Light Relations in Plant Canopies S. B. Idso and C. T. de Wit A theory of light relations in plant canopies is presented which has potential applications in remote sens- ing and photosynthetic modeling of plant canopies. Predictions of the model are compared with field measurements of light reflection and transmission in a corn crop. Both reflection at the top of the canopy and transmission at the bottom are predicted within 1% of the measured values. Profiles connecting these upper and lower limits are equally well approximated. Variations in the predictions with altitude angle of the sun are confirmed by the observation of several investigators. 1. Introduction The interaction of direct solar radiation and diffuse skylight with the leaves of a plant canopy is a phe- nomenon which requires detailed understanding before proper analyses can be made of remotely acquired electrooptical data obtained by aircraft or spacecraft passing over cropped surfaces. Early theoretical studies of this problem were based on a one-parameter relation equivalent to the Bouguert-Lambert law;' these works have been reviewed briefly by Myers and Allen. 2 A recent advancement has been the applica- tion of the Kubelka and Munk' (K-M) two-parameter theory to the interaction of light with a plant com- munity by Allen and Richardson. 4 Whereas the Bouguer-Lambert law makes no prediction regarding the reflectance of a medium, Allen and Richardson were able to use K-M theory to predict accurately both reflectance and transmittance from two, four, six, and eight stacked cotton leaves in a spectrophotometer. They indicated also that their procedures were suffi- ciently general to apply to an actual plant canopy, al- though this application was not made. In studying plant canopies, however, it becomes apparent that the extension of K-M theory proposed by Allen and Richardson is not precisely valid, the reason being that K-M theory requires a uniform dis- tribution of leaf inclinations. Nichiporovich 5 and de Wit 6 both have experimental evidence indicating that probably no plant canopy has leaves distributed S. B. Idso is with the U.S. Water Conservation Laboratory, Phoenix, Arizona 85040; C. T. de Wit is with the Agricultural University, Wageningen, The Netherlands. Received 23 June 1969. in such a manner. Thus, in a real canopy, the K-M scattering and absorption coefficients would be vari- ables dependent upon the altitude angle of the sun. Or, viewing the situation from a different perspective, the K-M extension proposed by Allen and Richardson would predict a canopy reflectance that would not vary over the day. It is well known, however, that re- flectance from a cropped surface usually exhibits a diurnal variation. This effect has been observed with alfalfa, barley, wheat, oats, cotton, and sorghum by Fritschen,7 with plant cane and pangola grass by Chia,' with short grass and kale by Monteith and Szeicz, 9 and with sod by Idso et al.1 0 to name a few specific examples. In these studies, reflectance was found to increase by 4-6% at low solar elevations, compared to values obtained at solar noon. It is the purpose of this paper to describe a theory of light relations in plant canopies which overcomes the limitations of K-M theory. The new theory will be applied to an actual crop growing in a field situation, and its prediction of crop reflectance and transmittance plus vertical profiles of these two parameters will be compared with the results of field measurements. 11. Theory Since the distribution of leaf inclinations in a plant canopy will influence the light relations therein, it is necessary to somehow characterize this factor. A useful convention for this representation is the leaf distribution function, introduced by de Wit. 6 It is defined as the frequency distribution of leaf inclinations from the horizontal. Figure 1 depicts six examples: planophile canopies, where horizontal leaves are most frequent; erectophile canopies, where vertical leaves are most frequent; plagiophile canopies, where leaves at some oblique inclination are most frequent; ex- tremophile canopies, where leaves are least frequent at oblique inclinations; spherical canopies, where the relative frequency of leaf inclinations is the same as the January 1970 / Vol. 9, No. 1 / APPLIED OPTICS 177
Transcript
Light Relations in Plant Canopies
S. B. Idso and C. T. de Wit
A theory of light relations in plant canopies is presented which has potential applications in remote sens-ing and photosynthetic modeling of plant canopies. Predictions of the model are compared with fieldmeasurements of light reflection and transmission in a corn crop. Both reflection at the top of the canopyand transmission at the bottom are predicted within 1% of the measured values. Profiles connecting theseupper and lower limits are equally well approximated. Variations in the predictions with altitude angleof the sun are confirmed by the observation of several investigators.
