ORIGINAL PAPER

Modeling of light transmission under heterogeneous forestcanopy: an appraisal of the effect of the precision levelof crown description

David Da Silva & Philippe Balandier & Frédéric Boudon &

André Marquier & Christophe Godin

Received: 18 April 2011 /Accepted: 13 September 2011# INRA and Springer-Verlag, France 2011

Abstract& Context Light availability in forest understory is essentialfor many processes; it is, therefore, a valuable informationregarding forest management. However, its estimation is oftendifficult and direct measurements are tedious. Models can beused to compute understory light, but they often require a lotof field data to accurately predict light distribution, particu-larly in the case of heterogeneous canopies.& Aims The influence of the precision level of crowndescription was studied with a model (MμSLIM) that canbe used with both detailed and coarse parameterization withthe aim of reducing field data requirements to a minimum.

& Methods We analyzed the deterioration of the predic-tion quality of light distribution to the reduction of inputsby comparing simulations to transmitted light measure-ments in forests of increasing complexity in threedifferent locations.& Results With a full set of parameters to describe the treecrown (i.e., crown extension in at least eight directions,crown height and length), the model accurately simulatedthe light distribution. Simplifying crown description by ageometric shape with a mean radius of crown extension ledto deteriorated but acceptable light distributions. Allometricrelationships used to calculate crown extension from trunkdiameter at breast height seriously reduced light distributionaccuracy.

Keywords Light modeling . Forest . Mixed stand .

Uneven-aged stand . Canopy description

1 Introduction

Light in forest understory is a fundamental resource drivingmany processes related, for example, to regenerationgrowth, vegetation cover and composition, and animalhabitat (Balandier et al. 2009). Light quality is fundamentalfor morphogenetic processes, whereas light quantity drivesprocesses linked to carbon acquisition. In this article, onlylight quantity is considered.

For a long time, light in forest has only been consideredas a factor controlling tree growth, especially in the case ofregular even-aged stands in the temperate area. In thatcontext, tree density has been managed to get a very darkunderstory with often bare soil, a sign that trees absorbedthe maximum of radiations (Perrin 1963). Nowadays, asilviculture closer to nature (also named continuous cover

Handling Editor: Erwin Dreyer

D. Da Silva (*)Department of Plant Sciences, University of California Davis,237 Hunt Hall, One Shields Avenue,Davis, CA 95616, USAe-mail: dodasilva@ucdavis.edu

P. BalandierResearch Unit on Forest Ecosystems, Cemagref,45290 Nogent-sur-Vernisson, France

F. BoudonCIRAD, Virtual Plants INRIA Team, UMR AGAP,TA A-108/02, Avenue Agropolis,Montpellier Cedex 5, 34398, France

P. Balandier :A. MarquierUMR 547 (INRA–Université B. Pascal) PIAF, INRA,Site de Crouel,234 Avenue du Brézet,63039 Clermont-Ferrand Cedex 2, France

C. GodinINRIA, Virtual Plants INRIA Team, UMR AGAP,TA A-108/02, Avenue Agropolis,Montpellier Cedex 5, 34398, France

Annals of Forest ScienceDOI 10.1007/s13595-011-0139-2

forestry) with natural regeneration is promoted or redis-covered (Hale et al. 2009). Irregular uneven-aged standsoften with mixed species are managed with partial cuttingto create gaps in the forest canopy. These gaps favor somepatch of light essential to tree regeneration, but alsopromote the development of understory vegetation thatcan compete with young trees and compromise forestdynamics (Balandier et al. 2009). Under these conditions,in order to sustain young tree growth while avoiding toodense understory vegetation, it is essential to proportionlight by the size of the gaps (Gaudio et al. 2011).

However, estimating light quantity in forest understory is notso easy (Lieffers et al. 1999). On one hand, visual assessmentsare strongly biased by the meteorological conditions, the hourof the day, and the operator himself. On the other hand, directmeasurements by sensors are often tedious, expensive, requiretechnical competences, and their results also depend on themeteorological conditions and the solar pathway on the day ofmeasurement (Pukkala et al. 1993).

Since the pioneer work of Monsi and Saeki (1953), manymodels simulating light interception and transmission byplant canopies have been developed (for reviews, seeMyneni et al. 1989 and Sillion and Puech 1994) and offeran operative alternative. They are based on either statisticalrelationship between vegetation characteristics such as cover,leaf area or leaf area index (LAI), and light transmission orexplicit description of the canopy topology and geometry aselements intercepting light. When forest canopy is homoge-neous, i.e., even-aged pure regular stands, statistical modelspredict mean light quantity in forest understory withacceptable accuracy (Balandier et al. 2006b; Hale et al.2009; Sonohat et al. 2004). However, prediction quality ofthat type of models decreases drastically with the increase inheterogeneity of the stand structure (Balandier et al. 2010).In particular, in irregular uneven stands, a statistical meanhas no sense because it does not reveal the light distributionbetween understory and gaps which is of most interest forthe forester (Courbaud et al. 2003; Song and Band 2004).With that information, local light availability to favor treeregeneration, plant biodiversity, or animal habitat, for exam-ple, can be estimated (Balandier et al. 2006a).

