( θ ), the macro-gaps between

Multiscale modelling of gap fractions within stands

J. Dauzat1_{, N. Lamanda}2_{, Y. Nouvellon}3

1 _{UMR AMAP, TA 40 / PS2, 34398 Montpellier cedex 05, France} 2 _{Cirad cp-cocotier, TA 80/01, 34398 Montpellier 5, France}

3 _{Cirad forêt-plantations, BP 1264 - Pointe-Noire - République démocratique du Congo}

Introduction

In a homogenous canopy with randomly dispersed leaves, the directional gap fractions can be calculated as

P

( )

θ

₌

e

−G( )θ LAI/cosθ_{(eq.1), where θ is a zenithal angle and G(θ) a function converting}

the actual Leaf Area Index into its projection area. However most actual stands do not fit this formula which has to be corrected with a clumping index _{Ω as proposed by Nilson [4]:}

( )

_θ

_e

G( )θ (θ)LAI/cosθ

P

₌

− Ω _{(eq. 2). This directional clumping index can be estimated from gap fractions}

measurements, but its further use is limited to the cases where the canopy structure remains comparable. When this condition is not fulfilled –for instance in the case of higher plant density- the gaps between neighbouring plants are strongly affected and the stand clumpiness may be significantly modified. For this reason, we introduced a distinction between

P

_M

( )

θ

, the “macro-gaps” between plant crowns and P_µ

( )

θ

, the “micro-gaps” between foliage elements within plant crowns. The first goal of this paper was to analyse through simulations on coconut stands in what extend a “micro-clumping index”, _Ωµ, defined from P_µ

( )

θ

analysis is actually independent of plant density. Once the above assumption is validated, one can apply the micro clumping index, Ω_µ, for the stand LAI inversion. This implies that the macro and micro-gap fractions can be properly discriminated. The second goal of this paper was to test a practical procedure for this discrimination through simulation experiments on a theoretical mixed forest stand.

Material and methods

Model definition

The total stand gap fraction

P

(

_θ

)

, can be written as the sum of the macro-gaps fraction between the tree crowns,

P

_M

( )

θ

and the micro-gaps fraction,P_µ

( )

θ

inside the tree crowns:

( )

θ

θ

θ

P Pµ

P( )= _M( )+ (eq.3).

According to this model, the efficient LAI restricted to the crown projection area is:

[

1 ( )

]

/

_θ

µ LAI PM

LAI = − , (eq.4).

and the corresponding micro-clumping index _Ωµ

( )

θ

is such that:

( )

( )θ θ θ µ

θ

µ µ cos / ) ( LAI G

e

P

₌

− Ω (eq. 5)

It was hypothesised that Ω_µ

( )

θ

computed in that way is less dependent on stand canopy structure (tree density) than the stand clumping index

_Ω

( )

θ

. If so, _Ωµ

( )

θ

derived for a given canopy could be used for LAI inversion on other canopies of the same species.

In order to test this hypothesis, a simulation experiment was carried out, using realistic “architectural” mock-ups to estimate

P

(

_θ

)

, and opaque ellipsoidal crown representations to

within a subplot will correspond in most cases either to

P

_M

(

_θ

)

or to P_µ(

_θ

). This procedure is tested over theoretical mixed forest stands of the same LAI but variable

P

_M

(

_θ

)

resulting from tree patterns.

Tree crown representations

The model described above has been tested on coconut stands. The architectural description of 15 years old trees was done in Vanuatu and data were modelled as described in Dauzat and Eroy [2]. Stochastic 3D coconut mock-ups (fig. 1a) were then simulated with the “Lubi” coconut simulator of the Archimed simulation platform. These “architectural” mock-ups represent individual leaflets with 6 polygons. In order to get simplified coconut crowns representations and shorten the simulations of light interception, alternative representations of fronds without individual leaflets are output by the “Lubi” program (fig. 1b). These representations are obtained by joining the base and tips of some leaflets selected along the rachis (e.g. every 10 leaflets). A frond is thus represented with a single lamina but its global geometry is kept. Last, semi-ellipsoidal envelopes were fitted to the coconut crowns (fig. 1c)..

Figure 1:

a: “architectural” representation of a coconut with each leaflet represented by 6 polygons;

b: wire-frame view of a simplified coconut mock-up in which fronds are represented with a single lamina;

c: semi-ellipsoidal crown representation (in yellow) fitted on its architectural representation (in red). The procedure for LAI estimation from field data was tested on a mixed canopy with conical and ellipsoidal crowns (figure 2). Leaves were positioned randomly or with clumping within the crowns and a spherical leaf angle distribution (LAD) was used.

Figure 2:

Simulation of a mixed forest stand with truncated ellipsoidal or conical crown shapes. Leaves are positioned randomly or with clumping within the crowns. The leaf angle distribution (LAD) is set as spherical.

Simulation of multiscale clumping indices in coconut stands

A coconut stand was constituted with architectural or geometrical trees in a triangular pattern. At the regular density (143 trees ha-1_{), neighbouring trees are 9m apart. For the ¼ density, trees are more}

widely spaced keeping the same pattern.

The simulation of gap fractions is performed with the MIR model of the Archimed simulation platform [2]. This program calculates images by projecting the stand elements on a horizontal plane In this case, the images figure the shade cast by trees on a horizontal ground. The gap fraction for a given viewing angle is therefore the ratio of the non-shaded pixels of the image over its total pixels number. Note that the MIR model is run with a toricity option simulating an infinite coconut stand.

