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Mathematical characterization of maize canopies
Agricultural and Forest Meteorology, 66 (1993) 2 4 7 - 2 6 5
247
0168-1923/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved
Mathematical characterization of maize canopies 1 D.W. Stewart*, L.M. Dwyer Centre for Land and Biological Resourees Research, Research Branch, Agriculture Canada, Ottawa, Ont. K1A 0C6, Canada (Received 7 October 1992; revision accepted 5 April 1993)
Abstract Leaf area and its distribution in space largely determine canopy photosynthesis and dry matter accumulation. Methods were developed to characterize the morphology of maize plants in order to calculate leaf area and leaf angle distributions. Plants were placed against a grid where ligule heights and leaf curvature characteristics were measured. Leaf widths were then measured every 10 cm along the length of the leaf. The curvature of each leaf was characterized by a general quadratic equation expressed in terms of the initial leaf angle, the eoordinates of the m a x i m u m height of the leaf and the coordinates of the leaf tip. A third order polynomial, expressed in terms of leaf width at the ligule, the length of the leaf and two empirical shape coefficients, was used to describe leaf shape. From these equations leaf areas were calculated as a function of plant height and distance between rows. Leaf angle distributions were also calculated. In order to reduce the number of coefficients needed to describe a canopy, the various parameters involved with leaf curvature and shape and internode length were fitted by least squares to second order polynomial functions of leaf number. These second order polynomial functions were used to generate leaf area and angle distributions which compared favourably with the original measurements.
Introduction The distribution of leaf area in space is a fundamental aspect of plant growth because it determines radiation interception and plant photosynthesis. Therefore, a great deal of effort has been devoted to measuring and characterizing leaf area distribution, particularly as a function of depth in the canopy (Ross, 1981). Sinoquet et al. (1991) have categorized this work as follows: (i) The stratified clipping method (Monsi and Saeki, 1953) which was destructive but provided leaf area distribution. (ii) The nondestructive method using a mechanical spatial coordinate apparatus (Warren Wilson, 1965; Lang, 1973) which determined leaf orientation and areal. *C o r r e s p o n d i n g a u t h o r . I C L B R R c o n t r i b u t i o n n u m b e r 92-167.
248
D.W. Stewart, L.M. Dwyer Agricultural and Forest Meteorology 66 (1993) 247 205
(iii) The plant profile method (Loomis et al., 1968; Loomis and Williams, 1969; Stewart and Lemon, 1969; Daynard, 1971) which has been used to estimate leaf angle distribution and vertical and horizontal distribution of leaf area. Sinoquet et al. (1991) used an electronic spatial coordinate method with a three-dimensional ultrasonic digitizer. It had the advantage of tracing leaf patterns electronically. However, it required a still atmosphere, making it necessary to place plants in an enclosed space. The present paper provides a modification of the plant profile method. The plant profile method has been restricted to two-dimensional growth. Stewart and Lemon (1969) and Daynard (1971) placed maize plants against vertical flat surfaces. Daynard (1971) traced the patterns of leaf curvature onto paper while Stewart and Lemon (1969) noted the coordinates on a grid marked in 10 x 10 cm squares where leaves intersected with grid lines. Leaf width as a function of leaf length was measured in each study. From these measurements, leaf area was calculated as a function of depth in the canopy and of distance from the row. Stewart and Lemon (1969) also calculated leaf angle distributions. In both studies the plant profile method tended to be slow and laborious. Pr~vot et al. (1991 ), also using a plant profile method, mathematically described the curvature and shape of maize leaves and from these equations calculated leaf area and leaf angle distributions. The modification presented in the present study is simpler than that of Pr~vot et al. (1991) and less laborious than that of Daynard (1971) and Stewart and Lemon (1969). Methods
Two maize hybrids, an experimental leafy hybrid developed by Glenn Seeds Ltd. (Leafy), and a normal hybrid (Check, Pioneer 3790), were grown at three planting densities (40 000, 60 000 and 80 000 plants ha --1) with three replications in a complete randomized block design at the Central Experimental Farm, Ottawa, Canada (45 ~' lat., 76 ° long.). Rows were 76 cm apart with precise spacing within the rows. No attempt was made to align plants in adjacent rows. Planting1 occurred on May 9, 1991. Fertility was based on field tests; 200 kg ha N was added when corn was knee-high. Shortly after tasselling (beginning July 30, 1991) a plant from each plot was placed against a vertical grid. The following measurements were made on each leaf: (i) the height of the leaf ligule; (ii) the angle between the leaf and the horizontal plane at the leaf ligule (00); (iii) the coordinates of the maximum height of the leaf (xM, YM); (iv) the coordinates of the leaf tip (XL, YL) (Fig. 1). Individual leaf measurements also included leaf width at the ligule and every 10 cm along the leaf, and total leaf length. Plant height and stem
D.W. Stewart, L.M. Dwyer / Agricultural and Forest Meteorology 66 (1993) 247-265
249
XM YM
/
>-
~
X
LYL
Fig. 1. Leaf shape parameters; 00 represents the angle between the leaf and the horizontal plane at the leaf ligule, XM, YM are coordinates of the maximum height of the leaf and XL, YL are the coordinates of the leaf tip.
diameter at the top, middle and bottom of the plant were also measured. It should be noted that modulations of the leaf lamina were ignored and leaves were assumed to be ribbonlike with equal length main veins and leaf margins. Edmeades and D a y n a r d (1979) used a four coefficient quadratic to describe leaf curvature but had to assume that the angle subtended by the stem and lamina was zero at the origin (0,0) or point of leaf insertion, and 180 ° at the leaf tip. In this study we developed a quadratic expression without any leaf angle assumptions. The general expression for a quadratic is A x 2 + By 2 + Cxy + D x + Ey + G ----0
(1)
where x is the horizontal distance from the origin or ligule, y is the vertical distance and A, B, C, D, E and G are empirical leaf curvature coefficients. There are five restrictions to Eq. (1) (i) (ii) (iii) (iv) (v)
atx=0, y=0 at x = 0, ( d y / d x ) = tan00 at x = XM, y = YM at x = XM, ( d y / d x ) = 0 atx=xL, y=yL
250
D.W. Stewart, L.M. Dwver
Agricultural and Forest Meteorology 66 ( 1993,, 247 265
The first restriction implies that G equals zero. We can also divide by E because the right-hand side of Eq. (1) is zero. This is equivalent to assuming E equals 1.0. Then Eq. (1) becomes d x 2 + By 2 + Cxv + Dx + t.... {}
(2i
Differentiating Eq. (2) with respect to x resulted in 2 A x + 2By(dy/dx) + C x ( d y / d x ) .4. ('!' +- D x ( d y / d x ) = 0
{3;
OF
dy dx
(2Ax + (),+ D) \ ~ y ~ (7l, 4 i
Applying restrictions (i) and (ii) to Eq. (3) meant that D was equal to - t a n G . This eliminated another coefficient leaving only A, B and C. Applying restrictions (iii) and (iv) resulted in 2 A x M + Cy M - tan00 2By M + Cx M + I
:-.0
(4)
We assumed that 2By M + C x M +
I ¢0
('i
resulting in 2Ax M
q- G y M - tan00 .... 0
(6!
or C =
(tan00
-
(7)
2AXM)/): M
Substituting XM, YM and C from Eq. (7) in Eq. (2), A was expressed as 9
2
A = By~/x M +
),'M/XM
"~q"
By substituting x L and YL (restriction v) in Eq. (2) and combining it with Eqs. (7) and (8) we produced B =
2XMXLy L -Jr-t a n O o x ~ x L - - . V M X I - -
tanOO/YM)X2 XLYL - x ~ ) . , l .
