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e c o l o g i c a l m o d e l l i n g 2 0 3 ( 2 0 0 7 ) 257–269

avai lab le at www.sc iencedi rec t .com

journa l homepage: www.e lsev ier .com/ locate /eco lmodel

hanging scale in ecological modelling: A bottom uppproach with an individual based vegetation model

icolas Boulaina,∗, Guillaume Simionib, Jacques Gignouxc

Laboratoire HydroSciences Montpellier Universite Montpellier II, Case Courrier MSE, Place Eugene Bataillon,4095 Montpellier Cedex 5, FranceForest Products Commission, CRC Greenhouse Accounting, Locked Bag 888, Perth Business Centre, WA 6849, AustraliaBIOEMCO, UMR 7573, Ecole Normale Superieure, 46 rue d’Ulm, 75230 Paris, Cedex 05, France

r t i c l e i n f o

rticle history:

eceived 18 July 2006

eceived in revised form

0 November 2006

ccepted 21 November 2006

ublished on line 8 January 2007

eywords:

AR

AI

D model

eer–Lambert

eterogeneous

patial structure

a b s t r a c t

Net primary production (NPP) depends crucially on the sun radiation absorption, and hence

its prediction on the modelling of the radiation. In heterogeneous systems such as savan-

nas, the photosynthetically active radiation (PAR) absorption is more difficult to predict

than in homogeneous systems, due to the presence of tree clumps and open grass areas.

Using a detailed 3D model, we propose new formulations of PAR absorption derived from

the Beer–Lambert law for heterogeneous vegetation. To account for intra-plot vegetation

heterogeneity we assume an intra-plot partition of space into two or three zones, based on

tree location. We develop two models derived model from the Beer–Lambert law: (1) a simple

partition of space below and outside trees, (2) a partition in three zones, including the tree

shading area. The only model to predict PAR interception for all types of spatial patterns was

the most complex one (including tree + tree shading zone + open grass areas). In this model,

extinction coefficients can be derived from vegetation structural parameters like tree cover,

and tree shading area. Tree shading area could be computed from geometric site parame-

ters (latitude, longitude, tree height). With increasing satellite resolution it can be possible

to obtain these structural parameters of vegetation (cover, overlap, tree height) needed by

this Beer–Lambert derived model. Such an absorption model could then be used as a basis

for a Monteith-style model of vegetation functioning for heterogeneous vegetation.

like forests or crops, but we lack such a straightforward for-

. Introduction

caling up models developed at the fine scale of most eco-ogical processes into larger scale models (water catchment,andscape, regions) is a major issue of current environmental

odelling (e.g. Grimm et al., 2005). Unfortunately, no generalgreement exists on how to proceed. In this paper, we focusn the scaling up of the photosynthetically active radiation

PAR) absorption to demonstrate a way to construct a coarseesolution model from a detailed, individual based modellingf vegetation functioning.

∗ Corresponding author. Tel.: +33 04 67 14 90 78; fax: +33 04 67 14 47 74.E-mail address: boulain@msem.univ-montp2.fr (N. Boulain).

304-3800/$ – see front matter © 2006 Elsevier B.V. All rights reserved.oi:10.1016/j.ecolmodel.2006.11.024

© 2006 Elsevier B.V. All rights reserved.

PAR is one of the primary resources for vegetation:absorbed PAR determines for each plant the dry matter thatcan be synthesized (Monteith, 1972). A good model for veg-etation production depends on a good prediction of PARinterception by the cover (Friborg et al., 1997; Sellers et al.,1997; Daly et al., 2000; Nouvellon et al., 2001). Beer–Lambertlaw is generally used with success for homogeneous systems

mulation for heterogeneous vegetation like savannas.In mixed tree–grass systems, trees have a direct effect

on the grass layer through the shadow cast by their crowns

i n g

when included in TREEGRASS.

258 e c o l o g i c a l m o d e l l

(Mordelet, 1993; Ludwig, 2001). Below tree crowns, incidentPAR and thermal infrared radiation (TIR) decrease, and con-sequently soil and vegetation temperature too (Belsky et al.,1989; Mordelet, 1993). Radiation decrease depends on treecrown structure: leaf area index (Lai et al., 2000), leaf den-sity, leaf angular distribution, leaf optical parameters, whichmodify the radiation absorbed by crowns and the radiationtransmitted to the lower strata (Sinoquet and Bonhomme,1991; Lappi and Stenberg, 1998).

A mixture of trees and grasses characterizes savannaecosystems. Trees are spread randomly or in patches abovea continuous layer of grass, with bare soil areas in dry savan-nas (Menaut, 1983). Savanna ecosystems represent 20% of theworld continental land cover and 40% of the tropical zone(Solbrig et al., 1990; Scholes and Hall, 1996). They have a sig-nificant impact on global carbon and water fluxes through ahigh primary production (Ciret et al., 1999; Daly et al., 2000)and frequent burnings (Lacaux et al., 1995).

Fine scale heterogeneity is a characteristic of savannasboth at the scale of grasses (Abbadie et al., 1992) and at thescale of trees (Barot et al., 1999). We will focus here on treespatial heterogeneity. Tree clumping is promoted by fire andgrazing (Menaut et al., 1990; Jeltsch et al., 1998), while regular-ity in tree spatial pattern results from resource competition(Belsky et al., 1989). Short distance seed dispersal may also beresponsible for tree clumping.

