Three dimensional model of the interception of light by a canopy

Agricultural and Forest Meteorology 90 Ž1998. 215–233

Three dimensional model of the interception of light by a canopy Francisco de Castro ) , Ned Fetcher Dept. of Biology, UniÕersity of Puerto Rico, P.O. Box 23360, Rio Piedras, 00931-3360, Puerto Rico Received 27 December 1996; accepted 23 November 1997

Abstract We present a computer model that simulates, in three dimensions, the interception of light by a canopy. The model divides the canopy into cubic cells, each one characterized by mean leaf angle and the leaf area index. The model calculates the probability that a beam will penetrate to any given cell without being intercepted by the foliage in the path, using an exponential extinction function. Penetration of direct and diffuse light is calculated separately. The model also considers the reflection and transmission of light. The model has been tested against real measurements taken in an artificial plantation, reaching an r 2 of 0.714 between predicted and observed values. A sensitivity test of the model is also presented. The parameters analyzed are: total Leaf Area Index ŽLAI., vertical distribution of LAI, incident radiation, mean angle of leaves, transmission coefficient of leaves and elevation of the sun. A stepwise regression analysis of the results let us identify the variables which most affect the model outputs: Leaf Area Index, elevation of the sun, mean leaf angle, and incident radiation. The transmission coefficient of leaves does not exert significant influence on the intercepted radiation. The proportions of direct and diffuse components in incoming radiation have a significant influence on the results. q 1998 Elsevier Science B.V. Keywords: Model; Canopy; Radiation interception; Puerto Rico; Tropical forest

1. Introduction Light can often limits plant recruitment and growth in shaded environments. If nutrients or water are not limiting, then radiation is usually the most important factor influencing the growth and development of plants ŽChazdon and Fetcher, 1984; Chazdon, 1986; Comeau et al., 1993; Oberbauer et al., 1989; Popma and Bongers, 1991; Russell et al., 1989.. Also, the quality of the light Žits spectral composition. can affect the morphology of vegetation ŽEndler, 1993.. )

Corresponding author. Fax: q1-787-764-3875; e-mail: [email protected]

The interception of light by a canopy and the radiation environment in the understory depends on many factors: latitude, position of the sun, LAI, the three dimensional distribution of LAI, extinction coefficient of the canopy, transmission and reflection coefficients of the leaves, etc. ŽCraig DeLong, 1991; Pukkala et al., 1991.. The radiation environment within the canopy in a tropical forest is very heterogeneous. The density of plants, usually very high, combined with tree falls and canopy gaps, produces a highly variable distribution of light in space and time ŽCanahm et al., 1990; Chazdon, 1988; Fernandez and Myster, 1995; ´ Washitani and Tang, 1991., not only in the under-

0168-1923r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII S 0 1 6 8 - 1 9 2 3 Ž 9 7 . 0 0 0 9 7 - X

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F. de Castro, N. Fetcherr Agricultural and Forest Meteorology 90 (1998) 215–233

story, but throughout the volume of the canopy. The dynamics of canopy gaps and their influence on species composition and growth is analyzed by several authors in ‘Treefall gaps and forest dynamics’ Ž1989.. To describe adequately the heterogeneity of light environments within forest canopies, a three dimensional model is needed. Most models of light interception in the literature deal with one, vertical, dimension Že.g., Wang and Baldocchi, 1989. or two dimensions, usually applied to crops where an horizontal homogeneity can be assumed ŽYang et al., 1990; Federer, 1971; Jarvis and Leverenz, 1983; Oker-Blom et al., 1991.. Some models which consider three dimensions use idealized trees or geometrical shapes arranged in a regular array ŽBegue, ´ ´ 1993; Charles-Edwards and Thorpe, 1976; CharlesEdwards et al., 1986; Pukkala et al., 1991; Ryel et al., 1993.. This type of model, although useful for theoretical purposes, does not adequately represent the actual structure of a tropical forest canopy. West and Welles Ž1992. model the structure of individual trees Ž Eucalyptus sp.. as sets of spheres and then apply a light interception model to that arrangement of spheres. This may be regarded as a more realistic representation of the canopy structure, but it is difficult to apply to tropical forests due to the usually irregular form of tree crowns. The model MAESTRO developed by Wang and Jarvis Ž1990. based on the ellipsoidal model pioneered by Norman and Welles Ž1983. is a comprehensive approach to the problem, including physiological traits of the trees being simulated and able to simulate photosynthesis and transpiration as well as radiation interception, although, to our knowledge, the last two processes has not been tested against real measurements up to now. On the other hand, MAESTRO requires detailed measurements of so many parameters that it would be difficult to apply to a large area. Baldocchi and Collineau Ž1994. have thoroughly reviewed the subject recently. Our approach takes three dimensions into account, although it does not consider individual trees separately but rather the distribution of leaves throughout the canopy volume. It does not require the description or modelling of tree crowns, but instead the direct measurement of LAI and leaf angles. The model can work in one, two or three

dimensions depending upon the data provided. The model is envisaged to simulate the radiation absorption over relatively large stands, this is why the calculations and measurements of parameters are maintained as simple as possible, and why more sophisticated models have not been used. In the development of a model, it is important to know the influence of each parameter in the operation of the model, and in the results obtained ŽBegue, ´ ´ 1993; Wang and Baldocchi, 1989.. In complex models it is not obvious what changes are to be expected in the results by a given change in one or several parameters. First, the relationship of a parameter with the result may not be linear, but exponential, parabolic, or other, or may not be uniform throughout the range of the parameter values. Second, changes in a parameter can affect the relationship between other parameters and the result. The best way to obtain a clear picture of the relationships between parameters and results, and among the parameters, is a sensitivity test. This test also shows to what parameters the model is most sensitive, and thus, which needs to be measured more accurately when applying the model to a real situation. We have applied a sensitivity test to the model to study the variability of the relationship between the absorbed radiation and the parameters. Also we establish which parameters have the strongest influence on the results. We have used a stepwise regression analysis to examine the influence of the variations in each parameter on the result, and a non-parametric comparison test ŽWilcoxon. to detect if there are significant differences in the predictions of the model in different scenarios. The simulation of a system or process gives us a powerful tool to understand its nature. If the model works properly, its parameters can be changed to learn more about the response of the system to different situations. Also, the validation of the model gives us confidence in its theoretical background and allows us to generalize our conclusions. 2. Description of the model 2.1. Position of the sun Since the model uses three dimensions, the geometrical relationships between the foliage elements