(91
.V~X[ + .X~I.!'I_ --2A'MYMXLYl.
Therefore, A, B and C could be calculated directly from measurements of 0{} XM, YM, XL and YL. By rearranging Eq. (2), y was expressed as (.....................................
) .... ( - ( C x + 1) -~
Cx + 1) 2 - - 4 B x ( A x - t a n O o ) ) / 2 B
(1(}}
There could be a negative sign in front of the square root as well, but from experience we found that it was not needed for leaf area calculations. This
D.W. Stewart, L.M. Dwyer / Agricultural and Forest Meteorology 66 (1993) 247 265
251
method used only one equation and five easily measured parameters to calculate leaf curvature, compared with the two equations with one discontinuity and nine parameters of Pr6vot et al. (1991)• To calculate leaf area and leaf angles we divided the x axis into small increments ( x i - xi-1) and used Eq. (10) to calculate the corresponding (.Fi -- Yi 1). The length (z) of the leaf increment was calculated from Zi--Zi-1 =
v / ( X i - - X i - I / +(Yi-Yi
1) 2
(11)
The angle (0i) of this leaf increment with the horizontal was calculated from 0i = A R C T A N ( ( y i - y i - 1 ) / ( x i - xi-1))
(12)
To determine the area of this leaf increment the following function was needed W -= W 0 + blZ + b2 z2 + b3 z3
(13)
where W is leaf width, W0 is the leaf width at the ligule, z is the distance along the leaf and bl, b2 and b3 are empirical leaf shape coefficients. Because W equalled zero when z equalled L (the leaf length) then
b 3 = - ( W 0 + bl L + b2L2/L 3)
(14)
and
W=- W 0 + blz + b2z2 - ( W o + b l L + b2L2)z3/L 3
(15)
Because W0 and L were measured, only bl and b2 were unknown. These were calculated for each leaf by nonlinear least-squares regression (Marquardt, 1963) from the leaf width measurements every 10 cm along each leaf. Leaf area segments Si were calculated by integrating W with z or
Wdz
S i = Zi
(16)
I
By repeating these procedures for each leaf we calculated leaf area in a'two dimensional plane. Then we rotated this plane from - 8 5 ° to + 85 ° in 5° increments where 0 ° represented the plane perpendicular to the row of plants. Leaf area in each of these planes was projected mathematically onto the plane perpendicular to the row and then averaged for the 35 positions. Stern areas were calculated as one-half the surface area of stem cylinders using the three measurements of stem diameters and interpolating with height. Onehalf of the stem area was considered equivalent to leaf area because leaf area is the area of only one surface of the leaf. This resulted in average leaf and stem area densities in the two-dimensional plane perpendicular to the maize row. We also calculated leaf angle distributions restricting ourselves to only one dimension, i.e. leaf angle distributions at different depths in the canopy. To characterize a canopy using this method we needed leaf parameters W0, bl, b2, A, B, C, L, 00, and the height of each ligule. This resulted in 108-144
D.W. Stewart, L.M. Dwyer ; Agricultural and Forest Meteorology 66 ! 1993J 247 265
252
Table 1 Measured plant parameters for Leafy and Check hybrids at three planting densities Parameter
Leafy (plants'ha) 40 000
Plant height (m) Ear height (m) Leaf number L (m) W0 (m) xt, (m) 00 (degrees) Leaf area per plant (m ~') Average stem diameter (m)
Check (plants:ha~
60 000
80 000
40 000
60 000
80 00()
2.240
2.070
0.693 0.160 0.572 (/.075 (L312 0.573 0.596 0.025
0.753 0.147 0,542 0.070 0.299 0.524 0.488 0.022
2.227 0.678 0.160 0.539 0.065 0.291 0.525 0.526 0.021
1.833 0.8t0 0.1 l 7 0.680 0.070 0.456 0.499 0.502 0.025
1,872 0.883 0, I I ~ 0.670 0.069 (/.442 0,48q 0.485 0.025
........ 0.803 0. I(!7 0.06/ O. 17(i 0.558 045 ~0,453 0 D I ~,)
parameters for plants having 12 16 leaves. To reduce the number of parameters required to characterize the canopy these leaf parameters were regressed against leaf number using second-order polynomials. Leaf area was then calculated as a function of leaf number, depth in the canopy and distance from the row 0.08
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0.06 E C,J
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80000
~ check60000 , check 40000 • leafy80000 - teafy60000 • leafy40000 i 0.03
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: 0.08
LEAF AREA METHOD l(m 2) Fig. 2. A c o m p a r i s o n o f leaf area calculated using leaf curvature, leaf width relationships (method 21 and calculated simply by using Eq. (18) (method 1 ).
D.W. Stewart, L.M. Dwyer / Agricult,,~al and Forest Meteorology 66 (1993) 247 265
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ROW
INCREMENT
Fig. 3. L e a f area index for the (a) Leafy a n d (b) C h e c k hybrids at three densities. T h e leaf area index is s h o w n in 0.048 m i n c r e m e n t s between rows n u m b e r e d f r o m the row.
Results and discussion Some plant and leaf characteristics of the two hybrids are given in Table 1. The Leafy hybrid was taller and had a lower ear height, more leaves and a larger leaf area than the normal Check. Individual leaves of the Check were
D.W. Stewart, L.M. Dwyer / Agricultural and Forest Meteorology 66 (1993) 247 26.5
254
longer and branched out further from the row. There was little difference in W0, 00 and stem diameter. Few, if any, density effects are evident in Table 1. To check the leaf area calculations we summed leaf area increments (S,) from Eq. (16) for each leaf (method 2). We compared these values with leaf area calculated by using values of Wo, bl, D2, and L for each leaf and integrating W over leaf length (method 11. That is
A=
(Wo+blc+h2=~--(~~+blL +h2L2)z3/L:~)dz
{17!