While it is easy to measure or compute the effect of trees ongrasses at the scale of a small tree clump, it becomes difficultto do it at a larger scale (e.g. a 1 ha plot) due to the complexityof the tree spatial patterns encountered in savannas. Simioniet al. (2003) have shown that these small scale patterns hada significant effect on radiation absorption, water and carbonfluxes at the plot-scale. Model aggregation methods exist thatenable to scale up from detailed models to coarser resolutionmodels, either in time or in space, or both. Model aggregationis performed through explicit formulation, when mathemati-cally possible, of coarse-scale model parameters as a functionof fine-scale model parameters (e.g. Delage et al., 1999; Augeret al., 2000). However, although scaling up in time is relativelywell understood and modelled (Luan et al., 1996; Wirtz, 2000),scaling up in space is much more difficult: apparently, there isa gap between individual-based (or distance-dependent), spa-tially explicit models and space independent or distributionbased models (Picard and Franc, 2001). For forests, Lischkeet al. (1998) and Loffler and Lischke (2001) did such a spatialupscaling of a gap model, Deutschman et al. (1999) studied theinfluence of small scale heterogeneity with a location depen-dent model. Clearly, there is a lack of a commonly acceptedmethod for scaling up in space in models of spatially hetero-geneous vegetation.

Here, we set up an aggregation method for scaling upthese results, focusing on PAR absorption as the primaryprocess determining plant production, which were obtainedwith the detailed spatially explicit savanna functioning modelTREEGRASS-2 (Simioni, 2001). In this model, radiation physicsin heterogeneous foliage, competition for light and water are

treated mechanistically to simulate water fluxes and primaryproduction for small savanna plots of ∼1 ha. Processes aredescribed in detail with biophysical laws. This enables to testand reproduce the small-scale spatial structure effects, but

2 0 3 ( 2 0 0 7 ) 257–269

prevents scaling to larger areas (km2) where processes likehydrology, animal migration, fire spread and soil heterogene-ity arise. Modelling 1 km2 of savanna implying thousands oftrees with TREEGRASS-2 or any other detailed 3D radiationabsorption model would take too much computation time. Tomodel these larger areas, vegetation spatial structure mustbe summarized. This is usually done rather empirically, andoften the vertical vegetation structure is well represented butnot the horizontal heterogeneity within strata. Some modelslike the SAVANNA model (Coughenour, 1993) use a good com-promise to represent spatial structure effects, but rely on anempirical parametrisation.

The aim of this paper is to derive a simple PAR absorptionmodel from a mechanistic 3D model in a bottom-up approach.This is a first step in the process of scaling up an ecosys-tem model from the plot scale to the landscape scale, usinga mechanistic bottom-up approach.

2. Methods

Intercepted radiation is well described for homogeneous veg-etation by the Beer–Lambert law (BL) (Monsi and Saeki, 1953):

tPAR = iPAR × e(−kLAI) (1)

where tPAR and iPAR are, respectively, the transmitted andincident photosynthetically active radiation, LAI the site treeleaf area index (measured as total tree leaf area divided by thewhole site surface) and k the extinction coefficient of PAR. Allthese variables are measured over a vegetation area consid-ered as homogeneous (Fig. 1a).

Our aim is to adapt a modified version of this lawaccounting for within-plot heterogeneity adapted to sparseor heterogeneous vegetation. We use a bottom-up simula-tion experiment approach, fitting models derived from theBeer–Lambert law to results of TREEGRASS-2 simulations ofplots with contrasted tree spatial patterns.

2.1. The TREEGRASS-2 model

2.1.1. Model descriptionTREEGRASS-2 is a spatially explicit, individual based ecosys-tem model that simulates water fluxes and photosynthesisfor small tree–grass areas (100–10,000 m2) over one to a fewvegetation cycles, with a daily time step for plant physiol-ogy and water fluxes and a sub-daily time step for radiationcomputations. Competition for light and water are treatedmechanistically.

It includes submodels derived from:

(i) The 3D RATP model (radiation absorption, transpirationand photosynthesis) (Sinoquet et al., 2001) that com-putes radiation and energy budgets within vegetationcanopies. No modification of this model has been done

(ii) The PEPSEE model (production efficiency and phenologyin savanna ecosystems) (Le Roux et al., 1996) for primaryproduction, phenology and soil water balance simulation.

e c o l o g i c a l m o d e l l i n g 2 0 3 ( 2 0 0 7 ) 257–269 259

Fig. 1 – Graphical representation of the three simplified models. A part of the incident PAR (iPAR) above the vegetation LAI isabsorbed; the transmitted PAR (tPAR) becomes the incident PAR for the next layer, down to the soil. Three models wereapplied on three layers (tree, grass and soil): (a) Beer–Lambert law; (b) BLD model with two zones (tree and open) and (c)3ZBLD model with three zones (tree, shadow and open). With c and c’ the tree and the shading zone fractional cover, LAItand LAIg, respectively, the tree and the grass LAI, � the proportion of tree leaf area needed to produce the same effect ong d ˛′

t

(

2TIu

rass as lateral shading in a vertically structured model, ˛ anree and the shading zone.