F. de Castro, N. Fetcherr Agricultural and Forest Meteorology 90 (1998) 215–233

and the incident radiation are of primary importance. The model calculates the position of the sun for the latitude, longitude, date and hour of the simulation. First, the declination of the sun Žin degrees., d , which depends on the date, is calculated through the following empirical formula, which is claimed to have less than 1% of error ŽMorris, 1989.:

217

Another situation that yields erroneous results is when the declination is negative and the latitude is smaller than the absolute value of declination. In that case, the above correction has to be applied irrespective of the value of h e . 2.2. Incoming radiation

sin d s 0.39785 sin 278.9707 q 0.9856 Dy q 1.9163 =sin Ž 356.6153 q 0.9856 Dy .

Ž 1.

where Dy is day of year: January 1st is 1 and December 31st is 365. Then the model calculates the elevation Ž b . and azimuth Žangle in the horizontal plane measured from north eastward, a ., of the sun through ŽGates, 1980; List, 1984; West and Welles, 1992.: sin b s sin f sin d q cos f cos d cos h

Ž 2.

sin a s ycos d sin hrcos b

Ž 3.

where f is the latitude, d the declination and h the ‘hour angle’ of the sun: the angle between the meridian of the observer and that of the sun ŽGates, 1980; List, 1984; West and Welles, 1992.. We used the method described by Gates Ž1980. to estimate local hour for the longitude of the observer and time of the day. At some combinations of latitude and time of the year, the formula for azimuth gives erroneous results. The reason is the cos d function in the formula of the azimuth of the sun. The declination can take positive or negative values from y238 to q238, but the cosines of two angles with different signs and equal absolute value are the same. If the sun crosses the east line Žan imaginary line pointing from the observer to the east. the calculated azimuth is not correct. To calculate the east hour angle Ž h e ., the hour angle at which the sun crosses the east line, we use the expression ŽIER, 1981.: h e s cosy1 Ž tan drtan f . .

Ž 4.

If h is less than h e the azimuth of the sun is corrected Ž a X . as:

a X s 180 y a

The amount of incoming radiation is one of the most important parameters in the model. By ‘incoming radiation’ we mean: total downward solar radiation flux density Ž R s .. This is divided into: downward direct-beam solar radiation flux density Ž R b ., and downward diffuse solar radiation flux density Ž R d .. For brevity we will call them: direct and diffuse radiation, respectively. The diffuse and direct components of incident radiation are treated separately in the calculations, so it is necessary to know the values of both components. On the other hand, it is not usual to have measurements of both components, but only of global radiation, so it is necessary to estimate the proportion of direct and diffuse radiation in the global incident radiation. There are several references on the subject ŽBecker, 1987; Liu and Jordan, 1960; Weiss and Norman, 1985., but no general agreement on the method to be used to estimate the proportion. There are several methods to calculate the proportion of direct radiation in global radiation, both theoretical and empirical. Theoretical methods are preferable because they are more general, but no one method is completely satisfactory for all latitudes and seasons. Many factors, including clouds, aerosols, etc., affect the scattering of radiation in the atmosphere and therefore the proportion of diffuse light. Another problem is the variety of methods used for the measurement of diffuse radiation and the calculation of its proportion to global radiation ŽBecker, 1987.. The DiffuserGlobal ratio also changes depending on the time period considered: hours, days or months ŽSpitters et al., 1986.. The method of Spitters is based on the examination of many other sources and the ratios he suggests Žas recommended by de Jong, 1980. seem to be the best estimation. West and Welles Ž1992. use the same method, parameterized for the location of their interest. Usually empirical methods give better adjustment, but their

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validity is limited to a particular site and time. Theoretical methods are more general and allow to apply the model to a wider range of sites and conditions with less input data needed. We have used the method described in Spitters et al. Ž1986., which is claimed to be a good compromise for most latitudes. Alternatively, the model is prepared to read measured Žor user-estimated. values of direct and diffuse incident radiation from a file if they are available. First, the incoming radiation outside the atmosphere Ž R o . is estimated as ŽGates, 1980.: R o s Cs

d

ž/ d

2

Ž sin f sin d q cos f cos d cos h . Ž 5 .

where Cs is the solar constant Ž1370 J my2 sy1 . and the factor ŽŽ drd . 2 . is the ratio between the average distance sun–Earth and its actual value at a given moment. It ranges between 0.965 and 1.035, and so is neglected in the model ŽWest and Welles, 1992.. R o is given in J my2 s I1 ŽW my2 .. The ratio between diffuse radiation Ž R d . and total radiation Ž R s . is related to the ratio R srR o as shown in Table 1 Žfrom Spitters et al., 1986.. The model estimates R o and, if the values of incident R s are given, it calculates diffuse incoming radiation Ž R d . using the ratios in Table 1. Then Direct Radiation Ž R b . is calculated as R b s R s y R d .

is assigned a value for the mean elevation angle of the leaves inside the cell Ž b l .. 2.4. Direct radiation interception The formulae that were used in this section were taken from Roberts and Miller Ž1977.. The basic equation of the model calculates the probability that a direct light beam entering a cell exits from the cell without being intercepted by the leaves the cell contains. This probability P bX Žthe subscript b means direct. in each cell is given by: P bX s e Žy g k L. .

