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Plant architectural parameters of a greenhouse cucumber row crop
Agricultural and Forest Meteorology, 51 (1990) 93-105
93
Elsevier Science Publishers B.V., Amsterdam - - Printed in The Netherlands
P L A N T A R C H I T E C T U R A L P A R A M E T E R S OF A G R E E N H O U S E C U C U M B E R ROW C R O P
XIUSHENG YANG, TED H. SHORT, ROBERT D. FOX and WILLIAM L. BAUERLE
Ohio Agricultural Research and Development Center, Ohio State University, Wooster, OH 44691 (U.S.A.) (Received May 13, 1989; revision accepted November 1, 1989)
ABSTRACT Yang, X., Short, T.H., Fox, R.D. and Bauerle, W.L., 1990. Plant architectural parameters of a greenhouse cucumber row crop. Agric. For. Meteorol., 51: 93-105. A group of parameters which describe the plant architecture of a greenhouse cucumber row crop were experimentally quantified. Leaf area index was found to be linearly related to plant height. Spatial distributions of the foliage area were described by a beta and a periodic cosine distribution function in the vertical and cross-row direction, respectively. The parameters of the distribution functions were also determined as functions of plant height. Each architectural parameter was chosen to be useful in greenhouse microclimate studies, especially in modeling the radiation transfer in a row crop canopy.
INTRODUCTION
A quantitative description (or model) of a physical plant stand is usually termed as plant architecture. It typically delineates the shape, size, geometric and orientational distributions of plant elements, and is a fundamental subject of plant microclimatic studies. According to Ross (1981), plant architectures should normally include: (1) a quantitative description of the structural characteristics of both individual plants and a stand as a whole; (2) quantitative descriptions of plant growth and development. The second criterion normally necessitates a plant growth model which was beyond the scope of this study. The first criterion was the focus of this research to quantify the plant architectural parameters for a row crop of cucumber plants in a greenhouse. These architectural parameters were then used to model and predict the climate variables within a production greeiahouse (Yang, 1988; Yang et al., 1989). 0168-1923/90/$03.50
© 1990 Elsevier Science Publishers B.V.
X. YANGET AL.
94
THE STAND MODEL
Figure 1 is a sketch of the plant stand model used in this study, where h designates the height of canopy, W represents the distance between rows and w is the width of an individual plant stand. The canopy was assumed to have an infinite horizontal dimension and the distribution of plant elements in the cross-row direction was assumed to be strictly periodic. Within the plant rows however leaf area was assumed to be uniformly distributed along the row direction. A rectangular cross-section of the plant stand was chosen based on observations of mature greenhouse cucumber crops. A right-handed Cartesian coordinate system was chosen in such a way that the y axis was always pointing to the row direction, x to the cross-row direction and z upwards. The origin of the coordinate system was placed at the bottom center of an arbitrarily chosen row in order to take advantage of symmetry. The basic geometric parameters for describing the plant stand model included the leaf area index (LAI), the average leaf dimension (LD), and the three structural parameters, h, W and w as shown in Fig. 1. LAI is defined as the ratio of the total single-side leaf area of a greenhouse plant stand to the total ground area, and LD is the geometric average of the mean leaf length and width. For most greenhouse vegetable crops, h is the same as the foliage height which is measured as the distance from the ground to the top level of the plants whenever no nursery benches or containers are used. While W is independent of plants and is normally determined by a grower based on growing experience, all the remaining geometric parameters are functions of plant growth. If the plant height is chosen to be the basic index of growth stage and the plants are pruned for only upward growth, LAI, LD and w can all be expressed as functions of h.
\
\ \
\
J I
k
w
>1
Fig. 1. Sketch of the structural model of a row-crop stand, where h a n d w are the average height a n d width of the row p l a n t stand, respectively, and W is the distance between rows.
PLANT ARCHITECTURAL PARAMETERS
95
THEORY OF THE DISTRIBUTION FUNCTIONS The vertical distribution of plant leaves is normally characterized by a vertical foliage area density function, az, with units of m-1. It can be defined as the leaf area per unit volume or the leaf area index per unit vertical dimension at a height of z. As a consequence h
f a~(z)dz=LAI
(1)
Based on statistical measurements on field agricultural crops, a~ has been expressed in numerous ways (Ross and Magi, 1971; Allen, 1974; Ross, 1975; among others). Unfortunately, none of these field crop descriptions was considered appropriate for this study, either because the required initial parameters were too complicated to determine as in the model of Ross and his colleagues, or because the approximations were too simple to represent the reality of a greenhouse crop, as were those of Allen's study. A beta distribution was chosen to describe the vertical density function of the foliage area, as carried out by Goel and Strebel (1984) in representing leaf orientational distributions, where
az(z) =LAIfl(z)
(2)
The beta distribution is expressed as
~(a-1)(1 - ~)(b-1) O < ~<-1 fl(~) - ~F(a+b) )~ 1
(3)
f fl(~)d~= 1 0
where F represents the standard gamma function. The beta function is a general one which can fit a wide variety of shapes between any two values yet needs only two parameters, namely a and b, to be determined (Giffin, 1971 ). This flexibility of form makes it a very useful distribution to fit empirical data when a theoretical justification for any other distribution is hard to define. Parameters a and b in eq. 3 can be easily determined by solving the following relationships provided that the mean and variance of the distribution are known tl
/~=a+b a s-
ab (a+b)2(a+b+l)
Applying eqs. 3-5 to a variable z defined on (0,h), one has
(4) (5)
96
x. YANG ET AL.