0 or
,4 -- 0.75 W0 L + 0.5hi L 2 4 - 0 . 3 3 3 b 2 L ~
ill S
There was close agreement between the two methods (Fig. 2), indicating that the numerical integration of Eq. (16) along the natural curvature of the leaf
a)
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LEAFY
CHECK
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ROW INCREMENT Fig. 4. Leaf area densities (m2/m 3) for the (a) Leafy and (b) Check hybrids at 60000 plants ha by depth in the canopy for each 0.048 m row increment numbered from the row to midway between tu, c~ rOWS.
D.W. Stewart, L.M. Dwyer / Agricultural and Forest Meteorology 66 (1993) 247-265
a)
1.0
255
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UJ 0.6 rr IJ_ IJJ > ~__ 0.4 < ,,_1 ::3 0.2 O
---.-.~ --'i~
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LEAF ANGLE b)
>O Z W
,, -- -- -- 8 0 0 0 0 plants/ha
1.0 -
• .....
6 0 0 0 0 plants/ha
o - -
4 0 0 0 0 plants/ha
~
1
--,z/~/
0.8
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>
t< ...J
0.4
--
i¢ ~
0.2 O 0.0
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20
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70
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I 90
LEAF ANGLE Fig. 5. Cumulative frequency of leaf angles over the whole canopy for the (a) Leafy and (b) Check hybrids at three densities.
D.W. Stewart, L.M. Dwyer i Agricultural and Forest Meteorology 66 {1993j 247 265
256
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D.W. Stewart, L.M. Dwyer / Agricultural and Forest Meteorology 66 (1993) 247 265
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D.W. Stewart, L.M. Dwyer / Agricultural and Forest Meteorology 66 (1993) 247-265
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D.W. Stewart, L.M. Dwyer / Agricultural and Forest Meteorology 66 (1993) 247-265
259
did not introduce serious errors and that this method accurately calculated leaf area with height and between rows on a per plant basis. The 1991 year was relatively dry. Leaf areas were smaller than usual and the measurements showed that leaf area index (LAI) decreased markedly from the plant stems to the middle of the row (Fig. 3(a,b)). As would be predicted from average values of XL and leaf area (Table 1), the Leafy hybrid had higher LAI values at the stem but smaller values midway between rows for comparable planting densities. Calculated leaf area densities as a function of depth in the canopy and distance from the row are shown in Fig. 4. This clearly shows the highest densities near the stem and at middle canopy depths. Although the Leafy hybrid had higher leaf area densities near the row than the Check, the Check hybrid filled the inter-row area more evenly than the Leafy hybrid. The assumption in these leaf area calculations is that azimuth angles of plant planes are evenly distributed. There is, in fact, some evidence that maize plants adjust azimuth angles to fill inter-row spaces (Loomis and Williams, 1969; P. Girardin and M. Tollenaar, University of Guelph, personal communication, 1992). Leaf area in the field is therefore probably more uniformly distributed across the row than calculated and presented in Fig. 2. When azimuth angles become available for Canadian conditions we can insert them into the calculations. Cumulative leaf angle distributions calculated from Eq. (I 2) showed differences as well (Fig. 5(a,b)). No real pattern with depth in the canopy emerged from these measurements of leaf angles, probably because only three plants were measured per treatment. However, it was evident that the Leafy hybrid had more leaves between 35 and 65 °. Above 20 °, the Check had a very even distribution of leaf angles. Polynomial functions of leaf number were fitted by least squares to all the plant parameters (Table 2). It should be noted that the ear leaf number was set equal to zero with leaves above the ear being positive and those below negative. This means that the a 0 coefficients in Table 2 are the values of the parameters at the ear leaf. Parameters such as ligule height, leaf length and leaf width at the ligule had well defined patterns with leaf numbers with correlation coefficients greater than 0.8 (Table 2, Fig. 6). Leaf shape coefficients (bl and b2) were less well correlated with leaf number (r >10.6) but this could result from less absolute variation in their value with leaf number. That is, shape patterns vary less with leaf number than leaf width or ligule height. However, the parameters involved with leaf curvature were poorly correlated with leaf number and this was because of plant variation in curvature patterns of individual leaves. Data plus the function for the 'A' parameter are shown in Fig. 7. The plant to plant variation in A, B and C resulted in poor plant reconstruction when the polynomial functions were used. Therefore we fitted values of XM, YM, XL, and YL to polynomial functions instead. This
D.W. Stewart L.M. Dwyer ,, Agricultural and Forest Meteorology 66 (1993 i 247-265
260
a) 1.0 7",
I"1-
(.9
0.8
u.I "ILLI J'm
0.6
(.9 J £3 1.11 N J ,<
•
0.4
-- 80000 plants, ha
~s
rr
O
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~
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/
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t
0
~
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•
60000 plants, ha
c. . . .
40000 plants, ha
I
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i
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LEAF N U M B E R b) 1.0
I-T
(.9
0.8
LIJ "r" W ._1 :E:)
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< rr
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t -4
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LEAF N U M B E R Fig. 6. Polynomial regressions of ligule height normalized with respect to total plant height against leaf number for the (a) Leafy and (b) Check hybrids at three densities. Symbols represent data points and lines represent fitted curves for each density.
D.W. Stewart, L.M. Dwyer / Agricultural and Forest Meteorology 66 (1993) 247 265
261
a) 0,24
0.20 O O
0,16 0
,<
z~
•
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0.12 ,x •
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LEAF NUMBER
b)
-- -- -- 80000 plants/ha
0.24
• .....
60000 plants/ha
o
40000 plants/ha
0.20
0.16
, ~ 0.12
0.08
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0.04
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LEAF NUMBER Fig. 7. Polynomial regressions of the leaf curvature coefficient, A, against leaf number for the (a) Leafy and (b) Check hybrids at three densities. Symbols represent data points and lines represent fitted curves for each density.
D . W . Stewart, L . M . D w v e r , Agricultural and Forest Meteorology 66
262
19()3 ) 247 265
a) 50
-
40
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X " •
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LEAF NUMBER b) 50
-
80000 p l a n t s h a -
60000 plants/ha 40000 p l a n t s h a
40 •
3
30 "
X
•
T,,
•'
0
20
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10
0 -8
I -4
0
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LEAF NUMBER Fig. 8. Polynomial regressions of the XM leaf parameter against leaf number for the (a) Leafy and (b) Check hybrids at three densities. Symbols represent data points and lines represent fitted curves for each density.