2.3. Distribution of foliage For the purposes of the model, we consider the whole canopy enclosed in a three dimensional array of cells ŽFig. 1.. Cell size may be varied to adjust spatial accuracy in the simulations. The lengths of the three sides of the cells do not have to be the same. Each cell i contains a given amount of leaves, expressed as its leaf area index Ž Li .. Also each cell Table 1 Relationship between the ratio R d r R s and the ratio R s r R o for hourly radiation values ŽSpitters et al., 1986. R d r R s s1 R d r R s s1–6.4 Ž R g r R o 0.22. 2 R d r R s s1.47–1.66 R g r R o Rd r Rs s R

Fig. 1. Representation of the three dimensional array as considered in the model.

for for for for

R s r R o - 0.22 0.22 - R s r R o - 0.35 0.35- R s r R o - K K - Rs r Ro

Rs 0.847y1.61 sin b q1.04 sin2 b , and K s Ž1.47y R .r1.66 ŽSpitters et al., 1986..

Ž 6.

g is a scaling factor that Roberts and Miller Ž1977. used in order to include only that part of the foliage actually penetrated by the solar beam as it passes through the canopy to any particular cell. To calculate g , first the entering and exiting points of the beam in the cell are calculated, followed by the length of the trajectory of the beam inside the cell. g is the ratio between that trajectory and the vertical dimension of the cell. The symbol k is the extinction coefficient of the leaves in the cell, which depends on the relationship between the mean leaf angle Ž b l . and that of the entering beam Ž b .. Following Warren Wilson Ž1960., k is: ks

½

cos b l

if b G b l

cos b l w 1 q 2 Ž tan u y u . rp

if b - b l

where u s cosy1 Žtan b cot b l ..

Ž 7.

F. de Castro, N. Fetcherr Agricultural and Forest Meteorology 90 (1998) 215–233

The negative exponential function used assumes a random distribution of foliage within the cell. Nilson Ž1971. and Lemeur and Blad Ž1974., examined the different functions to be used for the calculation of the probability of interception depending on the assumed distribution of foliage: random, regular or clumped. Eq. Ž6. is applied to all the cells in the array and a new array P is generated storing the results. In a second step, the model calculates the probability of penetration to the center of a cell i by a direct beam from outside the canopy Ž P b, i . as: n

P b ,i s P bY ,i Ł P bX ,i

Ž 8.

is1

where n is the number of cells in the path of the beam from outside the canopy to the cell of interest. P b,X i is defined by Eq. Ž6. and is previously calculated for each cell in the path, and P bY is the probability of penetration from outside the cell of interest to its center, and is also calculated by Eq. Ž6.. The values P b, i represent the probability that a beam of direct light reaches the center of the cell i from outside the canopy.

Diffuse radiation does not come from a single source as does direct radiation. It comes from all directions with variable intensity, depending on the elevation and other factors. The penetration of diffuse radiation is approximated by summing the penetration of direct radiation over the whole hemisphere at four different azimuths Ž a s 08, 908, 1808 and 2708. and four different elevations Ž b s 08, 308, 608 and 808. and then taking the mean of the 16 values. The maximum elevation is not 908 to avoid an excessive influence of the vertical direction in the calculation of the mean. Thus, the probability of penetration of diffuse light to any cell i Ž Pd, i . is:

Pd ,i s

1

Ý Ý P b ,i 16 a

b

and P b, i is the probability of direct beam penetration to cell i at azimuth a and elevation b , calculated as explained previously. 2.6. Total radiation in each cell The calculated values P b and Pd for each cell represent the probability that direct or diffuse light reaches it from outside the canopy. They should be transformed into absolute radiation values multiplying them by the respective values of incident radiation: direct Ž R b . and diffuse Ž R d .. Since the probability of interception does not depend on the wavelength, the model can be applied to any portion of the incident light: total, PAR, etc., by using the appropriate value for incident radiation. The radiation reaching the cell i is calculated as: Direct radiation: R b ,i s P b ,i R b

Ž 10 .

Diffuse radiation: R d ,i s Pd ,i R d

Ž 11 .

2.7. Transmission and reflection From Eq. Ž6., we can estimate the probability of interception of radiation entering a cell i ŽPi i . as: Pi i s 1 y P bX ,i s 1 y e Žy g k L.

2.5. Diffuse radiation interception

Ž 9.

219

Ž 12 .

That probability can be considered the proportion of radiation intercepted within the cell. The amount of radiation actually intercepted in the cell i ŽRi i . is the product of that proportion by the total radiation reaching the cell: Ri i s Ž R b ,i q R d ,i . Pi i

Ž 13 .

A portion of that radiation is reflected and another portion is transmitted, depending on the mean transmission and reflection coefficients of the vegetation Ž rt and rr , respectively.. Both coefficients are parameters of the model and are considered constant for all the cells. We can calculate the amount of radiation reflected Ž R r . and transmitted Ž R t . as: R r s Ri rr

Ž 14 .

R t s Ri rt

Ž 15 .

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F. de Castro, N. Fetcherr Agricultural and Forest Meteorology 90 (1998) 215–233

Reflected radiation is considered to be evenly distributed in all directions and to be entirely diffuse, so one sixth of the value of R r is added to the diffuse component Ž R d, i . of the six neighbors of cell under consideration. The transmitted radiation is entirely added to the diffuse component of the cell beneath. Although these processes are included in the model they have not been applied in this work.

3. Field measurements Data to test the model were collected in an plantation located in Toa Baja, in the north coast of Puerto Rico Ž188N, 66830X W.. The plantation contains only three species, and the ground is level with paths through the trees, so the cherry picker needed to measure the data at several heights could enter easily. Also the plantation is well documented from its origin, and several studies have been carried out there. Finally the plantation is not too close to other trees, so the measurements are not much disturbed by them. The plantation is rectangular, 120 m long and 55 m wide. It is composed by 18 plots Ž6 = 3. of 15 = 15 m, with 5 m wide paths between them. The longer axis is oriented almost North–South. The height of the trees ranges between 10 and 15 m. There are three species: Leucaena leucocephala, Eucalyptus robusta and Casuarina equisetifolia. Nine of the plots are monocultures and the other nine are mixed stands of two species. 3.1. Leaf area index We measured the leaf area index and mean leaf angle every 20 m in the east–west axis, and each 10 m in the north–south axis, as depicted in Fig. 2. Measurements were made with a LAI-2000 system ŽLI-COR. which estimates simultaneously both variables. We measured all the points at ground level, 5 m and 10 m high. The measurements at 5 and 10 m were taken from a cherry picker with a reach of 15 m. Each measuring point represented one cell, with dimensions of: 20 = 10 = 5 m. The different dimensions of the cells in the three directions were imposed by the resolution of the measuring device. As

