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A simple model of light penetration into row crops
Agricultural and Forest Meteorology, 36 (1986) 297--315 Elsevier Science Publishers B.V., Amsterdam -- Printed in The Netherlands
A SIMPLE MODEL OF LIGHT PENETRATION INTO ROW CROPS D.M. WHITFIELD
Irrigation Research Institute, Department of Agriculture and Rural Affairs, Tatura 3616, Victoria (Australia) (Received November 1, 1984; revision accepted July 26, 1985) ABSTRACT Whitfield, D.M., 1986. A simple model of light penetration into row crops. Agric. For. Meteorol., 36: 297--315. A simple model of light penetration into row crops was derived by considering the projections of the foliage elements to be confined to a potentially-shaded region (R*) of a horizontal plane subtending the canopy. Individual plants were modelled as optionally truncated ellipsoids. The extent and area of R* were estimated from the extent of the ellipsoids and their distribution on the soil surface in relation to the position of the sun. Uniform spatial distributions of projected foliage elements between the sun and R*, and empirical, ellipsoidal distributions were appraised. The ellipsoidal distributions were based on the structure of the plants comprising the canopies. Apart from crops with extreme gradients in vertical distribution of leaf area density, appraisal against a relatively complex numerical method showed the simple model incorporating ellipsoidal distributions of foliage over R* reliably estimated profiles of light penetration into a range of theoretical crops with contrasting structures. The simple model accounts for differences in extent of individual plants in a readily envisaged manner, for distinct and overlapping plants within rows, and for differences in light penetration attributable to angle of sight. Canopy light interception may be analysed in terms of the light intercepted by the isolated plants constituting the canopy, the extent of eompetition for light over R*, and the extent o f light penetration direct to the soil surface through failure to intercept the canopy volume.
INTRODUCTION
The diversity of canopy geometries and the dependence of light penetration on both solar elevation and azimuth have hampered development of a general, efficient means of estimatin$ light penetration into row crops. Structural change during development is an additional problem when light penetration is to be modelled over the life of the crop. An efficient means of estimating light penetration into row crops is required. It should be based on readily measured or estimated parameters of canopy extent and should naturally encompass the progression from spaced plants early in growth to the continuous and discontinuous canopies of mature crops. The utility of estimates of light interception, the c o m p l e m e n t of penetration, is firmly established. Current analyses of crop growth are based on estimates or measures of light interception, and of the efficiency of utilisation of the intercepted light (Warren-Wilson, 1967; Charles-Edwards
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© 1986 Elsevier Science Publishers B.V.
298 and Fisher, 1980; Fisher et al., 1980; Monteith, 1981; Charles-Edwards, 1981, 1982). Warren-Wilson (1981a, b) generalised the growth analysis of isolated plants and crops by considering their respective abilities to intercept light. His analysis is particularly appropriate to row crops where light interception typically depends on the foliage display of individual plants during early growth and becomes increasingly reliant on density and distribution of plants on the soil surface with time. Several authors have adequately modelled crops with continuous rows (e.g., Jackson and Palmer, 1972; Allen, 1974; Charles-Edwards and Thorpe, 1976; Palmer, 1977; Goudriaan, 1977), b u t less restrictive models accounting for a wider range of geometries are necessarily more complex and more demanding of computing facilities (see Norman and Welles, 1983). Mann et al. (1980) alleviated the computational load of numeric models with an analytic approximation of light penetration to the base of row crops comprising plants confined to entire ellipsoidal envelopes. Individual plants had uniform light intercepting characteristics and were considered to be randomly distributed within rows. In the simple model developed here, crops are assumed to be comprised of arrays of equally-spaced identical plants in parallel rows on a horizontal soil surface. The origin of plants in each row varies randomly. Individual plants are confined to optionally-truncated, ellipsoidal foliage envelopes. These are flexible and appropriate to a diverse range of plant shapes. Based on these assumptions, light penetration is estimated from the structure and light intercepting characteristics of a single member of the crop. Thus, the model offers the potential to analyse a wide range of canopy structures, accounting for both distinct and overlapping plants within rows. It can also be employed to estimate profiles of light penetration. However, it does n o t apply where plants are distributed on strict rectangular or diagonal planting patterns because the added regularity of such crops necessitates a more detailed treatment than that presented here. The extent and area of the potentially shaded region, R * , on a horizontal plane, R, subtending the c a n o p y were calculated from the area of the shadow cast by an opaque representation of an individual plant and the extent of this shadow in relation to the distribution o f plants on the soil surface. The sun was taken to be a point source at a positive angle above the horizon. The method therefore parallels aspects of the approaches of Jackson and Palmer (1972, 1981), Goudriaan (1977) and Mann et al. (1980) in this respect. Vertical heterogeneity in the light intercepting attributes of the foliage was also considered. In one case, and in parallel with the approach of Goudriaan (1977), foliage elements were assumed to be uniformly distributed over R* despite spatial variation in projected foliage density. Alternatively, an ellipsoidal spatial distribution of projected foliage elements over R* was assumed. The two variants of the simple model, one incorporating the uniform distribution of foliage elements over R* and the other the ellipsoidal
299
distribution of foliage elements, were applied to a range of theoretical 'crops' with contrasting architectural features. Their performance was appraised against a relatively complex numerical m e t h o d (Mutsaers, 1980; Whitfield and Connor, 1980a, b; Norman and Welles, 1983) based on earlier work of Charle~Edwards and Thornley (1973), Allen (1974) and Mann et al. (1979). Whilst experimental verification of the numeric procedure can only be regarded as reasonable (Charles-Edwards and Thorpe, 1976; Whitfield and Connor, 1980a; Norman and Welles, 1983), it is the logical alternative to the use of the simple model presented here. The numerical method considers the probability of penetration of numerous parallel rays. However, accounting for the passage of numerous beams through all ellipsoids located between the sun and the region of interest on R* is a major computing task. Furthermore, and in contrast to crops with strict planting patterns, the crops considered in this study have a random c o m p o n e n t in their structure. To account for this, repeated computations of light penetration for a range of canopies are required of the numerical procedure. An application of the simple model for the analysis of relationships between canopy light interception and light interception by the constituent isolated plants is also proposed. This quantifies the extent of competition between plants in row crops, and the extent of light wasted through failure to intercept the canopy volume. METHODS Mann et al. (1977) showed that the beam-lit fraction (or gap frequency), e, of the region, R, of a horizontal plane subtending an arbitrary distribution of foliage can be generally estimated using
e =
~(x,y)dxdy/AR R
= ~exp[--F(x,y)]dxdy/Aa
(1)
R
~(x, y) is the fractional light penetration to a point on R specified by coordinates, x and y, and F(x, y) is the projected leaf area index b e t w e e n R(x, y) and the sun. A a is the area of the region, R, under consideration. Dramatic variations in F(x, y) in discontinuous canopies cause the need for numerical integration of eq. 1, although analytic methods m a y still be appropriate in some circumstances (e.g., Mann et al., 1980).
Canopy structure The ellipsoidal envelope describing the extent of an individual plant was taken as (cf. Charles-Edwards and Thornley, 1973; Whitfield, 1980; Whitfield and Connor, 1980b) x 2 / a 2 + y 2 / b 2 -b (z - - h ) 2 / c 2 --- 1,
h -- c < ZL < Z < Zu < h + c
(2)
300 optionally truncated at heights, z L and z U . The origin of a Cartesian coordinate system, O, was taken as the intersection of the plant axis with the soil
surface, xOy. Oy was taken along the row and Ox across the row. ZL, Zu, a, b, and c are parameters of plant extent, a, b and c are the lengths of the semi-axes of the entire ellipsoid on Ox, Oy and Oz, respectively. The horizontal extent of the entire foliage envelope is a m a x i m u m at z = h, the height of the centre of the ellipsoid. The volume of foliage, V, is calculated from these parameters. The extent of the crop is then given by (cf. Allen, 1974; Charles-Edwards and Thorpe, 1976; Thorpe, 1978; Whitfield and Connor, 1980a)
[X --nxdx]2/a 2 4- [y --(ny q- (5)dy]2/b 2 q- [z --h]2/c 2 = 1
(3)
dy is the distance between plants within rows and d x is the distance between rows. These are also fixed parameters. In both the simple and numerical models, R was taken as the rectangular area, dy • d x . This is the ground area per plant, A R . 6, a random value between 0 and 1, was varied between rows to force the crops under study to deviate from those with strict planting patterns, ny and n~ are integers, each taking a value of zero for the plant sited on R. (ny + 6) "dy and n~ "dx define the displacement of the nyth plant in the n~th row from that sited on R (see Fig. 1).
I I I
.........
G
y
d dy
- - - -
--d×--
nx=-i
n×=O
nx=l
nx=2
6=0.2
8=0
8=0.6
,8=0. £
Fig. 1. Plan v i e w o f the distribution o f plants o n the soil surface o f m o d e l canopies. Four r o w s are d e p i c t e d (n x = - - 1 . . . 2), w i t h corresponding changes in the value o f 5, the r a n d o m o f f s e t varied b e t w e e n rows, ~ is t h e r o w - s u n angle. Eqs. 2 and 3 give details.
301 ny and nx may be negative. However, ¢, the angle between the rows and the projection of a ray on the soil surface, was restricted to the range, 0~b~90 ° , and a and b were confined to the ranges, 0 ~ a ~ d x and 0 ~ b ~ dy, without loss of generality. The minimum value of both ny and nx was therefore - - 1 . In the extensive crops considered here, nx and ny tend to infinity. This creates the need to consider the passage of each beam through a large number of plants at low solar elevations when using the numeric method.