fl(z)-hF(a)F(b) 1F(a+b) (h)'a-1)( 1-~) Z'~(b-~'O
}
h
(6)
f
/~(z)dz=l
0
where ~h [ ,Uh ( h -- ,Uh ) __ ff2 ]
(7)
haah b=a(h-#h)
a=
(8)
tth The parameters tth and a~ are the mean and variance, respectively, of the vertical foliage area distribution function defined by eq. 2. The cross-row leaf area distribution for a greenhouse row crop can also be described with the same concept as that of vertical foliage area distribution. A cross-row density function of foliage area, ax, is defined by 1/2
f ax(x)dx=LAI
(9)
--1/2
ax
Similar to as, the cross-row foliage area density function, has units of m - 1 and can be interpreted as the per unit cross-row dimension at x within the domain [ - ½, ½]. A more complete definition ofa~ can be given as a distribution function of plant elements in the cross-row direction within an arbitrary domain [ ] by
LAI
L/2,L/2
L/2
1
f ax(x)dx=LAI
(10)
--L/2
However, it is more explicit with eq. 9 to understand the similarities between the two density functions a~ and a~ in conjunction with the definition of For a horizontally non-uniform row plant stand, the cross-row leaf area distribution has not been fully investigated. Allen (1974) proposed, but did not validate, a parabolic representation for an isolated row in his model of radiation transfer. A comparison between his approximation and some available measurements of a maize crop cited by Allen (1974) indicated that the parabolic distribution was not a good representation. In his paper, Allen pointed out that a Gaussian distribution would be a better approximation for both vertical and cross-row foliage area density functions. Since the vertical leaf area distribution of most agricultural crops does not necessarily possess sym-
LAL
PLANT ARCHITECTURAL PARAMETERS
97
metrical characteristics, his claim about vertical distribution is questionable. However, the cross-row foliage area of certain crops is indeed observed to be somewhat normally distributed. Thus, the use of a Gaussian distribution in some cases might be a good choice. For most greenhouse vegetable crops however, especially cucumbers, the foliage area does not show a pronounced concentration near the center of the row. Therefore the simplification from a general Gaussian distribution to a much simpler cosine distribution was assumed acceptable, if not better. Denoting the cosine distribution defined for an isolated row as n Cs(x) =
n cos
wx
----
---2
w
w
w
W
(11)
w/2
f -
Cs(x)dx= 1
W/2
the cross-row foliage area density function, ax, can t h a n be simply expressed as
ax(x)= WLAICs(x)
W
W
---<_x<-2 2
(12)
The most important advantage of the above representation is that it can be easily expanded to an infinite range of x with a Fourier series. By treating eq. 11 as a periodic function with a period of W, the expanded cosine distribution function can be written as
[ 2nn "~ Cs(x)=ao+ ~__lancos~--~-x]
(13)
where 1
ao= W
(14)
2 an= W [ l _ ( ~ ) 2 ]
nwn c°s W
(15)
Another very important parameter in describing the plant stand architecture, especially in modeling radiative transfer, is the so called G-function which can be interpreted as the mean projection of a unit foliage area in a given
98
X. YANG ET AL.
direction (Ross, 1975 ). Computation of the G-function based on its definition, however, is tedious and time consuming. Goel and Strebel (1984) proposed using a beta distribution function to represent the leaf orientation of vegetable crops in a universal way. However to use it one needs to determine the mean and variance of leaf orientations; practically, that is not an easy task. Some empirical equations for computing the G-function from the leaf inclination index, X, and the zenith angle of a radiation ray, 0, are available. According to Goudriaan (1977), the G-function could be approximated as
G=Go+O.877(1-2Go) cos 0
(16)
where G(~= 0.5 - 0.633X- 0.33X2
{17)
However information on relationships between X and z was missing in these studies. Observations indicated that mature cucumber crops tended to have more horizontal leaves at the upper level and inclined ones at the lower part. A linear distribution ofx on the vertical direction was assumed to incorporate this phenomenon into consideration, where
X(Z) =Xo+ kxz
(18)
EXPERIMENTATIONWITH A GREENHOUSECUCUMBERCROP Five randomly chosen cucumber (mustang variety) plants were sampled twice a week to measure the parameters necessary to describe the plant architecture. The measurements were taken in a computer-controlled horticultural greenhouse at the Ohio Agricultural Research and Development Center, The Ohio State University at Wooster, OH, beginning 27 March 1987 (the day in which plants were transplanted into the house) and continuing until 7 June of the same year. The greenhouse cucumber plants were grown according to Smith (1987) and Bauerle (1984) in rows on rockwool blocks; the blocks were placed on reflective plastic mulch. For each sample, as shown in Fig. 2, the following parameters were measured: (1) number of leaves (nL); (2) distance of each leaf from the ground (z); (3) zenith (0a) and azimuth (~L) (from north) angles of each leaf's normal; (4) horizontal projection of the petiole of each leaf (lp); (5) length (lL) and width (WL) of each leaf. The average distance between rows, W, and that between any two nearby plants along the row direction were measured to be 0.9 and 0.6 m, respectively.