Fig. 2. Distribution of the LAI and leaf angles measuring points within the plantation. Each square represents a plot within the plantation. The spaces between plots are trails.

stated by the manufacturer, the LAI-2000 measures a circumference of radius roughly equal to the height of the vegetation, which is 10 m, so each measure represents a circular area of 20 m of diameter. Since we used a view restriction cap of 1808 to avoid direct sun in the sensor ŽLICOR, 1990, Fig. 3., the measurement area was restricted in the axis north–south to only 10 m. In the calculations of the LAI values from the original files, we eliminated the outer ring of the LAI-2000 sensor to reduce the radius of the measured area ŽLICOR, 1990; Welles and Norman, 1991.. The vertical dimension was set to 5 m, so the array had 12 = 3 = 3 cells. The LAI values measured at levels 0 and 5 m ŽL 0 and L 5 . had to be corrected to include only that foliage inside the corresponding cell ŽL 0 – 5 and L 5 – 10 ., and not all the foliage above the measuring point, which is the primary result. That values were calculated subtracting the value of the upper cell from the inferior one: L 5 – 10 s L 5 y L 10

Ž 16 .

L 0y 5 s L 0 y L 5

Ž 17 .

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Mta ,1 s Mta ,T Ž n1 q n 2 . rn1 y Mta ,2 Ž n 2rn1 .

Ž 28 . Assuming that the ratio wL 1rL 2 x is equal to the ratio w n1rn 2 x, then: Mta ,1 s Mta ,5 – 10 s Mt a ,T Ž L 1 q L 2 . rL 1 Fig. 3. Area measured by the LAI-2000 in each measurement. It depends on the height of the canopy and the view restriction cap used. In this case, area is half a circle of 10-m radius.

The same thing was to be taken into account for the average leaf angles. When measuring, say, at 5 m level, the LAI-2000 estimates the Mean Tilt Angle Ž Mta , following Lang, 1986. of all the foliage elements placed above it, not just those included in the 5 m directly above. Thus, it is necessary to estimate the Mta for each level Žfrom 5 to 10 and from 10 to 15. separately. The same is true for ground level. We assume that the Mta can be expressed as: Mta s ÝTarN

Ž 18 .

where Ta is tilt angle for individual leaves and N is the number of leaves. Let the lower level Ž5–10. be denoted 1, the upper level Ž10–15. be denoted 2 and together Ž5–15. be denoted T. Mta ,T s ÝTa ,TrN

Ž 19 .

Mta ,1 s ÝTa ,1rn1

Ž 20 .

Mta ,2 s ÝTa ,2rn 2

Ž 21 .

ÝTa ,T s ÝTa ,1 q ÝTa ,2

Ž 22 .

N s n1 q n 2

Ž 23 .

Substituting Eqs. Ž22. and Ž23. in Eq. Ž19.: Mta ,T s Ž ÝTa ,1 q ÝTa ,2 . r Ž n1 q n 2 .

Ž 24 .

Thus: ÝTa ,1 s Mta ,T Ž n1 q n 2 . y ÝTa ,2

Ž 25 .

From Eq. Ž21.: ÝTa ,1 s Mta ,T Ž n1 q n 2 . y Mta ,2 n 2

Ž 26 .

Substituting Eq. Ž26. in Eq. Ž20.: Mta ,1 s Ž Mta ,T Ž n1 q n 2 . y w Mta ,2 n 2 x . rn1

Ž 27 .

y Mt a ,2 Ž L 2rL 1 .

Ž 29 .

The same procedure is applied to the calculation of the Mta,0 – 5 , from Mta,5 – 10 and Mta,10 – 15 . In some of the points, a slightly higher value of LAI was obtained at 10 m than at 5 m. In these cases, the value of LAI in the cell from 5 to 10 m was considered to be 0. This may have occurred because of the imprecision in the placement of the sensor, since it was not always possible to place the arm of the cherry picker directly above the point of measurement at the ground level. Another source of variation was a change of sky conditions between the two measurements Žsometimes more than 10 min.. Since the LAI-2000 estimates the LAI indirectly based on the transmittance of light by the canopy, there may be some doubts about the independence of the measurements and the results predicted by the model. In fact, the calculations of the device and those of the model are based on similar principles, but we are confident that the differences between them are strong enough to consider the application of the model as an independent test. First, the actual formula included in the LAI-2000 software is not the same than that used in the model. Besides, the LAI-2000 uses only diffuse radiation in the calculations Žin fact, direct radiation should be avoided to hit the sensor., while the model predicts both diffuse and direct radiation. Finally, the measurements of LAI Žbased on light transmittance. were made in different dates than that of light used to test the model, so the light conditions, position of the sun, cloud cover, etc., were different. Unfortunately, a direct measurement of LAI in three dimensions is not feasible in our case. It would imply to manually harvest all the leaves, branches and trunks in cubes of 20 = 10 = 5 m, 108 times, 36 of them at 15 m of height. Besides, that would mean to completely destroy the plantation, and there were

222

F. de Castro, N. Fetcherr Agricultural and Forest Meteorology 90 (1998) 215–233

significant differences Ž p - 0.01, t-test., while in the second case Žfour rings vs. two rings. the differences were higher Ž p ) 0.05. but the relationship between the two series of results is still very good ŽFig. 4.. So we think that the use of four rings will provide adequate estimates of LAI. Besides, due to the high density of the plantation Žaround 6000 trees hay1 . the distribution of foliage is markedly uniform. We are confident that the problem of leaf clumping is attenuated with respect to temperate natural forests. 3.2. Incident radiation