The gap fractions are simulated for the 46 directions defined by the Turtle model [3] with several azimuths in 8 elevation angles. For the analysis, gap fractions obtained for the same elevation in different azimuths are averaged. The simulations run with the architectural mock-ups give the total gap-fractions

P

(

_θ

)

whereas the simulations run with the ellipsoidal coconuts give the macro-gap fraction

P

_M

(

_θ

)

. The micro-gaps fraction P_µ (

_θ

) is obtained by difference from eq. 3. Note thatP_µ(

_θ

) is dependent on the crown representation. If coconut crowns are represented with single lamina fronds instead of semi-ellipsoidal shapes, the intra-crown micro-porosity will be much smaller (because it doesn’t integrate the inter-fronds gaps). We thus refer later on P_µ (

_θ

) as the crown porosity,

P

_c

(

θ

)

, when deduced from projection of ellipsoidal crowns and as the frond porosity,P_f (

_θ

), when deduced from the projection of single lamina fronds. The associated clumping indices derived from eq. 5 correspond to two clumping scales and are consequently noted

_Ω

_c

( )

θ

and

( )

θ

f

Ω .

LAI inversion in a mixed forest stand

LAI estimation from gap fractions was tested over theoretical mixed forest stands with regular or clumped tree patterns of the same LAI of 2 (fig. 3). Averaging simulated gap fractions at the scale of square subplots gives a set of P_subplot

( )

θ

values sampled over the stand area. The theoretical laisubplot

associated to a given value of P_subplot

( )

θ

is given by inversion of eq. 2:

) ( ) ( / cos )) ( ln(

_θ

_θ

_θ

Ω

_θ

− = P G

lai _subplot eq. 6

When a subplot is included within a macro-gap, Psubplot

( )

θ

=1 and lai = 0. In other cases,

( )

θ

subplot

P is assumed to correspond to the micro-porosity so that _Ωµ

( )

θ

(supposedly determined from previous experiments) can be applied. The stand LAI and the foliage LAD parameter _{χ can then} be fitted in such way that the integration of

( )

θ

G( )θ µ(θ)LAIµ/cosθ

subplot

e

P

=

− Ω is as close as possible to the simulated gap fraction at the stand scale.

Figure 3: Maps of local gap fractions in virtual mixed forest stands with the same trees in a regular pattern

(left) and a clumped pattern (right). Results

Clumping index in coconut stands

Figure 4 presents the stand clumping index _{Ω calculated for a regular tree density and ¼ of it. As} expected, the _{Ω value is strongly modified when changing the density. At the opposite, the crown} clumping index, _Ωc, appears only slightly dependent on the tree density. The frond clumping index has

a higher value (less clumpiness) and can be considered as independent of the tree density.

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 30 60 90 zenithal angle cl u m ping i n de x stand reg. density stand 1/4 density crown, reg. density crown, 1/4 density frond, reg. Density frond, 1/4 density Figure 4:

Clumping index calculated at stand scale (lines), crown scale (circles) and frond scale (x,+) for 2 coconut densities

LAI inversion from gap fractions in a mixed stand

LAI estimation is expressed as a function of the spatial integration area of P(_{θ) (i.e. the subplots} size). One can check (fig. 5) that the estimated LAI and _{χ values tend toward their actual value (2 and} 1 respectively) for small areas.

0.0 0.5 1.0 1.5 2.0 2.5 0.1 1 10 100 1000 subplot size (m2₎ fitte d L A I 5ag 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.1 1 10 100 1000 subplot size (m2) fitte d c 5ag

Figure 5: Estimation of LAI and χ (parameter of the ellipsoidal LAD) as a function of the integration area of

gap fractions. Actual values are 2 and 1 respectively. Discussions and conclusions

The stand clumping index is strongly dependent on the tree density and its improper use may lead to important errors in LAI estimation from gap fractions. Simulation experiments on coconut models show that a clumping index, _Ωµ

( )

θ

, derived from the micro-gap fraction is far less dependent on the tree density and can therefore be used for LAI inversion in different coconut stands provided that the macro and micro-gap fractions can be discriminated.

The macro and micro-gaps components can be inferred from simulations but an operational alternative procedure is proposed when using field data. Results show that this procedure allows a correct stand LAI estimation. Despite the fact that simulations were done without foliage clumping within tree crowns1_{, the procedure is expected to give similar results whenever the proper crown}

clumping index is used.

From these two simulation experiments, it appears that a multiscale gap model allows the definition of a crown clumping index specific of a tree species whereas the stand clumping that integrates macro-gaps and micro-macro-gaps is strongly dependent on the stand structure and density. Further work will be needed to check the stability of the crown clumping index in different situations for a given species.

References

[1] Chen, J. M. and J. Cihlar, 1995b. Plant canopy gap size analysis theory for improving optical measurements of leaf area index. Applied Optics, 34:6211-6222.

[2] Dauzat, J., Eroy, M.N., 1997. Simulating light regime and intercrop yields in coconut based farming systems. Europ. J. Agron. 7, 63–74.

[3] Den Dulk J.A., 1989. The interpretation of remote sensing, a feasibility study. Thesis, Wageningen.

[4] Nilson, T., 1971. A theoretical analysis of the frequency of gaps in plant stands. Agric. For. Meteorol. 8, 25–38.