99
PLANT ARCHITECTURAL PARAMETERS 0L
Z
Fig. 2. Leaf parameters measured in determining the plant architecture, where z is the height of a leaf from ground, 0Land eL are the zenith and azimuth angles of the leaf normal, ILand WLare the length and width of the leaf, and lp is the horizontal projection of the leaf petiole. RESULTS AND DISCUSSION
Leaf area index
T w e n t y - t h r e e h e a l t h y leaves were c h o s e n to d e t e r m i n e the r e l a t i o n s h i p bet w e e n the area o f a c u c u m b e r leaf, LA, a n d its length, la, a n d width, WL, as defined in Fig. 2. B y precisely m e a s u r i n g t h e area of each leaf with a leaf area meter, t h e r e l a t i o n s h i p t h e n was d e t e r m i n e d as L A = 0.739/LWL-- 0.00104
(19)
where L A was in m 2 a n d IL a n d WL were b o t h in m. T h e c o r r e l a t i o n coefficient R 2 for eq. 19 was 0.99. T h e L A I of a p l a n t was o b t a i n e d b y s u m m i n g up t h e area o f all its leaves a n d t h e n dividing t h e s u m m a t i o n b y the c o r r e s p o n d i n g g r o u n d area (0.54 m 2). L A I was s h o w n to be linearly related to t h e p l a n t height, h, as s h o w n in Fig. 3. A statistical analysis gave t h e following regression e q u a t i o n with a n R 2 value of 0.99 L A I = 0 . 8 8 6 h - 0.0965
(20)
100
X. YANGET AL. 2.00
1.50
o/"
Y
Jo
J
1,00
,d
/ J
0,50
0.00 0.00
~-0.60
1.20
'
~
1.80
'
2.40
h (m) Fig. 3. Measurements of LAI (mean values of five sample plants) in relation to plant height. The solid line represents the corresponding regression equation (eq. 20 ).
Average leaf dimension and stand width Whereas the average leaf dimension, LD, was obtained by geometric averaging of the mean leaf length and width, the stand width, w, came from the arithmetic average of the horizontal (radial) dimensions of leaves with their petioles. Their dependence on h was expressed exponentially as
LD = LDmax [ 1 - e x p ( - kLDh) ]
(21)
w=wmax[1-exp(-k~h ) ]
(22)
where LDm~x and wm~xwere the maximum values of LD and w for mature plants and were assigned to be 0.30 and 0.80 m, respectively, based on observation. The values of kLD and k~ were determined to be 1.964 and 1.449, respectively, by linear regressions on a logarithmic scale. The corresponding R 2 values for the logarithmic transformations of eqs. 21 and 22 were 0.93 and 0.94, and the plots of LD and w vs. h are shown in Fig. 4 (A) and (B), respectively.
Mean and variance of az For employing the beta distribution to describe the vertical foliage area density function, the mean and variance of the distribution need to be determined. They were computed according to Giffin (1971) by 1
/~h= ~
nL
i~1 zilaii
(23)
101
PLANT ARCHITECTURAL PARAMETERS .32
E f..., d c" o
.24
g E .~6 .J
i
.08
<
I
O.G~
.6
1.~2 h (m)
1.8
2.4
E
"E)
0 b3
E o E_
l i
•
1.2
1.8
2.4
h (m~
Fig. 4. Dependence of average leaf dimension (A) and plant stand width (B) on plant height, where the scattered points are measured means and the smooth curves are the regression lines of eqs. 21 and 22, respectively. 2
1
n~
{Th=~-I 2 (Zi--, uh)21aii
(24)
i=1
where nL was the number of leaves of a sample plant, zi was the vertical position and/aii was the leaf area index contributed by the ith leaf. Both Hh and ah 2 were shown to be functions of growth stage. Relationships between them and plant height, h, were determined, both with an R 2 value of 0.99, as being Hh= 0 . 4 3 4 h + 0.106
(25)
a ~ = ( 0 . 2 6 4 h - 0.033) 2
(26)
Once Hh and a~ are determined, the parameters a and b of the vertical foliage area density function can be readily computed by eqs. 7 and 8. Table 1 shows
102
X. YANGET AL.
TABLE 1 Plant architectural parameters of a greenhouse cucumber row crop at selected growing heights h
LAI
(m)
LD
W
(m)
(m)
a
b
0.2431 (0.009)2
0.102 (0.002)
0.124 (0.003)
0.420 (0.016)
_3 -
0.453 (0.005)
0.274 (0.003)
0.157 (0.001)
0.522 (0.009)
2.559 (0.143)
1.086 (0.004)
1.003 (0.019)
0.890 (0.008)
0.203 (0.007)
0.737 (0.010)
1.731 (0.099)
1.364 (0.036)
1.540 (0.042)
1.320 (0.065)
0.205 (0.003)
0.752 (0.003)
1.631 (0.066)
1.616 (0.072)
2.037 (0.066)
1.658 (0.054)
0.204 (0.001)
0.783 (0.011 )
1.439 (0.076)
1.648 (0.082)
'Mean value of five randomly selected sample plants. 2Standard error of the corresponding means. 3Not determined.
the experimentally determined architectural properties of the cucumber row crop at selected growing stages, among which w, a and b are the basic parameters for theoretically evaluating the cross-row and vertical density functions. Density functions of foliage area
The empirical vertical density function of leaf area, az, was first computed
by az(Zi) -
/aii
(27)
Zi + l / 2 - - Z i - - 1 / 2
where z~+1/2 and zi-1/2 represented the midpoints between the ith and the ( i + 1 )th or the ( i - 1 )th leaf. The results were then averaged over each layer of interest (0.1 m) to obtain a histogram. Experimental determination of the horizontal density function of foliage area was somewhat complex. First, the projection of each leaf on the x axis was identified. A step function was used for each leaf to describe its contribution to ax, assuming that the LAI contributed by a leaf was evenly distributed on its cross-row projection. The total density function was then obtained by summing all the step functions posed by the leaves. Histograms were finally drawn with the averages over subdomains of 0.05 m length along the x direction.