Main features of TREEGRASS-2 are:

(i) Space is divided into a 3D grid of cells. Cell size canbe adapted to fit the scale of vegetation components.Aboveground cells can contain only grass, only tree, orgrass and tree foliage elements. Cells containing foliageare called ‘vegetation cells’. Stems and branches are nottaken into account for radiation transfer. Leaf area of asingle individual tree is spread across cells according tothe proportion of the tree canopy cylinder overlappingeach cell.

(ii) The grass layer is treated as continuous living material atthe scale of grid cells. Trees are treated individually. Eachindividual tree canopy is assumed to occupy a cylindricalvolume which can span many cells, referred to as a foliagecrown.

iii) The radiation absorption submodel (RATP) calculates theamounts of PAR, thermal and infrared radiation absorbedor emitted at each time step in each vegetation cell, andin each soil surface cell. This submodel accounts for thedaily and yearly variations of solar height and azimuth,diffuse radiation, and reflection/transmission by foliagetypes and soil. Radiation balance depends on leaf areadensity and leaf angle distributions in each vegetationcell, and species-specific leaf optical properties. We con-sider two foliage types per grass species (living and deadleaves), and one per tree species (living leaves). Radiationabsorption is computed five times a day, accounting fordiurnal variations in sun position and radiation intensity.Night-time water and energy transfers are neglected.

.1.2. ParametersREEGRASS-2 has been parametrised on the Lamto savanna in

vory Coast (Menaut and Cesar, 1979). Validation tests on sim-lated grass biomass and necromass, soil water in two layers,

, respectively, the coefficient of grass LAI reduction in the

and tree PAR absorption for this site can be found in (Simioni,2001; Simioni et al., 2000, 2003). The model has demonstratedits ability to reproduce observed data for this site, includingspatial pattern effects on grass net primary production, PARabsorption, and evapotranspiration. Regarding PAR absorp-tion, the RATP model has been validated on various systems(Sinoquet et al., 2001).

For this study we retained one type of grass and one treespecies chosen among the dominant species in Lamto. Forthe herbaceous layer we chose a generic grass with com-mon parameters measured on two very close species of theAndropogoneae tribe: Hyparrhenia diplandra and Andropogoncanaliculatus. These two grasses represent more than 45% ofthe standing biomass of the herbaceous layer (Abbadie, 1983;Cesar, 1992; Le Roux, 1995) and have a synchronous phenol-ogy. Crossopteryx febrifuga is one of the dominant tree speciesin Lamto, present either in clumps or as isolated individuals,and tree phenology is different when isolated or in tree clumps(Simioni, 2004). For our simulations, we used the same param-eters, initial values of state variables, and forcing climatic dataas Simioni (2001).

2.1.3. SimulationsSimulations encompassed a range of tree spatial distributions,involving different levels of tree density and clumping. Asa compromise between individual tree size, density, spatialstructure, and computing power (Simioni et al., 2000), we sim-ulated areas of 400 m2 (refered to as “plots”) for a completeyear, with basal grid cells of 1 m2 and four vertical strata. Rel-evance of cell size had been tested in Simioni et al., 2000.About 1 m2 was chosen as the best compromise between spa-

tial detail and computation time. This cell size is sufficient tocapture the spatial variability of tree foliage over a 100–1000 m2

plot. Number of trees per plot varied from 2 to 96 (i.e. from50 to about 2400 trees per hectare), matching the observed

i n g

260 e c o l o g i c a l m o d e l l

range of densities in the field (Menaut and Cesar, 1979). Wedecided to ignore the effects of tree size variability because:(1) a previous study demonstrated no significant effect of treesize distribution on carbon and water fluxes (Simioni et al.,2003); (2) at the scale of our plots, tree location in space islikely more important than tree size in determining fluxes; (3)in savanna landscapes, trees are often of comparable sizes;(4) with regards to PAR absorption, which is a mainly physics-driven process, large trees and small tree clumps might playa comparable role. To focus on spatial pattern effects only, allmodelled trees had the same size and shape: 3 m height, 1.5 mcrown height and 1 m crown radius. Grass covered 100% of thesite surface (no bare soil).

TREEGRASS was initialized with computer-generated mapsof tree communities. Maps were generated with the SAS soft-ware (SAS, 1990). For each tree density, nine different types ofpatterns were generated as initial conditions for simulationsusing standard spatial point process generators (Fig. 2):

(i) ‘Random’ patterns, according to a Poisson process withintensities (average densities) 50, 100, 200, 400, 800, 1600and 2400 trees ha−1.

Fig. 2 – Maps of the spatial patterns of the simulation experimenclumped, regular), total number of trees on the plot and numberpositions are depicted on maps, all trees have the same size anddensity (2, 4, 8, 16, 32, 64 and 96 trees per plot) nine different typaggregated and two regular patterns). Long and short are the inhfor aggregated patterns and trees for regular patterns.