D.W. Stewart, L.M. Dwyer / Agricultural and Forest Meteorology 66 (1993) 247-265
263
proved more successful, and comparable values for XM are shown in Fig. 8 for canopy reconstruction. We then calculated values of A, B and C from the leaf number polynomials of XM, YM, XL, YL, and 00 using Eqs. (7)-(9). This procedure proved surprisingly successful. Very good agreement was found between measured and reconstructed LAI variation between the rows (Table 3). There was a tendency for reconstructed plants to have lower LAIs midway between rows. This is understandable, as polynomial fitting averages all leaves and tends to reduce extreme values of leaves which project further from the stem. In other years with larger leaf areas and longer leaves, this discrepancy would not be as large. Reconstructed leaf area and leaf angle distribution also correlated well with measured values (Table 4). While not perfect, leaf number polynomial functions of these various parameters make it easier to mathematically characterize a plant canopy. In summary, we have demonstrated a method of characterizing plant phenotype which clearly showed differences between hybrids in leaf area distribution and leaf angle distribution. The Leafy hybrid had a larger leaf area but the leaf area was not as evenly distributed between the rows as for the Check. This would probably mean less light interception for the Leafy hybrid, although this was not measured in our experiment. Leaf angles were also different; most of the leaf angles for the Leafy hybrid were between 35 and 65 ° while the Check hybrid had leaf angles relatively evenly distributed. It is anticipated that the analysis described in this study will be useful for future modelling studies. The one advantage of this method for mathematiTable 3 Leaf area index between rows, with row increments numbered from the row calculated using (a) measured values and (b) regression coefficients Hybrid
Leafy
Check
Planting density (plants/ha)
Row increment
40000
(a) (b)
60000
(a) (b)
1
2
7
r
RMSE
3
4
5
6
8
5.93 4.35 5.87 4.35
3.04 2.97
2.21 2.07
1.59 1.47
1 . 1 6 0.84 1 . 0 9 0.74
0.65 0.36
0.999
6.69e - 02
7.87 5 . 7 1 7.80 5.65
3.92 3.87
2.79 2.76
1.80 2.00
1 . 1 4 0.76 1 . 3 5 0.67
0.56 0.19
0.998
1.83e - 01
80000
(a) 10.75 8 . 0 1 (b) 10.59 8.15
5.55 5.65
4.05 4.00
2.91 2.87
1.94 2.09
1.33 1.32
1.18 0.52
0.997
2 . 5 5 e - 01
40000
(a) (b)
4.22 3.46 4 . 0 1 3.43
2.55 2.56
1.96 1.91
1.57 1.46
1.31 1.22
1.16 1.12
1.12 1.08
0.998
6.17e - 02
60000
(a) (b)
6.25 5.16 6.00 5.02
3.79 3.71
2.84 2.75
2.27 2.08
1.89 1.71
1.65 1.58
1.54 1.54
0.999
6.62e - 02
80000
(a) (b)
8.04 6.33 7.96 6.38
4.51 4.51
3.26 3.22
2.48 2.36
2.01 1.95
1.88 1.84
1.89 1.83
0.999
4.75e
02
264
D . W . Stewart, L . M . D w v. e r
Agricultural and Forest Meteorology 66 / 1993 •J ~~4 , ~ _~6 ~
Table 4 Correlations and root mean square errors between measured and reconstructed leaf area densities and cumulative leaf angle distributions for the Leafy and Check hybrids t.eafy (plants/ha)
Check
(plants/ha)
40 000
60 000
80 000
40 000
60 000
80 000
Leaf density r RMSE (m -~ )
0.938 0362
0963 0.468
(i).757 Iql 8
0.946 0.336
0.875 0.745
0.972 0,601
Leaf angle r RMSE (degrees)
0.965 0.432
0.968 0,420
0.815 0.835
0.958 0.413
0.917 0.691
0.96 t) 0.26(!
cally characterizing the canopy 1s that leaf area for individual leaves can bc stored in a three-dimensional array (leaf number, height and distance between rows). This makes possible the calculation of photosynthesis for individual leaves in detailed soil-plant-atmosphere models (Stewart and Lemon, 1969), These calculations should prove useful in trying to assess the effect of contrasting architecture on canopy photosynthesis and yield, to take into consideration changes of leaf photosynthesis with depth (Dwyer et al., 1989), and to assess the contribution of individual leaves when seeking to define the ideal leaf number for a given climatic region. Acknowledgements We appreciate the efforts of A. Neamtz and B. Wilson for much of the data analysis and of L. Evenson, J.-A. Dugas, D. Balchin and B. D o w and staff for technical assistance. References Daynard, T.B., 1971. Characterization of corn (Zea m a y s L.) canopies from measurements of individual plants. Agron. J., 63:133 135. Dwyer, L.M., Stewart, D.W., Balchin, D., Houwing, L., Marur, C.J. and Hamilton, R.1, 1989. Photosynthetic rates of six maize cultivars during development. Agron. J., 81:597 602 Edmeades, G.O. and Daynard, T.B., 1979. The relationship between final yield and photosynthesis at flowering in individual maize plants. Can. J. Plant Sci., 59:585 601. Lang, A.R.G., 1973. Leaf orientation of a cotton crop. Agric. For. Meteorol., 11 : 37 5 I. Loomis, R.S. and Williams, W.A., 1969. Productivity and morphology in crop stands: patterns with leaves. In: J.O. Eastin, Z.A. Haskins, O.G. Sullivan, C.H.M. Van Basvel (Editors), Physiological Aspects of Crop Yields. American Society of Agronomy, Madison, WI. Loomis, R.S., Williams, W.A., Duncan, W.G., Dovrat, A. and Nuzez, A., 1968. Quantitative descriptions of foliage display and light absorption in field communities of corn plants. Crop Sci., 8:352 356.
D.W. Stewart, L.M. Dwyer / Agricultural and Forest Meteorology 66 (1993) 247 265
265
Marquardt, D.W., 1963. An algorithm for least squares estimation of non-linear parameters. J. Soc. Ind. Appl. Math., 11:431 441. Monsi, M. and Saeki, I., 1953. Uber den Lichtfaktor in den Planzengesellscheften und seine Bedentung fur die Stroffproduction. Jpn. J. Bot., 14:22 52. Pr6vot, L., Aries, F. and Monestiez, P., 1991. Mod61isation de la structure geometrique du mais. Agronomie, 11:491 503. Ross, J., 1981. The Radiation Regime and Architecture of Plant Stands. Junk, The Hague, 391 pp. Sinoquet, H., Moulia, B. and Bonhomme, R., 1991. Estimating the three-dimensional geometry of a maize crop as an input of radiation models: comparison between three-dimensional digitizing and plant profiles. Agric. For. Meterol., 55: 233-249. Stewart, D.W. and Lemon, F.R., 1969. A simulation of net photosynthesis of field corn. U.S. Army ECOM Technical Report 2-681-6, U.S. Army Electronic Command, Fort Huachuca, Arizona. Warren Wilson, J., 1965. Stand structure and light penetration. I. Analysis by point quadrats. J. Appl. Ecol., 2:383 390.