1. Introduction
The interaction of direct solar radiation and diffuseskylight with the leaves of a plant canopy is a phe-nomenon which requires detailed understanding beforeproper analyses can be made of remotely acquiredelectrooptical data obtained by aircraft or spacecraftpassing over cropped surfaces. Early theoreticalstudies of this problem were based on a one-parameterrelation equivalent to the Bouguert-Lambert law;'these works have been reviewed briefly by Myers andAllen.2 A recent advancement has been the applica-tion of the Kubelka and Munk' (K-M) two-parametertheory to the interaction of light with a plant com-munity by Allen and Richardson. 4 Whereas theBouguer-Lambert law makes no prediction regardingthe reflectance of a medium, Allen and Richardsonwere able to use K-M theory to predict accurately bothreflectance and transmittance from two, four, six, andeight stacked cotton leaves in a spectrophotometer.They indicated also that their procedures were suffi-ciently general to apply to an actual plant canopy, al-though this application was not made.
In studying plant canopies, however, it becomesapparent that the extension of K-M theory proposedby Allen and Richardson is not precisely valid, thereason being that K-M theory requires a uniform dis-tribution of leaf inclinations. Nichiporovich5 andde Wit6 both have experimental evidence indicatingthat probably no plant canopy has leaves distributed
S. B. Idso is with the U.S. Water Conservation Laboratory,Phoenix, Arizona 85040; C. T. de Wit is with the AgriculturalUniversity, Wageningen, The Netherlands.
Received 23 June 1969.
in such a manner. Thus, in a real canopy, the K-Mscattering and absorption coefficients would be vari-ables dependent upon the altitude angle of the sun.Or, viewing the situation from a different perspective,the K-M extension proposed by Allen and Richardsonwould predict a canopy reflectance that would not varyover the day. It is well known, however, that re-flectance from a cropped surface usually exhibits adiurnal variation. This effect has been observedwith alfalfa, barley, wheat, oats, cotton, and sorghumby Fritschen,7 with plant cane and pangola grass byChia,' with short grass and kale by Monteith andSzeicz,9 and with sod by Idso et al.10 to name a fewspecific examples. In these studies, reflectance wasfound to increase by 4-6% at low solar elevations,compared to values obtained at solar noon.
It is the purpose of this paper to describe a theory oflight relations in plant canopies which overcomes thelimitations of K-M theory. The new theory will beapplied to an actual crop growing in a field situation, andits prediction of crop reflectance and transmittanceplus vertical profiles of these two parameters will becompared with the results of field measurements.
11. Theory
Since the distribution of leaf inclinations in a plantcanopy will influence the light relations therein, it isnecessary to somehow characterize this factor. Auseful convention for this representation is the leafdistribution function, introduced by de Wit.6 It isdefined as the frequency distribution of leaf inclinationsfrom the horizontal. Figure 1 depicts six examples:planophile canopies, where horizontal leaves are mostfrequent; erectophile canopies, where vertical leavesare most frequent; plagiophile canopies, where leavesat some oblique inclination are most frequent; ex-tremophile canopies, where leaves are least frequent atoblique inclinations; spherical canopies, where therelative frequency of leaf inclinations is the same as the
January 1970 / Vol. 9, No. 1 / APPLIED OPTICS 177
A1.0
_J 0.8
o 0.6C~
2 0.4U-
E 0.20
0.00 30 60 90
IL, Degrees
1.0
J0.8
°0.6
, 0.4
E0.2
0.00 30 60
IL, Degrees90
Fig. 1. (a) The four general types of leaf distribution functions:a. planophile; b. extremophile; c. plagiophile; d. erectophile.(b) Two special leaf distribution functions: e. corn canopy (mea-
sured); f. spherical.
relative frequency of the inclinations of the surfaceelements of a sphere; and a measured leaf distributionfunction for corn.
The first step of what we may call the de Wit-Idso(D-I) theory is to derive what de Wit6 termed the lightdistribution function of a crop from its leaf distributionfunction. Since the incident light on a leaf due todirect light is proportional to the sine of the angle (LS)between the leaf and the rays of the sun, this lightdistribution function is defined as the cumulativefrequency distribution of intercepted light as a functionof sin(LS).