2 0 3 ( 2 0 0 7 ) 257–269

(ii) Six types of aggregated patterns, according to a clus-ter Poisson process (Ripley, 1977; Diggle, 1979) withmean clump radii of 0.765 or 1.53 m, and 4, 16, or32 trees per clump. No pattern was generated when totaltree number was lower than the number of trees requiredto form a single clump.

(iii) Two types of regular patterns, according to a sequen-tial inhibition process (Ripley, 1977; Diggle, 1979) withan inter-trunk inhibition distance of 1.5 m (short) or2 m (long) and the same intensities as the randompatterns.

2.1.4. Vegetation structure descriptorIn natural systems, the parameters of plant spatial distribu-tion are often unknown and difficult to estimate. In our case,we used a standardized measure of tree crown overlap (˝) tosummarize tree spatial pattern, with a small number of plant

parameters:

˝ = nS − c

nS − Smax(2)

t. All plots are arranged by tree distribution (random,of trees per clump. Plots are 20 m × 20 m. Only treesshape: 3 m height and 1.5 m crown radius. For each treees of patterns were generated (one random, six types ofibition distance, respectively, 2 and 1.5 m, between clumps

g 2 0

tt˝

l

otc

˝

2

Ipoaea

2Fmgbctcicctdst

Ll

gt(dcda

t

tfgo

[−{1/

e−(1

tree height from all sun elevations, removing the early morn-ing and late afternoon sun position that would yield a too largezone (Fig. 3). The criterion used to reject sun positions wasthat the total of positions removed represented less than 10%

Fig. 3 – Variation of the shading zone extent (expressed asthe ratio of the shade zone extent (a) to the tree total height(Ht) in meters) from canopy edge for various latitudes (indegrees). Computation was based on standard astronomiccalculations of sun elevation, day length, and incomingsolar energy at the top of the atmosphere (McCullough andPorter, 1971), using a 1 min time step. For the study site

◦

e c o l o g i c a l m o d e l l i n

With S the tree canopy mean area (m2), Smax the maximalree canopy area (i.e. area of the largest tree of the plot in m2), che tree cover (m2) and n the number of trees on the plot. When

= 0, the distribution is either very regular or with a veryow density (i.e. tree crowns almost never overlap), while an

verlap equal to 1 means a very aggregated distribution (i.e., allree crowns are covered by the crown of the largest tree). In ourase, with identical trees, S = Smax and the formula becomes:

= nSmax − c

(n − 1)Smax(3)

.2. Simple PAR interception model

n order to model PAR interception at the plot scale, we com-are the performance of three simplified models, an originalne and two already proposed models (the Beer–Lambert lawnd a simple, two-zones Beer–Lambert derived model (Jacksont al., 1983; Dolman, 1993)). All these models only require vari-bles and parameters measurable at the whole plot scale.

.2.1. A simple Beer–Lambert derived (BLD) modelor a mixed tree–grass vegetation as in a savanna, a minimalodel is to consider a partition of space into two areas, the

rass layer with no trees (“open area”) and the grass layereneath trees (“tree area”) (Fig. 1b). Tree fractional cover (c)an be used to quantify the proportion of these two areas onhe plot, so the incoming PAR can be expressed in each zone as×iPAR (tree area) and (1 − c) ×iPAR (open area). Since our goal

s to build a simple plot-scale model, we have to use plot-scaleomputed LAI. Hence LAIs appearing in equations are alwaysomputed over the whole plot surface. However, partitioninghe plot into two zones with and without trees means twoifferent local LAIs must be considered for physiological con-istency in this plot. The local and whole-plot LAIs are relatedhrough the tree cover parameter:

LAI = c ×LAIc + (1 − c) ×LAI1−c where LAI is the whole plotAI, LAIc the local LAI in the tree covered zone and LAI1−c theocal LAI in the open zone.

In the following the t and g indices stand for trees andrasses. aPAR, tPAR and iPAR are, respectively, the absorbed,ransmitted and incident photosynthetically active radiationW m−2). LAIt, LAIg and dLAIg are the tree, living grass andead grass leaf area index (m2 m−2) over the entire plot andthe proportion of tree cover (0 ≤ c ≤ 1). Parameters kt,, kg andkg are, respectively, light extinction coefficient for trees, grassnd dead grass outside and under the tree canopy.

For the tree layer the model becomes:

PARt = iPAR(1 − c) + iPAR × c × e(ktLAIt/c) (4)

For the grass layer, we assume that there is a reduc-

tPARg = [iPAR(1 − c)e

+ [iPAR × c ×

ion (˛) of grass LAI under tree canopy due to competitionor light between trees and grass. We therefore write therass LAI in the tree zone (LAIc) as: LAIc = ˛LAI1−c. Grass LAIn the plot (LAIg) corresponds to the sum of LAI in each

3 ( 2 0 0 7 ) 257–269 261

zone (under and outside tree canopy), computed over thesurface of the zone, weighted by the corresponding area:LAIg = cLAIg,c + (1 − c)LAIg,1−c. For simplicity, we assumed thesame reduction (˛) of the dead LAI under tree canopy.