In that case, complex models with an explicit 3Ddescription of elements intercepting light give moreaccurate results. The approach of these models usuallyinvolve the use of numerous tracing (sun) rays coming fromdifferent points in the sky to a particular point within thestand and compute transmission along those rays, takinginto account foliage characteristics, thus generating adetailed 2D map of light availability in the understory(Cescatti 1997b; Bartelink 1998; Brunner 1998; Stadt andLieffers 2000; Courbaud et al. 2003; Mariscal et al. 2004).However, these models require abundant field measure-ments to parameterize them (Gersonde et al. 2004; Song

and Band 2004). These measurements, like a map of treelocation on the site, the 3D geometry of each tree crownwith information on leaf area density (LAD), distribution,and clumping inside the crown, are tedious and oftendifficult to obtain. Moreover, for management purposes, adetailed 2D map of light availability in the understory is notalways required. For the forest manager, a histogram oflight transmittance at the plot level (i.e., percentage of soilper class of transmitted light, without knowing the spatialdistribution of the light transmittance) can be sufficient.

From a scientific point of view, there is a need to identifythe key elements of plant architecture that are necessary toconsider to correctly predict different processes, includinglight transmission, at different scales. Indeed, simplifica-tions which are unacceptable at a detailed level ofrepresentation can become acceptable at a more integratedlevel (Tardieu 2010). However, the difficulty is to definewhat the key elements are and the emerging propertieslinked to them related to such or such a scale of description.

MμSLIM is a model especially designed to bridge the gapbetween the statistical and explicit approach. It can be seen asa multiscale mixed model in the sense that the lightinterception estimation method can be changed at each scale(Da Silva et al. 2008). Furthermore, MμSLIM was designedto work with any type of envelope encompassing thevegetation for which light attenuation is to be computed,thus allowing precise description of complex canopy andcrown shapes. This feature allowed overcoming the restric-tion to analytically defined envelope used in other models(Norman and Welles 1983; Cescatti 1997a). Although it wasinitially dedicated to study light interception by isolatedtrees, its versatile design and multiscale approach made itsadaptation for use at the canopy scale easy. Therefore, wewere able to use it to analyze how the deterioration of thecanopy description influences the accuracy of the lighthistogram simulated at the plot level. Our objective was tospecify the minimum field data needed by the model to beable to accurately simulate the light distribution. Ourexperimental design was as follows: (1) compare thesimulated light distribution using the more detailed descrip-tion of the canopy with measured data on the field and (2)analyze the statistical distance between histograms simulatedfor decreasing levels of description and measured data.

2 Material and method

2.1 Forest sites and plots

Three forest sites located in France with a total of eightplots were used with increasing canopy structure complexity;five monospecific even-aged stands made up of Pinussylvestris L. or Pinus nigra Arnold, two mixed Quercus

D. Da Silva et al.

petraea L.–P. sylvestris L. uneven-aged stands, and onemixed Abies alba–P. sylvestris–Fagus sylvatica L. uneven-aged stands (Table 1).

The Scots pine stands are located in the Chaîne desPuys, a mid-elevation volcanic mountain range at a placenamed Fontfreyde (Table 1). The elevation is 900 ma.s.l.,mean annual rainfall is about 820 mm, and mean annualtemperature is about 7°C. The soil is a volcanic brown soilat pH 6.0 with no mineral deficiency. The pines were 30 to50 years old at time of measurement, with a density rangingfrom 500 to 2,300 stems ha−1 (Table 1). They are mainlyregular even-aged stands. Some other species can be locallypresent such as Betula pendula.

The mixed oak–pine stands are located in the NationalForest of Orleans, in the subpart named Lorris. They aretypical stands of temperate plain forest with a mild climate(mean annual rainfall of 720 mm; mean annual temperatureof 10.8°C) and acidic sandy poor soils temporarily floodedin winter or spring. They are irregular uneven-aged stands.Median diameter at breast height (DBH) is about 18 cm,whereas some individuals, in particular of Scots pine, canreach 70 cm. Some other species are locally present such as

Betula sp., Carpinus betulus, Sorbus torminalis, Malussylvestris, or Pyrus communis.

The third site is located in the South-East of France, in themountainous area named Mont Ventoux. One plot, Ven-touxC7, on the southern side of the mountain, 1,100 ma.s.l., isan old even-aged P. nigra plantation, almost regular, withsome pine and beech seedlings, on a flat area. The secondplot, Ventoux34, on the northern side of the mountain,1,400 ma.s.l., is a mixed irregular stand with advancedregeneration of A. alba and F. sylvatica with some tallersaplings almost reaching the open overstory of old planted P.sylvestris and some other scattered species (Sorbus aria,Acer opalus, etc.). The climate is mountainous withMediterranean influences (mean annual temperature is 8.6°Cand 6.7°C and mean annual rainfall is 1,080 and 1,360 mm forVentouxC7 and Ventoux34, respectively). The soils are acalcosol (VentouxC7) and a colluvial rendosol (Ventoux34),20 and 50 cm deep in average, respectively, with a high stonecontent (50% and 75%, respectively), both with a silty claytexture on a fissured limestone. The median DBHs approxi-mate 38 and 15 cm (for the first and the second plot,respectively) with some trees reaching 50 cm.

