by ROMC. Moreover, ROMC proposes eleven mea-sures such as the single scattering contribution from foliage, from soilbackground, multiple-scattered and total BRDF in principal and crossplanes as well as the spectral albedo of the canopy, i.e., the directional–hemispherical reflectance and foliage absorption, etc. Moreover, a lot ofnew measures were recently added in the RAMI-IV exercise.

As the canopy heterogeneity is not modeled in our algorithm, thepresent study deals with homogeneous vegetation only. Moreover,since we are interested in the validation of a hot spot model, only mea-sures dealing with the single scattering BRDFs are presented. Althoughthe hot spot appears only in the discrete medium case, simulations forthe turbid case are presented to study the impact of the hot spot effect.

In the following, some detailed results concerning the purist cor-ner case are presented and then global results about all numerical ex-periments are shown.

As only the single scattering reflectances are considered, the resultsof purist corner, near-infrared or red domains are almost equivalent. Infact, for all of them the bidirectional gap probability is the same (since itdoes not depend on leaf albedo) and the difference between them con-cerns the bi-directional area scattering coefficient, w [cf. Eqs. (47) and(48)]. We test the six possible experiments, corresponding to three

(a) Single scattering from foliage reflectance, Principle plane

(b) Single scattering from foliage reflectance, Cross plane

(c) Single scattering from soil background reflectance, Principle plane

(d) Single scattering from soil background reflectance, Cross plane

Fig. 8. Reflectance simulations of the LSDM, NADIM and Kuusk models plotted against the ROMC reference (ROMCREF) reflectance for the twenty six discrete ROMC experimentspresented in Table 1.

201A. Kallel, T. Nilson / Remote Sensing of Environment 130 (2013) 188–204

values of LAI: 1, 2 and 5 and two leaf distributions: planophile anderectophile. They correspond to the experiments HOM11, HOM12 andHOM15 as presented in Table 1. To be short, only one sun zenith angleis considered (θs=30°).

Fig. 6 shows the single scattering from foliage reflectance variationin the principal plane for LSDM, NADIM and Kuusk as well as theROMC reference (ROMCREF) model. Due to the discretization stepof the observation zenith angle (θo) in ROMC (2°), the reflectance inthe exact backscattering direction is not shown. Indeed, (θo) couldnot be equal to (θs=30°). By comparing Fig. 6a, c, e or b, d, f, wecan see that the reflectance increases along with an increase in LAI.Moreover, the reflectance in the planophile case is higher than theerectophile case as already explained in Section 1. The hot spot peakis the highest for the largest LAI values and also in the planophilecase it is wider than the erectophile one. In fact, for such a high sunelevation (θs=30°), the effective sunlit leaf area (i.e. the area of theprojected leaf in a plane normal to the sun direction) is the highestin the planophile case which yields to the largest mutual shadowing(cf. Section 1).

In order to make a deeper comparison study, we test the global per-formance of the models in both the turbid and the discrete cases. Fig. 7shows the single scattering from foliage reflectance simulated by theLSDM, NADIM and Kuusk models plotted against the referenceROMCREF reflectances for the six experiments in the discrete mediumcase presented in Fig. 6 (cf. Fig. 7.b) and the corresponding simulationsin the turbid case (cf. Fig. 7.a). The closer to the 1:1 line the results are,the closer to the reference ROMCREF reflectances the simulations areand therefore could be considered better. The difference betweenFig. 7a and b is the hot spot effect. Fig. 7.a shows that all the modelsgive rather similar results. Particularly, LSDM and Kuusk are equivalentand provide very small error (RMS=9×10−4). When the hot spot istaken into account, themodel errors increase in general. The best resultsare by LSDM with RMS=4.3×10−3, while for NADIM RMS increasesslightly to reach 5.9×10−3. It is noted here that such a performance dif-ference could not be extremely significant if taking into account thatROMCREF simulations are not totally free from error. Due to the hotspot peak underestimation (cf. Section 1), Kuusk model performancesare largely decreased from the turbid to the discrete case with RMS=

202 A. Kallel, T. Nilson / Remote Sensing of Environment 130 (2013) 188–204

0.0119. In terms of angular variation in the principal plane, Fig. 6 showsthat our model is always inside the ROMCREF gray accuracy region(error lower than 5%). It can be also noticed that as the hot spotpeak is narrower in the erectophile case, the error is lower in thiscase. Besides, compared to Kuusk's model, we see that, in general,our model performs better. The hot spot peak width is generallyunderestimated by the Kuusk's model. This phenomenon is morepronounced in the erectophile case as already explained inSection 1. NADIM code provides better results than Kuusk's model,its curves are slightly higher than the LSDM ones but in general with-in the ROMCREF gray region. However, NADIM simulation is justabove the gray region close to the hot spot area for experimentHOM15_DIS_PLA_NR1 (cf. Fig. 6.e).