All this yields for the grass layer:

(1−c)}][{kg(LAIg/1−(1−˛)c)+dkg(dLAIg/1−(1−˛)c)}]]

/c){[ktLAIt+kg˛(LAIg/1−(1−˛)c)]+[dkg˛(dLAIg/1−(1−˛)c)]}] (5)

2.2.2. A more complex model, the three zonesBeer–Lambert derived model (3ZBLD)We obtain this model by simply adding a third area to theBLD model to represent the effect of lateral shading by trees(Fig. 1c). Because of scattering and reflection, lateral tree shadeeffects are to be expected. The shading zone is represented asa constant width area around tree crowns. The shape of theshading zone is normally more asymmetric (i.e., it should be acircle on the equator line only, an egg-shaped area within thetropical zone, and a crescent-shaped area outside), but here wefocus only on the size of this zone and neglect the shape effect,considering that most savannas occur in the tropical zone.TREEGRASS computes the real solar positions, hence the exactshape of the shading zone. We computed the shading zoneextent by averaging over a whole year the ground-projected

used in simulations, latitude is 5.13 N, the threshold ofincoming radiation is ∼580 W m−2 (for a maximumincoming solar radiation of ∼1360 W m-2) which yields a0.85 coefficient, hence a 2.55 m shade zone extent for 3 mtall trees.

262 e c o l o g i c a l m o d e l l i n g 2 0 3 ( 2 0 0 7 ) 257–269

Fig. 4 – Example of a TREEGRASS simulation showing thedecrease in grass LAI (gray levels, the darker meaning thegreater reduction in LAI) related to tree crown positions(inner circles) and shading zone extent (outer circles). Allthe significant grass biomass reduction is included within

kg{(dLAIg)/(1−c′−c)+(˛c)+(˛′c′)}]

Fig. 5 – Relation between tree–grass competitionparameters and vegetation structural parameters. With ˛ isthe reduction of grass LAI under the tree crown, ˛′ is thereduction of grass LAI under the tree shadow, ˝ is the treecrown overlap and LAImax the maximum LAI over theseason. (a) ˛ as a function of ˝; (b) ˛′ as a function ofLAImax. Dots are the values and lines represent best fit: (a)

dataset and reduce the number of parameters to estimate

the shading zone.

of the total incoming solar radiation. This procedure yieldeda ∼2.5 m zone from the canopy edge for 3 m tall trees. Thisextent closely matches the observed grass LAI decrease nearbytrees simulated by TREEGRASS (Simioni et al., 2000 and Fig. 4).In this area, lateral shade is simulated by adding a virtualfraction ˇ of total tree LAI over the area. ˇ represents the pro-portion of tree leaf area needed to produce the same effect ongrass as lateral shading in a vertically structured model. Thefake tree LAI over the shading area is ˇ LAIt. Parameter c′ isthe fractional area of the tree shading zone similar to c.

The equation for trees becomes:

tPArt = iPAR(1 − c′ − c) + iPAR × c′ × e(−ktˇLAIt/c′)

+ iPAR × c × e(−ktLAIt/c) (6)

For the grass layer, there is a partition according to c and c′;˛′ is the coefficient of grass LAI reduction in the shading zonesimilar to ˛ (under trees in the BLD model). Grass LAI in thetree zone (LAIc) is: LAIc = ˛LAI1−c−c′ and in the shading zoneLAIc′ = ˛′LAI1−c−c′ . As in the BLD model, grass LAI on the plot(LAIg) corresponds to the sum of LAI in each zone weighted byeach area LAIg = c LAIc + c′LAIc + (1 − c − c′)LAI1−c−c′ . The incom-ing PAR is partitioned into c iPAR under tree canopy, c′ iPARin the shading zone and (1 − c − c′) iPAR in the open area. Thesame applies to dead grass, yielding:

′ [−{1/(1−c′−c)}][kg{(LAIg)/(1−c′−c)+(˛c)+(˛′c′)}+d

tPArg = iPAR(1 − c − c) × e

+ iPAR × c′ × e(−1/c′)[ktˇLAIt+kg˛′{(LAIg)/(1−c′−c)+(˛c)+(˛′c′)}+dkg˛′{(dLA

+ iPAR × c × e(−1/c)[ktLAIt+kg˛{(LAIg)/(1−c′−c)+(˛c)+(˛′c′)}+dkg˛{(dLAIg)/

˛ = −043 � + 0.71 (r2 = 0.85, p < 0.05, d.f. = 40) and (b)˛′ = −0.20 log(LAImax) + 0.72 (r2 = 0.80, p < 0.05, d.f. = 40).

2.2.3. Fitting the BL, BLD and 3ZBLD models toTREEGRASS-2 results˛ and ˛′ were directly computed for each map from simulation3D detailed outputs (Fig. 5). In those detailed outputs, we dis-tinguished grass pixels under tree crown, in the shading zoneand in the open area. ˛ is the ratio between the mean of grassLAI tree crown pixels and the maximum grass LAI in the openarea. Similarly, ˛′ is the ratio between the mean of grass LAItree shadow pixels and the maximum grass LAI in the openarea.