Table 1 Plot description (each plot was approximately 50×50 m)

Site Latitude;longitude

Elevation(m)

Density(stems ha−1)

Mean height (m)(min–median–max)

DBH (cm)(min–median–max)

Basal area(m2 ha−1)

Tree species Basalarea

Opacity

Fonfreyde1 47°42′ N 920 2,353 7.3 7 54 Pinus sylvestris 99.4 0.835

02°58′ E 14.5 16 Other species 0.6 0.75–0.8618.4 33

Fonfreyde2 47°42′ N 920 1,103 7.2 7 33 Pinus sylvestris 73.6 0.835

02°58′ E 14.5 17 Other species 26.4 0.75–0.8618.3 42

Fonfreyde3 47°42′ N 920 563 6.3 9 22 Pinus sylvestris 90.4 0.835

02°58′ E 14.2 21 Other species 9.6 0.75–0.8618.2 37

Fonfreyde5 45°41′ N 910 710 11.3 9 42 Pinus sylvestris 96.2 0.835

02°59′ E 18.3 28 Other species 3.8 0.75–0.8623 43

Lorris38 47°47′ N 150 664 4 7 32 Pinus sylvestris 56.7 0.79

02°34′ E 16.1 17 Quercus petraea 34.2 0.867

29.3 71 Other species 9.1 0.75–0.86

Lorris255 47°49′ N 150 607 3.7 7 24 Pinus sylvestris 36.7 0.79

02°26′ E 17 19 Quercus petraea 57.5 0.867

22.3 47 Other species 5.8 0.75–0.86

VentouxC7 44°07′ N 1,102 231 7.4 7 14 Pinus nigra 98.9 0.84

05°19′ E 17 38 Other species 1.1 0.75–0.8620.8 48

Ventoux34 44°10′ N 1,410 2,205 5 7 34 Abies alba 43.2 0.835

05°13′ E 9.4 15 Pinus sylvestris 28.2 0.835

16 34 Fagus sylvatica 27.3 0.85

Other species 1.3 0.75–0.86

Scale of description effects on light modeling

2.2 Canopy structure measurements

All trees in a square area, the experimental unit (Fig. 1a), ofsize approximately three times the height of dominant trees,were identified (species), located by their x, y coordinates,and measured for their total height and DBH. Top andbottom crown heights were also measured. Lateral crownextents were assessed by visually projecting to the soil itscharacteristic points (i.e., the points that better describe thecrown irregularities) in, at least, four directions, butsometimes with more than eight directions when necessary.The azimuth and distance of those points from trunk werethen measured (Fig. 1b, c).

2.3 Light measurements

The light measurements were conducted in the centralzone of the experimental unit. The surrounding zonewhere dendrometric measures were done acts as a bufferzone accounting for the attenuation of the encompassingforest (Fig. 1a). The light measurements were achievedusing a square grid composed of 64 PAR sensors, namely,Solems PAR/CBE 80 sensors. Each sensor was separatedfrom its neighbor in both directions by 2 to 3.5 mdepending on the considered plot (i.e., depending on themean size of the trees). This distance was set by previouscomputation based on geostatistics (data not shown) andso that two contiguous measurements (points) can beconsidered as independent. Light measurements were donefor each plot during 24 h during summer to take the fullsun course into account. One additional PAR sensor,Sunshine sensor BF2, was positioned outside of theexperimental unit in full light to measure global (total)and diffuse incident radiation. This extra sensor uses anarray of photodiodes with a computer-generated shadingpattern to measure incident solar radiation and the D/Gratio, i.e., the diffuse to global radiation ratio. Transmit-tance for each point in the understory is then computed asthe ratio of light measured in the understory divided by theincident value. The D/G ratio is taken into account forsimulating the transmittance with the model.

2.4 Model description

Estimation of light transmission is a process that requiresfirst the stand reconstruction. Then, the light attenuationwithin each crown must be defined so that the light modelcan compute the transmittance under the entire canopy.

2.4.1 Canopy reconstruction

From the dendrometric measurements on the field, amockup of the stand is generated. This generation involvesthe construction of a crown (and trunk) for each tree of thestand and its positioning. To reconstruct the 3D envelopesof the trees from the field measurements, we used thePlantGL library (Pradal et al. 2008). This library containsseveral geometric models, including different types ofenvelopes and algorithms, to reconstruct the geometry ofplants at different scales. In this work, we used the skinnedsurface which is a generalization of the surface ofrevolution with varying profiles being interpolated. Theenvelope of a skinned hull is a closed skinned surfacewhich interpolates a set of profiles {Pk, k=0, …, K}positioned at angle {αk, k=0, …, K} around the z axis,where K is the number of profiles used to generate theskinned surface. This surface is thus built from any numberof profiles with associated direction. In such case, a profileis supposed to pass through common top and bottom pointsand at an intermediate point of maximum radius, i.e., theheight of the crown maximum width. Two shape factors, CT

and CB, were used to describe the shape of the profilesabove and below the maximum width. Mathematically, twoquarters of the superellipse of degree CT and CB were usedto define the top and bottom part of the profiles; for details,see Da Silva (2008). Note that our envelopes can be viewedas extension of Cescatti’s (1997a) asymmetric hull withprofiles in any direction instead of the restricted cardinaldirections. Flexibility of our model enabled us to measurethe most adequate profiles in case of irregular crowns. The3D envelope of a tree was thus obtained using a skinnedsurface generated from profiles defined from an angle, amaximum width and its associated height, and two shapes

a bc

Fig. 1 a Experimental unit.b Field data: each dot locates atree (its trunk) and each arrowdefines a specific azimuth anddistance from the trunk,characterizing the crown extend.c Zoom in on the interest zonewhere light measurements whereconducted

D. Da Silva et al.

factors. The trunk of the tree was represented as a cylinderof DBH, the height of the cylinder being set to the crownbottom height. The 3D representation of an experimentalunit is presented in Fig. 2.