Fig. 4. Comparison of the LAI estimated by the LAI-2000 ŽLICOR. using four rings Ž1–4., and: three rings Župper plot. and two rings Žlower plot.. The regression, and the t-test Žnon-equal variances assumed. are indicated.

administrative reasons that prevented it. The placement of baskets to collect litterfall would not be achievable neither. It should be pointed out that the most important feature of the model is its three dimensionality, and the placement of a sufficient number of baskets in the required locations within the volume of the canopy, and their frequent sampling, would be impossible. Several studies ŽChason et al., 1991; Gower and Norman, 1991. have reported that the LAI-2000 may underestimate the true LAI as much as 35–40% ŽGower. or 45% ŽChason. due to the clumping of foliage, which is not considered in the underlying theory of the device. Despite of that, Chason et al. Ž1991. found that eliminating the three outer rings of the sensor in the calculations of LAI, the estimated values was no significantly different from the true value Ž4.17 " 0.73 and 4.89 " 0.95, respectively.. We have compared the results obtained using four rings with those using two and three rings. In the first case Žfour rings vs. three rings. there are no

The model predicts the radiation reaching each cell of the array, so to test the model we compared the predicted values with observed ones. We measured incident radiation Žas Photosynthetic photon flux density, PPFD. in an open area close to the plantation with a LI-COR quantum sensor connected to a data logger. Measurements were taken at 15 s intervals. At the same time, we measured the radiation inside the plantation with the same kind of sensor at 40 points regularly spaced within each of the 20 = 10 m2 cells at ground level. It was not possible to measure the radiation in the same way at 5 m and 10 m, so we only took one point at each cell at 5 m and 10 m. The two sensors were previously calibrated against each other in the open. For each of the cells the mean of the 40 data points was calculated and assigned as the value of incident radiation for that cell. To test the predictions of the model, we compared the average of the probabilities of direct and diffuse light penetration, as calculated by the model, for those cells at the ground level Ž0 to 5 m., with the ratio: total incident PAR vs. average PAR at the ground level for each cell inside the plantation. Both variables ranged from 0 to 1. We did not use the radiation values taken at higher levels Ž5 to 10 m and 10 to 15 m. in the comparison because only one datum was taken in each cell at that heights, which was not considered a representative sample of the light level within an area of 20 = 10 m. The program C2000 ŽLI-COR. was used to match the measurements of radiation taken outside the trees with those taken at the same time inside the trees. The outside values were registered at 15 s intervals, and the

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program then chose the outside value closest in time to each inside value to calculate the ratio.

4. Sensitivity test The parameters included in the test are: total Leaf Area Index ŽLAI., vertical distribution of LAI, incident radiation, mean angle of leaves, transmission coefficient of leaves, and elevation of the sun. The values assigned to each parameter are shown in Table 2. The values for each parameter were varied, one at a time, while the others were maintained constant, and the model was run for each combination of values. The output of the model is the mean interception of radiation, defined as: the difference of incident radiation and the mean radiation reaching the ground level. The sensitivity test was carried out with an array of 30 = 30 = 20 cells. The dimensions of the cells were: 3 = 3 = 1 m, thus the total simulated canopy represents an area of 1 ha, and a height of 20 m. For each run of the model, the values of LAI and mean angles of leaves were calculated and stored in the foliage array. In the calculation of the intercepted radiation, the cells of the five rows closer to the edges of the array were not taken into account. Only a square of 20 = 20 cells at the center of the array was included. The Leaf Area Index varied from 1 to 8, which we considered as a reasonable interval. Two different distributions of LAI were considered: uniform and non-uniform. In the first case, all the cells have the same value, which is calculated as Total LAI divided by 20 Žthe height of the array.. In the second case, only those cells at the upper part of the array Žthe

Fig. 5. Simulation of the incident radiation in the sensitivity test of the model. The function used to generate the values is: R s s R max sin Ž b ., where b is the elevation of the sun. R ma x can take two values: 1500 or 750.

uppermost third, six layers of cells. contains leaves, while the rest are empty. The value of LAI for each cell is higher in this case: Total LAI)3r20. The two contrasting distributions were intended to test the effect of vertical LAI distribution, and not only its absolute value, on the results. The elevation of the sun Ž b . varied from 158 to 908, at 158 intervals ŽTable 2.. These are the elevations corresponding to latitude 08, from 6:00 AM to 12:00 PM when declination is 0. To exclude another source of variation in the test, the azimuth of the sun is maintained constant at 908 ŽEast.. The daily variation of incident radiation Ž R s . was simulated as R s s R max sin Ž b . Ž m mol my2 sy1 ., where R max can take two values: 1500 or 750. This function produces a bell-shaped curve similar to real

Table 2 Values of the parameters for the sensitivity test Parameter

Values

Vertical distribution of LAI Total Leaf Area Index Mean angle of the leaves Transmission coefficient of leaves Incident radiation

Uniform vs. Concentrated at the upper most thir 1, 2, 3, 4, 5, 6, 7, 8 108, 308, 508, 708 0, 0.02, 0.04, 0.06, 0.08, 0.1 Variable 1500 P sin Žsun elevation. 158, 308, 458, 608, 758, 908

Elevation of the sun

The model is run for every combination of input values.

Constant 1500 or 750

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incident radiation with a cloudless sky ŽFig. 5.. This method establishes a correlation between two of the parameters: elevation of the sun and incident radiation. To avoid the effect of this correlation in the analysis of the results, two additional sets of tests were made maintaining the incident radiation constant at the same values: 1500 and 750 m mol my2 sy1 . To test the effect of the proportion of diffuse radiation in global radiation we made some of the tests twice: first, with the global radiation divided in

direct and diffuse components and, second, considering all the incoming radiation as diffuse, an assumption that approximate the situation of a uniformly clouded sky. The angle of leaves is directly included in the main formula ŽEq. Ž6.. through the extinction coefficient Ž k .. The relationship between the elevation of the sun and the angle of the leaves has a definite influence in the probability of penetration ŽFig. 6.. Thus, we have varied the angle of leaves from 108 to 708, for each of the sun elevation angles.