247
0168-1923/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved
Mathematical characterization of maize canopies 1 D.W. Stewart*, L.M. Dwyer Centre for Land and Biological Resourees Research, Research Branch, Agriculture Canada, Ottawa, Ont. K1A 0C6, Canada (Received 7 October 1992; revision accepted 5 April 1993)
Abstract Leaf area and its distribution in space largely determine canopy photosynthesis and dry matter accumulation. Methods were developed to characterize the morphology of maize plants in order to calculate leaf area and leaf angle distributions. Plants were placed against a grid where ligule heights and leaf curvature characteristics were measured. Leaf widths were then measured every 10 cm along the length of the leaf. The curvature of each leaf was characterized by a general quadratic equation expressed in terms of the initial leaf angle, the eoordinates of the m a x i m u m height of the leaf and the coordinates of the leaf tip. A third order polynomial, expressed in terms of leaf width at the ligule, the length of the leaf and two empirical shape coefficients, was used to describe leaf shape. From these equations leaf areas were calculated as a function of plant height and distance between rows. Leaf angle distributions were also calculated. In order to reduce the number of coefficients needed to describe a canopy, the various parameters involved with leaf curvature and shape and internode length were fitted by least squares to second order polynomial functions of leaf number. These second order polynomial functions were used to generate leaf area and angle distributions which compared favourably with the original measurements.
Introduction The distribution of leaf area in space is a fundamental aspect of plant growth because it determines radiation interception and plant photosynthesis. Therefore, a great deal of effort has been devoted to measuring and characterizing leaf area distribution, particularly as a function of depth in the canopy (Ross, 1981). Sinoquet et al. (1991) have categorized this work as follows: (i) The stratified clipping method (Monsi and Saeki, 1953) which was destructive but provided leaf area distribution. (ii) The nondestructive method using a mechanical spatial coordinate apparatus (Warren Wilson, 1965; Lang, 1973) which determined leaf orientation and areal. *C o r r e s p o n d i n g a u t h o r . I C L B R R c o n t r i b u t i o n n u m b e r 92-167.
248
D.W. Stewart, L.M. Dwyer Agricultural and Forest Meteorology 66 (1993) 247 205
(iii) The plant profile method (Loomis et al., 1968; Loomis and Williams, 1969; Stewart and Lemon, 1969; Daynard, 1971) which has been used to estimate leaf angle distribution and vertical and horizontal distribution of leaf area. Sinoquet et al. (1991) used an electronic spatial coordinate method with a three-dimensional ultrasonic digitizer. It had the advantage of tracing leaf patterns electronically. However, it required a still atmosphere, making it necessary to place plants in an enclosed space. The present paper provides a modification of the plant profile method. The plant profile method has been restricted to two-dimensional growth. Stewart and Lemon (1969) and Daynard (1971) placed maize plants against vertical flat surfaces. Daynard (1971) traced the patterns of leaf curvature onto paper while Stewart and Lemon (1969) noted the coordinates on a grid marked in 10 x 10 cm squares where leaves intersected with grid lines. Leaf width as a function of leaf length was measured in each study. From these measurements, leaf area was calculated as a function of depth in the canopy and of distance from the row. Stewart and Lemon (1969) also calculated leaf angle distributions. In both studies the plant profile method tended to be slow and laborious. Pr~vot et al. (1991 ), also using a plant profile method, mathematically described the curvature and shape of maize leaves and from these equations calculated leaf area and leaf angle distributions. The modification presented in the present study is simpler than that of Pr~vot et al. (1991) and less laborious than that of Daynard (1971) and Stewart and Lemon (1969). Methods
Two maize hybrids, an experimental leafy hybrid developed by Glenn Seeds Ltd. (Leafy), and a normal hybrid (Check, Pioneer 3790), were grown at three planting densities (40 000, 60 000 and 80 000 plants ha --1) with three replications in a complete randomized block design at the Central Experimental Farm, Ottawa, Canada (45 ~' lat., 76 ° long.). Rows were 76 cm apart with precise spacing within the rows. No attempt was made to align plants in adjacent rows. Planting1 occurred on May 9, 1991. Fertility was based on field tests; 200 kg ha N was added when corn was knee-high. Shortly after tasselling (beginning July 30, 1991) a plant from each plot was placed against a vertical grid. The following measurements were made on each leaf: (i) the height of the leaf ligule; (ii) the angle between the leaf and the horizontal plane at the leaf ligule (00); (iii) the coordinates of the maximum height of the leaf (xM, YM); (iv) the coordinates of the leaf tip (XL, YL) (Fig. 1). Individual leaf measurements also included leaf width at the ligule and every 10 cm along the leaf, and total leaf length. Plant height and stem
D.W. Stewart, L.M. Dwyer / Agricultural and Forest Meteorology 66 (1993) 247-265
249
XM YM
/
>-
~
X
LYL
Fig. 1. Leaf shape parameters; 00 represents the angle between the leaf and the horizontal plane at the leaf ligule, XM, YM are coordinates of the maximum height of the leaf and XL, YL are the coordinates of the leaf tip.
diameter at the top, middle and bottom of the plant were also measured. It should be noted that modulations of the leaf lamina were ignored and leaves were assumed to be ribbonlike with equal length main veins and leaf margins. Edmeades and D a y n a r d (1979) used a four coefficient quadratic to describe leaf curvature but had to assume that the angle subtended by the stem and lamina was zero at the origin (0,0) or point of leaf insertion, and 180 ° at the leaf tip. In this study we developed a quadratic expression without any leaf angle assumptions. The general expression for a quadratic is A x 2 + By 2 + Cxy + D x + Ey + G ----0
(1)
where x is the horizontal distance from the origin or ligule, y is the vertical distance and A, B, C, D, E and G are empirical leaf curvature coefficients. There are five restrictions to Eq. (1) (i) (ii) (iii) (iv) (v)
atx=0, y=0 at x = 0, ( d y / d x ) = tan00 at x = XM, y = YM at x = XM, ( d y / d x ) = 0 atx=xL, y=yL
250
D.W. Stewart, L.M. Dwver
Agricultural and Forest Meteorology 66 ( 1993,, 247 265
The first restriction implies that G equals zero. We can also divide by E because the right-hand side of Eq. (1) is zero. This is equivalent to assuming E equals 1.0. Then Eq. (1) becomes d x 2 + By 2 + Cxv + Dx + t.... {}
(2i
Differentiating Eq. (2) with respect to x resulted in 2 A x + 2By(dy/dx) + C x ( d y / d x ) .4. ('!' +- D x ( d y / d x ) = 0
{3;
OF
dy dx
(2Ax + (),+ D) \ ~ y ~ (7l, 4 i
Applying restrictions (i) and (ii) to Eq. (3) meant that D was equal to - t a n G . This eliminated another coefficient leaving only A, B and C. Applying restrictions (iii) and (iv) resulted in 2 A x M + Cy M - tan00 2By M + Cx M + I
:-.0
(4)
We assumed that 2By M + C x M +
I ¢0
('i
resulting in 2Ax M
q- G y M - tan00 .... 0
(6!
or C =
(tan00
-
(7)
2AXM)/): M
Substituting XM, YM and C from Eq. (7) in Eq. (2), A was expressed as 9
2
A = By~/x M +
),'M/XM
"~q"
By substituting x L and YL (restriction v) in Eq. (2) and combining it with Eqs. (7) and (8) we produced B =
2XMXLy L -Jr-t a n O o x ~ x L - - . V M X I - -
tanOO/YM)X2 XLYL - x ~ ) . , l .