2.4.2 Estimation of light transmission

The light transmission under such reconstructed stand wascomputed using the multiscale light interception model(MμSLIM) presented in Da Silva et al. (2008). For eachdirection of incoming light, a set of beams was cast and theattenuation of each beam through the canopy was estimat-ed. In the present study, the model was a simple two-scalemodel where the porous envelopes of crowns constitutedthe finest scale and the stand constituted the coarser scale.In replacement of the delicate estimation of LAD, a globalopacity value, pgl, was associated with each envelope of thereconstructed stand; the trunks were considered opaqueorgans, thus their opacity was 1. Crowns’ global opacitywas estimated from field photographs shooting verticallydifferent parts of the crown. Each image was digitized,segmented to black and white, and then the pixels werecounted using PiafPhotem software (Adam et al. 2006).The ratio of black pixels to the total number of pixelsyielded the opacity of the crown, while the ratio of whitepixels to the total number of pixels produced the crown

porosity, hence porosity=1−opacity. However, the opacityof each individual crown was not directly estimated usingthis method because the segmentation procedure stillrequired human intervention. Instead, a global opacityvalue per species was estimated over a sample of pictures(Table 1). MμSLIM allowed then to compute the lighttransmission of the canopy by considering the lightattenuation of each beam. The light attenuation for onebeam is computed according to the opacities of the crownsencountered along its trajectory and according to the pathlength within each of these crowns. The porosity expressionthat takes the beam traveling distance into account is relatedto G (Chen and Black 1992; Ross 1981; Stenberg 2006),the extinction coefficient that depends on light directionand leaf inclination distribution, and LAD, the leaf areadensity. It can be expressed as:

p ¼ 1

b

X

b2B1� exp �G� LAD� lbð Þð Þ ð1Þ

where B is the set of beams b, β is its cardinality, and lb isthe path length of the beam into the crown. To take thisdistance into account, we thus need to determine thequantity G×LAD.

The global opacity value can be regarded as the result ofopacity estimation using the Beer–Lambert law in aninfinite horizontally homogeneous layer:

pgl ¼ 1� exp �G� LAIð Þ: ð2ÞThis relation yields an expression for G×LAI, where

LAI is the leaf area index. In the case of crowns with afinite volume, V, the usual definition of LAI can beextended to be expressed as a function of the total leafarea, TLA, and the projected envelope area, PEA (Sinoquetet al. 2007):

LAI ¼ TLA

PEAð3Þ

and since LAD is the ratio of total leaf area to crown volume:

LAD ¼ TLA

V; ð4Þ

we can express G×LAD as:

G� LAD ¼ G� LAI� PEA

V¼ � log 1� pgl

� �� PEA

V:

ð5ÞUsing this G×LAD value with Eq. 1 naturally leads to a

smaller envelope opacity value than pgl. This is due to thefact that the negative exponential of a mean value is lessthan the mean of negative exponential values. Therefore, anumerical approximation was carried out using the valuefrom Eq. 5 as the starting value to speed up convergence.

a b

c d

Fig. 2 3D reconstruction of the Lorris255 experimental unit using theasymmetric crowns, side view with opaque wall visualization (a) andtop view (b), and using the crowns from the allometric radiusapproach, side view (c) and top view (d). Colors are used to visuallydifferentiate between tree species: dark green for pine (P. sylvestrisL.), green for oak (Q. petraea L.), orange for birch (Betula L.), andbrown for hornbeam (C. betulus L.)

Scale of description effects on light modeling

The process stopped when the opacity computed usingEq. 1 was equal to the global opacity within a user-definederror, ε. This G×LAD approximation was done once foreach crown and for each direction and stored for furtherusage. The crown opacity for each beam, pb, was thencomputed using the expression:

pb ¼ 1� exp �G� LAD� lbð Þ: ð6Þ

The light transmission of the canopy along one directionwas estimated by computing, for each beam, its total opacityresulting from its traveling through the canopy, i.e., possiblygoing through multiple crowns; for algorithmic and compu-tational details, see Da Silva et al. (2008). Experimentalconditions were simulated by using two sets of incomingradiation directions. The first set, for diffuse light, discretizedthe sky hemisphere into 46 solid-angle sectors of equal area,according to the Turtle sky proposed byDenDulk (1989). Thedirections used were the central direction of each solid-anglesector. The second set, for direct light, was used to simulatethe trajectory of the sun, and the directions were dependentof the location (latitude, longitude), the day of the year, andon the time step used for the sun course discretization(approximately 25 directions in this work). Each of thesedirections was associated with weighting coefficients derivedfrom the standard overcast distribution of sky radiance(Moon and Spencer 1942). The porosity is directly the ratioof transmitted light to incident light; a value of 1 means thatall light goes through, while a value of 0 means that no lightgoes through. The gray-level image constituted by the beamporosity values is, therefore, a shadow map of the canopytransmittance. For each defined direction, MμSLIM wasused to compute the opacity values of the beams and toproduce such an image in a plane orthogonal to the light. Toensure both results precision and fast computation, a lineicdensity of 150 was used for the beam sampling (Da Silva etal. 2008), thus generating 150×150-pixel images. The imageswere then projected on the ground and rotated according to thelight directions azimuth. Images from each set of incomingradiation directions were merged using the weighting coef-ficients, producing two intermediate images, one for theintegrated transmittance of diffuse light and one for the directlight. Finally, these two images were merged using the diffuseto total ratio, D/G, measured from the extra sensor in thefield. The gray-level values of pixels, from 0 (black) to 1(white), represented the transmittance classes simulated byMμSLIM. The simulated light transmittance of the centralzone could then be compared to the field measurements.

2.5 Virtual experiments

To assess the effect of the deterioration of canopydescription on the simulated light histogram, we generated

five different types of mockups for each stand withdecreasing amount of information on canopy architecture.