Light in tercep ring properties of individual plants F(x, y) (see eq. 1) varies with the distance traversed within the foliage by a ray in its path to R, and the foliage density and projection of unit area of foliage along the ray. The relative dispersion of foliage elements along the beam m a y also be taken into account (see Acock et al., 1970; Lang and Shell, 1976; Whitfield, 1980). Foliage density, projection of unit leaf area and relative dispersion have a multiplicative effect (Acock et al., 1970), and are subsumed into a variable, p, the projected leaf area density. In both the simple and numeric models, each ellipsoid was divided horizontally into five layers of equal depth to simulate vertical variation in p; p was assumed uniform within layers. Five layers were considered sufficient for the present purpose, p was varied between 1 m-1 and 16 m-1 with approximate variations in projection of unit leaf area and leaf area density ranging between 0.1--0.9 and 0.5--15 m-1 respectively.
The numeric method Equation 1 is the basis of the numeric method, where multiple estimates of ~(x, y) are averaged over the area of interest, A R , to derive estimates of ~. e was taken as the mean of 100 estimates of ~(x, y) at regularly spaced points on R. ' Given that p varies only with height, z, within an individual plant, F(x, y) was estimated as 5
F(x, y) = ~ piAz(x, y ) j c o s 0
(4)
i
The index, i references the ith layer of foliage subtended by R, and Az(x, y) is the vertical c o m p o n e n t of beam penetration in a layer. 0 is the zenith angle of the beam. The summation is only effective within an ellipsoid, b u t includes contributions from all plants between R and sun as each ray passes in and o u t of successive ellipsoids in its path to R. The minimum fractional light penetration to R (~mm) was estimated as the proportion of points where ~(x, y) was 1 (i.e., F(x, y) = 0).
The simple model The assumption that crops are comprised of identical plants spaced at equal distances along parallel rows implies that the area of shadow cast b e y o n d R
302 by the plant centred on R is equal to that cast by neighbouring plants back on R. Thus, the foliage of an individual plant was considered to project only over R. The potentially shaded region of R, R*, was further considered to be confined to a subregion of R accounting for the fraction, f*, of AR (0 ~ f * ~ 1). B e y o n d this region, F(x, y) = 0 and ~ ( x , y ) = 1. Taking ~* as the beam-lit fraction of R*, the beam-lit fraction of R (~) is given by = (1 -- f*) + f*~*
(5)
and, applying eq. 1, exp [-- F(x, y)] dxdy/f*AR
(6)
R* Equations 5 and 6 require estimates of f* and of the spatial distribution of projected foliage elements over R*. The lower b o u n d (~m~) is clearly ~mm = l - - f *
(7)
Estimation o f the area o f the potentially-shaded region The total area of shadow (Amax) cast on a horizontal plane by the foliage envelope representing an individual plant, and its extents on Oy (ey) and Ox (ex), were calculated analytically. Details are available on request. Four cases, relating ex to dx and ey to dy, were considered: (a) Shadow envelope confined to the region, R (e x ~ dx, ey ~ dy ): here, f* = Amax/A R
(8a)
(b) Shadow envelope extends beyond R along the row but not into adjacent rows (e~ ~ dx, ey ~ dy ): in this instance, shadows of area equal to that cast b e y o n d R were assumed to be cast back on R by adjacent plants. These are confined within ex on Ox. f* is given by f* = minimum (exdy, Amax)/AR
(8b)
on the assumption that the area of shadow extending b e y o n d R is additively translated into the region, exdy. In cases (c) and (d), the shadow envelope extends into adjacent rows. The elliptical shadow envelope was replaced by a rectangular one of the same area The rectangular envelope was assumed to extend across e~/dx row widths with a mean extent, ey ', of Amax/ex on Oy. (c) Shadow envelope extends into adjacent rows but mean extent along row less than the distance between plants (ex ~ d~, ey ' ~ dy ): the a r e a o f s h a d o w b e y o n d R was considered to be translated randomly over AR as a consequence of the random variation in 0 b e t w e e n rows. f* is then given by
f* = [ 1 -- (1 -- ey '/dy ) (exldx) ]
(8c)
303
(d) Shadow envelope encloses R (ex ~ dx, ey' ~ dy): here f* = 1
(8d)
Spatial distributions of pro]ected foliage elements over R* On the basis of uniform p within layers, the average projected leaf area index subtended by R* is given b y 5
F* = ~ viPi/f*A a
(9)
i
vi is the volume of the ith layer o f each plant. Two spatial distributions of foliage elements over R* were appraised: (a) Uniform distribution, where F(x, y) is a constant, F*/cos 0. Division by cos 0 accounts for increased pathlengths when the sun is not directly overhead. Here ~* = exp (-- F * / c o s 0)
(10)
Substituting for e* in eq. 5, the first estimate of ~ (~v) is ~u = ( l - - f * )
+ f * exp (-- F* /cos O)
(11)
This reduces to the c o m m o n expression, ~ = exp (-- F / c o s 0), only where the shadow envelope completely encloses R (i.e., case (d), f * = 1). (b) Empirical ellipsoidal (or hemiellipsoidal) distribution based on the ellipsoidal geometry of the individual plants. This takes some account of the non-uniform spatial distribution of foliage elements over R*. The second estimate of ~* is given by (see Appendix) ~* = 211 -- (Q + 1) exp (-- Q)]/Q2 where
(12)
Q = 3F*/2 cos0 ~E is given by ~E = [ l - - f * ]
+ 2f*[1--(Q
+ 1) e x p ( - Q ) ] / Q 2
When f* = 1, ~E is greater than ~v in the range, 0.75 < F * / c o s 0 ~ 7.5, which corresponds approximately with 2% ~ dv ~ 50%. The maximum difference is approximately 4.5%.
Appraisal of the simple model Both the simple and numeric models were used to estimate the profile, ~(z), in three 'crops' of contrasting structure, and to estimate effects of angle of sight on light penetration to the base of two other canopies. Structural characteristics of the five crops are presented in Table I. z U was arbitrarily taken as 1.0 m in each, and c and h were taken as 0.5 m. Entire plants of Crop A were tall in relation to their horizontal extent, and were distinct
304 TABLE I Structural Characteristics of the five basic 'crops' used to appraise the simple model. In each instance, c = h = 0 . 5 m , and z U = 1.0m. z L was varied between 0 and 0.8m in Crops A, B and C to generate vertical profiles of ~. Parameters. are defined in eqs. 2 and 3. Crop
a (m)
b (m)
dy (m)
d x (m)
A B C D E
0.1 0.5 0.5 0.1 0.25
0.1 0.5 0.5 0.25 0.1
0.5 1.0 0.5 0.5 0.5
0.5 1.0 1.5 0.5 0.5
TABLE II Combinations of projected leaf area density, p (m-l), in plants constituting Crops A, B and C. Subscripts reference layers of 0.2m depth above the soil surface with higher values referring to upper layers. Combination
P1
P2
P3
P4
Ps
1 2 3 4 5 6 7 8 9 10
1 16 1 16 2 8 16 4 1 4
2 8 4 4 4 4 8 8 2 2
4 4 16 1 8 2 4 16 4 1
8 2 4 4 4 4 8 8 2 2
16 1 1 16 2 8 16 4 1 4
f r o m their n e i g h b o u r s . T h e plants c o m p r i s i n g Crops B and C were spherical, e x t e n d i n g t o the limits o f AR in Crop B b u t m a k i n g distinct r o w s in Crop C. Crop A represents a sparse c a n o p y , Crop B a p p r o x i m a t e s a c o n t i n u o u s canopy and Crop C approximates a hedgerow. F o r each o f these, influences o f five u n i f o r m values o f p w i t h i n the p l a n t (p = 1, 2, 4, 8 and 16 m - l ) , and ten c o m b i n a t i o n s o f these values (Table II), were investigated. ~ and ~m~ were e s t i m a t e d at 21 angles o f sight o n five transects at heights o f 0, 0.2, 0.4, 0.6 and 0.8 m above soil level. These profile d a t a also r e p r e s e n t estimates o f light p e n e t r a t i o n into a diverse range o f c a n o p y s t r u c t u r e s associated with v a r y i n g severities o f t r u n c a t i n g t h e ellipsoidal evelope. Angles o f sight were 0 -- 0 ° and f o u r r o w - s u n angles at each o f 0 = 18 °, 33 °, 45 °, 57 ° and 72 °. R o w - s u n angles (~b) were 0 ° and 90 ° (along a n d across rows, respectively), a n d t w o at r a n d o m in the range, 0 ° < ¢ ~ 90 °. T h e ability o f t h e simple m o d e l to estimate ~m~n was first appraised. F o r
305
each combination of values of p, the mean differences, ~ - ~, were then calculated for each crop over the range of angles of sight for each transect. Crops D and E were included to assess the ability of the model to cope with plants with noncircular cross-sections, and to investigate attendant effects of angle of sight. Crop D represents another hedgerow, whereas plants with identical physical dimensions were aligned across rows to form a relatively dispersed canopy in Crop E. The random offset, b, precluded a hedgerow distribution.
RE S UL TS
Estimates o f the poten tially-shaded area Comparisons between simple and numeric estimates o f ~mm indicate deficiencies in estimates of f* (eqs. 7, 8). Mean differences, ~ i . - - ¢ ~ m , over all transects and angles of sight, were (with standard errors in parentheses) 0.9% (0.29%), 0.5% (0.41%) and --2.8% (0.19%) for Crops A, B and C respectively. Respective mean differences over angles of sight on individual transects ranged from 0 to 1.4%, from 0.3% to 0.7%, and from --5.0% to 1.8% (Table III). Data for all transects are shown in Fig. 2. -
-
10@ r
T
or)
co
40
20
r ,
°
!