X Fig. 6. Shape of the function P b s e Žy g k L. for different values of the three parameters: L ŽLeaf Area Index. and extinction coefficient Ž k ., which depends on sun elevation, and leaf angle. The parameter g is set to 0.5. The X-axis stands for the leaf angle and the Y-axis for the sun elevation. Each subfigure depicts the form of the function for a given LAI value, indicated beside it.

F. de Castro, N. Fetcherr Agricultural and Forest Meteorology 90 (1998) 215–233

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The output of the model is the percentage of incident radiation intercepted by the canopy Ž Ir ., calculated as the difference between incident radiation and the mean radiation at ground level. The intercepted radiation is thus: Ir s 100 y 100 P

Ri Rs

where R i is the radiation in the cell i, and R s is the incident radiation.

5. Results 5.1. Prediction of intercepted radiation The LAI values obtained ranged from 1.0 to 4.12 Žmean 2.77 " 0.77.. With most of the LAI in the upper part of the canopy ŽFig. 7.. These values of LAI can be considered as normal for a forest and so are adequate to test the model. The variation of incident PAR outside the canopy along the measuring period Ždivided into two days. is shown in Fig. 8. The agreement between calculated and observed values was tested with linear regression. Ideally, the slope of the regression should be 1, the intercept should be 0 and the r 2 as close to 1 as possible. The slope is 1.01, the intercept is 0.002 and the r 2 is 0.714 ŽFig. 9..

Fig. 7. Vertical profile of the mean LAI values in the study area.

Fig. 8. Incident PAR radiation along the measurement period Žtwo days..

5.2. SensitiÕity analysis The results of the sensitivity test have been analyzed through a stepwise regression. The dependent variable is the intercepted radiation, and the independent variables the parameters included in the test: LAI, leaves mean angle, incident radiation, sun elevation and leaf transmission coefficient. Stepwise regression selects independent variables Žone at a time.. The variable that produces the best prediction of the dependent variable Žbased on a F-test. is

Fig. 9. Comparison between expected and observed values of radiation. Observed values are the ratio between radiation inside a cell Žmean of 40 observations. and the outside Žincoming. radiation. Expected values are the probabilities of penetration for direct and diffuse radiation Žmean of both values.. Each point represents one cell. Only those cells at ground level Ž0 to 5 m. are included.

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Table 3 Scenarios to which the stepwise regression have been applied separately

Variable Incident radiation Constant Incident radiation

Uniform LAI

Non-uniform LAI

Max: 1500 1500

Max: 1500 1500

Max: 750 750

Max: 750 750

entered into the equation first, the independent variable that adds the next largest amount of information is entered second, and so on. After each variable is entered, the F-value of each variable already entered into the equation is checked, and any variables with small F-values are removed. This process is repeated until adding or removing variables does not significantly improve the prediction of the dependent variable. We applied the analysis separately for data included in each scenario presented in Table 3. Each group represent a different combination of conditions: uniform LAI vs. non-uniform LAI, variable vs. constant incident radiation and, finally, two levels of maximum incident radiation. The stepwise regression establishes those parameters with stronger influence in the dependent variable. The comparison of the results of the regression analysis for each group allows us to establish the relative importance of each parameter in different situations. A second set of runs were made with two different assumptions about the nature of incoming radiation. In the first case, the model calculates the proportion of direct and diffuse radiation in incident radiation. In the second case, all the incoming radiation is considered diffuse. The two series of results were compared with each other to test the effect of the directrdiffuse ratio in the calculated absorbed radiation. The values for the parameters in these runs were the same than those previously explained, except for the transmission coefficient of leaves, which was maintained constant at 0.04 Ž4%.. 5.2.1. Uniform LAI distribution The first parameter included in the regression is LAI in all cases. The rest of the parameters are: sun elevation, mean leaf angle and incident radiation ŽTable 4.. The result is the same for the two levels of

Table 4 Stepwise regression of the results of the model Step

r2

M.S.E.

1 2 3 4 5

0.00 0.33 0.64 0.69 0.69

433.57 291.10 155.09 135.78 135.41

Variable

Coefficient STD error Student’s t P

CONSTANT 12.9 b1 0.197 b 0.292 LAI y5.211 Rs 0.006

T y25.66 34.32 13.85 2.15

1.096 11.74 0.014 13.86 0.046 6.28 0.138 y37.62 0.0026 2.15

Variables included LAI LAI, b LAI, b , b l LAI, b , b l , R s

0.00 0.00 0.00 0.00 0.032

Dependent variable is the absorbed radiation, and independent variables are the parameters included in the test: LAI, sun elevation Ž b ., leaf angle Ž b l . and Incident radiation Ž R s .. Incident radiation is variable, maximum is 1500 and the distribution of LAI is uniform. ns1344. F to enter s F to exits 4.00.

maximum incident radiation: 1500 and 750, when it is variable. If incident radiation is set constant it is not chosen as predictive variable, as expected, but

Table 5 Stepwise regression of the results of the model Step

r2

M.S.E.

1 2 3 4 5

0.00 0.267 0.449 0.621 0.651

651.61 477.73 359.71 247.03 227.99

Variable

Coefficient STD Error Student’s t P

CONSTANT 86.11 b1 0.196 b 1.554 LAI y4.743 Rs y0.111

T y22.13 y21.01 24.75 10.63

1.422 60.55 0.018 10.63 0.060 25.76 0.180 y26.39 0.0034 y32.39

Variables included Rs R s , LAI R s , LAI, b , b l R s , LAI, b , b l

0.00 0.00 0.00 0.00 0.00

Dependent variable is the absorbed radiation, and independent variables are the parameters included in the test: LAI, sun elevation Ž b ., leaf angle Ž b l . and Incident radiation Ž R s .. Incident radiation is variable, maximum is 1500 and the distribution of LAI is non-uniform. ns1344. F to enter s F to exits 4.00.