1. The asymmetric mockup where all available dendro-metric data are used to generate asymmetric crowns.

2. The simple mean mockup where crowns are repre-sented by simple geometric shapes, cones for pines,spheres otherwise, and where the radius of each shapeis the mean of the measured radii.

3. The simple max mockup where crowns are representedby simple geometric shapes, cones for pines, spheresotherwise, and where the radius of each shape is themaximum of the measured radii.

4. The allometric radius mockup where crowns arerepresented by simple geometric shapes, cones forpines, spheres otherwise, and where the radius of eachshape is obtained from allometric relation with DBH.

5. The allometric mockup where crowns are representedby simple geometric shapes, cones for pines, spheresotherwise, and where the radius of each shape and theheight of cones are obtained from allometric relationwith DBH.

For every type of mockup, the values for the height ofthe crown base and for the height of the cones (except forthe allometric type) are the measured ones. Allometricrelations were defined as the linear model between DBHand mean radius and, for pines, between DBH and height.The relations obtained from field measurements data aredescribed in Table 2, and Fig. 2 shows 3D reconstructionsusing two different types of crown shapes.

To complement the attenuation effect of the buffer zonefor low elevation radiations, it was necessary to add anartificial cylindrical wall surrounding the zone of interest.The wall was centered on the zone of interest with a radiusof the zone of interest size. The height of the wall was set tothe mean base crown height of trees in the zone of interest.The coefficient of transmission of the wall (i.e., wallopacity) was calibrated using in silico simulation experi-ments with the stands of one site and then validated usingthe other stands from independent sites. Adequacy of thismethod will be discussed further on.

2.6 Model estimation

The model outputs were qualitatively and quantitativelycompared to experimental data. The qualitative estimationswere carried out by comparing the light transmittancecumulative distribution functions (CDF) to determine over-and/or under-represented light transmittance class. The quan-titative estimations were achieved with the Kolmogorov–Smirnov test (K-S test) and the absolute discrepancy (AD)index (Gregorius 1974; Pommerening 2006). The K-S test isa nonparametric test for the equality of continuous, one-

D. Da Silva et al.

dimensional probability distributions that can be used tocompare two samples. It quantifies the maximum distancebetween the empirical CDF of two samples. The nulldistribution of this statistic is calculated under the nullhypothesis that the samples are drawn from the samedistribution. The p value of the K-S test represents the αlevel at which the null hypothesis can be rejected (i.e.,a p value>0.1 means that the null hypothesis cannot berejected at 10% or lower level).

The AD index is defined as:

AD ¼ 1

2

Xn

i¼1

si � s0i

�� �� AD 2 0; 1½ � ð7Þ

where n is the number of classes, si is the relative frequencyin class i of the first distribution, and s'i is the relativefrequency in class i of the second distribution. AD isdefined as the relative proportion that needs to beexchanged between the classes if the first distribution wereto be transformed into the second distribution. Correspond-ingly, 1−AD is the proportion common to both distribu-tions, a value of AD=1 means that both distributions haveno common class, whereas AD=0 signifies that thedistributions are absolutely identical (Pommerening 2006).

Both quantitative analyses were carried out with lightmeasurements as reference values and the results are shownin Table 3.

3 Results

The comparison between the CDF of the measured lighttransmittance and the ones obtained with the differentmockups, as shown in Fig. 3, were used to qualitatively

assess the quality of the model prediction and the effect ofdeterioration of crown shape description. The CDF pro-vides an alternate representation to light transmittancehistograms that facilitates comparison between differentdistributions. The importance of a specific light transmit-tance class is given by the slope of the CDF at that point, i.e., anonrepresented class will yield a null slope and the moresubstantial the class will yield the steeper the slope (Fig. 3).The classes of high importance will be titled as main classes,whereas the classes of low significance will be designated asminor classes.

3.1 Simulation with asymmetric mockups

The light model using the asymmetric mockups was able tosimulate light transmission distribution similar to themeasurement with a level of confidence of the K-S testabove 5% for all stands except Fonfreyde5 (3%) andFonfreyde1 (<1%). In that case, the mean comparisonyielded a statistically significant difference for the same twostands. However, the AD values were below 0.3 for allstands with a mean of 0.21 (Table 3).

In the case of Fonfreyde1, the measured transmitted lightwas almost evenly distributed in only two light transmit-tance classes, [0–5] and [5–10]. The simulated distributionshowed the same two main classes with an unbalance infavor of the [0–5] class. Although the p value of the K-Stest allowed rejecting the hypothesis that the measured andsimulated light distributions were similar, the AD value wasbelow the mean value of all stands. The simulatedtransmittance of Fonfreyde2 were almost identical to themeasured ones despite the missing [0–5] class in thesimulation (i.e., simulated CDF slope is null through entire[0–5] class). The best values for both the p value and AD

Table 2 Allometric relationbetween DBH and crown meanradius and height (linear modelsof the form a×DBH+b wereused for both mean radiusand height)