I
A
28
48
NUMERICRL
88
8@
ESTIMATE
[%)
i~8
Fig. 2. Simple estimates of minimum light penetration into Crops A (~), B (o) and C (A) ascompared with numeric~ estimates. Thesolidline depictsthe 1:1 relationship.
306 T A B L E III Changes in t h e m e a n d i f f e r e n c e , ~ m i n - drain, over t h e range o f angles o f sight and h e i g h t s o f t r a n s e c t in c r o p s A, B and C. S t a n d a r d errors o f t h e d i f f e r e n c e s , and t h e range o f ~min, are given in p a r e n t h e s e s . Height o f transect (m)
Crop A (%)
Crop B (%)
Crop C (%)
0 0.2 0.4 0.6 0.8
1.1 1.4 1.0 0.8 0
0.4 0.4 0.7 0.6 0.3
- - 1.8 (0.37, 0 - - 3 6 ) --1.8(0.37, 0--36) - - 2.4 (0.31, 0 - - 3 6 ) - - 2.9 (0.37, 0--38) - - 5.0 (0.32, 3 6 - - 5 2 )
(0.65, 0 - - 8 8 ) (0.75, 0 - - 8 8 ) (0.80, 2 8 - - 8 8 ) (0.60, 2 9 - - 8 8 ) (0.42, 6 8 - - 9 2 )
(1.07, 0 - - 2 0 ) (1.07, 0 - - 2 0 ) (1.13, 0 - - 2 0 ) (0.88, 1--25) (0.29, 3 4 - - 5 0 )
Light penetration in crops with uniform light intercepting characteristics In Crop A, differences in p caused mean differences between simple and numeric estimates o f ~ to range bet w e e n - - 1 . 4 % and - - 0 . 1 % with the u n i f o r m distribution, and f r om - - 0 . 1 % to 0.1% with the ellipsoidal distrib u t i o n (Table IV). F o r Crop B, the respective ranges were -- 2.7% to -- 0.9% and -- 0.5% to 0.4%. Associated with the differences in estimates of ~mm, the T A B L E IV Mean d i f f e r e n c e s , ~ - - ~, over t h e range o f angles o f sight and h e i g h t s o f t r a n s e c t for c r o p s A, B and C u n d e r t h e u n i f o r m and ellipsoidal d i s t r i b u t i o n s w i t h u n i f o r m p. S t a n d a r d errors, and t h e m a x i m u m d i f f e r e n c e s b e t w e e n ~ and ~ achieved b y varying z L b e t w e e n 0 and 0.8 m, are given in p a r e n t h e s e s . p ( m - 1)
Uniform distribution m e a n d i f f e r e n c e (%)
Ellipsoidal d i s t r i b u t i o n m e a n d i f f e r e n c e s (%)
Range o f t (%)
Crop A
1 2 4 8 16
------
77--99 59--99 36--98 13--97 4--96
Crop B
1 2 4 8 16
--0.9 - - 1.9 - - 2.7 - - 2.6 - - 1.7
Crop C
0.1 0.3 0.9 1.4 1.4
(0.04, - (0.06, - (0.09, - (0.16,-(0.23, - -
0.2) 0.5) 1.3) 2.2) 2.2)
0 0 --0.1 0 0.1
(0.13, (0.20, (0.34, (0.45, (0.47,
1.5) 2.4) 4.0) 5.2) 5.0)
0.2 0.4 0.2 --0.3 --0.5
0.12, 0.21, 0.33, 0.41, - 0.43, - -
0.7) 1.3) 1.4) 2.2) 2.9)
19--95 4--90 0--83 0--73 0--62
1 2
- - 2.5 (0.17, - - 3.5) --4.2(0.23,--5.3)
--0.9 --1.8
0 . 1 4 , - - 1.3) 0.20,--2.4)
12--93 2--88
4
-- 4.8 (0.28, -- 6.8)
--2.6
0.22, -- 3.9)
0--79
8 16
- - 4.4 (0.29, - - 6.8) --3.8(0.29,--7.5)
--3.0 --3.0
0.23, - - 4.5) 0.23,--5.4)
0--68 0-59
------
(0.04, 0.2) (0.05, - - 0.1) (0.08, - - 0.5) (0.17, 0.8) (0.24, 1.2)
307
model generally underestimated ~ in Crop C. Differences in p caused deviations ranging from - - 4 . 8 % to - - 2 . 5 % with the uniform distribution, and from -- 3.0% to -- 0.9% with the ellipsoidal distribution. In Crop A, the largest mean deviation associated with variations in both p and height of transect was only 2.2% (Table IV). In crop B, the largest deviation within transects was --5.2% with the uniform distribution, and -- 2.9% with the ellipsoidal distribution. In Crop C, respective values were -- 7.5% and -- 5.4%. The ellipsoidal distribution therefore provided the more acceptable estimates in Crops B and C.
Light penetration in crops with varying light intercepting characteristics ~v and ~E generally underestimated ~ when p was varied within plants (Table V). Comparing mean differences over transects and angles of sight in Crop A, simple model estimates were up to 3.7% and 2.6% less than numeric estimates using the uniform and ellipsoidal distributions, respectively, for the range of combinations of p shown in Table II. Respective values were 8.3% and 5.3% for Crop B, and 7.8% and 5.7% for Crop C. The largest discrepancies were generally attributable to combination 4, an extreme in the vertical distribution of p. Considering means over angles of sight at different heights of transect in Crop A, the simple model underestimated ~ by up to 6.0% and 4.2% using the uniform and ellipsoidal distributions, respectively (Table V). These occurred with combination 3. In Crop B, worst-case deviations achieved by varying the height of transect were -- 12.9% and -- 9.3%, caused by combination 4. Otherwise, the largest deviations within transects were -- 7.0% and -- 3.9% for the t w o distributions. In Crop C, combination 4 again produced the largest worst-case deviations (mean deviations of - - 1 1 . 9 % a n d - 9.3% with the uniform and ellipsoidal distributions, respectively), but the uniform distribution also produced large within-transect deviations with combinations 1, 6, 7 and 10. Thus, differences were marked where the variation in p within the plant was most pronounced, and were most serious where high values of p were confined to a small volume of the plant. Use of an average value for F* (eq. 9) is clearly inappropriate under these circumstances. In general, the ellipsoidal distribution produced more reliable estimates than the uniform distribution.
Estimates of light penetration into Crops D and E Figure 3 demonstrates effects of varying angle of sight on light penetration to ground level in Crops D and E. Looking d o w n the rows (~ = 0°), ~ decreased with increasing zenith angle in both stands, b u t the effect was most marked in Crop E. At @ = 90 °, the influence of zenith angle on Crop D was very similar to that for Crop E at ~ = 0 ° because of essentially
3O8 TABLE V Mean differences, ~ -- ~, over the range of angles of sight and heights of transect for crops A, B and C under the uniform and ellipsoidal distributions with non-uniform combinations of p. Standard errors, and the maximum differences between ~ and ~ achieved by varying z L between 0 and 0.8 m, are given in parentheses Comb n"
Crop A
Crop B
CropC
Uniform distribution mean difference (%)
Ellipsoidal distribution mean difference (%)
Range o f t (%)
1 2
-- 3.0 0.18, - - 1 . 5 0.18,
-- 4.1) --4.0)
-- 1.7 ( 0 . 1 4 , - - 2.4) --0.8(0.13,--2.3)
30--96 29--99
3
--3.7
0.40,
--6.0)
--2.5(0.32,--4.2)
25--99
4 5
-- 3.7 0.21, - - 1 . 6 0.18,
-- 4.9) --2.6)
-- 2.6 (0.17, -- 3.8) --0.7 (0.14,--1.4)
28--96 31--99
6 7 8 9
--1.5 --2.3 --2.3 --0.7
0.09, 0.16, 0.26, 0.09,
--1.9) --3.4) --3.8) --1.2)
--0.7 (0.09,--1.1) --0.8(0.16,--1.8) --0.9(0.21,--2.0) --0.3(0.07,--0.5)
34--97 13--96 11--98 55--99
10
- - 0 . 7 0.06,
--0.9)
--0.3(0.05,--0.4)
58--98
1 2 3
-- 5.2 0.44, -- 2.0 0.33, - - I . I (0.38,
-- 7.0) -- 3.9) --2.2)
-- 2.5 ( 0 . 4 0 , - - 3.9) 0.2 (0.30, 1.7) 0.7 (0.36, 1.4)
0--62 0--95 0--95
4
-- 8.3 (0.53, -- 12.9)
-- 5.3 (0.48, -- 9.3)
5
--1.6(0.35,
--2.8)
6 7 8
--5.0(0.39, -- 4.0 (0.48, -- 1.5 (0.43,
--6.9) -- 6.0) -- 4.1)
0.9 (0.34,
0--63
1.7)
0--90
--2.0(0.37,--3.7) -- 1.6 (0.44, -- 2.9) 0.2 (0.40, -- 1.3)
0--73 0--63 0--83
9
-- 1.3 (0.22,
-- 1.8)
1.0 (0.23,
1.9)
2--95
I0
--3.8(0.37,
--4.6)
--1.4(0.25,
2.4)
4--83
1 2 3 4 5 6 7 8 9 I0
--6.1(0.31, --7.5) -- 3.9 (0.30, -- 6.0) --3.8(0.32, --6.1) -- 7.8 (0.44, -- 11.9) --4.1(0.29, --6.2) -- 6.3 (0.33, -- 8.4) --5.3(0.33, --7.5) --3.6(0.27, --6.9) --4.1(0.30, --6.0) -- 5.4 (0.26, -- 6.8)
--4.5(0.26,--5.4) -- 2.2 (0.22, -- 3.2) --2.7 (0.26,--3.6) -- 5.7 (0.39, -- 9.3) --2.4(0.21,--3.4) -- 3.9 (0.28, -- 5.5) --3.9(0.28,--5.4) --2.5(0.20,--4.2) --1.9(0.21,--2.8) -- 2.8 (0.24, -- 3.9)
0--59 0--93 0--93 0--59 0--88 0--68 0--59 0--79 0--93 2--79
e q u i v a l e n t g e o m e t r i e s . H o w e v e r , a t ¢ = 9 0 ° i n C r o p E, ~ first d e c r e a s e d s l o w l y w i t h i n c r e a s e i n 0 a n d t h e n fell m o r e r a p i d l y . U n d e r t h e s e c o n d i t i o n s , t h e s t o c h a s t i c e l e m e n t , 5, p l a y e d a s i g n i f i c a n t r o l e i n n u m e r i c e s t i m a t e s o f ~. T h u s , t h e s i m p l e m o d e l p r o d u c e d g o o d e s t i m a t e s o f ~ f o r t h e r a n g e o f a n g l e s o f s i g h t i n t h e s e c r o p s e x c e p t f o r t h e i n s t a n c e , 0 -- 57 °, ¢ -- 9 0 °, i n C r o p E.