Eqs. (1) and (4)–(7) were fitted to TREEGRASS-2 results, withiPAR, tPAR, LAIg and LAIt, as input variables, and kt, kg and dkg

as unknown parameters. In order to use efficiently the whole

Ig)/(1−c′−c)+(˛c)+(˛′c′)}]

(1−c′−c)+(˛c)+(˛′c′)}] (7)

g 2 0

iwiufpaa

dv

2

Maptcfi

3

3T

3Al>fitT

FT(

e c o l o g i c a l m o d e l l i n

n each model, fitting for complex models (BLD and 3ZBLD)as performed in two steps: first tree interception, then grass

nterception. For the 3ZBLD model, there is one additionalnknown parameter, ˇ, since c, ˛, c′ and ˛′ were computedor each map. All fits were performed using the PROC NLINrocedure of the SAS software (SAS, 1990). Goodness of fit wasssessed through the root mean square error (RMSE) statistics,nd through a graphical analysis (Figs. 6 and 7) with:

RMSE =√

(obs2 − pred2) with obvious notations for pre-icted (TREEGRASS-2) and observed (BL, BLD and 3ZBLD)alues.

.3. Comparing model results to real data

easurements of radiative exchange for the 1994 year arevailable for our study site (Gautier, 1994). The real site is alot of 400 m2, with 36 trees (Fig. 8). The structural parame-ers are c = 77 m2, S = 2.32 m2, Smax = 21.11 m2 and ˝ = 0.11. Weomputed the shading zone of trees from the map of the realeld plot, tree height and site latitude as explained above.

. Results

.1. Fitting BL, BLD and 3ZBLD models toREEGRASS-2: selection of the best model

.1.1. Tree PAR interceptionll non-linear model adjustments were significant at the 0.01

evel (tests based on approximate F statistics), and r2 were

0.99 in all cases. The 3ZBLD model was the only one tot the outputs of TREEGRASS-2 for all types of spatial pat-erns (Figs. 6 and 7), especially for the clumped distribution.he BL model showed a good fit only for random and regular

ig. 6 – Model evaluation: root means square error (RMSE) of theREEGRASS-2 PAR absorption and ‘predictions’ are PAR absorptio�), BLD (�) and 3ZBLD (�).

3 ( 2 0 0 7 ) 257–269 263

spatial patterns. The BLD model overestimated PAR transmis-sion in almost all the low density clumped spatial patterns(Figs. 6 and 7). The temporal pattern of tree PAR intercep-tion was correctly simulated by all models in spite of theirdifferences (Fig. 9).

3.1.2. Grass PAR interceptionAll non-linear model adjustments were significant at the 0.01level (tests based on approximate F statistics), and r2 were>0.99 in all cases. In a few cases (maps 7, 39, 40, 46 and 47, cf.Fig. 2) where the total tree influence area (c + c′) was equal to1, the BL model fitted the data better than the others. On mostof the LAI range, the three models fitted the data equally well.The temporal pattern of grass PAR interception was correctlysimulated by all models in spite of their differences (Fig. 9). Athight LAI, the absorption of PAR by dead grass leaves causeda hysteresis in the absorption curve in all models (includingTREEGRASS).

3.2. Variation of estimated model parameters as afunction of vegetation structural parameters

3.2.1. Extinction coefficientsThe kt coefficients ranked in average as follows: lower rangewith the BL model (0.05–0.5), intermediate range with the3ZBLD model (0.4–0.7), and higher range with the BLD model(0.4–0.8) (Fig. 10). The kt of the 3ZBLD model varied significantlywith tree overlap (Fig. 10), but not with tree cover or shadingzone area. The same pattern was observed for both BLD and

BL models.

kg coefficients for living leaves were lowest for the 3ZBLD(0.2–0.5) and BLD (0.2–0.55) models, and were higher for the BLmodel (0.55–0.65). They varied significantly with tree cover and

three models for all spatial patterns (‘observation’ isn by Beer–Lambert, BLD and 3ZBLD models): Beer–Lambert

264 e c o l o g i c a l m o d e l l i n g 2 0 3 ( 2 0 0 7 ) 257–269

Fig. 7 – Four examples of TREEGRASS-2 outputs fitted by the Beer–Lambert, BLD and 3ZBLD models for tree (middle column)and grass (right column) layer with the A16 random pattern, the G16s4 compact clump, the G32s32 loose clump and theR64l regular pattern. Maps represent the tree influence on the grass layer (left column). Maps show the grass LAI for eachpixel of the simulated plot and pixel level of grey depicts the tree impact on the grass LAI: white if there is no LAI reduction

n cycD. x-

and black for the greatest reduction (for the whole vegetatioBeer–Lambert; grey soild line: BLD and black solid line: 3ZBL

shading zone for the BLD and 3ZBLD model, but were constantin all other cases (Fig. 10).

3.2.2. Tree–grass competition parameters (˛ and ˛′)˛ and ˛′, which represent the reduction of grass LAI in the zoneunder tree influence, were only related to the tree structuralparameters: ˛ was linearly related (r2 = 0.85) to the tree overlap(Fig. 5); and ˛′ was a logarithmic function of the maximum treeLAI (r2 = 0.80) (Fig. 5).