In obtaining the light distribution function of aparticular crop, it is first necessary to calculate for ninedifferent altitude angles of the sun (IS = 5, 150, . . ..850) and nine different leaf inclinations (IL = 5,150, . . . , 85') the probabilities of having a light rayintercepted by a leaf with the sine of its angle to thelight equal to or smaller than sin(LS). The equationsfor accomplishing this feat were first derived by deWit.6 Since his development was very abbreviated, amore thorough derivation has been made and is in-cluded in Appendix A of this paper. Since the eighty-one sets of calculations which must be made for all pairs(IS, IL) are quite tedious, this work is done by compu-ter. Since they need only be done once, however, andthe answers are identically useful for all problems ofthis type, results are included for the benefit of otherinvestigators in Appendix B.
The calculation of a crop light distribution functionfrom the data in Appendix B and the crop's leaf dis-
tribution function now proceeds on the basis that thelight intercepted by leaves in any leaf inclination classis proportional to the number of leaves in this class andto the projected area in the direction of the sun of oneunit leaf area of the class. Thus, by computer, nineintegrated crop light distribution functions for sunaltitudes of 5, 15°, . . . , 850 are obtained. The re-sults for a corn crop are contained in Table I. Forpurposes which will be apparent later, it is also desirableto transform these fractions of intercepted light intofractions of leaves which receive the light within thevarious sin(LS) intervals. The formula which ac-complishes this transformation is
FR(J) = R(J)/SR, (1)
where
R(J) = [S(J) - S(J - 1)]/SN(J),10
SR = E R(J),J=1
(2)
(3)
and S(J) = cumulative frequency (fraction) of in-tercepted light, with J = 1, 2, . . . , 10 correspondingto light intercepted at values of sin(LS) <0.1, 0.2,
, 1.0. SN(J) = 0.05, 0.15, . . . , 0.95, correspond-ing to J = 1, 2, . . .,10. Results of these calculationsfor a corn crop are contained in Table II.
Up to this point, D-I theory has been concerned withwhat in essence is the uppermost layer of leaves in theplant canopy; and the next step is to consider theextinction of the solar beam as it penetrates the canopyfoliage. In this context, leaf area index (LAI) andcanopy density () are introduced. The latter ofthese two parameters was first used by Monsi andSaekill to describe the penetration of light into canopiesof perfectly absorbing horizontal leaves. De Wit,6
however, generalized it to the case of distributivelyinclined leaves that do reflect and transmit light.Its meaning then became somewhat more vague.Nevertheless, it may be considered to be characteristicof the extent to which the leaves are uniformly dis-tributed in inclination; a random distribution havingS = 0, and S = 1 implying all leaves positioned hori-zontally. A good value for corn is 0.1.
Considering first direct light incident at an inclina-tion IS, the interception of this light is proportional tothe projection of one unit leaf area in the direction of the
Table I. The Light Distribution Functions for a Corn Crop for Sun Inclinations IS = 50, 15°, . . ., 850
light, OP(IS), averaged for the crop as a whole, andinversely proportional to the projection of one unitsoil area in the same direction. Since the latter projec-tion is equal to the sine of IS, the direct light thatpenetrates the canopy without interception may bedescribed by the relation
I = I I -S sn(IS)] S (4)
where 10 is the amount of light arriving from the verti-cal direction at the canopy and LAI/S is the number oflayers defined for the crop foliage, e.g., for LAI = 3.5(in a case to be considered shortly) and S = 0.1, thecrop is considered to be composed of thirty-five verti-cally stacked layers.
If we let K represent the nine altitudinal positionsof the sun and N represent the number of canopylayers, we may calculate the set of numbers
X(K, N) = [-sin(IS)2 * (5)
Since Eq. (4) describes the penetration of direct lightinto the canopy, these X(K, N) values thus represent thefractions of direct light not intercepted by each canopylayer N at time K (sun inclination IS). This being so,it is readily seen that X(K, N - 1) is the fraction of leavesin layer N receiving direct sunlight; for the fractionX(K, N - 1) of light not intercepted by layer N - 1 mustimpinge upon the fraction X(K, N- 1) of the total area oflayer N, and if it is assumed that the leaves are ran-domly distributed spatially throughout the planar ex-tent of their respective layers, the fraction X(K, N - 1)of the leaves of layer N must be illuminated, too.As a corollary to this reasoning, it is evident that thoseleaves in any layer N receiving diffuse skylight only is
DF(K, N) = 1- X(K, N -1). (6)
Furthermore, considering the distribution of directlight interception by the crop, we may write the ex-
pression for the fractions of leaves in each layer whichin addition to diffuse light also receive direct lightwithin specified sin(LS) or J intervals as
DS(K, N, J) = FR(K, J) X X(K, N - 1). (7)
All of these calculations are done by computer. Toconserve space, however, these intermediate resultswill not be presented.