309
7~
j
78
68
4I
68
°z 5e
4
58
~-- 48 ~
q
48
-•"<
y=
,
\\
i
4 ~ ,
'
4
o m',,
~
\,
4 ",
4
,
18 ~- C r o p
I
D
4
I
® 28
48 ZENITH
58
Crop
18 ~
E
\
S
! 8~
8
2~
ANGLE OF SIGftT
4~
68
88
(degr'e~as]
Fig. 3. Simple model estimates of light p e n e t r a t i o n below Crops D and E with change in zenith angle of sight at r o w - - s u n angles of 0 ( ) and 90 ° ( . . . . ). The leaf area density of individual plants was 4 m - 1. Symbols depict single estimates from the numeric model at r o w - - s u n angles of 0 ° (•) and 90 ° (o). Simple model estimates were based o n an ellipsoidal distribution of foliage elements. •
O
Influence of non-rectangularplanting patterns Light penetration to the soil surface of Crop A was strongly dependent on angle of sight and the light intercepting properties of the foliage. At low values of p, row-sun angle had little influence on ~ over a range of zenith angles, but ~b played a greater role at higher values of p (Fig. 4). The influence of b on estimates from the numeric model is demonstrated in Fig. 5, where effects of changes in ¢ at a zenith angle of 57 ° are shown. At p = 1 m - i , there was little change in ~ with increase in ~b, and the m a x i m u m standard error of five numeric estimates of ~ at each value of ~b was only 0.4%. However, Fig. 5 shows t h a t ~ first declined to a m i n i m u m of approximately 20% at ~b = 20 °, then rose to a m a x i m u m of 35% a t e = 90 ° f o r p = 16 m-1. Beyond the minimum, standard errors of the numeric estimates increased from 0.6% to a m a x i m u m of 5.6% at ¢ = 70 °. These data demonstrate the need for multiple estimates of ~ when using the numeric model with even a relatively minor random c o m p o n e n t in the structure of the canopy. The data a l s o demonstrate t h a t the simple model provided satisfactory estimates of the means.
310 188
--~
.
.
.
.
~
i08
80 - ~ --
--,
, ~
. . . . . .
-
88
78
~
60
CD
, "\
58
7[3
~
68
~
5B
2 28
28
',
"
18 i
0
i
28
i
i
48
i
i
68
]~
~
~
88
ZENITH RNGLE OF SIGHT [degrees)
I I
L
0
I
i
i
38
i
L
I I
I
i
J
68
i
I i
98
ROW-SUN RNGLE [degreesl
F i g . 4 . N u m e r i c e s t i m a t e s o f light p e n e t r a t i o n b e l o w Crop A w i t h change in z e n i t h angle
o f sight at r o w - - s u n angles o f 0 ° densities o f 1 m - 1 (©, e ) a n d 1 6 m row--sun angles o f 0 ° ( ) and based on an ellipsoidal distribution
( o p e n s y m b o l s ) and 9 0 ° (solid s y m b o l s ) at leaf area - 1 ( ~ , m). The curves depict simple m o d e l e s t i m a t e s at 9 0 ° ( . . . . ), r e s p e c t i v e l y . S i m p l e m o d e l e s t i m a t e s were
o f foliage elements.
F i g . 5. N u m e r i c e s t i m a t e s o f light p e n e t r a t i o n b e l o w Crop A w i t h change in r o w - - s u n angle at a z e n i t h angle o f 57 °. Leaf area densities in c o n s t i t u e n t plants were 1 m - 1 ( 0 ) and 1 6 m - 1 ( e ) . Standard errors o f five e s t i m a t e s at each r o w - - s u n angle are s h o w n for the latter. The curves d e p i c t simple m o d e l e s t i m a t e s based on an ellipsoidal distribution o f foliage elements.
DISCUSSION
The basic canopy structures considered in this study varied from the more-or-less isolated plants of Crop A to the continuous (Crop B) and hedgerow (Crop C) systems characteristic of many well-developed r o w crops (Table I). The profile data generated by varying ZL in these crops represent the equivalent of 15 crops comprised of plants with varying severities of truncation, and indirectly encompass an additional 12 crops when it is considered that truncation achieved by varying zL is equivalent to that achieved by inverting entire ellipsoids and varying Zv. The range of angles of sight and the effects of truncation on canopy structure were sufficient to vary ~mm over the range, 0--92% (Table III, Fig. 2). The diversity of crop-sun geometries considered here therefore provides adequate scope for appraising the flexibility and general application of the simple model. The simple model was appraised at two levels. The first concerned its ability to estimate ~ n and the second focused on its ability to estimate
311 1.o
:. . . .
. ....
!
Z E (AA
z
0.8
Z
u
0.6
F--
~
O-
0.4
~
0.2
CO Z (.3
8.0
,_ i
L 2
L 4
,
,
,
,
,
5
6
7
8
9
10
LERF RRER INDEX OVER SHRDED REGION
Fig. 6. Change in ~ per u n i t change in ~min as a f u n c t i o n of leaf area index from u n i f o r m ( ) and ellipsoidal ( . . . . ) distributions of foliage elements over the potentially shaded region, R*.
light penetration given varying light intercepting properties of the foliage. Figure 6 demonstrates the general sensitivity of ~ to estimates of ~m~ for all model crops. It clearly shows the need for accurate estimates of ~min (and hence f*, see eq. 7) in the region where the average projected leaf area index over R* is greater than approximately 1.5. For example, ~ is increased by approximately 0.6% for a 1% overestimate of ~min with a leaf area index of 2 over R*. Figure 6 also demonstrates that estimates of e are essentially determined by the estimate of emm when the leaf area index over R* is greater than approximately 4.5. Appraisal of several alternatives to eqs. 8 showed that estimates of emm were generally more sensitive to the estimate of shadow extent across rows than that within rows (data not presented). The simple model provided reliable estimates of emm (Fig. 2). When individual plants were assumed to exhibit uniform light intercepting characteristics, the simple model provided satisfactory estimates of e even when a uniform distribution of leaf elements over R* was assumed (Table IV). Mann et al. (1979) have demonstrated that light penetration is underestimated if uniform distributions of foliage elements are used to approximate non-uniform distributions in single plants confined to eUipsoidal envelopes. This was evident here, where the assumption of a uniform distribution of foliage elements over R* was c o m m o n l y associated with low estimates of e. The ellipsoidal spatial distribution partly removed this bias, and the simple model was found inadequate only with extreme
312 variations in the vertical distribution of foliage (Table V). The use of an average value for F* where relatively high values of p are confined to a small volume of the individual plants (combinations 4, 6, 7 and 10, Tables II and V) is clearly inappropriate. Otherwise, the simple model incorporating the ellipsoidal distribution of foliage elements over R* provided reliable estimates of ~ over a range of canopies. The model provides a simple framework for analysing the efficiency of light interception b y row crops in terms of three parameters: light interception by isolated or spaced members of the crop, the fraction of light wasted through failing to intercept the canopy volume and the extent of interplant competition for the light intercepted by the canopy. The analysis derives from eq. 11, which may be rewritten as ~c = ( l - - f * )
+ f* exp
(--F*/cosO + F'/cosO) exp ( - - F ' / c o s 0)
Here, ~c is the fraction of light transmitted by the canopy and F' is the projected leaf area index of the isolated plant over Amax, the area of shadow cast b y a solid representation of the plant. Isolated plants transmit the fraction, ~s (= exp [-- F'/cos 0 ] ), of the incoming light to the soil surface within the extent of the shadow and intercept the fraction, 1 - ~s. The proportion of light wasted by failing to impinge on the foliage volume is l - f*. Thus, the extent of interplant competition or interference over the potentially shaded region can be taken as C = 1 - - exp
(--F*/cosO + F'/cosO).