103
PLANT ARCHITECTURAL PARAMETERS
A 2_4
0.8~n 0"~).0
1.0
2.0 3.0 4.0 5.0•0.0 1.0 2.0 3.0 4.0 VERTICAL FOUAGE AREA DENSITY FUNCTION(m-ll
5.0
B T-
h=Zlm
h=Z0m
o
o ".5
0.0
L .5/-.5 x(m}
0.0
Fig. 5. Examples of vertical (A) and cross-row (B) foliage area density functions of the cucumber crop.
Examples of the histograms of the vertical and cross-row foliage area density functions are presented in Fig. 5 (A) and (B), respectively, where the smooth curves were the theoretical predictions of eqs. 2 and 12 with the empirical values of a, b and w. The R 2 values for the theoretical representations were in the range 0.64-0.89 for az and 0.61-0.83 for ax, respectively, indicating that the beta and cosine distributions did match the experimental data reasonably well.
Leaf inclination index of mature plants The leaf inclination index, X, was used to represent the average inclination of leaves in the plant canopy. The value of X could range from - 1 to + 1 with the maximum value representing horizontal leaves and the minimum value representing vertical (drooping) leaves. A very strong correlation was found between the inclination of each leaf and
104
x. YANG ETAL. ,
1.000 '
o
o
ooo
, o o o
o
o
o o
'
oo
u
oo
o
o
oooo o
x
0
O.500
o o
©
o
oo
o
o
o
oo
~
o
oo
o
0.000
0
© o
-0.500
-
o
1.000
ooo
o
oo
o
~
0.000
0.200
o
i
i
0.400
0.600
Normalized height,
J
0.800
1.000
z/h
Fig. 6. Vertical distribution of the leaf inclination index of mature cucumber plants with the regression line ofeq. 18.
its vertical location on mature cucumber plants. Figure 6 shows a plot of leaf inclination index, X, versus normalized height, z/h. The solid line w~is the proposed distribution function, eq. 18, calibrated through a linear regression. The intercept, X0, was determined as - 0.566 while the slope, kx was given by
kx=l.549/h
(28)
The R 2 value of the linear regression was 0.88. The weighted average ,~, of eq. 28, defined as h
1 f %=L-A-I
X(z)azdz
(29)
0
was 0.38 for mature plants (h = 2.0 m ), very close to the data of leaf inclination index for cucumber crops (0.39) given by Nichiporovich {Ross, 1981 ). SUMMARYAND CONCLUSIONS A group of parameters which describe the plant architecture of a greenhouse cucumber row crop were quantitatively determined. These included geometric descriptions of a canopy model, spatial distributions of foliage elements and leaf orientational characteristics. LAI of the greenhouse cucumber crop was found to be linearly related to the plant height. Stand width and leaf dimension could also be well represented by exponential functions of plant height. Spatial distributions of foliage area were described by a beta distribution and a periodic
PLANT ARCHITECTURAL PARAMETERS
105
cosine distribution in the vertical and cross-row direction, respectively. Measurements of the foliage area density functions supported well the above proposed distribution functions. A linear vertical distribution of leaf inclination index was also experimentally justified. The determined architectural parameters were anticipated to be useful in greenhouse microclimate studies, especially in modeling the radiation transfer into a row crop stand. ACKNOWLEDGEMENT
Salaries and research support was provided by State and Federal Funds appropriated to the Ohio Agricultural Research and Development Center, The Ohio State University. Manuscript number 136-89.
REFERENCES Allen, L.H.J., 1974. Model of light penetration into a wide-row crop. Agron. J., 66: 41-47. Bauerle, W.L., 1984. Bag culture production of greenhouse tomatoes. Special Circular 108, Ohio Agricultural Research and Development Center, Ohio State University, Wooster, OH. Giffin, W.C., 1971. Introduction to Operations Engineering. Irwin, Homewood, IL, 632 pp. Goel, N.S. and Strebel, D.E., 1984. Simple beta distribution representation of leaf orientation in vegetation canopies. Agron. J., 76: 800-802. Goudriaan, J., 1977. Crop Micrometeorology - A Simulation Study. Simulation Monographs, Pudoc, Wageningen, 249 pp. Ross, J., 1975. Radiative transfer in plant communities. In: J.L. Monteith (Editor), Vegetation and the Atmosphere. I. Principles. Academic Press, New York, pp. 13-56. Ross, J., 1981. The Radiation Regime and Architecture of Plant Stands. Junk, Boston, MA, 391 pp. Ross, J. and Magi, H., 1971. A calculation method for determining leaf area and its vertical distribution in the barley crop. Est. Contrib. Int. Biol. Programme III, Tartu, pp. 101-112. Smith, D.L., 1987. Rockwool in Horticulture. Grower Books, London, pp. 148-150. Yang, X., 1988. Greenhouse microclimate: transport processes, plant responses and dynamic modeling. PhD Dissertation, Ohio State University, Columbus, OH, 290 pp. Yang, X, Short, T.H., Fox, R.D. and Bauerle, W.L., 1989. A computer model of the bioclimate of a greenhouse. ASAE Paper No. 89-4016, American Society of Agricultural Engineers, St. Joseph, MI.