At this stage, the canopy has been divided intoN layers of J radiation reception classes for each of Ksolar altitudes. Since the theory beyond this pointcalls for actual radiation data, it is now necessary tocompute the times at which the sun inclination of 50,150, . . . , 850 occur for the specific time and place inquestion. This information is readily obtained fromthe relation
T (K) = 1 arc cos sin(IS) - sin(L) sin(D)115 a cos(L) cos(D) J (8)
where T(K) is the time in hours before and after solarnoon when a specified altitude angle is obtained by thesun and L and D are, respectively, the latitude of thespecific location and the declination of the sun for thespecified date. Since the sun may not achieve an alti-tude angle of 850 for all times and places, it is alsonecessary to solve the equation
when the hour angle (H) equals zero, for the maximumvalue of IS.
Since light is both reflected and transmitted by thecanopy leaves, we must calculate the intensities of lightintercepted by each layer and thus available for thisscattering. The first step is to calculate the incidentdiffuse skylight intercepted at each layer. It is basedupon the same general principles used in the calculationof the penetration of direct sunlight. The sky isdivided into zones of 10 deg width, centered at inclina-tions of 50, 15°, . . , 850. Then, from Table III, the
January 1970 / Vol. 9, No. 1 / APPLIED OPTICS 179
Table ll. The Relative Contribution to the Illuminance of a Horizontal Surface of 10 deg Zones from a Sky of Uniform Brightness'
Table IV. Diffuse Skylight and Direct Solar Radiation, in cal cm- 2 min-', as Reconstructed for 1 August 1961 at Ithaca, New York,for Sun Inclinations IS = 5, 150, . . . , 65°
penetration of diffuse light from these sections is de-termined as fractions of that incident upon a horizontalsurface above the crop. These results are added to-gether to give the penetration of diffuse light from theentire sky. The solar times available from calculationsof the previous paragraph then allow the proper in-tensities of diffuse skylight measured above the crop tobe utilized in calculating the actual intensities of diffuseskylight incident on each canopy layer. By subtractingthe results for each layer from those of the layer im-mediately preceding them, the diffuse skylight in-tercepted by each layer is then obtained; and multiply-ing these numbers by the sum of the reflectance andtransmittance of the leaf material for diffuse skylight,the actual intensities of diffuse skylight scattered byeach layer become available. A similar treatment ofthe intercepted direct sunlight added to these diffuseskylight results then gives the final distribution of thefirst set of scattered light sources in the canopy. Also,a source of scattered light is the underlying soil sur-face; and it is given a strength equivalent to twice itsreflectance.
If we now represent the strength of the scatteredlight source of a layer N by ST(N), and if we let X(N)represent the penetration of diffuse light through Nsuccessive layers (calculated in the previous paragraph),we can rapidly complete the theory. First, since thetransmitted and reflected light are practically ideallyscattered,' 2 approximately half of the light scattered ata layer goes up and the other half down, thus making itpossible to express the contribution of layer 1 to theilluminance of layer N as 0.5 ST(1)Z(N - 1). Simi-larly, the contribution from layer 2 may be written as0.5 ST(2)Z(N - 2), the contribution from layer N - 1as 0.5 ST(N - 1)Z(1), the contribution from layer N+ 1 as 0.5 ST(N + )Z(1), the contribution from thelast layer, N1MAX, as 0.5 ST(NVIAX)Z(N1M1AX - N),and the contribution from layer N'IAX + 1, the soilsurface, as 0.5 ST(NMIIAX + 1)Z(NMIIAX + 1 - N).(The use of the factor 0.5 here is what necessitatedmultiplying the reflectance of the soil by two in theprevious paragraph to obtain its scattering strength.)
Writing these expressions in summation form, we gettwo separate relations:
N-1D(N) = 0.5 E ST(L) X Z(N - L),
L=1
NMAX+1U(N) = 0.5 E ST(L) X Z(L - N).