This has a minimum of 0 with no interference or competition between plants (i.e., F* =F', f*AR =Amax), and a maximum of 1. Now, ~s, ~c and f* are directly available from the model. An indirect measure of interplant competition or interference over R* is therefore C -- [ ( l - - f * )
q-f*~s--~c]/f*~s
and the simple equation, ~c = ( 1 - - f * ) ÷ f * ( 1 - - C ) ~ s describes canopy interception (1 -- ~c ) as a function of wastage (1 -- f* ), the ability of the isolated, single plant to intercept light ( 1 - ~s), and interference, C. Combined with its flexibility, the results show that the simple model applies to many crops which would otherwise require extensive mathematical treatment. Furthermore, the m e t h o d of estimating the area of the potentially shaded region subtending the canopy and superimposing an appropriate distribution of foliage applies generally. The approach used in this paper should therefore apply with other distributions of plants on the soil surface and alternatives to the ellipsoidal approximation of single plant extent. Appraisal of similar models over a more diverse range of canopy structures warrants attention, as the present analysis indicates that direct
313 application of Beer's Law is only appropriate where the area of the shadow envelope cast by an individual plant is commensurate with the ground area per plant, and where the distribution of foliage elements over the shadow envelope can be considered as uniform. LIST OF MAJOR SYMBOLS a, b, c, h dx dy ex
parameters describing extent o f ellipsoidal foliage envelope distance between rows distance between plants within rows extent of the shadow cast by a solid representation of an individual plant across the rows ey extent of the shadow cast by a solid representation of an individual plant along the rows f* potentially shaded fraction of the plane subtending the canopy Ox, Oy, Oz axes of Cartesian coordinate system ZL, ZU heights of truncating the ellipsoidal foliage envelope Amax area of shadow cast by a solid representation of an individual plant AR ground area per plant F projected leaf area index F* projected leaf area index over the potentially shaded region R horizontal plane subtending part or all of canopy R* potentially shaded region of a plane subtending the canopy b random offset preventing a rectangular distribution of plants on the soil surface 0 zenith angle of a beam ¢ row-sun angle p projected leaf area density fraction of light penetrating the canopy e* beam-lit fraction of the potentially shaded region ~mm minimum light penetration to the plane subtending the canopy ~E, ~V model estimates of light penetration using ellipsoidal and uniform distributions of projected foliage elements, resp. APPENDIX
Derivation o f eq. 12 The foliage elements subtended by R* were assumed to be confined to an ellipsoidal envelope (semiaxes ~, ~ and 7) at a constant leaf area density, p'. With beams from the zenith, the normalised density function o f projected leaf elements on the plane subtending the ellipsoid is given by g(z) = 2p'z/(47r~fJTp'/3 ) = 3z/27A*
(A.1)
314
within the elliptical shadow envelope. Here, A* = ~r~. Beyond the ellipse, g(z)
=
0
Equation A.1 describes an ellipsoidal spatial distribution of foliage elements over the ellipse with g(x, y) = 3(1 -- x2/c~ 2 -- y 2 / ~ 2 ) l / 2 / 2 " y A * Now A, the area of a subellipse concentric with the elliptical shadow envelope (0 ~
= A*(1--z2/'y
2)
Rearranging and substituting for z in eq. A1, g(A) is given by g(A) = 3 ( 1 - - A / A * ) l n / 2 A * Applying eq. 1, 5" is given by A*
5*
J exp [ - - g ( A ) ( F ' A * ) ] d A / A * 0
where F is the mean projected leaf area index over the elliptical shadow envelope, and is given by F = 47p'/3A*
Thus A*
5* =
j exp [-- (3F/2) ( 1 - - A / A * ) I / 2 ] d A / A * 0
Integration and substitution of F*/cos 0 for F yields eq. 12. The same expression is generated when foliage elements are assumed to be confined to an hemiellipsoidal foliage envelope. REFERENCES Acock, B., Thornley, J.H.M. and Warren Wilson, J., 1970. Spatial variation of light in the canopy. In: Prediction and Measurement of Photosynthetic Productivity. Pudoc, Wageningen, pp. 91--102. Allen Jr., L., 1974. Model of light penetration into a wide-row crop. Agron. J., 66: 41-47. Charles-Edwards, D.A. 1981. The Mathematics of Photosynthesis and Productivity. Academic Press, London, 127 pp. Charles-Edwards, D.A., 1982. Physiological Determinants of Crop Growth. Academic Press, London, 161 pp. Charles-Edwards, D.A. and Thornley, J.H.M., 1973. Light interception by an isolated plant -- a simple model. Ann. Bot., 37 : 919--928. Charles-Edwards, D.A. and Thorpe, M.R., 1976. Interception of diffuse and direct-beam radiation by a hedgerow apple orchard. Ann. Bot., 40: 603--613.
315 Charles-Edwards, D.A. and Fisher, M.J., 1980. A physiological approach to the analysis of crop growth data. I. Theoretical considerations. Ann. Bot., 46: 413--423. Fisher, M.J., Charles-Edwards, D.A. and Campbell, N.A., 1980. A physiological approach to the analysis of crop growth data. II. Growth of Stylosanthes humilis. Ann. Bot., 46: 425--434. Goudriaan, J., 1977. Crop Micrometeorology: A Simulation Study. Pudoc, Wageningen, pp. 54--66. Jackson, J.E. and Palmer, J.W., 1972. Interception of light by model hedgerow orchards in relation to latitude, time of year, and hedgerow configuration and orientation. J. Appl. Ecol., 9: 341--357. Jackson, J.E. and Palmer, J.W., 1981. Light distribution in discontinuous canopies: calculation of leaf angles and canopy volumes above defined 'irradiance contours' for use in productivity modelling. Ann. Bot., 47 : 561--565. Lang, A.R.G. and Shell, G.S.G., 1976. Sunlit areas and angular distributions of sunflower leaves for plants in single and multiple rows. Agric. Meteorol., 16: 5--15. Mann, J.E., Curry, G.L., Hartfiel, D.J. and DeMichele, D.W., 1977. A general law for direct sunlight penetration. Math. Biosci., 34: 63--78. Mann, J.E., Curry, G.L. and Sharpe, P.J.H., 1979. Light interception by isolated plants. Agric. Meteorol., 20: 205--214. Mann, J.E., Curry, G.L., DeMichele, D.W. and Baker, D.N, 1980. Light penetration in a row-crop with random plant spacing. Agron. J., 72: 131--142. Monteith, J.L., 1981. Does light limit crop production? In: C.B. Johnson (Editor), Physiological Processes Limiting Plant Productivity. Butterworths, London, pp. 23-38. Mutsaers, H.J.H., 1980. The effect of row orientation, date and latitude on light absorption by row crops. J. Agric. Sci., Cambridge, 95: 381--386. Norman, J.M. and Welles, J.M., 1983. Radiative transfer in an array of canopies. Agron. J., 75: 481--488. Palmer, J.W., 1977. Diurnal light interception and a computer model of light interception by hedgerow apple orchards. J. Appl. Ecol., 14: 601--614. Thorpe, M.R., 1978. Net radiation and transpiration of apple trees in rows. Agric. Meteorol., 19: 41--57. Warren-Wilson, J., 1967. Ecological data on dry-matter production by plants and plant communities. In: E.F. Bradley and O.T. Denmead (Editors), The Collection and Processing of Field Data. Wiley, New York, pp. 77--123. Warren-Wilson, J., 1981a. Analysis of light interception by single plants. Ann. Bot., 48: 501--505. Warren-Wilson, J., 1981b. Analysis of growth, photosynthesis and light interception for single plants and stands. Ann. Bot., 48: 507--512. Whitfield, D.M., 1980. Interaction of single tobacco plants with direct-beam light. Aust. J. Plant Physiol., 7 : 435--447. Whitfield, D.M. and Connor, D.J., 1980a. Penetration of photosynthetically active radiation into tobacco crops. Aust. J. Plant Physiol., 7 : 449--461. Whitfield, D.M. and Connor, D.J., 1980b. Architecture of the individualplants in a fieldgrown tobacco crop. Aust. J. Plant Physiol., 7: 415--433.
A SIMPLE MODEL OF LIGHT PENETRATION INTO ROW CROPS D.M. WHITFIELD
Irrigation Research Institute, Department of Agriculture and Rural Affairs, Tatura 3616, Victoria (Australia) (Received November 1, 1984; revision accepted July 26, 1985) ABSTRACT Whitfield, D.M., 1986. A simple model of light penetration into row crops. Agric. For. Meteorol., 36: 297--315. A simple model of light penetration into row crops was derived by considering the projections of the foliage elements to be confined to a potentially-shaded region (R*) of a horizontal plane subtending the canopy. Individual plants were modelled as optionally truncated ellipsoids. The extent and area of R* were estimated from the extent of the ellipsoids and their distribution on the soil surface in relation to the position of the sun. Uniform spatial distributions of projected foliage elements between the sun and R*, and empirical, ellipsoidal distributions were appraised. The ellipsoidal distributions were based on the structure of the plants comprising the canopies. Apart from crops with extreme gradients in vertical distribution of leaf area density, appraisal against a relatively complex numerical method showed the simple model incorporating ellipsoidal distributions of foliage over R* reliably estimated profiles of light penetration into a range of theoretical crops with contrasting structures. The simple model accounts for differences in extent of individual plants in a readily envisaged manner, for distinct and overlapping plants within rows, and for differences in light penetration attributable to angle of sight. Canopy light interception may be analysed in terms of the light intercepted by the isolated plants constituting the canopy, the extent of eompetition for light over R*, and the extent o f light penetration direct to the soil surface through failure to intercept the canopy volume.
INTRODUCTION
The diversity of canopy geometries and the dependence of light penetration on both solar elevation and azimuth have hampered development of a general, efficient means of estimatin$ light penetration into row crops. Structural change during development is an additional problem when light penetration is to be modelled over the life of the crop. An efficient means of estimating light penetration into row crops is required. It should be based on readily measured or estimated parameters of canopy extent and should naturally encompass the progression from spaced plants early in growth to the continuous and discontinuous canopies of mature crops. The utility of estimates of light interception, the c o m p l e m e n t of penetration, is firmly established. Current analyses of crop growth are based on estimates or measures of light interception, and of the efficiency of utilisation of the intercepted light (Warren-Wilson, 1967; Charles-Edwards
0168/1923/86/$03.50
© 1986 Elsevier Science Publishers B.V.