(91
.V~X[ + .X~I.!'I_ --2A'MYMXLYl.
Therefore, A, B and C could be calculated directly from measurements of 0{} XM, YM, XL and YL. By rearranging Eq. (2), y was expressed as (.....................................
) .... ( - ( C x + 1) -~
Cx + 1) 2 - - 4 B x ( A x - t a n O o ) ) / 2 B
(1(}}
There could be a negative sign in front of the square root as well, but from experience we found that it was not needed for leaf area calculations. This
D.W. Stewart, L.M. Dwyer / Agricultural and Forest Meteorology 66 (1993) 247 265
251
method used only one equation and five easily measured parameters to calculate leaf curvature, compared with the two equations with one discontinuity and nine parameters of Pr6vot et al. (1991)• To calculate leaf area and leaf angles we divided the x axis into small increments ( x i - xi-1) and used Eq. (10) to calculate the corresponding (.Fi -- Yi 1). The length (z) of the leaf increment was calculated from Zi--Zi-1 =
v / ( X i - - X i - I / +(Yi-Yi
1) 2
(11)
The angle (0i) of this leaf increment with the horizontal was calculated from 0i = A R C T A N ( ( y i - y i - 1 ) / ( x i - xi-1))
(12)
To determine the area of this leaf increment the following function was needed W -= W 0 + blZ + b2 z2 + b3 z3
(13)
where W is leaf width, W0 is the leaf width at the ligule, z is the distance along the leaf and bl, b2 and b3 are empirical leaf shape coefficients. Because W equalled zero when z equalled L (the leaf length) then
b 3 = - ( W 0 + bl L + b2L2/L 3)
(14)
and
W=- W 0 + blz + b2z2 - ( W o + b l L + b2L2)z3/L 3
(15)
Because W0 and L were measured, only bl and b2 were unknown. These were calculated for each leaf by nonlinear least-squares regression (Marquardt, 1963) from the leaf width measurements every 10 cm along each leaf. Leaf area segments Si were calculated by integrating W with z or
Wdz
S i = Zi
(16)
I
By repeating these procedures for each leaf we calculated leaf area in a'two dimensional plane. Then we rotated this plane from - 8 5 ° to + 85 ° in 5° increments where 0 ° represented the plane perpendicular to the row of plants. Leaf area in each of these planes was projected mathematically onto the plane perpendicular to the row and then averaged for the 35 positions. Stern areas were calculated as one-half the surface area of stem cylinders using the three measurements of stem diameters and interpolating with height. Onehalf of the stem area was considered equivalent to leaf area because leaf area is the area of only one surface of the leaf. This resulted in average leaf and stem area densities in the two-dimensional plane perpendicular to the maize row. We also calculated leaf angle distributions restricting ourselves to only one dimension, i.e. leaf angle distributions at different depths in the canopy. To characterize a canopy using this method we needed leaf parameters W0, bl, b2, A, B, C, L, 00, and the height of each ligule. This resulted in 108-144
D.W. Stewart, L.M. Dwyer ; Agricultural and Forest Meteorology 66 ! 1993J 247 265
252
Table 1 Measured plant parameters for Leafy and Check hybrids at three planting densities Parameter
Leafy (plants'ha) 40 000
Plant height (m) Ear height (m) Leaf number L (m) W0 (m) xt, (m) 00 (degrees) Leaf area per plant (m ~') Average stem diameter (m)
Check (plants:ha~
60 000
80 000
40 000
60 000
80 00()
2.240
2.070
0.693 0.160 0.572 (/.075 (L312 0.573 0.596 0.025
0.753 0.147 0,542 0.070 0.299 0.524 0.488 0.022
2.227 0.678 0.160 0.539 0.065 0.291 0.525 0.526 0.021
1.833 0.8t0 0.1 l 7 0.680 0.070 0.456 0.499 0.502 0.025
1,872 0.883 0, I I ~ 0.670 0.069 (/.442 0,48q 0.485 0.025
........ 0.803 0. I(!7 0.06/ O. 17(i 0.558 045 ~0,453 0 D I ~,)
parameters for plants having 12 16 leaves. To reduce the number of parameters required to characterize the canopy these leaf parameters were regressed against leaf number using second-order polynomials. Leaf area was then calculated as a function of leaf number, depth in the canopy and distance from the row 0.08
,~dp"' ~r
0.07
0.06 E C,J
CI O
0.05
I
w
0.04
w G~
0.03 4
w _D
0.02
check
,
4~ 0.01
o ~
--
0.00
0.00
~
i 0.01
i 0.02
80000
~ check60000 , check 40000 • leafy80000 - teafy60000 • leafy40000 i 0.03
i 0.04
J 0.05
I 0.06
i 0.07
: 0.08
LEAF AREA METHOD l(m 2) Fig. 2. A c o m p a r i s o n o f leaf area calculated using leaf curvature, leaf width relationships (method 21 and calculated simply by using Eq. (18) (method 1 ).
D.W. Stewart, L.M. Dwyer / Agricult,,~al and Forest Meteorology 66 (1993) 247 265
12-
10-
8
a) A----4k 80000 plants/ha O - - Q 60000 plants/ha O- - 0 40000 plants/ha
~', ",,,,
-
%
•
~
~
%
6
~, %
%
< u..I I:E < LL
< u..I
/
/ o°°°
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2
11
~," , @
'
I
"*,s /S SI S
j
d
f
..
"" .1_
f
/
"~o .x
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iS
,"
o,~\ %
4
?
~
,
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253
.~
n.
'
I
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I
d[~"
'
I
'
I
b)
10
n
~,.
?
,.,. %
,,
,,/
\
",, h
4
~^
~ /
o.'..
2
,..,e o. _
",O-_...l~:...a....~....:IF-.:@
"0~.,.,
"
-.@---@--
"~"
~ 0.......0
/
~. .o
I
I
I
I
I
I
I
I
2
4
6
8
10
12
14
16
ROW
INCREMENT
Fig. 3. L e a f area index for the (a) Leafy a n d (b) C h e c k hybrids at three densities. T h e leaf area index is s h o w n in 0.048 m i n c r e m e n t s between rows n u m b e r e d f r o m the row.