93
Elsevier Science Publishers B.V., Amsterdam - - Printed in The Netherlands
P L A N T A R C H I T E C T U R A L P A R A M E T E R S OF A G R E E N H O U S E C U C U M B E R ROW C R O P
XIUSHENG YANG, TED H. SHORT, ROBERT D. FOX and WILLIAM L. BAUERLE
Ohio Agricultural Research and Development Center, Ohio State University, Wooster, OH 44691 (U.S.A.) (Received May 13, 1989; revision accepted November 1, 1989)
ABSTRACT Yang, X., Short, T.H., Fox, R.D. and Bauerle, W.L., 1990. Plant architectural parameters of a greenhouse cucumber row crop. Agric. For. Meteorol., 51: 93-105. A group of parameters which describe the plant architecture of a greenhouse cucumber row crop were experimentally quantified. Leaf area index was found to be linearly related to plant height. Spatial distributions of the foliage area were described by a beta and a periodic cosine distribution function in the vertical and cross-row direction, respectively. The parameters of the distribution functions were also determined as functions of plant height. Each architectural parameter was chosen to be useful in greenhouse microclimate studies, especially in modeling the radiation transfer in a row crop canopy.
INTRODUCTION
A quantitative description (or model) of a physical plant stand is usually termed as plant architecture. It typically delineates the shape, size, geometric and orientational distributions of plant elements, and is a fundamental subject of plant microclimatic studies. According to Ross (1981), plant architectures should normally include: (1) a quantitative description of the structural characteristics of both individual plants and a stand as a whole; (2) quantitative descriptions of plant growth and development. The second criterion normally necessitates a plant growth model which was beyond the scope of this study. The first criterion was the focus of this research to quantify the plant architectural parameters for a row crop of cucumber plants in a greenhouse. These architectural parameters were then used to model and predict the climate variables within a production greeiahouse (Yang, 1988; Yang et al., 1989). 0168-1923/90/$03.50
© 1990 Elsevier Science Publishers B.V.
X. YANGET AL.
94
THE STAND MODEL
Figure 1 is a sketch of the plant stand model used in this study, where h designates the height of canopy, W represents the distance between rows and w is the width of an individual plant stand. The canopy was assumed to have an infinite horizontal dimension and the distribution of plant elements in the cross-row direction was assumed to be strictly periodic. Within the plant rows however leaf area was assumed to be uniformly distributed along the row direction. A rectangular cross-section of the plant stand was chosen based on observations of mature greenhouse cucumber crops. A right-handed Cartesian coordinate system was chosen in such a way that the y axis was always pointing to the row direction, x to the cross-row direction and z upwards. The origin of the coordinate system was placed at the bottom center of an arbitrarily chosen row in order to take advantage of symmetry. The basic geometric parameters for describing the plant stand model included the leaf area index (LAI), the average leaf dimension (LD), and the three structural parameters, h, W and w as shown in Fig. 1. LAI is defined as the ratio of the total single-side leaf area of a greenhouse plant stand to the total ground area, and LD is the geometric average of the mean leaf length and width. For most greenhouse vegetable crops, h is the same as the foliage height which is measured as the distance from the ground to the top level of the plants whenever no nursery benches or containers are used. While W is independent of plants and is normally determined by a grower based on growing experience, all the remaining geometric parameters are functions of plant growth. If the plant height is chosen to be the basic index of growth stage and the plants are pruned for only upward growth, LAI, LD and w can all be expressed as functions of h.
\
\ \
\
J I
k
w
>1
Fig. 1. Sketch of the structural model of a row-crop stand, where h a n d w are the average height a n d width of the row p l a n t stand, respectively, and W is the distance between rows.
PLANT ARCHITECTURAL PARAMETERS
95
THEORY OF THE DISTRIBUTION FUNCTIONS The vertical distribution of plant leaves is normally characterized by a vertical foliage area density function, az, with units of m-1. It can be defined as the leaf area per unit volume or the leaf area index per unit vertical dimension at a height of z. As a consequence h
f a~(z)dz=LAI
(1)
Based on statistical measurements on field agricultural crops, a~ has been expressed in numerous ways (Ross and Magi, 1971; Allen, 1974; Ross, 1975; among others). Unfortunately, none of these field crop descriptions was considered appropriate for this study, either because the required initial parameters were too complicated to determine as in the model of Ross and his colleagues, or because the approximations were too simple to represent the reality of a greenhouse crop, as were those of Allen's study. A beta distribution was chosen to describe the vertical density function of the foliage area, as carried out by Goel and Strebel (1984) in representing leaf orientational distributions, where
az(z) =LAIfl(z)
(2)
The beta distribution is expressed as
~(a-1)(1 - ~)(b-1) O < ~<-1 fl(~) - ~F(a+b) )~ 1
(3)
f fl(~)d~= 1 0
where F represents the standard gamma function. The beta function is a general one which can fit a wide variety of shapes between any two values yet needs only two parameters, namely a and b, to be determined (Giffin, 1971 ). This flexibility of form makes it a very useful distribution to fit empirical data when a theoretical justification for any other distribution is hard to define. Parameters a and b in eq. 3 can be easily determined by solving the following relationships provided that the mean and variance of the distribution are known tl
/~=a+b a s-
ab (a+b)2(a+b+l)
Applying eqs. 3-5 to a variable z defined on (0,h), one has
(4) (5)
96
x. YANG ET AL.