Tree species Site Slope (a) Intercept (b) R2

Main radius Pinus Fonfreyde 2.62 13.91 0.72

Lorris 2.29 50.64 0.57

Ventoux 1.86 69.28 0.83

Quercus petraea Lorris 2.73 129.69 0.60

Fagus sylvatica Ventoux 3.98 75.56 0.82

Acer opalus Ventoux 10.10 9.68 0.66

Betula pendula Fonfreyde 2.35 79.12 0.67

Lorris 2.13 94.36 0.29

Carpinus betulus Lorris 7.18 74.78 0.36

Populus tremula Lorris 4.86 19.90 0.86

Sorbus torminalis Lorris 3.19 135.01 0.67

Ventoux 2.88 85.38 0.71

Height Pinus Fontfreyde 0.08 10.09 0.47

Lorris 0.10 10.27 0.64

Ventoux 0.12 2.25 0.93

Scale of description effects on light modeling

were obtained for this stand. In the case of Fonfreyde3, thedifference in the standard deviation (Table 3) between themeasured and simulated light distributions was a goodindicator of the fact that the lower and higher classes, [15–25] and [50–60], respectively, were not simulated to theprofit of the median classes [30–40] as shown by theAsymmetric CDF being below and then above the CDF ofmeasurements. On the contrary, in the case of Fonfreyde5,the Asymmetric CDF started below and ended above themeasurement one, indicating that the median classes ([10–20]) were undersimulated to the benefit of the [30–35] classthat did not appear in the measurements and the [5–10] onethat has very low substance. The highest AD values wereobtained for these two stands.

The simulated distributions for Lorris38 correctly ren-dered both the main and minor classes with, however, a lesspronounced peak for the [5–10] class expressed by themore abrupt slope of the measured CDF. There were nomissing or extra classes simulated and the AD was justabove the mean value. Similarly to Lorris38, the main andminor classes of Lorris255 were correctly simulatedwithout any missing or extra classes. Within each group,main and minor, some classes were under-represented to thebenefit of the over-represented ones. However, these

Table 3 Simulation results for the different mockups reconstruction

Stand/mockup Measures Asymmetric Simple mean Simple max Allometric radius Allometric

Fonfreyde1 Mean 5.15a 4.19b 5.06a 0.44c 4.29b 4.14b

SD 1.3 1.71 2.41 0.57 2 2.02

p value 1.3e−5 9.8e−3 <1e−5 7.1e−5 3.1e−5AD 0.203 0.063 0.516 0.141 0.172

Fonfreyde2 Mean 14.7a 14.69a 15.72a 7.19c 21.37b 20.32

SD 3.58 3.18 4.62 3.14 4.18 3.89

p value 0.987 0.05 <1e−5 <1e−5 <1e−5AD 0.078 0.219 0.75 0.594 0.531

Fonfreyde3 Mean 38.08a,b 37.05a 39.58 25.81c 48.09d 43.18e

SD 9.46 5.73 8.66 9.3 7.71 7.35

p value 0.08 0.52 <1e−5 1.3e−5 9.8e−5AD 0.281 0.234 0.453 0.406 0.329

Fonfreyde5 Mean 16.89a 19.29b 18.87b,d 9.52c 16.81a,b,d 18.32a,d

SD 4.37 6.68 7.94 6.35 7.43 8.32

p value 0.03 0.017 <1e−5 9.8e−3 0.05

AD 0.281 0.344 0.578 0.406 0.406

Lorris38 Mean 8.43a 8.41a 8.61a 0.645c 15.79b 16.09b

SD 6.85 5.54 5.99 1.31 7.66 7.91

p value 0.126 0.029 <1e−5 1.9e−5 <1e−5AD 0.219 0.172 0.781 0.344 0.359

Lorris255 Mean 20.8a 21.33a 25.36b 5.49c 21.1a 21.1a

SD 7.06 7.14 7.01 4.99 11.54 11.42

p value 0.758 0.029 <1e−5 0.16 0.203

AD 0.172 0.25 0.844 0.281 0.266

VentouxC7 Mean 53.34a 54.3a,b 57.38d 47.72c 55.66a,b 54.87b,d

SD 7.06 5.91 7.5 9.9 7.74 7.68

p value 0.52 2e−4 7e−5 0.005 0.017

AD 0.172 0.359 0.375 0.266 0.281

Ventoux34 Mean 12.25a,b 8.87a 10.54a 3.44c 16.85b 17.05b

SD 15.29 11.35 13.52 7.9 19.59 19.74

p value 0.05 9.8e−3 <1e−5 0.274 0.188

AD 0.25 0.266 0.391 0.281 0.281

The p value of the K-S test and the AD index were obtained with the measured light values as reference. Means with the same letter indicate thatthe difference between the means are not statistically significant at α=10% for an independent two-sample t test

Fig. 3 CDF of light transmittance for each stand and for everymockup type. Note that, for legibility purposes, the x-axis scale wasadapted for each stand

b

D. Da Silva et al.

Scale of description effects on light modeling

discrepancies were subsidiary as the very high p value andthe below-the-mean AD attest. In the case of VentouxC7,the simulated distribution reproduced the bell shape of themeasured one, but with slightly heavier tails due to the twoextra classes on the sides of the distribution, [35–40] and[65–70]. Note that the isolated [10–15] measured class wasnot simulated, as shown by the advent of a proportiondifference between measurements and simulation. Similarlyto Lorris255, a high p value and a low AD were observeddespite of these differences. Finally, the simulated distribu-tion for Ventoux34, whereas correctly simulating the twomain classes, [0–5] and [5–10] accounting for almost 70%of the transmittance distribution, increased the unbalancebetween them in favor of the lowest one. The minor classeswere adequately simulated but without respecting thediscontinuity shown in the measurements as indicated bythe above-mean AD.