Fig. 8 shows the variation of the LSDM, NADIM and Kuusk modelsingle scattering reflectances versus those simulated by ROMCREF inthe principal and cross planes for all the discrete medium experi-ments presented in Table 1. In general, we see that LSDM is the closestto the 1:1 line. In terms of global performance, LSDM RMS is alwayslower than for the others. NADIM and Kuusk model results are gen-erally above and below the 1:1 line, respectively, leading that theyoverestimate and underestimate the hot spot peak, respectively.NADIM and Kuusk's model RMS comparison shows that NADIM (re-spectively Kuusk) provides better estimation of the single scatteringfrom foliage (respectively from soil background) reflectance.

When a bias occurs between the actual reflectance and simulation,the hot spot parameter (rl) can be adjusted in order to minimize thebias. In particular, the Kuusk's model has been used by Kallel(2010b) and Kallel (2012) to model the hot spot effect and rl wastaken equal to 1.5 and 3 times the leaf radius for planophile anderectophile leaf distributions, respectively. Using, LSDM such calibra-tion is not needed. However, for actual vegetation canopies, leaves arenot necessarily disk-shaped and do not have the same size. In thiscase, and for all the models, one has to calibrate the radius of the as-sumed disk-shaped leaf in order to represent a certain reality. How-ever, even if the hot spot parameter adjustment is needed for all themodels, less parameter adjustment is needed for better physicallybased models. For instance, the adjustment for LSDM does not de-pend on the leaf orientation distribution.

4. Conclusion

A physical hot spot model called LSDM was presented in thispaper. A point within the vegetation canopy is sunlit or observed, ifit belongs to a certain cylinder free from leaf centers. The base ofthe cylinder is centered on the considered point and axis orientedto the sun and observation directions, respectively. It was shownthat the bidirectional gap probability for the point to be sunlit andobserved is none other than the probability that both cylinders arefree from leaf centers at the same time. To a great extent, the hotspot feature is determined by the common volume of these twocylinders. A new algorithm to calculate the common volume of thecylinders via elliptic integrals was proposed. In addition to thePoisson leaf spatial distribution as assumed by classical hot spotmodels, regular (positive Binomial distribution) or clumped (nega-tive Binomial distribution) distributions were modeled usingLSDM. Different model versions corresponding to each distributionwere compared. Particularly, it was shown that compared to thePoisson distribution, the regular (respectively the clumped) distri-bution increases (respectively decreases) the single scattering fromfoliage reflectance, since it increases (respectively decreases) the ap-parent LAI value.

The model validationwas done based on the ROMCweb-based tool.The two single scattering reflectances as simulated by the LSDMmodelwith Poisson distribution, NADIM code (corresponding to the Semi-Discrete model) and the Kuusk model were compared with the ROMCreference simulations. Almost for all the experiments, our model

showed the best agreement with the reference. In general, Kuuskmodel underestimated the hot spot peak whereas NADIM code slightlyoverestimated the reflectance.

Future research will deal with the integration of our approach inthe FDM radiative transfer model (Kallel, 2010a, 2010b, 2012) inorder to enhance its performances. In this case, a decomposition ofthe LSDM fluxes into a number of virtual sub-fluxes with constantextinctions is needed. Also, the development of new RTM for Binomi-al leaf spatial distribution is among our perspectives.

Acknowledgments

The authors thank the Estonian Research Foundation for a post-doctoral grant (2008/2009). Sincere thanks are also extended toA. Kuusk for numerous constructive discussions and J. Pisek forEnglish-language improvement. The authors also wish to thankN. Gobron for making the NADIM code available online. Finally, theauthors would like to thank the anonymous reviewers for their valu-able comments and suggestions to improve the quality of the paper.

Appendix A. Integral numerical computation

Appendix A.1. vso computation

Assume in the following that rs≤ ro.To estimate vso, all the integrals in Eq. (27) should be computed.