fl(z)-hF(a)F(b) 1F(a+b) (h)'a-1)( 1-~) Z'~(b-~'O
}
h
(6)
f
/~(z)dz=l
0
where ~h [ ,Uh ( h -- ,Uh ) __ ff2 ]
(7)
haah b=a(h-#h)
a=
(8)
tth The parameters tth and a~ are the mean and variance, respectively, of the vertical foliage area distribution function defined by eq. 2. The cross-row leaf area distribution for a greenhouse row crop can also be described with the same concept as that of vertical foliage area distribution. A cross-row density function of foliage area, ax, is defined by 1/2
f ax(x)dx=LAI
(9)
--1/2
ax
Similar to as, the cross-row foliage area density function, has units of m - 1 and can be interpreted as the per unit cross-row dimension at x within the domain [ - ½, ½]. A more complete definition ofa~ can be given as a distribution function of plant elements in the cross-row direction within an arbitrary domain [ ] by
LAI
L/2,L/2
L/2
1
f ax(x)dx=LAI
(10)
--L/2
However, it is more explicit with eq. 9 to understand the similarities between the two density functions a~ and a~ in conjunction with the definition of For a horizontally non-uniform row plant stand, the cross-row leaf area distribution has not been fully investigated. Allen (1974) proposed, but did not validate, a parabolic representation for an isolated row in his model of radiation transfer. A comparison between his approximation and some available measurements of a maize crop cited by Allen (1974) indicated that the parabolic distribution was not a good representation. In his paper, Allen pointed out that a Gaussian distribution would be a better approximation for both vertical and cross-row foliage area density functions. Since the vertical leaf area distribution of most agricultural crops does not necessarily possess sym-
LAL
PLANT ARCHITECTURAL PARAMETERS
97
metrical characteristics, his claim about vertical distribution is questionable. However, the cross-row foliage area of certain crops is indeed observed to be somewhat normally distributed. Thus, the use of a Gaussian distribution in some cases might be a good choice. For most greenhouse vegetable crops however, especially cucumbers, the foliage area does not show a pronounced concentration near the center of the row. Therefore the simplification from a general Gaussian distribution to a much simpler cosine distribution was assumed acceptable, if not better. Denoting the cosine distribution defined for an isolated row as n Cs(x) =
n cos
wx
----
---2
w
w
w
W
(11)
w/2
f -
Cs(x)dx= 1
W/2
the cross-row foliage area density function, ax, can t h a n be simply expressed as
ax(x)= WLAICs(x)
W
W
---<_x<-2 2
(12)
The most important advantage of the above representation is that it can be easily expanded to an infinite range of x with a Fourier series. By treating eq. 11 as a periodic function with a period of W, the expanded cosine distribution function can be written as
[ 2nn "~ Cs(x)=ao+ ~__lancos~--~-x]
(13)
where 1
ao= W
(14)
2 an= W [ l _ ( ~ ) 2 ]
nwn c°s W
(15)
Another very important parameter in describing the plant stand architecture, especially in modeling radiative transfer, is the so called G-function which can be interpreted as the mean projection of a unit foliage area in a given
98
X. YANG ET AL.
direction (Ross, 1975 ). Computation of the G-function based on its definition, however, is tedious and time consuming. Goel and Strebel (1984) proposed using a beta distribution function to represent the leaf orientation of vegetable crops in a universal way. However to use it one needs to determine the mean and variance of leaf orientations; practically, that is not an easy task. Some empirical equations for computing the G-function from the leaf inclination index, X, and the zenith angle of a radiation ray, 0, are available. According to Goudriaan (1977), the G-function could be approximated as
G=Go+O.877(1-2Go) cos 0
(16)
where G(~= 0.5 - 0.633X- 0.33X2
{17)
However information on relationships between X and z was missing in these studies. Observations indicated that mature cucumber crops tended to have more horizontal leaves at the upper level and inclined ones at the lower part. A linear distribution ofx on the vertical direction was assumed to incorporate this phenomenon into consideration, where
X(Z) =Xo+ kxz
(18)
EXPERIMENTATIONWITH A GREENHOUSECUCUMBERCROP Five randomly chosen cucumber (mustang variety) plants were sampled twice a week to measure the parameters necessary to describe the plant architecture. The measurements were taken in a computer-controlled horticultural greenhouse at the Ohio Agricultural Research and Development Center, The Ohio State University at Wooster, OH, beginning 27 March 1987 (the day in which plants were transplanted into the house) and continuing until 7 June of the same year. The greenhouse cucumber plants were grown according to Smith (1987) and Bauerle (1984) in rows on rockwool blocks; the blocks were placed on reflective plastic mulch. For each sample, as shown in Fig. 2, the following parameters were measured: (1) number of leaves (nL); (2) distance of each leaf from the ground (z); (3) zenith (0a) and azimuth (~L) (from north) angles of each leaf's normal; (4) horizontal projection of the petiole of each leaf (lp); (5) length (lL) and width (WL) of each leaf. The average distance between rows, W, and that between any two nearby plants along the row direction were measured to be 0.9 and 0.6 m, respectively.