Results and discussion Some plant and leaf characteristics of the two hybrids are given in Table 1. The Leafy hybrid was taller and had a lower ear height, more leaves and a larger leaf area than the normal Check. Individual leaves of the Check were
D.W. Stewart, L.M. Dwyer / Agricultural and Forest Meteorology 66 (1993) 247 26.5
254
longer and branched out further from the row. There was little difference in W0, 00 and stem diameter. Few, if any, density effects are evident in Table 1. To check the leaf area calculations we summed leaf area increments (S,) from Eq. (16) for each leaf (method 2). We compared these values with leaf area calculated by using values of Wo, bl, D2, and L for each leaf and integrating W over leaf length (method 11. That is
A=
(Wo+blc+h2=~--(~~+blL +h2L2)z3/L:~)dz
{17!
0 or
,4 -- 0.75 W0 L + 0.5hi L 2 4 - 0 . 3 3 3 b 2 L ~
ill S
There was close agreement between the two methods (Fig. 2), indicating that the numerical integration of Eq. (16) along the natural curvature of the leaf
a)
b)
LEAFY
CHECK
2.40
2.40
L-.2 . . . . o~,-. 2.00
E
1.6o
"-
/,v
. ~
1.2o
.......
2.00
J +, ..
1,60
"'-
)
""
/ i/ ¢.,I.. w~-I I 2~@
i .# t \
\
, I ;/ / r
,
I
) ~ ) ) , /22
0.8
'
.+ (
uJ "1-
'L
\ ' \ \',
"" \~ '\ I . - i/¢
I--r-
- -\
"'-'-,"
~
/ /
1.20
/
{
/
",~,. ...-
"~ ~ ( / ' , - - . . . . . . . . 227.72 \%
\ t\_ _ \
~. . . . -.
t/1
I
0.8
0.4
0.4
0.0
<2,
?
,~ / 7
¢ I I I/ I ¢ 2 @/ o.",I
I
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'
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2.0
'
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40
'
I
'
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6.0
'
I
'
1
8.0
0.0
I
I '
I ' 2.0
I ' I ' 4.0
I '
' 6,0
I'
I 8.0
ROW INCREMENT Fig. 4. Leaf area densities (m2/m 3) for the (a) Leafy and (b) Check hybrids at 60000 plants ha by depth in the canopy for each 0.048 m row increment numbered from the row to midway between tu, c~ rOWS.
D.W. Stewart, L.M. Dwyer / Agricultural and Forest Meteorology 66 (1993) 247-265
a)
1.0
255
f~
>O 0.8 Z l.U
O
7
UJ 0.6 rr IJ_ IJJ > ~__ 0.4 < ,,_1 ::3 0.2 O
---.-.~ --'i~
0.0
10
1"-- o-- I -- • 20
30
1
1
I
l
I
40
50
60
70
80
I 90
LEAF ANGLE b)
>O Z W
,, -- -- -- 8 0 0 0 0 plants/ha
1.0 -
• .....
6 0 0 0 0 plants/ha
o - -
4 0 0 0 0 plants/ha
~
1
--,z/~/
0.8
O
W 0.6 cr 1.1. ill
>
t< ...J
0.4
--
i¢ ~
0.2 O 0.0
10
20
30
40
50
60
70
80
I 90
LEAF ANGLE Fig. 5. Cumulative frequency of leaf angles over the whole canopy for the (a) Leafy and (b) Check hybrids at three densities.
D.W. Stewart, L.M. Dwyer i Agricultural and Forest Meteorology 66 {1993j 247 265
256
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D.W. Stewart, L.M. Dwyer / Agricultural and Forest Meteorology 66 (1993) 247-265
258
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D.W. Stewart, L.M. Dwyer / Agricultural and Forest Meteorology 66 (1993) 247-265
259
did not introduce serious errors and that this method accurately calculated leaf area with height and between rows on a per plant basis. The 1991 year was relatively dry. Leaf areas were smaller than usual and the measurements showed that leaf area index (LAI) decreased markedly from the plant stems to the middle of the row (Fig. 3(a,b)). As would be predicted from average values of XL and leaf area (Table 1), the Leafy hybrid had higher LAI values at the stem but smaller values midway between rows for comparable planting densities. Calculated leaf area densities as a function of depth in the canopy and distance from the row are shown in Fig. 4. This clearly shows the highest densities near the stem and at middle canopy depths. Although the Leafy hybrid had higher leaf area densities near the row than the Check, the Check hybrid filled the inter-row area more evenly than the Leafy hybrid. The assumption in these leaf area calculations is that azimuth angles of plant planes are evenly distributed. There is, in fact, some evidence that maize plants adjust azimuth angles to fill inter-row spaces (Loomis and Williams, 1969; P. Girardin and M. Tollenaar, University of Guelph, personal communication, 1992). Leaf area in the field is therefore probably more uniformly distributed across the row than calculated and presented in Fig. 2. When azimuth angles become available for Canadian conditions we can insert them into the calculations. Cumulative leaf angle distributions calculated from Eq. (I 2) showed differences as well (Fig. 5(a,b)). No real pattern with depth in the canopy emerged from these measurements of leaf angles, probably because only three plants were measured per treatment. However, it was evident that the Leafy hybrid had more leaves between 35 and 65 °. Above 20 °, the Check had a very even distribution of leaf angles. Polynomial functions of leaf number were fitted by least squares to all the plant parameters (Table 2). It should be noted that the ear leaf number was set equal to zero with leaves above the ear being positive and those below negative. This means that the a 0 coefficients in Table 2 are the values of the parameters at the ear leaf. Parameters such as ligule height, leaf length and leaf width at the ligule had well defined patterns with leaf numbers with correlation coefficients greater than 0.8 (Table 2, Fig. 6). Leaf shape coefficients (bl and b2) were less well correlated with leaf number (r >10.6) but this could result from less absolute variation in their value with leaf number. That is, shape patterns vary less with leaf number than leaf width or ligule height. However, the parameters involved with leaf curvature were poorly correlated with leaf number and this was because of plant variation in curvature patterns of individual leaves. Data plus the function for the 'A' parameter are shown in Fig. 7. The plant to plant variation in A, B and C resulted in poor plant reconstruction when the polynomial functions were used. Therefore we fitted values of XM, YM, XL, and YL to polynomial functions instead. This
D.W. Stewart L.M. Dwyer ,, Agricultural and Forest Meteorology 66 (1993 i 247-265
260
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D.W. Stewart, L.M. Dwyer / Agricultural and Forest Meteorology 66 (1993) 247 265
261
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D . W . Stewart, L . M . D w v e r , Agricultural and Forest Meteorology 66
262