F. de Castro, N. Fetcherr Agricultural and Forest Meteorology 90 (1998) 215–233 Table 6 Results of the comparison ŽWilcoxon test. of absorbed radiation with the two assumed distributions of incident radiation: direct plus diffuse, and only diffuse Incident radiation

Direct and diffuse Only diffuse

LAI uniform

227

previous case, the transmission coefficient of leaves does not exert significant influence.

LAI non-uniform

W

p

n

W

p

n

4.614 4.453

0 0

47 53

5.295 5.504

0 0

37 40

the rest of the parameters included and their precedence are the same as with variable incident radiation. The transmission coefficient of leaves is not selected as predictive variable in any case. 5.2.2. Non-uniform LAI distribution The first parameter included is the incident radiation if it is variable, and then are included: LAI, sun elevation and mean leaf angle ŽTable 5.. As in the

5.2.3. Effect of diffuse Õs. global ratio If the LAI distribution is uniform, the absorbed radiation is significantly greater if the incident radiation is divided in its components, direct and diffuse, than if it is considered all diffuse. If the LAI distribution is not uniform the result is the opposite ŽTable 6.. The absorbed radiation shows a hyperbolic response with respect to the LAI. This response is more or less pronounced depending on the values of other parameters, mainly sun elevation and leaf mean angle. All the graphics in this section are presented vs. the LAI, because it is the primary source of variation. In some cases the absorption reaches a constant maximum value at LAI around 5, while in

Fig. 10. Sensitivity of the relationship between radiation absorption and LAI to the leaf angle Žindicated in the box. for several sun elevations Žindicated as b .. The results correspond to a uniform LAI distribution and variable incident radiation Ž R ma x s 1500..

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other cases no stabilization of the curve is observed, even at high LAI. 5.2.4. Effect of mean leaf angle The most evident effect of increasing the mean leaf angle is to accentuate the differences due to the sun elevation angle ŽFig. 10.. The minimum and maximum absorption Žfor low and high LAI, respectively. increases for low sun elevations and decreases for high sun elevations, when the leaf angle increases. The differences in absorption between 108 and 708 for leaf angle at high values of LAI Ž8. and high sun elevations Ž758 or 908. can reach up to a 22%. Another effect of increase leaf angle is that the response of absorption to LAI becomes more linear and less hyperbolic. In the extreme case, with low sun elevation Ž158. and high leaf angle Ž708., the response of absorption to LAI is almost constant at

100%. If the sun elevation is higher Ž758 or 908. the response to LAI varies from 20% to 75%, but is still almost linear. If the distribution of LAI is not uniform the effect of leaf angle is similar, but less visible. In this case the absorption is, in general, lower than with uniform LAI, that is, the penetration of light to the ground is higher. For low sun angles Ž158 or 308. the absorption is almost constant around 55%. This effect is due to the penetration of radiation through the sides of the array, since there are no leaves intercepting it. 5.2.5. Effect of sun eleÕation For very low elevation of the sun Ž158., the absorption is very high for all LAI and leaf angle ŽFig. 11.. The increase of sun elevation promotes a descent in the absorption in the whole range of LAI values, but especially for low LAI, which makes the

Fig. 11. Sensitivity of the relationship between radiation absorption and LAI to the elevation of the sun Žindicated in the box., for several leaf angles Žindicated as b l .. The results correspond to a uniform LAI distribution and variable incident radiation Ž R max s 1500..

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relationship between absorption and LAI closer to a hyperbola. This effect is more visible for high leaves angles. For non-uniform LAI distribution the effects are similar, except for very low sun elevation. In that case the absorption is also constant but much lower than with a uniform LAI distribution, around 55%. 5.2.6. Shape of the function of penetration probability From the analysis of the main equation of the model ŽEq. Ž6.., the relationship is established between the probability of light penetration in a single cell and the parameters of the equation: LAI, sun elevation, and leaf angle ŽFig. 6.. At low LAI values Žless than 2., the relationship of leaf angle with P bX is almost constant, that is, the leaf angle does not exert a significant influence in the value of the function. When LAI increases this relationship becomes exponential, above all for high sun elevations. The relationship between the function value and sun elevation is the opposite. For low LAI values Žless than 4. the relationship is hyperbolic, while for high LAI values it becomes almost constant.

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fit precisely with the assumed shape of the cells in the model, a rectangular prism. Third, a smaller cell size Ž20 = 10 = 5 m. would have increased the precision of model predictions, but the characteristics of the LAI-2000 prevented more detailed measurements, since the resolution of the device, in the study site, was 10-m radius. The distribution of light at the ground level, compared with the model predictions shows a good general agreement, except in two zones ŽFig. 12.. The higher expected value at the southern part of the plantation is probably due to the presence of a stand of trees some distance apart, which are not included in the calculations, and which are intercepting part of the incident radiation. The lack of adjustment at the middle-west part is harder to explain. The expected value is lower than the observed one. That can be due to a poorer estimation of the LAI or mean leaf angles at the cells above that point. The model described has no adjustable parameters, unlike that of Roberts and Miller Ž1977., it is

6. Discussion 6.1. Prediction of intercepted radiation The model works well. Both the predicted average interception of PAR and its spatial distribution agree with the observed values. The r 2 of 0.714 is quite good considering that reflection and transmission of light were not included, as well as the limited resolution of LAI-2000 and the big cell size imposed by it. The lack of fit can be due to several reasons. First, the device used to estimate the LAI requires some conditions to give accurate estimations, conditions that are rarely met in the study site. The sky brightness should be uniform, a very uncommon condition in tropical climate, usually with very irregular cloud distribution. The sun should be at low elevation, or even under the horizon, which, in our latitude Ž188., is restricted to a short period after sunrise or before sunset. Second, the LAI-2000 measurement area is a conic like volume, which does not

Fig. 12. Two dimensional distribution of the PAR values inside the plantation at the ground level. Expected and observed values are shown.