99
PLANT ARCHITECTURAL PARAMETERS 0L
Z
Fig. 2. Leaf parameters measured in determining the plant architecture, where z is the height of a leaf from ground, 0Land eL are the zenith and azimuth angles of the leaf normal, ILand WLare the length and width of the leaf, and lp is the horizontal projection of the leaf petiole. RESULTS AND DISCUSSION
Leaf area index
T w e n t y - t h r e e h e a l t h y leaves were c h o s e n to d e t e r m i n e the r e l a t i o n s h i p bet w e e n the area o f a c u c u m b e r leaf, LA, a n d its length, la, a n d width, WL, as defined in Fig. 2. B y precisely m e a s u r i n g t h e area of each leaf with a leaf area meter, t h e r e l a t i o n s h i p t h e n was d e t e r m i n e d as L A = 0.739/LWL-- 0.00104
(19)
where L A was in m 2 a n d IL a n d WL were b o t h in m. T h e c o r r e l a t i o n coefficient R 2 for eq. 19 was 0.99. T h e L A I of a p l a n t was o b t a i n e d b y s u m m i n g up t h e area o f all its leaves a n d t h e n dividing t h e s u m m a t i o n b y the c o r r e s p o n d i n g g r o u n d area (0.54 m 2). L A I was s h o w n to be linearly related to t h e p l a n t height, h, as s h o w n in Fig. 3. A statistical analysis gave t h e following regression e q u a t i o n with a n R 2 value of 0.99 L A I = 0 . 8 8 6 h - 0.0965
(20)
100
X. YANGET AL. 2.00
1.50
o/"
Y
Jo
J
1,00
,d
/ J
0,50
0.00 0.00
~-0.60
1.20
'
~
1.80
'
2.40
h (m) Fig. 3. Measurements of LAI (mean values of five sample plants) in relation to plant height. The solid line represents the corresponding regression equation (eq. 20 ).
Average leaf dimension and stand width Whereas the average leaf dimension, LD, was obtained by geometric averaging of the mean leaf length and width, the stand width, w, came from the arithmetic average of the horizontal (radial) dimensions of leaves with their petioles. Their dependence on h was expressed exponentially as
LD = LDmax [ 1 - e x p ( - kLDh) ]
(21)
w=wmax[1-exp(-k~h ) ]
(22)
where LDm~x and wm~xwere the maximum values of LD and w for mature plants and were assigned to be 0.30 and 0.80 m, respectively, based on observation. The values of kLD and k~ were determined to be 1.964 and 1.449, respectively, by linear regressions on a logarithmic scale. The corresponding R 2 values for the logarithmic transformations of eqs. 21 and 22 were 0.93 and 0.94, and the plots of LD and w vs. h are shown in Fig. 4 (A) and (B), respectively.
Mean and variance of az For employing the beta distribution to describe the vertical foliage area density function, the mean and variance of the distribution need to be determined. They were computed according to Giffin (1971) by 1
/~h= ~
nL
i~1 zilaii
(23)
101
PLANT ARCHITECTURAL PARAMETERS .32
E f..., d c" o
.24
g E .~6 .J
i
.08
<
I
O.G~
.6
1.~2 h (m)
1.8
2.4
E
"E)
0 b3
E o E_
l i
•
1.2
1.8
2.4
h (m~
Fig. 4. Dependence of average leaf dimension (A) and plant stand width (B) on plant height, where the scattered points are measured means and the smooth curves are the regression lines of eqs. 21 and 22, respectively. 2
1
n~
{Th=~-I 2 (Zi--, uh)21aii
(24)
i=1
where nL was the number of leaves of a sample plant, zi was the vertical position and/aii was the leaf area index contributed by the ith leaf. Both Hh and ah 2 were shown to be functions of growth stage. Relationships between them and plant height, h, were determined, both with an R 2 value of 0.99, as being Hh= 0 . 4 3 4 h + 0.106
(25)
a ~ = ( 0 . 2 6 4 h - 0.033) 2
(26)
Once Hh and a~ are determined, the parameters a and b of the vertical foliage area density function can be readily computed by eqs. 7 and 8. Table 1 shows
102
X. YANGET AL.
TABLE 1 Plant architectural parameters of a greenhouse cucumber row crop at selected growing heights h
LAI
(m)
LD
W
(m)
(m)
a
b
0.2431 (0.009)2
0.102 (0.002)
0.124 (0.003)
0.420 (0.016)
_3 -
0.453 (0.005)
0.274 (0.003)
0.157 (0.001)
0.522 (0.009)
2.559 (0.143)
1.086 (0.004)
1.003 (0.019)
0.890 (0.008)
0.203 (0.007)
0.737 (0.010)
1.731 (0.099)
1.364 (0.036)
1.540 (0.042)
1.320 (0.065)
0.205 (0.003)
0.752 (0.003)
1.631 (0.066)
1.616 (0.072)
2.037 (0.066)
1.658 (0.054)
0.204 (0.001)
0.783 (0.011 )
1.439 (0.076)
1.648 (0.082)
'Mean value of five randomly selected sample plants. 2Standard error of the corresponding means. 3Not determined.
the experimentally determined architectural properties of the cucumber row crop at selected growing stages, among which w, a and b are the basic parameters for theoretically evaluating the cross-row and vertical density functions. Density functions of foliage area
The empirical vertical density function of leaf area, az, was first computed
by az(Zi) -
/aii
(27)
Zi + l / 2 - - Z i - - 1 / 2
where z~+1/2 and zi-1/2 represented the midpoints between the ith and the ( i + 1 )th or the ( i - 1 )th leaf. The results were then averaged over each layer of interest (0.1 m) to obtain a histogram. Experimental determination of the horizontal density function of foliage area was somewhat complex. First, the projection of each leaf on the x axis was identified. A step function was used for each leaf to describe its contribution to ax, assuming that the LAI contributed by a leaf was evenly distributed on its cross-row projection. The total density function was then obtained by summing all the step functions posed by the leaves. Histograms were finally drawn with the averages over subdomains of 0.05 m length along the x direction.