As solution, we propose here to transform them (integration by sub-stitution) to elliptic integrals (Abramowitz & Stegun, 1964). Twokinds will be used,

F�ϕ mj Þ ¼ ∫ϕ

0dϕ′ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1−m2 sin2 ϕ′� �q

E�ϕ mj Þ ¼ ∫ϕ

0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1−m2 sin2 ϕ′

� �qdϕ′

;

ðA:1Þ

with F and E the first and second kinds of the incomplete elliptic inte-gral, respectively (Abramowitz & Stegun, 1964). Optimal numericalcodes to estimate these integrals are proposed in Abramowitz andStegun (1964).

The three integrals in Eq. (27) are separately transformed basedon integration by substitution.

Let ϕ ¼ arccos 12rs

r2s−r2ou þ u

� �h i. Therefore,

u ¼ rs cos ϕð Þ þ ro

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1− rs

ro

� �2sin2 ϕð Þ

s: ðA:2Þ

By substitution ϕ for u in the first integral of Eq. (27),

∫ba arccos

12rs

r2s−r2ou

þ u

!" #du ¼

ϕ rs cos ϕð Þ þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2o−r2s sin

2 ϕð Þq� �

−rs sin ϕð Þ−roE ϕrsro

������ �

ϕ að Þϕ bð Þ:

ðA:3Þ

Similarly, for the last integral of Eq. (27),

∫bau

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1− 1

2rs

r2s−r2ou

þ u

!" #2vuut du ¼23rsro sin ϕð Þ cos ϕð Þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1− rs

ro

� �2sin2 ϕð Þ

s−2

3r2s sin

3 ϕð Þ

− ro3rs

r2s þ r2o� �

E ϕrsro

�����þ ro3rs

r2o−r2s� �

F ϕrsro

������ �ϕ bð Þ

ϕ að Þ:

ðA:4Þ

Let us deal with the second integral of Eq. (27).

203A. Kallel, T. Nilson / Remote Sensing of Environment 130 (2013) 188–204

Let ϕ′ ¼ arccos 12ro

r2o−r2su þ u

� �h i. Therefore,

u ¼ro cos ϕ′

� �−rs

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1− ro

rs

� �2sin2 ϕ′

� �s; if u∈ a;u�½ �

ro cos ϕ′� �

−rs

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1− ro

rs

� �2sin2 ϕ′

� �s; if u∈ u�

; b½ �;

8>>>><>>>>:

ðA:5Þ

with

u� ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffir2o−r2s

q: ðA:6Þ

By substitution ϕ′ for u in the second integral of Eq. (27).

∫u�

a arccos12ro

r2o−r2su

þ u

!" #du ¼ ϕ′ rocos ϕ′

� �−

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2s−r2osin

2 ϕ′� �q� �

−rosin ϕ′� � �ϕ′ u�ð Þ

ϕ′ að Þ

þrs∫ϕ′ u�ð Þϕ′ að Þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1− ro

rs

� �2sin2 ϕ′

� �sdϕ′

;

ðA:7Þ

and,

∫bu� arccos

12ro

r2o−r2su

þ u

!" #du ¼ ϕ′ rocos ϕ′

� �þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2s−r2osin

2 ϕ′� �q� �

−rosin ϕ′� � �ϕ′ bð Þ

ϕ′ u�ð Þ

−rs ∫ϕ′ bð Þϕ′ u�ð Þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1− ro

rs

� �2sin2 ϕ′

� �sdϕ′

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}X

:

ðA:8Þ

Let ϕ″ such that ϕ″� �

¼rors sin ϕ′

� �. By substitution ϕ″ for ϕ′ in X ex-

pression,

X ¼ r2s−r2orsro

F ϕ″ rsro

�����þ ro

rsE ϕ″ rs

ro

������ �ϕ″ bð Þ

ϕ″ að Þ:

"ðA:9Þ

Appendix A.2. Single scattering from foliage reflectance estimation

As shown in Eq. (47), the single scattering reflectance from foliagereflectances derivation needs the estimation of a complex integralwhere an analytical expression is not trivial. An approximation ofthe exact solution is therefore needed.