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completely deterministic. The model considers the actual, not simulated, distribution of foliage in the canopy, and so can be applied to any kind of forest. We know of no other models that consider three dimensions and have been applied to real forests and tested against direct measurements of the parameters, except that of West and Welles Ž1992.. One of the main interests of the model is to use it as a tool to simulate the effect of different characteristics of the canopy on the interception and reflection of radiation. Different tree arrangements or diverse tree structures can be simulated and apply the model to them to establish the relationship of each characteristic in the final result. Also, the effect of changes in the reflection or transmission coefficients in the availability of PAR Žor total. radiation at the ground of the forest can be checked, or the dynamics of sunflecks in the understory at different dates, latitudes, orientations or canopy structures, which can be important for many species ŽChazdon, 1988; Pearcy and Yang, 1996.. The sensitivity test reveals that the two most important parameters are: LAI and leaf angles. These parameters are of very difficult and time consuming measurement or estimation in nature. In fact many of the models found in the ecological literature do not measure them at all, but derive them from reasonable theoretical assumptions Žsee for instance Myneni, 1991.. This situation is somewhat less common in the agronomic field, where canopies are usually more manageable. It should be kept in mind that one of the most important goals of modelling approach to scientific problems is the ability to predict unknown states of the system, in space or time. If a model requires parameters that are much more difficult to measure than the variable the model predicts, it will be of little use in the prediction task. If we intend to study the radiation environment over a large area, it would be easier Žand more precise. to measure it directly with radiation sensors connected to dataloggers than to measure LAI and leaf angles and other parameters and then apply a model to them to predict radiation values. To obtain the full potential of radiation interception models, it is needed a reliable method to reconstruct the structure of the canopy with relatively low effort. In the agronomic field, some methods have been developed, although they seem of difficult application to forest canopies. Andrienu et

al. Ž1995. for instance, described a method to reconstruct the structure of a maize canopy by stereo-plotting, and then calculate the directional and bi-directional gap fraction distributions associated with it. The method gave good agreement between measurements and simulations. Unfortunately, it is quite inadequate to apply in forest canopies since it required the placement of two photographic cameras high above the canopy and to clip out the leaves and stems in several steps from top to bottom. Pearcy and Yang Ž1996. used individual geometrical measurements taken in the field to reconstruct the projected image of a plant so that light absorption from any direction can be assessed. The method is successful in simulating small plants, but could be difficult to apply to large trees. Despite of that, as the authors point out, hypothetical leaf orientations, sizes and shapes, and branching patterns could provide a mean to simulate the structure of larger canopies. Probably the best approach would be a method derived from satellite images analysis. Maybe the present technology in this field available to civilians has not enough resolution to undertake such measurements, but it will in the future. 6.2. SensitiÕity test The primary result of the analysis is that the LAI is the most important parameter of the model. Unfortunately, as already stressed, the direct measurement of LAI in forests is difficult, and usually only extrapolations and broad estimations are available. The direct measurement in three dimensions is almost impossible. Also the distribution of LAI has a significant influence on the results. The presence of gaps or irregularities in the canopy can make the absorption of radiation vary to a large extent. This stresses the importance of three dimensional models, which are able to account for the variation of LAI in all the canopy volume. The second most important parameter of the model, sun elevation, can be calculated or measured accurately, and thus it is not subject to estimation errors. Finally, the test suggest that the angle of leaves has a significant influence in the calculated radiation interception. This agrees with the results of other

F. de Castro, N. Fetcherr Agricultural and Forest Meteorology 90 (1998) 215–233

studies ŽMorris, 1989; Wang and Baldocchi, 1989.. Although some works suggest precisely the opposite result ŽGoudriaan, 1988.. The penetration probability function Ž P bX . shows a weak effect of the leaf angle on the value of the function for low LAI values, but a strong, exponential relationship for high LAI values. The relationship of P bX with sun elevation and LAI is the opposite. For low LAI values the relationship is hyperbolic, while for high LAI values it becomes constant. This implies that in sparse canopies, with LAI values under 2, the position of the sun is a more important parameter than leaf angle. On the contrary, for dense, well developed canopies, with LAI above 3 or 4, leaf angle is a significant parameter, while the position of the sun becomes a less important parameter. The transmission coefficient of leaves does not have a significant influence on the results. This is probably due to the fact that only the quantitative aspect of the radiation is considered in this work. If we considered all the spectrum, or the PAR interval, and several transmission coefficients Žone for each section of the absorption spectrum. instead of only one single value, probably the composition of light would vary depending on the transmission coefficients. This aspect will be included in the model by substituting the single value of transmission coefficient by a vector with different coefficients for different wavelengths. This vector represents a simplification of the transmission spectrum of the leaves and will allow us to include the qualitative aspect of the radiation, the color of light, in the model. This can be an important aspect for the application of the model to the ecological study of forests, as pointed out by Endler Ž1993., or for the prediction of the spectrum of radiation reflected by the canopy.

Acknowledgements We thank the useful comments of Dr. Paul Bayman and Dr. Gary Toranzos of the University of Puerto Rico. Also, we thank the constructive criticism of two anonymous reviewers and the help in the field of Adisel Montana, ˜ and Sebastian, ´ the truck driver and to Mr. Camilo.

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Appendix A. List of symbols Only non-standard symbols, or those particular to this work, are included. For other symbols, see Reifsnyder et al. Ž1991., and the reference cited in each case. h Hour angle. The angle between the observer’s meridian and that of the sun he East hour angle. The hour angle at which the sun crosses the east line bl Mean leaf elevation angle Dy Day of year Mta Mean tilt angle of leaves as measured by the LAI-2000. More subscripts can be added, separated by commas, to constrain its meaning: T by Total, or numbers specifying height levels Ta Tilt angle of an individual leaf r t , rr Coefficients of transmission and reflection of leaves, respectively P bX Probability of non-interception of direct radiation in a cell PdX Same as P bX for diffuse radiation R max Maximum level of incident radiation in the sensitivity test R b, i Direct radiation received in cell i R d, i Diffuse radiation received in cell i Pi i Probability of interception of radiation in cell i Ri i Radiation actually intercepted in cell i R r , R t Radiation reflected and transmitted, re spectively, in every cell

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