L=N+1
(10)
(11)
D(N) represents the intensity of scattered light incidenton layer N from all layers above it; and U(N) repre-sents the scattered light incident on layer N from all
layers below it. The amounts of these fluxes that areintercepted by each layer may also be calculated andsecondary scattered light sources created. Two orthree such cycles are sufficient to account for essentiallyall scattered light. Thus, at the end, one has availablethe profile of light moving upward through the canopy,U(N), and by adding D(N) to the penetration of directsunlight and diffuse skylight, the profile of light movingdownward through the canopy. The upper limitof U(N) divided by incident solar radiation and skylightgives crop reflectance; and a similar operation uponthe lower limit of transmitted light gives crop trans-mittance.
Ill. ApplicationIn applying D-I theory, three of the most important
pieces of data required are the scattering coefficients ofthe individual plant leaf material for direct solar radia-tion, diffuse skylight, and light transmitted throughvegetation. From spectral measurements of reflec-tance and transmittance made of a corn leaf by Yocumet al. ' and spectral distributions of these three energyfluxes published by Gates,' 4 these factors calculated forcorn are 0.54, 0.21, and 0.62, respectively. The firsttwo of these three coefficients are used in computingthe strengths of the original scattering sources, and thethird is used in computing the strengths of succeedingscattering sources.
Also needed is the leaf distribution function of thecrop; and Fig. 1 includes this function for corn asmeasured by de Wit.6 Then, leaf area index and
10
ts
, 200z
25
30
/ II ' A ', , , II 5 10 20 50 100 20 50 100
REFLECTANCE (%) TRANSMISSION (%) R/T (%)
Fig. 2. Calculated and measured profiles of reflection R andtransmission T in a corn crop.' R is computed as upward movingradiation divided by incident radiation and T as downward mov-
ing radiation divided by incident radiation.
180 APPLIED OPTICS / Vol. 9, No. 1 / January 1970
24
wZz
UJ-
LIiw
20
6
2
84 6 8 10 12 14
SOLAR TIME (HOURS)
Fig. 3. Calculated reflectance of a corn croptime for clear day conditions.
I
16 18 20
, as a function of
canopy density are required. We have determined 0.1to be a good value of canopy density for corn; andleaf area index is determined for the specific stage ofgrowth by measurement. The application of D-Itheory presented will be for a corn crop growing atIthaca, New York, on 1 August 1961. The leaf areaindex measured for that situation by Baker and Mus-grave" was 3.5. This value, together with the canopydensity of 0.1, yields thirty-five crop layers for in-vestigation.
Finally, direct solar radiation and diffuse skylightdata are needed. Solving Eq. (9) it is found that sunelevations of 750 and 850 are not reached at Ithaca on1 August; so they are omitted from further considera-tion. Then, solving Eq. (8) yields values of 6.7, 5.7,4.8, 3.9, 3.0, 2.0, and 0.5 for the hours before and aftersolar noon when the sun occupies altitudinal positionsof 50, 150, . . . , 65°, respectively. At these times, datafrom Lemon and Wright,' 6 Allen et al."' and Threlkeld' 7
yield values of direct and diffuse radiation fluxes abovethe crop as contained in Table IV.
The results of using these data in the manner pre-scribed in Sec. II are shown in Figs. 2 and 3. Also in-cluded in Fig. 2 are some experimental measurementsmade by Allen and Brown'8 on this crop six weeks laterand Allen et al." about the same time. The three pairof values for reflection and transmission within thecrop and the one at the bottom of the crop are the workof the former authors; the pair at the top of the crop arethe work of the latter group. Originally, Allen andBrown had reported reflection above the crop, too.However, their value was not measured but estimatedfrom certain assumptions. The value of Allen et al.'9
was the mean of measurements carried out on severaldays around 10 September 1961.
IV. Discussion
The degree of correspondence between the measuredand calculated profiles of reflection and transmission inFig. 2 is truly remarkable, especially considering thefact that upper and lower limits of these two parameterswere not matched, as was done in the evaluation ofK-M theory by Allen and Brown." Consideringproblems associated with canopy radiation measure-ments of this type, the verification of D-I theory must
be considered virtually complete, particularly when thereflection coefficient for the crop in Fig. 3 is integratedover the day to give the same average value as mea-sured by Allen et al."