3.2 Simulation with the simple mean mockups

Replacing the asymmetric by the simple mean mockupssignificantly reduced the p values of the K-S test, exceptfor Fonfreyde1 and Fonfreyde3, but yielded AD valuesbelow 0.3, except for Fonfreyde5 and VentouxC7, thatwere similar to the ones obtained with the asymmetricmockups as shown by the slight increase of the mean ADto 0.24. The principal effect of this change of crownshape on the distributions of light transmittance com-pared to the distributions obtained with the asymmetriccrowns was to reduce the most represented classes to thebenefit of the minor ones. This was comparable to thecrushing effect obtained on a Gaussian function byincreasing its variance parameter. The variance increasewas indeed observed for all stands but Lorris255 and,interestingly, the two-sample t test indicated statisticallysignificant differences for Fonfreyde5, VentouxC7, andLorris255. This effect actually benefited Fonfreyde1,Fonfreyde3, and Lorris38 simulation results. The p valueincreased and AD decreased significantly for Fonfreyde1and Fonfreyde3, whereas for Lorris38, both valuesdecreased.

3.3 Simulation with the simple max mockups

The simple max approach yielded light transmittancedistributions that were significantly shifted to the lowertransmittance classes, i.e., left-shifted CDF (Fig. 3).This effect was visible on the simulation results of allstands but was less pronounced for Fonfreyde3 andVentouxC7. The dramatic increase of the AD meanvalue to 0.6 well illustrated this behavior, but meancomparison was sufficient to assess the importance ofthe discrepancies.

3.4 Simulation with allometric mockups

The use of the allometric radius and allometric mockupsyielded similar results as illustrated by comparable meansand variances and similar mean AD values, 0.345 and0.333, respectively. The effect of this change of crownshape on the distributions of light transmittance was similarto the simple mean effect with the addition of a slight shifttoward higher transmittance classes, indicated by right-shiftedCDF (Fig. 3). The p values were significantly reduced exceptfor Fonfreyde1, Fonfreyde5, and Ventoux34, but only forFonfreyde1, this increase of the p value was associated witha decrease of AD (Table 3).

3.5 Comparison between all simulations

The AD results (Fig. 4) show that the asymmetric mockupsyielded the distributions closest to the measurement, thesimple mean approach slightly deteriorated the results,whereas the simple max approach had a dramatic effect.Using allometric crowns yielded results in between butcloser to the simple mean ones. These results confirmed theclassification trend yielded by the two-sample t test analysisthat showed no statistically significant differences betweenthe measurements and the asymmetric and simple meanapproaches on one side and between the two allometricapproaches on the other side and showed the segregation ofthe simple max one.

4 Discussion

4.1 Statistical considerations

Our study on the impact of the deterioration of the accuracyof crown representation on simulated light transmission

Fig. 4 The points represent the mean of AD calculated for each typeof mockup over all stands, the bars being the standard deviation. Sameletter above the bars indicate that the difference between the meansare not statistically significant at α=10% for an independent two-sample t test

D. Da Silva et al.

highlighted some statistical issues. In spatially explicittransmittance models, a spatially point to point comparisonapproach between measured and modeled transmittancegenerally shows little agreement because small errors incrown location, often coming from errors during fieldmeasurements, yield large local differences in transmittance(Mariscal et al. 2004), particularly when the proportion ofdirect radiation is important (Groot 2004). However, onlyconsidering a mean and variation around this mean (e.g.,SD) could lead to some errors (e.g., underestimation of areafully lighted) due to distribution noncentered on mean withsometimes a long tail or several modes. To avoid thisproblem, we compared the simulated distribution of lighttransmittance to the measured ones using two differentnonparametric descriptors, the K-S test and the AD index.

Although commonly used and mathematically well-founded, the K-S test showed some limitations whenapplied to peculiar distributions as the one from Fon-freyde1. The problem stems directly from the test definitionwhere the accordance between two distributions is based onthe maximum distance between the CDFs. When thedistribution shows very low dispersion, a small differencein distribution can yield a big maximum distance betweenCDFs and, therefore, a rejection of the null hypothesis fromthe K-S test. On the contrary, the AD index that measuresthe changes required to transform one distribution into theother is robust in regard of the distribution dispersion. Eventhough AD is very well suited to quantify the differencebetween distributions or to classify results, the AD value isnot associated with any confidence index. Hence, it is up tothe user to define the threshold under which distributionswill be considered similar. Consequently, combining theuse of the K-S test and the AD index allowed us toaccurately compare and assess the differences betweensimulated and measured light distributions.

4.2 Model evaluation

The goal of the present study was not to validate, sensustricto, the model MμSLIM. Such an evaluation wouldrequire an independent data set of light and tree measure-ments. Our objective was to evaluate the quality of lightpredictions with increasing deterioration of the crowndescription. However, the comparison of light measure-ments (with 64 sensors regularly distributed on a squaregrid in the plot) and the simulation with full asymmetriccrown description (i.e., crown radius extension in 4 to8 directions+crown height and length) showed the goodquality of simulated light distribution (K-S p value>0.10 orAD<0.3). Results are in the range of models using similarscales of description (e.g., Brunner 1998; Cescatti 1997b;Gersonde et al. 2004; Groot 2004; Mariscal et al. 2004;Stadt and Lieffers 2000).

As pointed out by Norman and Welles (1983), thecomputation of beam path lengths is a crucial procedure.Due to the multiscale design of MμSLIM, in addition to thepossibility of using any type of envelope, an analyticalresolution as proposed by Norman and Welles (1983) orCescatti (1997a) was not an option. This task was insteadperformed by ray tracing algorithms that determine andanalyze the path of each cast ray among the canopy crownshapes (Wang and Jarvis 1990). With the ever-increasingcomputational power of graphic cards available to high-level operations, this formerly time-consuming procedurecan now be executed for many directions without impairingthe model performances.