103
PLANT ARCHITECTURAL PARAMETERS
A 2_4
0.8~n 0"~).0
1.0
2.0 3.0 4.0 5.0•0.0 1.0 2.0 3.0 4.0 VERTICAL FOUAGE AREA DENSITY FUNCTION(m-ll
5.0
B T-
h=Zlm
h=Z0m
o
o ".5
0.0
L .5/-.5 x(m}
0.0
Fig. 5. Examples of vertical (A) and cross-row (B) foliage area density functions of the cucumber crop.
Examples of the histograms of the vertical and cross-row foliage area density functions are presented in Fig. 5 (A) and (B), respectively, where the smooth curves were the theoretical predictions of eqs. 2 and 12 with the empirical values of a, b and w. The R 2 values for the theoretical representations were in the range 0.64-0.89 for az and 0.61-0.83 for ax, respectively, indicating that the beta and cosine distributions did match the experimental data reasonably well.
Leaf inclination index of mature plants The leaf inclination index, X, was used to represent the average inclination of leaves in the plant canopy. The value of X could range from - 1 to + 1 with the maximum value representing horizontal leaves and the minimum value representing vertical (drooping) leaves. A very strong correlation was found between the inclination of each leaf and
104
x. YANG ETAL. ,
1.000 '
o
o
ooo
, o o o
o
o
o o
'
oo
u
oo
o
o
oooo o
x
0
O.500
o o
©
o
oo
o
o
o
oo
~
o
oo
o
0.000
0
© o
-0.500
-
o
1.000
ooo
o
oo
o
~
0.000
0.200
o
i
i
0.400
0.600
Normalized height,
J
0.800
1.000
z/h
Fig. 6. Vertical distribution of the leaf inclination index of mature cucumber plants with the regression line ofeq. 18.
its vertical location on mature cucumber plants. Figure 6 shows a plot of leaf inclination index, X, versus normalized height, z/h. The solid line w~is the proposed distribution function, eq. 18, calibrated through a linear regression. The intercept, X0, was determined as - 0.566 while the slope, kx was given by
kx=l.549/h
(28)
The R 2 value of the linear regression was 0.88. The weighted average ,~, of eq. 28, defined as h
1 f %=L-A-I
X(z)azdz
(29)
0
was 0.38 for mature plants (h = 2.0 m ), very close to the data of leaf inclination index for cucumber crops (0.39) given by Nichiporovich {Ross, 1981 ). SUMMARYAND CONCLUSIONS A group of parameters which describe the plant architecture of a greenhouse cucumber row crop were quantitatively determined. These included geometric descriptions of a canopy model, spatial distributions of foliage elements and leaf orientational characteristics. LAI of the greenhouse cucumber crop was found to be linearly related to the plant height. Stand width and leaf dimension could also be well represented by exponential functions of plant height. Spatial distributions of foliage area were described by a beta distribution and a periodic
PLANT ARCHITECTURAL PARAMETERS
105
cosine distribution in the vertical and cross-row direction, respectively. Measurements of the foliage area density functions supported well the above proposed distribution functions. A linear vertical distribution of leaf inclination index was also experimentally justified. The determined architectural parameters were anticipated to be useful in greenhouse microclimate studies, especially in modeling the radiation transfer into a row crop stand. ACKNOWLEDGEMENT
Salaries and research support was provided by State and Federal Funds appropriated to the Ohio Agricultural Research and Development Center, The Ohio State University. Manuscript number 136-89.
REFERENCES Allen, L.H.J., 1974. Model of light penetration into a wide-row crop. Agron. J., 66: 41-47. Bauerle, W.L., 1984. Bag culture production of greenhouse tomatoes. Special Circular 108, Ohio Agricultural Research and Development Center, Ohio State University, Wooster, OH. Giffin, W.C., 1971. Introduction to Operations Engineering. Irwin, Homewood, IL, 632 pp. Goel, N.S. and Strebel, D.E., 1984. Simple beta distribution representation of leaf orientation in vegetation canopies. Agron. J., 76: 800-802. Goudriaan, J., 1977. Crop Micrometeorology - A Simulation Study. Simulation Monographs, Pudoc, Wageningen, 249 pp. Ross, J., 1975. Radiative transfer in plant communities. In: J.L. Monteith (Editor), Vegetation and the Atmosphere. I. Principles. Academic Press, New York, pp. 13-56. Ross, J., 1981. The Radiation Regime and Architecture of Plant Stands. Junk, Boston, MA, 391 pp. Ross, J. and Magi, H., 1971. A calculation method for determining leaf area and its vertical distribution in the barley crop. Est. Contrib. Int. Biol. Programme III, Tartu, pp. 101-112. Smith, D.L., 1987. Rockwool in Horticulture. Grower Books, London, pp. 148-150. Yang, X., 1988. Greenhouse microclimate: transport processes, plant responses and dynamic modeling. PhD Dissertation, Ohio State University, Columbus, OH, 290 pp. Yang, X, Short, T.H., Fox, R.D. and Bauerle, W.L., 1989. A computer model of the bioclimate of a greenhouse. ASAE Paper No. 89-4016, American Society of Agricultural Engineers, St. Joseph, MI.