Besides being more versatile than K-M theory inpredicting changes in canopy reflection and transmis-sion with solar altitude, D-I theory has another ad-vantage quite apart from remote sensing; and this isthat it is particularly well suited for inclusion in com-puter simulation models of canopy photosynthesis.From K-M theory, for instance, all a person can gleanabout the light reception at various canopy layers is itsaverage intensity. Monteith20 has demonstrated quitevividly, however, that photosynthesis calculated from amean light intensity spread over all the leaf area of agiven layer can be much different from that calculatedfrom the same amount of radiation concentrated in sun-flecks of a correspondingly smaller area. D-I theoryeven provides more information than this, giving thefraction of leaves in each layer receiving diffuse lightonly and the fractions which also receive direct light atvarious angles of inclination to the rays of the sun.Thus, D-I theory improves the interpretative capabili-ties of remote sensing and provides a sound theoreticalframework for the photosynthetic modeling of plantcanopies.
This is a joint contribution from the Soil and WaterConservation Research Division, Agricultural ResearchService, U.S. Department of Agriculture, the Agri-cultural University, Wageningen, and the Institute forBiological and Chemical Research, Wageningen, TheNetherlands.
Appendix A
The purpose of this appendix is to present a detailedderivation of the light distribution function from rela-tions in evidence in Fig. 4. Part A of Fig. 4 representsa plant leaf of given orientation with respect to the soilsurface, and the line TS a ray of the sun. Part B isthe same configuration seen from a different point ofview. The angles IS, IL, DA, and LS are, respectively,the inclination of the sun, the inclination of the leaf, thedifference between the azimuths of the leaf and the sun,and the angle between the leaf and the rays of the sun.
A B
Fig. 4. Geometrical representations of the relations between aplant leaf, the soil surface, and a ray of the sun.
January 1970 / Vol. 9, No. 1 / APPLIED OPTICS 181
I I I I II I I I I
The derivation is begun by noting that
TSh = TS X cos(IS),
OS), = TSh X sin(DA),
and
01 = OS, X sin(IL),
thus making OR also equivalent to
OR = TS X [cos(IS) sin(IL) sin(DA)].
Next, it can be seen that
SS, = TS X sin(IS),
and that
PS = SSh X sin(IQ),
thus making
PS = TS X [sin(IS) sin(IQ)].
However, since it is evident that sin(IQ) = cos(IL),
PS = TS X [sin(IS) cos(IL)].
DAO = arc sin(-A/B), (A15)
(Al) for IS < IL. Since no light falls on the undersides of the(A2) leaves if IS > IL, the boundary angle in this case is
defined as - r/2.To determine the value of W, we note that if the
integration is carried out to the maximum value of DA,(A3) that is 7r/2, we must have S(J) = 1, or
(A4) 1 _ -A/2
(A5)
(A6)
(A7)
(A8)
Then, since OR = SLP and SSL = SLP + PS, wehave
SSL = TS X [sin(IS) cos(IL) + cos(IS) sin (IL) sin(DA)]. (A9)
A second way of expressing SSL, which is evident fromFig. 4, is
SSL = TS X sin(LS). (A10)
Combining this equation with the previous one for SSLand solving for sin(LS) then yields
where
sin(LS) = A + B X sin(DA),
A = sin(IS) cos(IL),
B = cos(IS) sin(IL).
Now the light distribution function S(J) for a givenvalue of IS and IL is defined as the probability that alight ray is intercepted by a leaf with a sine of its angleto the light equal to or smaller than sin(LS). Sincethe amount of light intercepted by the leaves in a smallazimuth interval is proportional to the size d(DDA) ofthis interval and since this amount of intercepted lightis also proportional to the projection of the leaf surfaceelements in the direction of the sun, that is with sin(LS),the light distribution function S(J), for J = sin(LS),
(All)
sin(LS) X d(DDA)71
/2
s
+JDA sin(LS) X d(DDA)l X W. (A16)
Evaluating the first of these integrals, after substitut-ing for sin(LS) the expression in Eq. (Al1), we have
rDAO- I sin(LS) X d(DDA) = - A X [DAO + (r/2)]J
+ B X cos(DAO); (A17)
and evaluating the second integral in a similar fashionyieldsT /2
JDAOsin(LS) X d(DDA) = A X [(7r/2) - DAO)]
+ B X cos(DAO). (A18)
Our initial expression (A16) thus reduces to
1 = [-2 X A X DAO + 2 X B X cos(DAO)] X W, (A19)
from which the proportionality constant W is deter-mined to be
W = [2 X B X cos(DAO) -2 X A X DAO]-'. (A20)
Now, for DA < DAO, the light distribution functionS(J) will be given by the integral
DA(A12) S(J) = t- 2
(A13) ! 7/2[A + B X sin(DDA)] X d(DDA)} X W.