3.2.3. Shading intensity parameter (ˇ)� was not significantly related to tree cover, area of theshading zone, nor aggregation, although values tended to be

le). Grey dotted line: TREEGRASS-2; black dashed line:Axis: LAI (m2 m−2) and y-axis: log(tPAR/iPAR).

smaller for large cover values. All values ranged between 0 and0.1 (mean = 0.031, STD error = 0.027), meaning that the lateralshading effect of trees is equivalent to 3% of their foliage beingadded over the area of influence considered here (i.e. ranging2.5 m away from the canopy edge).

3.3. Comparing model results to real data

We compared, for the tree level, the absorbed PAR predicted bythe four models to real data (Fig. 8). Results show a good pre-diction of PAR absorption for TREEGRASS-2 (r2 = 0.89, p < 0.05,d.f. = 108), 3ZBLD (r2 = 0.88, p < 0.05, d.f. =108) and BLD (r2 = 0.84,

e c o l o g i c a l m o d e l l i n g 2 0 3 ( 2 0 0 7 ) 257–269 265

Fig. 8 – Comparison of PAR transmission modelled with TREEGRASS, BL, BLD and 3ZBLD to real data from Gautier (1994).Top left: map of tree crowns on the measured field plot (20 m × 20 m); top right: simulated map with canopies in black solidline. Bottom curves: simulated transmitted PAR (Mj) vs. measured transmitted PAR (Mj) for all the all the vegetation cycle fort line.

pi

4

4v

TuJF(12(hpv

he four models used in this paper. Solid grey line is the 1:1

< 0.05, d.f. = 108). For the Beer–Lambert model, the predictions worse (r2 = 0.52, p < 0.05, d.f. = 108).

. Discussion

.1. A single model able to handle all types ofegetation spatial patterns

he Beer–Lambert law was developed and is currently mostlysed in models of forests (Korzukhin and Ter-Mikaelian, 1995;

ohansson, 1996; Bartelink, 1998; Brunner, 1998; de Castro andetcher, 1998; Bergez et al., 1999; Zhang and Xu, 2002), cropsShuttleworth and Wallace, 1985; Leblon et al., 1991; Daamen,997; Brisson et al., 1998; Purcell, 2000; van Oosterom et al.,001; Mackay et al., 2003) or other homogeneous vegetations

Nouvellon et al., 2000; Hartz-Rubin and DeLucia, 2001). Foreterogeneous vegetation, various solutions have been pro-osed: interception proportional to the fractional cover of eachegetation component (Jackson et al., 1983; Dolman, 1993);

empirical correction coefficients depending on tree clumping(Coughenour, 1993); mechanistic models representing absorp-tion at the leaf level, and deducing the total canopy absorptionassuming heterogeneity in leaf vertical distribution, includ-ing leaf angle distribution (Larsen and Kershaw, 1996), or inleaf 3D distribution within canopy (Sinoquet et al., 2001), orin the distribution of canopies in space (Asner and Wessman,1997; Bartelink, 1998; Simioni et al., 2000); mechanistic mod-els, like AMAPs model (Godin and Caraglio, 1998), using anexplicit 3D representation of all plant leaves and branches.The strength of our work is to summarize results of a rathercomplex mechanistic model into a relatively simple empiricalmodel by simply considering the area of tree shading, and notonly the tree canopy area.

Our results demonstrate that to properly model PAR inter-ception of trees and grasses in savanna systems, one needs

to account not only for tree cover, but also for a shading zoneextending outside tree canopies. The BL model fails at mod-elling tree PAR interception as soon as some tree clumpingis present (Figs. 6 and 7), while the BLD model fails for the

266 e c o l o g i c a l m o d e l l i n g 2 0 3 ( 2 0 0 7 ) 257–269

Fig. 9 – Four examples of seasonal course of tree and grass PAR absorption simulated by TREEGRASS, BL, BLD and 3ZBLDmodels for the tree (middle column) and the grass (right column) layer with the A16 random pattern, the G16s4 compactclump, the G32s32 loose clump and the R64l regular pattern. Maps represent the tree influence on grass layer (left column).Maps show grass LAI for each pixel of the simulated plot and pixel level of grey depicts the tree impact on grass LAI: whiteif there is no LAI reduction and black for the greatest reduction (for the whole vegetation cycle). Grey dotted line:

ne: B

TREEGRASS-2; black dashed line: Beer–Lambert; grey solid liy-axis: log(tPAR/iPAR).

same types of spatial patterns, and, what is more important,whenever grass LAI is high (Fig. 7). Even if it fails only forLAI > 0.8 LAImax, this means that absorption will be misesti-mated during most of the phenological cycle. The 3ZBLD isthe only one to fit for all types of spatial patterns (Fig. 6).

All the spatial patterns where the 3ZBLD models was notthe best fit were patterns with c + c′ = 1, which means that allthe plot was under the influence of trees. And not surprisingly,

the BL model fitted well in those cases (all with random or reg-ular, high density patterns of trees), because they were muchcloser to the assumption of a homogeneous cover implied bythe BL model than the other cases.

LD and black solid line: 3ZBLD. x-Axis: LAI (m2 m−2) and

4.2. Plot-level PAR extinction coefficients

For different species of trees (with different leaf angle dis-tributions), reported Beer–Lambert extinction coefficients areusually comprised between 0.28 and 0.65 (Martens et al., 1993;Cournac et al., 2002). 3ZBLD kt are in the same range (0.35–0.7),which means that variation of kt due to tree aggregation is ofthe same order of magnitude as the interspecific variations

due to leaf angle distribution.