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D.W. Stewart, L.M. Dwyer / Agricultural and Forest Meteorology 66 (1993) 247-265
263
proved more successful, and comparable values for XM are shown in Fig. 8 for canopy reconstruction. We then calculated values of A, B and C from the leaf number polynomials of XM, YM, XL, YL, and 00 using Eqs. (7)-(9). This procedure proved surprisingly successful. Very good agreement was found between measured and reconstructed LAI variation between the rows (Table 3). There was a tendency for reconstructed plants to have lower LAIs midway between rows. This is understandable, as polynomial fitting averages all leaves and tends to reduce extreme values of leaves which project further from the stem. In other years with larger leaf areas and longer leaves, this discrepancy would not be as large. Reconstructed leaf area and leaf angle distribution also correlated well with measured values (Table 4). While not perfect, leaf number polynomial functions of these various parameters make it easier to mathematically characterize a plant canopy. In summary, we have demonstrated a method of characterizing plant phenotype which clearly showed differences between hybrids in leaf area distribution and leaf angle distribution. The Leafy hybrid had a larger leaf area but the leaf area was not as evenly distributed between the rows as for the Check. This would probably mean less light interception for the Leafy hybrid, although this was not measured in our experiment. Leaf angles were also different; most of the leaf angles for the Leafy hybrid were between 35 and 65 ° while the Check hybrid had leaf angles relatively evenly distributed. It is anticipated that the analysis described in this study will be useful for future modelling studies. The one advantage of this method for mathematiTable 3 Leaf area index between rows, with row increments numbered from the row calculated using (a) measured values and (b) regression coefficients Hybrid
Leafy
Check
Planting density (plants/ha)
Row increment
40000
(a) (b)
60000
(a) (b)
1
2
7
r
RMSE
3
4
5
6
8
5.93 4.35 5.87 4.35
3.04 2.97
2.21 2.07
1.59 1.47
1 . 1 6 0.84 1 . 0 9 0.74
0.65 0.36
0.999
6.69e - 02
7.87 5 . 7 1 7.80 5.65
3.92 3.87
2.79 2.76
1.80 2.00
1 . 1 4 0.76 1 . 3 5 0.67
0.56 0.19
0.998
1.83e - 01
80000
(a) 10.75 8 . 0 1 (b) 10.59 8.15
5.55 5.65
4.05 4.00
2.91 2.87
1.94 2.09
1.33 1.32
1.18 0.52
0.997
2 . 5 5 e - 01
40000
(a) (b)
4.22 3.46 4 . 0 1 3.43
2.55 2.56
1.96 1.91
1.57 1.46
1.31 1.22
1.16 1.12
1.12 1.08
0.998
6.17e - 02
60000
(a) (b)
6.25 5.16 6.00 5.02
3.79 3.71
2.84 2.75
2.27 2.08
1.89 1.71
1.65 1.58
1.54 1.54
0.999
6.62e - 02
80000
(a) (b)
8.04 6.33 7.96 6.38
4.51 4.51
3.26 3.22
2.48 2.36
2.01 1.95
1.88 1.84
1.89 1.83
0.999
4.75e
02
264
D . W . Stewart, L . M . D w v. e r
Agricultural and Forest Meteorology 66 / 1993 •J ~~4 , ~ _~6 ~
Table 4 Correlations and root mean square errors between measured and reconstructed leaf area densities and cumulative leaf angle distributions for the Leafy and Check hybrids t.eafy (plants/ha)
Check
(plants/ha)
40 000
60 000
80 000
40 000
60 000
80 000
Leaf density r RMSE (m -~ )
0.938 0362
0963 0.468
(i).757 Iql 8
0.946 0.336
0.875 0.745
0.972 0,601
Leaf angle r RMSE (degrees)
0.965 0.432
0.968 0,420
0.815 0.835
0.958 0.413
0.917 0.691
0.96 t) 0.26(!
cally characterizing the canopy 1s that leaf area for individual leaves can bc stored in a three-dimensional array (leaf number, height and distance between rows). This makes possible the calculation of photosynthesis for individual leaves in detailed soil-plant-atmosphere models (Stewart and Lemon, 1969), These calculations should prove useful in trying to assess the effect of contrasting architecture on canopy photosynthesis and yield, to take into consideration changes of leaf photosynthesis with depth (Dwyer et al., 1989), and to assess the contribution of individual leaves when seeking to define the ideal leaf number for a given climatic region. Acknowledgements We appreciate the efforts of A. Neamtz and B. Wilson for much of the data analysis and of L. Evenson, J.-A. Dugas, D. Balchin and B. D o w and staff for technical assistance. References Daynard, T.B., 1971. Characterization of corn (Zea m a y s L.) canopies from measurements of individual plants. Agron. J., 63:133 135. Dwyer, L.M., Stewart, D.W., Balchin, D., Houwing, L., Marur, C.J. and Hamilton, R.1, 1989. Photosynthetic rates of six maize cultivars during development. Agron. J., 81:597 602 Edmeades, G.O. and Daynard, T.B., 1979. The relationship between final yield and photosynthesis at flowering in individual maize plants. Can. J. Plant Sci., 59:585 601. Lang, A.R.G., 1973. Leaf orientation of a cotton crop. Agric. For. Meteorol., 11 : 37 5 I. Loomis, R.S. and Williams, W.A., 1969. Productivity and morphology in crop stands: patterns with leaves. In: J.O. Eastin, Z.A. Haskins, O.G. Sullivan, C.H.M. Van Basvel (Editors), Physiological Aspects of Crop Yields. American Society of Agronomy, Madison, WI. Loomis, R.S., Williams, W.A., Duncan, W.G., Dovrat, A. and Nuzez, A., 1968. Quantitative descriptions of foliage display and light absorption in field communities of corn plants. Crop Sci., 8:352 356.
D.W. Stewart, L.M. Dwyer / Agricultural and Forest Meteorology 66 (1993) 247 265
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Marquardt, D.W., 1963. An algorithm for least squares estimation of non-linear parameters. J. Soc. Ind. Appl. Math., 11:431 441. Monsi, M. and Saeki, I., 1953. Uber den Lichtfaktor in den Planzengesellscheften und seine Bedentung fur die Stroffproduction. Jpn. J. Bot., 14:22 52. Pr6vot, L., Aries, F. and Monestiez, P., 1991. Mod61isation de la structure geometrique du mais. Agronomie, 11:491 503. Ross, J., 1981. The Radiation Regime and Architecture of Plant Stands. Junk, The Hague, 391 pp. Sinoquet, H., Moulia, B. and Bonhomme, R., 1991. Estimating the three-dimensional geometry of a maize crop as an input of radiation models: comparison between three-dimensional digitizing and plant profiles. Agric. For. Meterol., 55: 233-249. Stewart, D.W. and Lemon, F.R., 1969. A simulation of net photosynthesis of field corn. U.S. Army ECOM Technical Report 2-681-6, U.S. Army Electronic Command, Fort Huachuca, Arizona. Warren Wilson, J., 1965. Stand structure and light penetration. I. Analysis by point quadrats. J. Appl. Ecol., 2:383 390.