298 and Fisher, 1980; Fisher et al., 1980; Monteith, 1981; Charles-Edwards, 1981, 1982). Warren-Wilson (1981a, b) generalised the growth analysis of isolated plants and crops by considering their respective abilities to intercept light. His analysis is particularly appropriate to row crops where light interception typically depends on the foliage display of individual plants during early growth and becomes increasingly reliant on density and distribution of plants on the soil surface with time. Several authors have adequately modelled crops with continuous rows (e.g., Jackson and Palmer, 1972; Allen, 1974; Charles-Edwards and Thorpe, 1976; Palmer, 1977; Goudriaan, 1977), b u t less restrictive models accounting for a wider range of geometries are necessarily more complex and more demanding of computing facilities (see Norman and Welles, 1983). Mann et al. (1980) alleviated the computational load of numeric models with an analytic approximation of light penetration to the base of row crops comprising plants confined to entire ellipsoidal envelopes. Individual plants had uniform light intercepting characteristics and were considered to be randomly distributed within rows. In the simple model developed here, crops are assumed to be comprised of arrays of equally-spaced identical plants in parallel rows on a horizontal soil surface. The origin of plants in each row varies randomly. Individual plants are confined to optionally-truncated, ellipsoidal foliage envelopes. These are flexible and appropriate to a diverse range of plant shapes. Based on these assumptions, light penetration is estimated from the structure and light intercepting characteristics of a single member of the crop. Thus, the model offers the potential to analyse a wide range of canopy structures, accounting for both distinct and overlapping plants within rows. It can also be employed to estimate profiles of light penetration. However, it does n o t apply where plants are distributed on strict rectangular or diagonal planting patterns because the added regularity of such crops necessitates a more detailed treatment than that presented here. The extent and area of the potentially shaded region, R * , on a horizontal plane, R, subtending the c a n o p y were calculated from the area of the shadow cast by an opaque representation of an individual plant and the extent of this shadow in relation to the distribution o f plants on the soil surface. The sun was taken to be a point source at a positive angle above the horizon. The method therefore parallels aspects of the approaches of Jackson and Palmer (1972, 1981), Goudriaan (1977) and Mann et al. (1980) in this respect. Vertical heterogeneity in the light intercepting attributes of the foliage was also considered. In one case, and in parallel with the approach of Goudriaan (1977), foliage elements were assumed to be uniformly distributed over R* despite spatial variation in projected foliage density. Alternatively, an ellipsoidal spatial distribution of projected foliage elements over R* was assumed. The two variants of the simple model, one incorporating the uniform distribution of foliage elements over R* and the other the ellipsoidal
299
distribution of foliage elements, were applied to a range of theoretical 'crops' with contrasting architectural features. Their performance was appraised against a relatively complex numerical m e t h o d (Mutsaers, 1980; Whitfield and Connor, 1980a, b; Norman and Welles, 1983) based on earlier work of Charle~Edwards and Thornley (1973), Allen (1974) and Mann et al. (1979). Whilst experimental verification of the numeric procedure can only be regarded as reasonable (Charles-Edwards and Thorpe, 1976; Whitfield and Connor, 1980a; Norman and Welles, 1983), it is the logical alternative to the use of the simple model presented here. The numerical method considers the probability of penetration of numerous parallel rays. However, accounting for the passage of numerous beams through all ellipsoids located between the sun and the region of interest on R* is a major computing task. Furthermore, and in contrast to crops with strict planting patterns, the crops considered in this study have a random c o m p o n e n t in their structure. To account for this, repeated computations of light penetration for a range of canopies are required of the numerical procedure. An application of the simple model for the analysis of relationships between canopy light interception and light interception by the constituent isolated plants is also proposed. This quantifies the extent of competition between plants in row crops, and the extent of light wasted through failure to intercept the canopy volume. METHODS Mann et al. (1977) showed that the beam-lit fraction (or gap frequency), e, of the region, R, of a horizontal plane subtending an arbitrary distribution of foliage can be generally estimated using
e =
~(x,y)dxdy/AR R
= ~exp[--F(x,y)]dxdy/Aa
(1)
R
~(x, y) is the fractional light penetration to a point on R specified by coordinates, x and y, and F(x, y) is the projected leaf area index b e t w e e n R(x, y) and the sun. A a is the area of the region, R, under consideration. Dramatic variations in F(x, y) in discontinuous canopies cause the need for numerical integration of eq. 1, although analytic methods m a y still be appropriate in some circumstances (e.g., Mann et al., 1980).
Canopy structure The ellipsoidal envelope describing the extent of an individual plant was taken as (cf. Charles-Edwards and Thornley, 1973; Whitfield, 1980; Whitfield and Connor, 1980b) x 2 / a 2 + y 2 / b 2 -b (z - - h ) 2 / c 2 --- 1,
h -- c < ZL < Z < Zu < h + c
(2)
300 optionally truncated at heights, z L and z U . The origin of a Cartesian coordinate system, O, was taken as the intersection of the plant axis with the soil
surface, xOy. Oy was taken along the row and Ox across the row. ZL, Zu, a, b, and c are parameters of plant extent, a, b and c are the lengths of the semi-axes of the entire ellipsoid on Ox, Oy and Oz, respectively. The horizontal extent of the entire foliage envelope is a m a x i m u m at z = h, the height of the centre of the ellipsoid. The volume of foliage, V, is calculated from these parameters. The extent of the crop is then given by (cf. Allen, 1974; Charles-Edwards and Thorpe, 1976; Thorpe, 1978; Whitfield and Connor, 1980a)
[X --nxdx]2/a 2 4- [y --(ny q- (5)dy]2/b 2 q- [z --h]2/c 2 = 1
(3)
dy is the distance between plants within rows and d x is the distance between rows. These are also fixed parameters. In both the simple and numerical models, R was taken as the rectangular area, dy • d x . This is the ground area per plant, A R . 6, a random value between 0 and 1, was varied between rows to force the crops under study to deviate from those with strict planting patterns, ny and n~ are integers, each taking a value of zero for the plant sited on R. (ny + 6) "dy and n~ "dx define the displacement of the nyth plant in the n~th row from that sited on R (see Fig. 1).
I I I
.........
G
y
d dy
- - - -
--d×--
nx=-i
n×=O
nx=l
nx=2
6=0.2
8=0
8=0.6
,8=0. £
Fig. 1. Plan v i e w o f the distribution o f plants o n the soil surface o f m o d e l canopies. Four r o w s are d e p i c t e d (n x = - - 1 . . . 2), w i t h corresponding changes in the value o f 5, the r a n d o m o f f s e t varied b e t w e e n rows, ~ is t h e r o w - s u n angle. Eqs. 2 and 3 give details.
301 ny and nx may be negative. However, ¢, the angle between the rows and the projection of a ray on the soil surface, was restricted to the range, 0~b~90 ° , and a and b were confined to the ranges, 0 ~ a ~ d x and 0 ~ b ~ dy, without loss of generality. The minimum value of both ny and nx was therefore - - 1 . In the extensive crops considered here, nx and ny tend to infinity. This creates the need to consider the passage of each beam through a large number of plants at low solar elevations when using the numeric method.
Light in tercep ring properties of individual plants F(x, y) (see eq. 1) varies with the distance traversed within the foliage by a ray in its path to R, and the foliage density and projection of unit area of foliage along the ray. The relative dispersion of foliage elements along the beam m a y also be taken into account (see Acock et al., 1970; Lang and Shell, 1976; Whitfield, 1980). Foliage density, projection of unit leaf area and relative dispersion have a multiplicative effect (Acock et al., 1970), and are subsumed into a variable, p, the projected leaf area density. In both the simple and numeric models, each ellipsoid was divided horizontally into five layers of equal depth to simulate vertical variation in p; p was assumed uniform within layers. Five layers were considered sufficient for the present purpose, p was varied between 1 m-1 and 16 m-1 with approximate variations in projection of unit leaf area and leaf area density ranging between 0.1--0.9 and 0.5--15 m-1 respectively.
The numeric method Equation 1 is the basis of the numeric method, where multiple estimates of ~(x, y) are averaged over the area of interest, A R , to derive estimates of ~. e was taken as the mean of 100 estimates of ~(x, y) at regularly spaced points on R. ' Given that p varies only with height, z, within an individual plant, F(x, y) was estimated as 5
F(x, y) = ~ piAz(x, y ) j c o s 0
(4)
i
The index, i references the ith layer of foliage subtended by R, and Az(x, y) is the vertical c o m p o n e n t of beam penetration in a layer. 0 is the zenith angle of the beam. The summation is only effective within an ellipsoid, b u t includes contributions from all plants between R and sun as each ray passes in and o u t of successive ellipsoids in its path to R. The minimum fractional light penetration to R (~mm) was estimated as the proportion of points where ~(x, y) was 1 (i.e., F(x, y) = 0).