For aggregation <0.7, kt did not vary with ˝ for theBLD and 3ZBLD models (Fig. 10). This means that aggre-gation is correctly accounted for by the structure of the

e c o l o g i c a l m o d e l l i n g 2 0

Fig. 10 – Variation of absorption coefficients withvegetation structural parameter. (a) kt as function of ˝; (b)kg as function of c + c′. With the Beer–Lambert ( ), BLD (�)and 3ZBLD (�) models. In (a) lines represent best fit with asegmented model, with break points at ˝ = (0.5164 (t = 26.71,p < 0.05, d.f. = 38), 0.6969 (t = 10.29, p < 0.05, d.f. = 38), 0.7139(t = 17.91, p < 0.05, d.f. = 38)) for the Beer–Lambert, BLD and3ZBLD models, respectively. In (b) lines represent best fit,with r2 = (na, 0.91, 0.79) for the Beer–Lambert, BLD and3ZBLD models, respectively. Black square correspond to as

Boptc

r−ismy∼

aiiopboa

able to compute PAR interceptions well related to TREEGRASS-2 PAR interceptions for a wide range of tree spatial patterns.

ame value for BLD (�) and 3ZBLD (�) models.

LD and 3ZBLD models over this range. An aggregationf 0.7 is a fairly high value (in average, every tree of thelot has 70% of its crown shaded by another), meaninghat our results hold for a probably wide range of naturalases.

When aggregation was >0.7, all models show a significantesponse of kt to aggregation, with a similar slope close to0.6. This can be due to self shading occurring within trees

n clumped areas: when leaf area density becomes very high,elf shading is likely to decrease the total absorption. Further-ore, a single mechanism is likely to explain that all models

ield the same slope and the same aggregation threshold of0.7.

Contrary to kt, kg showed a residual variation with covernd shading zone area (Fig. 10), but not with tree overlap. Thentercept of all regression curves tended to the value estimatedn the BL model when tree cover was close to 0. The decreasef kg with increasing tree cover is due to a decrease in the pro-

ortion of grass biomass under trees (˛ and ˛′ parameters);ut assuming a constant decrease of grass LAI under treesver the whole phenological cycle is a crude approximation,nd variations of this decrease along the seasonal cycle may

3 ( 2 0 0 7 ) 257–269 267

have a significant impact on the kg estimates and explain theirresidual variation with tree cover.

4.3. The area of tree influence on grasses

The ˇ parameter measures the lateral shading effect of treeson grasses. It does not vary with the surface of the shadingzone nor with the tree overlap. The average value of ˇ is ratherlow (3%), yet this parameter is particularly important since it isresponsible for the better fit of the 3ZBLD model compared tothe two simpler models, and affects PAR interception mainlywhen LAI is well developed.

The estimation of the tree influence area is based on: (1)trivial computations of sun position, (2) the assumption thatthe distribution of sun directions can be truncated to thoseyielding 90% of the total incoming solar energy, and (3) theassumption that the shading zone is symmetrical in all direc-tions (which is a crude approximation, only valid close to theequator). This computation closely matches preliminary runsof TREEGRASS-2 and observed impact of tree presence on thenearby grass cells (Fig. 4 and Simioni, 2001). A better estimatewould easily be obtained by computing solar azimuths andrepeating the sampling of sun directions on the same cri-teria, yielding the exact shape of the shading zone (i.e. thiswould relax the assumption of a circular shading zone aroundtree crowns, only valid in the tropics). This would provideparametrizable and possibly more accurate estimates of thesurface of the shading zone and of ˇ.

4.4. Our 3ZBLD model and large scale net primaryproduction (NPP) modelling

In a climate change context, it is important to improve theprediction of NPP of terrestrial ecosystems (Gower et al., 1999;Prince, 1991). A simple model relates NPP and aPAR directlyby a positive linear relation: NPP = εaPAR, where � is the effi-ciency of aPAR use in biomass production (Green et al., 2003;Monteith, 1972). 3ZBLD takes into account vegetation plot het-erogeneity to calculate intercepted PAR. Turner et al. (2002)tried to access to the spatial variability of ε to improve the NPPmonitoring. They concluded that the spatial variability of ε

was a good way to improve NPP prediction if it could be associ-ated with a good aPAR prediction. The present study provides amethod to take account the spatial heterogeneity on the aPARprediction, and so can be used with success on a simple NPPmodel.

5. Conclusion

The 3ZBLD model is a good trade-off between computationpower and realism, which is intended to facilitate the mod-elling of radiation absorption at the scale of whole pixels inlandscape models. Knowing tree cover and clumping, we were

We showed that the Beer–Lambert’s law is suitable for denserandom or regular spatial patterns, but that the tree shad-ing area must be considered for clumped spatial patterns andlow-density patterns.

i n g

r

268 e c o l o g i c a l m o d e l l

Acknowledgments

This work was made possible through a Ph.D. grant fromFrench MESR (Ministere de l’Enseignement Superieur et de laRecherche). We would like to thank gratefully the two anony-mous reviewers for their constructive criticism.

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