(A21)
Evaluating this expression,S(J) = {B X cos(DA) - A X [(,r/2) + (DA)]} X W. (A22)
For DA > DAO, the light distribution functionmust be evaluated from the sum of the two integrals
( f DAOS(J) = _-
! J -.7/2(DA
+J| [A +1J DAO
[A + B X sin(DDA)] X d(DDA)
B X sin(DDA)] X d(DDA)} X W. (A23)
sin(LS) X d(DDA)S- f O
S(J) = _-J/
rDA -+ J sin(LS) X d(DDA) X W,J DAO - (A14)
where W is a proportionality constant.The reason for doing this integration in two parts is
to distinguish between light falling on the uppersidesand on the undersides of the leaves. Since light isparallel to a leaf when sin(LS) = 0, the boundary angleDAO is defined as
The first of these integrals results in the same expres-sion as was obtained for the previous integral, exceptnow DA is replaced by DAO. The second integral,SJ2, results in the expression
SJ2 = [A X DA - A X DAO - B X cos(DA)
+ B X cos(DAO)] X W. (A24)
Combining these results together, we get, for the lightdistribution function for DA > DAO,S(J) = {B X [2 X cos(DAO) - cos(DA)] - A
X [2 X DAO + (r/2) - DAI X W. (A25)
182 APPLIED OPTICS / Vol. 9, No. 1 / January 1970
Appendix B. This Appendix Contains the Eighty-One Light Distribution Functions for All Combinations of the Nine Sun Inclinations(IS = 50,15, .. . , 850) and the Nine Leaf Inclinations (IL =50,15°, . . ., 85)
References 11. M. Monsi and T. Saeki, Japan. J. Botany 14, 22 (1953).. A. C. HnynFH.erTe12. E. I. Rabinowitch, Photosynthesis and Related Processes1. A. C. Handy and F. H. Perrin, The Principles of Optics (nesinePbihrNwYr,15) o.2 at1
(McGraw-Hill Book Co., New York, 1932), Chap. 1, p. 24. (Interscience Publishers, New York, 1951), Vol. 2, Part 1.2. V. I. Myers and W. A. Allen, Appl. Opt. 7, 1819 (1968). 13. C. S. Yocum, L. H. Allen, and E. R. Lemon, Agron. J. 56,3. P. Kubelka and F. Munk, Z. Tech. Physik 11, 593 (1931). 249 (1964).4. W. A. Allen and A. J. Richardson, J. Opt. Soc. Amer. 58, 14. D. M. Gates, in Agricultural Meteorology (American Meteo-
1023 (1968). rological Society, Boston, 1965), p. 1.5. A. A. Nichiporovich, Soviet Plant Physiol. 8, 428 (1961). 15. D. N. Baker and R. B. Musgrave, Crop Sci. 4, 127 (1964).6. C. T. de Wit, "Photosynthesis of Leaf Canopies," Agr. Res. 16. J. L. Wright and E. R. Lemon, Agron. J. 58, 265 (1966).
Rept. 663 (Wageningen, 1965), Chap. 4, p. 11 17. J. L. Threlkeld, Thermal Environmental Engineering (Pren-7. L. J. Fritschen, Agr. Meteorol. 4, 55 (1967). tice-Hall, Englewood Cliffs, N.J., 1962), Chap. 14, p. 329.8. Lin-Sien Chia, Quart. J. Roy. Meteorol. Soc. 93, 116 (1967). tice-HAlle n d Cliffs, N.J. 196 p .9. J. L. Monteith and G. Szeicz, Quart. J. Roy. Meteorol. Soc. 1 L
87, 159 (1961). 19. L. H. Allen, C. S. Yocum, and E. R. Lemon, Agron. J. 56,10. S. B. Idso, D. G. Baker, and B. L. Blad, Quart. J. Roy. 253 (1964).
Meteorol. Soc. 95, 244 (1969). 20. J. L. Monteith, Ann. Botany (London) 29, 17 (1965).
A. Garcia, a glass technologist at Optics Technology, Inc.