To do it, the solution proposed in Kuusk's model code is adopted.The interval [−h, 0] is divided into N intervals [ai,ai+1], such thata1=−h, aN+1=0. The length of the intervals follows a logarithmicincrease law from the top to the bottom. Then, it is possible to writethe integral in ρso,poi1 expression [cf. Eq. (46)] as

∫0z¼−h exp −nlvso zð Þ½ �dz ¼

XNi¼1

∫aiþ1z¼ai

exp −nlvso zð Þ½ �dz: ðA:10Þ

For each integral [ai,ai+1], the integral is approximated as

∫aiþ1z¼ai

exp −nlvso zð Þ½ �dz≈ aiþ1−ai� � exp −nlvso aið Þ½ �−exp −nlvso aiþ1

� �� �nlvso aiþ1

� �−nlvso aið Þ :

ðA:11Þ

In our case, N is taken equal to 60.Similarly, it is possible to estimate the values of ρso,+1 and ρso,−1 .

Appendix B. Model limit (rl≪1)

In this appendix, we will prove that the LSDM reflectance con-verges to the turbid reflectance as given in Verhoef (1984) when rltends to zero.

According to Eq. (11)

limrl→0

rs ¼ 0 & limrl→0

ro ¼ 0: ðB:1Þ

Next, let us prove that

limrl→0

DiffTur rlð Þ ¼ 0; ðB:2Þ

with

DiffTur rlð Þ ¼ vso zð Þ−πr2s zj j−πr2o jzjr2l

����������≥0: ðB:3Þ

According to Eq. (27)

limrl→0 DiffTur rlð Þ ¼ limrl→0 jπminGs

μs;Go

μo

� �rl

ffiffiffiffiGsμs

q−

ffiffiffiffiffiGoμo

q��������

Δ Ωs;Ωoð Þ

−Gsμs

Δ Ωs;Ωoð Þ∫min rsþro ;−zΔ Ωs ;Ωoð Þð Þmin rs−roj j;−zΔ Ωs ;Ωoð Þð Þ arccos

12rs

r2s−r2ou

þ u

!" #du

−Goμo

Δ Ωs;Ωoð Þ∫min rsþro ;−zΔ Ωs ;Ωoð Þð Þmin rs−roj j;−zΔ Ωs ;Ωoð Þð Þ arccos

12rs

r2o−r2su

þ u

!" #du

þffiffiffiffiGsμs

qΔ Ωs;Ωoð Þrl

∫min rsþro ;−zΔ Ωs ;Ωoð Þð Þmin rs−roj j;−zΔ Ωs ;Ωoð Þð Þ u

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1− 1

2rs

r2s−r2ou

þ u

!" #2vuut duj

≤ limrl→0

πminGs

μs;Go

μo

� �rl

ffiffiffiffiGsμs

q−

ffiffiffiffiffiGoμo

q��������

Δ Ωs;Ωoð Þ þGs

μs

2 þ Go

μo

2

Δ Ωs;Ωoð Þ rs þ ro− rs−roj jð Þπ2

þffiffiffiffiGsμs

qΔ Ωs;Ωoð Þrl

rs þ ro− rs−roj jð Þ rs þ roð Þ

≤ limrl→0

rlΔ Ωs;Ωoð Þ

�πmin

Gs

μs;Go

μo

� � ffiffiffiffiffiGs

μs

s−

ffiffiffiffiffiffiGo

μo

s����������þmin

ffiffiffiffiffiGs

μs

s;

ffiffiffiffiffiffiGo

μo

s !

� Gs

μsþ Go

μo

� �2π þ

ffiffiffiffiffiGs

μs

sþ

ffiffiffiffiffiffiGo

μo

s ! ffiffiffiffiffiGs

μs

s" #�

¼ 0:

It follows that for rl≪1,

vso zð Þ≈πr2s zj j þ πr2o jzj: ðB:5Þ

Then, the reflectance due to single scattering contribution from fo-liage in the Poisson case [cf. Eq. (47)] is

ρ1so;poi ¼ w Ωs;Ωoð Þ∫0

z¼−h exp −nlvso zð Þ½ �dz;

≈w Ωs;Ωoð Þ∫0z¼−h exp −nl πr2s jzj þ πr2o jzj

� �h idz;

≈w Ωs;Ωoð Þ∫0z¼−h exp

LAIhπr2l

πGs

μsr2l zþ π

Go

μor2l z

� �" #dz;

≈w Ωs;Ωoð Þ1−exp − kþ Kð Þh½ �kþ K

:

ðB:6Þ

The last reflectance expression corresponds to the turbid case asshown in Section 1. Similarly, it is possible to prove that the other sin-gle scattering reflectances for the different leaf distributions given byEqs. (47) and (48) converge to the turbid case when rl tends to zero.These results prove the validity of our model for small hot spotparameter.

204 A. Kallel, T. Nilson / Remote Sensing of Environment 130 (2013) 188–204

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