The simple model The assumption that crops are comprised of identical plants spaced at equal distances along parallel rows implies that the area of shadow cast b e y o n d R
302 by the plant centred on R is equal to that cast by neighbouring plants back on R. Thus, the foliage of an individual plant was considered to project only over R. The potentially shaded region of R, R*, was further considered to be confined to a subregion of R accounting for the fraction, f*, of AR (0 ~ f * ~ 1). B e y o n d this region, F(x, y) = 0 and ~ ( x , y ) = 1. Taking ~* as the beam-lit fraction of R*, the beam-lit fraction of R (~) is given by = (1 -- f*) + f*~*
(5)
and, applying eq. 1, exp [-- F(x, y)] dxdy/f*AR
(6)
R* Equations 5 and 6 require estimates of f* and of the spatial distribution of projected foliage elements over R*. The lower b o u n d (~m~) is clearly ~mm = l - - f *
(7)
Estimation o f the area o f the potentially-shaded region The total area of shadow (Amax) cast on a horizontal plane by the foliage envelope representing an individual plant, and its extents on Oy (ey) and Ox (ex), were calculated analytically. Details are available on request. Four cases, relating ex to dx and ey to dy, were considered: (a) Shadow envelope confined to the region, R (e x ~ dx, ey ~ dy ): here, f* = Amax/A R
(8a)
(b) Shadow envelope extends beyond R along the row but not into adjacent rows (e~ ~ dx, ey ~ dy ): in this instance, shadows of area equal to that cast b e y o n d R were assumed to be cast back on R by adjacent plants. These are confined within ex on Ox. f* is given by f* = minimum (exdy, Amax)/AR
(8b)
on the assumption that the area of shadow extending b e y o n d R is additively translated into the region, exdy. In cases (c) and (d), the shadow envelope extends into adjacent rows. The elliptical shadow envelope was replaced by a rectangular one of the same area The rectangular envelope was assumed to extend across e~/dx row widths with a mean extent, ey ', of Amax/ex on Oy. (c) Shadow envelope extends into adjacent rows but mean extent along row less than the distance between plants (ex ~ d~, ey ' ~ dy ): the a r e a o f s h a d o w b e y o n d R was considered to be translated randomly over AR as a consequence of the random variation in 0 b e t w e e n rows. f* is then given by
f* = [ 1 -- (1 -- ey '/dy ) (exldx) ]
(8c)
303
(d) Shadow envelope encloses R (ex ~ dx, ey' ~ dy): here f* = 1
(8d)
Spatial distributions of pro]ected foliage elements over R* On the basis of uniform p within layers, the average projected leaf area index subtended by R* is given b y 5
F* = ~ viPi/f*A a
(9)
i
vi is the volume of the ith layer o f each plant. Two spatial distributions of foliage elements over R* were appraised: (a) Uniform distribution, where F(x, y) is a constant, F*/cos 0. Division by cos 0 accounts for increased pathlengths when the sun is not directly overhead. Here ~* = exp (-- F * / c o s 0)
(10)
Substituting for e* in eq. 5, the first estimate of ~ (~v) is ~u = ( l - - f * )
+ f * exp (-- F* /cos O)
(11)
This reduces to the c o m m o n expression, ~ = exp (-- F / c o s 0), only where the shadow envelope completely encloses R (i.e., case (d), f * = 1). (b) Empirical ellipsoidal (or hemiellipsoidal) distribution based on the ellipsoidal geometry of the individual plants. This takes some account of the non-uniform spatial distribution of foliage elements over R*. The second estimate of ~* is given by (see Appendix) ~* = 211 -- (Q + 1) exp (-- Q)]/Q2 where
(12)
Q = 3F*/2 cos0 ~E is given by ~E = [ l - - f * ]
+ 2f*[1--(Q
+ 1) e x p ( - Q ) ] / Q 2
When f* = 1, ~E is greater than ~v in the range, 0.75 < F * / c o s 0 ~ 7.5, which corresponds approximately with 2% ~ dv ~ 50%. The maximum difference is approximately 4.5%.
Appraisal of the simple model Both the simple and numeric models were used to estimate the profile, ~(z), in three 'crops' of contrasting structure, and to estimate effects of angle of sight on light penetration to the base of two other canopies. Structural characteristics of the five crops are presented in Table I. z U was arbitrarily taken as 1.0 m in each, and c and h were taken as 0.5 m. Entire plants of Crop A were tall in relation to their horizontal extent, and were distinct
304 TABLE I Structural Characteristics of the five basic 'crops' used to appraise the simple model. In each instance, c = h = 0 . 5 m , and z U = 1.0m. z L was varied between 0 and 0.8m in Crops A, B and C to generate vertical profiles of ~. Parameters. are defined in eqs. 2 and 3. Crop
a (m)
b (m)
dy (m)
d x (m)
A B C D E
0.1 0.5 0.5 0.1 0.25
0.1 0.5 0.5 0.25 0.1
0.5 1.0 0.5 0.5 0.5
0.5 1.0 1.5 0.5 0.5
TABLE II Combinations of projected leaf area density, p (m-l), in plants constituting Crops A, B and C. Subscripts reference layers of 0.2m depth above the soil surface with higher values referring to upper layers. Combination
P1
P2
P3
P4
Ps
1 2 3 4 5 6 7 8 9 10
1 16 1 16 2 8 16 4 1 4
2 8 4 4 4 4 8 8 2 2
4 4 16 1 8 2 4 16 4 1
8 2 4 4 4 4 8 8 2 2
16 1 1 16 2 8 16 4 1 4
f r o m their n e i g h b o u r s . T h e plants c o m p r i s i n g Crops B and C were spherical, e x t e n d i n g t o the limits o f AR in Crop B b u t m a k i n g distinct r o w s in Crop C. Crop A represents a sparse c a n o p y , Crop B a p p r o x i m a t e s a c o n t i n u o u s canopy and Crop C approximates a hedgerow. F o r each o f these, influences o f five u n i f o r m values o f p w i t h i n the p l a n t (p = 1, 2, 4, 8 and 16 m - l ) , and ten c o m b i n a t i o n s o f these values (Table II), were investigated. ~ and ~m~ were e s t i m a t e d at 21 angles o f sight o n five transects at heights o f 0, 0.2, 0.4, 0.6 and 0.8 m above soil level. These profile d a t a also r e p r e s e n t estimates o f light p e n e t r a t i o n into a diverse range o f c a n o p y s t r u c t u r e s associated with v a r y i n g severities o f t r u n c a t i n g t h e ellipsoidal evelope. Angles o f sight were 0 -- 0 ° and f o u r r o w - s u n angles at each o f 0 = 18 °, 33 °, 45 °, 57 ° and 72 °. R o w - s u n angles (~b) were 0 ° and 90 ° (along a n d across rows, respectively), a n d t w o at r a n d o m in the range, 0 ° < ¢ ~ 90 °. T h e ability o f t h e simple m o d e l to estimate ~m~n was first appraised. F o r
305
each combination of values of p, the mean differences, ~ - ~, were then calculated for each crop over the range of angles of sight for each transect. Crops D and E were included to assess the ability of the model to cope with plants with noncircular cross-sections, and to investigate attendant effects of angle of sight. Crop D represents another hedgerow, whereas plants with identical physical dimensions were aligned across rows to form a relatively dispersed canopy in Crop E. The random offset, b, precluded a hedgerow distribution.
RE S UL TS
Estimates o f the poten tially-shaded area Comparisons between simple and numeric estimates o f ~mm indicate deficiencies in estimates of f* (eqs. 7, 8). Mean differences, ~ i . - - ¢ ~ m , over all transects and angles of sight, were (with standard errors in parentheses) 0.9% (0.29%), 0.5% (0.41%) and --2.8% (0.19%) for Crops A, B and C respectively. Respective mean differences over angles of sight on individual transects ranged from 0 to 1.4%, from 0.3% to 0.7%, and from --5.0% to 1.8% (Table III). Data for all transects are shown in Fig. 2. -
-
10@ r
T
or)
co
40
20
r ,
°
!
I
A
28
48
NUMERICRL
88
8@
ESTIMATE
[%)
i~8
Fig. 2. Simple estimates of minimum light penetration into Crops A (~), B (o) and C (A) ascompared with numeric~ estimates. Thesolidline depictsthe 1:1 relationship.