To avoid the problem of low elevation angles, i.e.,interception of light by very far elements close to thehorizon in the field, whereas not represented in the model,we had to come up with an alternative to the classicalsolution of canopy duplication or projection on torus. Theseapproaches are based on the strong assumption of canopyhomogeneity; using them with our strongly heterogeneousstands would introduce a noncontrolled approximation thatwould, in turn, induce bias in the results interpretation.Instead, we chose to estimate the radiative parameters of thesurrounding environment using an inverse modeling ap-proach through the addition of an opaque wall (Fig. 2a).The wall opacity was calibrated using Fonfreyde standsdata with full asymmetric crown description. The value wasthen used as a parameter to run the simulations for thestands from the two other independent sites at the samelevel of crown description. Finally, the model predictionswere compared to the field light measurements. The resultssuggest that the coefficient of transmission of the wallproperly represented the radiative properties of the envi-ronment, at least for low elevation angles. Moreover,considering our objective, the opaque wall approachprovided a simple solution saving significant computationaltime otherwise required by the canopy duplication. Thisapproach seems thus promising and simple to set up butwould benefit from more complete sensitivity analysis, inparticular on wall size or position.

One interest of our model is that crown porosity is simplyestimated by a vertical photograph of tree crown (extension ofthe method of Canham et al. 1999) and seems to supportaccurate results in light distribution; however, the extent towhich this parameter influenced the results needs to be moreadequately studied. As pointed out by Stadt and Lieffers(2000), determination of leaf and shading elements (branch,trunk) is often critical in light transmission modeling.

4.3 Effects of crown description deterioration

Our results corroborate that differences in crown shape andsize are key determinants of light transmittance, as stated by

Scale of description effects on light modeling

Vieilledent et al. (2010). Moreover, this study confirmedthe importance of crown shape when simulating spatializedlight transmittance and endorsed the sensitivity to variationsin the crown geometry parameters, especially the crownradius parameter, as already reported (Beaudet et al. 2002;Brunner 1998; Cescatti 1997b). However, a crown meanradius with crown height and spatialization seems to be agood alternate (Courbaud et al. 2003) to detailed measures(i.e., measurement of crown extension in four to eightdirections) even though simple shapes like spheres andcones are known to be inadequate (Mariscal et al. 2004).Other shapes should be tested, more in relation with treespecies architecture. Simplifying the crown representationin the tRAYci model to average values for species andcanopy strata resulted in little reduction in the modelperformance (Gersonde et al. 2004). Describing tree crownextension with four to eight radius is typically nonfeasiblein practice in management or inventory operations, whereasassessing mean crown diameter may be acceptable in somecases. Therefore, an approach that starts with simple shapesthat can later be deformed using an optimization process(Boudon and Le Moguedec 2007; Piboule et al. 2005)should be considered in further studies. Indeed, previousstudies showed the importance of crown asymmetricplasticity in response to local light availability and space,among other factors (Vieilledent et al. 2010 and referencesin it).

However, using an allometric approach would be morecomfortable. The problem is that the use of relationshipsbetween tree DBH and crown diameter or crown heightoften decreased transmittance distribution prediction,whereas not for all stands. With the allometric approach,the model generated smaller crown than the simple mean,thus the same effect with shift to higher transmittance. Weonly tested linear relationships, and nonlinear functionscould have led to slightly better results (Beaudet et al.2002); however, many authors pointed out that thepredictive functions for crown radius from, e.g., DBH hasproved elusive (Stadt and Lieffers 2000) as they are oftenhighly affected by uncontrolled factors such as climatehazards, stand density, or thinning operations (as probablyreflected in the Fontfreyde’s plots). There is often a highvariability in tree allometry from an individual to another(Vieilledent et al. 2010). Actually, the R2 of the relation-ships linking crown diameter to trunk diameter in that studyare not excellent but in the range of those commonly foundin other studies (e.g., about 0.7 in Pinno et al. 2001 orPukkala et al. 1993).

4.4 Effects of the stand complexity

High light variability is generally recorded in forests due totemporal variations of the sun path and heterogeneous

spatial arrangement of light-intercepting elements in irregularand/or mixed stands (Courbaud et al. 2003; Pukkala et al.1993). In stands with a clumped structure (i.e., tree clumpsalternating with large gaps), only an approach at tree scalewith spatialization can correctly predict transmittance,whereas in dense stands, a Beer–Lambert law can be appliedat the stand canopy scale (Balandier et al. 2010). It is,however, important to note that, in dense stands, importanceof small gaps within tree crowns due to different causes(diseases, broken branches, not taken into account in thesimulation) can lead to noticeable differences as such ofFontfreyde1 (Fig. 3) (Beaudet et al. 2002). Low-densitystands Fonfreyde3 and VentouxC7 are less affected bythe simple max approach, probably because the crown“increase” is not enough to fill the “big” gaps in thecanopy. Problems of crown overlaps, in fact very difficultto quantify in the field, are also probably less critical thanin dense stands.

As already pointed out by Courbaud et al. (2003) orBalandier et al. (2010), the problem is that it is verydifficult to generalize results recorded on a site for aparticular stand structure to other sites or stands with otherspecies or structures. This argues in favor of studying theeffect of stand structure in interaction with the scale ofstand description; this could be done for instance by the useof point process analysis.

Acknowledgements The authors would like to thank for their helpin the field or for information on the stands, Catherine Menuet,Gwenael Philippe, Philippe Dreyfus, and Yann Dumas.

Funding The study was partly supported by the French program“ECOGER”, sustainable management of mixed forests.

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