306 T A B L E III Changes in t h e m e a n d i f f e r e n c e , ~ m i n - drain, over t h e range o f angles o f sight and h e i g h t s o f t r a n s e c t in c r o p s A, B and C. S t a n d a r d errors o f t h e d i f f e r e n c e s , and t h e range o f ~min, are given in p a r e n t h e s e s . Height o f transect (m)
Crop A (%)
Crop B (%)
Crop C (%)
0 0.2 0.4 0.6 0.8
1.1 1.4 1.0 0.8 0
0.4 0.4 0.7 0.6 0.3
- - 1.8 (0.37, 0 - - 3 6 ) --1.8(0.37, 0--36) - - 2.4 (0.31, 0 - - 3 6 ) - - 2.9 (0.37, 0--38) - - 5.0 (0.32, 3 6 - - 5 2 )
(0.65, 0 - - 8 8 ) (0.75, 0 - - 8 8 ) (0.80, 2 8 - - 8 8 ) (0.60, 2 9 - - 8 8 ) (0.42, 6 8 - - 9 2 )
(1.07, 0 - - 2 0 ) (1.07, 0 - - 2 0 ) (1.13, 0 - - 2 0 ) (0.88, 1--25) (0.29, 3 4 - - 5 0 )
Light penetration in crops with uniform light intercepting characteristics In Crop A, differences in p caused mean differences between simple and numeric estimates o f ~ to range bet w e e n - - 1 . 4 % and - - 0 . 1 % with the u n i f o r m distribution, and f r om - - 0 . 1 % to 0.1% with the ellipsoidal distrib u t i o n (Table IV). F o r Crop B, the respective ranges were -- 2.7% to -- 0.9% and -- 0.5% to 0.4%. Associated with the differences in estimates of ~mm, the T A B L E IV Mean d i f f e r e n c e s , ~ - - ~, over t h e range o f angles o f sight and h e i g h t s o f t r a n s e c t for c r o p s A, B and C u n d e r t h e u n i f o r m and ellipsoidal d i s t r i b u t i o n s w i t h u n i f o r m p. S t a n d a r d errors, and t h e m a x i m u m d i f f e r e n c e s b e t w e e n ~ and ~ achieved b y varying z L b e t w e e n 0 and 0.8 m, are given in p a r e n t h e s e s . p ( m - 1)
Uniform distribution m e a n d i f f e r e n c e (%)
Ellipsoidal d i s t r i b u t i o n m e a n d i f f e r e n c e s (%)
Range o f t (%)
Crop A
1 2 4 8 16
------
77--99 59--99 36--98 13--97 4--96
Crop B
1 2 4 8 16
--0.9 - - 1.9 - - 2.7 - - 2.6 - - 1.7
Crop C
0.1 0.3 0.9 1.4 1.4
(0.04, - (0.06, - (0.09, - (0.16,-(0.23, - -
0.2) 0.5) 1.3) 2.2) 2.2)
0 0 --0.1 0 0.1
(0.13, (0.20, (0.34, (0.45, (0.47,
1.5) 2.4) 4.0) 5.2) 5.0)
0.2 0.4 0.2 --0.3 --0.5
0.12, 0.21, 0.33, 0.41, - 0.43, - -
0.7) 1.3) 1.4) 2.2) 2.9)
19--95 4--90 0--83 0--73 0--62
1 2
- - 2.5 (0.17, - - 3.5) --4.2(0.23,--5.3)
--0.9 --1.8
0 . 1 4 , - - 1.3) 0.20,--2.4)
12--93 2--88
4
-- 4.8 (0.28, -- 6.8)
--2.6
0.22, -- 3.9)
0--79
8 16
- - 4.4 (0.29, - - 6.8) --3.8(0.29,--7.5)
--3.0 --3.0
0.23, - - 4.5) 0.23,--5.4)
0--68 0-59
------
(0.04, 0.2) (0.05, - - 0.1) (0.08, - - 0.5) (0.17, 0.8) (0.24, 1.2)
307
model generally underestimated ~ in Crop C. Differences in p caused deviations ranging from - - 4 . 8 % to - - 2 . 5 % with the uniform distribution, and from -- 3.0% to -- 0.9% with the ellipsoidal distribution. In Crop A, the largest mean deviation associated with variations in both p and height of transect was only 2.2% (Table IV). In crop B, the largest deviation within transects was --5.2% with the uniform distribution, and -- 2.9% with the ellipsoidal distribution. In Crop C, respective values were -- 7.5% and -- 5.4%. The ellipsoidal distribution therefore provided the more acceptable estimates in Crops B and C.
Light penetration in crops with varying light intercepting characteristics ~v and ~E generally underestimated ~ when p was varied within plants (Table V). Comparing mean differences over transects and angles of sight in Crop A, simple model estimates were up to 3.7% and 2.6% less than numeric estimates using the uniform and ellipsoidal distributions, respectively, for the range of combinations of p shown in Table II. Respective values were 8.3% and 5.3% for Crop B, and 7.8% and 5.7% for Crop C. The largest discrepancies were generally attributable to combination 4, an extreme in the vertical distribution of p. Considering means over angles of sight at different heights of transect in Crop A, the simple model underestimated ~ by up to 6.0% and 4.2% using the uniform and ellipsoidal distributions, respectively (Table V). These occurred with combination 3. In Crop B, worst-case deviations achieved by varying the height of transect were -- 12.9% and -- 9.3%, caused by combination 4. Otherwise, the largest deviations within transects were -- 7.0% and -- 3.9% for the t w o distributions. In Crop C, combination 4 again produced the largest worst-case deviations (mean deviations of - - 1 1 . 9 % a n d - 9.3% with the uniform and ellipsoidal distributions, respectively), but the uniform distribution also produced large within-transect deviations with combinations 1, 6, 7 and 10. Thus, differences were marked where the variation in p within the plant was most pronounced, and were most serious where high values of p were confined to a small volume of the plant. Use of an average value for F* (eq. 9) is clearly inappropriate under these circumstances. In general, the ellipsoidal distribution produced more reliable estimates than the uniform distribution.
Estimates of light penetration into Crops D and E Figure 3 demonstrates effects of varying angle of sight on light penetration to ground level in Crops D and E. Looking d o w n the rows (~ = 0°), ~ decreased with increasing zenith angle in both stands, b u t the effect was most marked in Crop E. At @ = 90 °, the influence of zenith angle on Crop D was very similar to that for Crop E at ~ = 0 ° because of essentially
3O8 TABLE V Mean differences, ~ -- ~, over the range of angles of sight and heights of transect for crops A, B and C under the uniform and ellipsoidal distributions with non-uniform combinations of p. Standard errors, and the maximum differences between ~ and ~ achieved by varying z L between 0 and 0.8 m, are given in parentheses Comb n"
Crop A
Crop B
CropC
Uniform distribution mean difference (%)
Ellipsoidal distribution mean difference (%)
Range o f t (%)
1 2
-- 3.0 0.18, - - 1 . 5 0.18,
-- 4.1) --4.0)
-- 1.7 ( 0 . 1 4 , - - 2.4) --0.8(0.13,--2.3)
30--96 29--99
3
--3.7
0.40,
--6.0)
--2.5(0.32,--4.2)
25--99
4 5
-- 3.7 0.21, - - 1 . 6 0.18,
-- 4.9) --2.6)
-- 2.6 (0.17, -- 3.8) --0.7 (0.14,--1.4)
28--96 31--99
6 7 8 9
--1.5 --2.3 --2.3 --0.7
0.09, 0.16, 0.26, 0.09,
--1.9) --3.4) --3.8) --1.2)
--0.7 (0.09,--1.1) --0.8(0.16,--1.8) --0.9(0.21,--2.0) --0.3(0.07,--0.5)
34--97 13--96 11--98 55--99
10
- - 0 . 7 0.06,
--0.9)
--0.3(0.05,--0.4)
58--98
1 2 3
-- 5.2 0.44, -- 2.0 0.33, - - I . I (0.38,
-- 7.0) -- 3.9) --2.2)
-- 2.5 ( 0 . 4 0 , - - 3.9) 0.2 (0.30, 1.7) 0.7 (0.36, 1.4)
0--62 0--95 0--95
4
-- 8.3 (0.53, -- 12.9)
-- 5.3 (0.48, -- 9.3)
5
--1.6(0.35,
--2.8)
6 7 8
--5.0(0.39, -- 4.0 (0.48, -- 1.5 (0.43,
--6.9) -- 6.0) -- 4.1)
0.9 (0.34,
0--63
1.7)
0--90
--2.0(0.37,--3.7) -- 1.6 (0.44, -- 2.9) 0.2 (0.40, -- 1.3)
0--73 0--63 0--83
9
-- 1.3 (0.22,
-- 1.8)
1.0 (0.23,
1.9)
2--95
I0
--3.8(0.37,
--4.6)
--1.4(0.25,
2.4)
4--83
1 2 3 4 5 6 7 8 9 I0
--6.1(0.31, --7.5) -- 3.9 (0.30, -- 6.0) --3.8(0.32, --6.1) -- 7.8 (0.44, -- 11.9) --4.1(0.29, --6.2) -- 6.3 (0.33, -- 8.4) --5.3(0.33, --7.5) --3.6(0.27, --6.9) --4.1(0.30, --6.0) -- 5.4 (0.26, -- 6.8)
--4.5(0.26,--5.4) -- 2.2 (0.22, -- 3.2) --2.7 (0.26,--3.6) -- 5.7 (0.39, -- 9.3) --2.4(0.21,--3.4) -- 3.9 (0.28, -- 5.5) --3.9(0.28,--5.4) --2.5(0.20,--4.2) --1.9(0.21,--2.8) -- 2.8 (0.24, -- 3.9)
0--59 0--93 0--93 0--59 0--88 0--68 0--59 0--79 0--93 2--79
e q u i v a l e n t g e o m e t r i e s . H o w e v e r , a t ¢ = 9 0 ° i n C r o p E, ~ first d e c r e a s e d s l o w l y w i t h i n c r e a s e i n 0 a n d t h e n fell m o r e r a p i d l y . U n d e r t h e s e c o n d i t i o n s , t h e s t o c h a s t i c e l e m e n t , 5, p l a y e d a s i g n i f i c a n t r o l e i n n u m e r i c e s t i m a t e s o f ~. T h u s , t h e s i m p l e m o d e l p r o d u c e d g o o d e s t i m a t e s o f ~ f o r t h e r a n g e o f a n g l e s o f s i g h t i n t h e s e c r o p s e x c e p t f o r t h e i n s t a n c e , 0 -- 57 °, ¢ -- 9 0 °, i n C r o p E.
309
7~
j
78
68
4I
68
°z 5e
4
58
~-- 48 ~
q
48
-•"<
y=
,
\\
i
4 ~ ,
'
4
o m',,
~
\,
4 ",
4
,
18 ~- C r o p
I
D
4
I
® 28
48 ZENITH
58
Crop
18 ~
E
\
S
! 8~
8
2~
ANGLE OF SIGftT
4~
68
88
(degr'e~as]
Fig. 3. Simple model estimates of light p e n e t r a t i o n below Crops D and E with change in zenith angle of sight at r o w - - s u n angles of 0 ( ) and 90 ° ( . . . . ). The leaf area density of individual plants was 4 m - 1. Symbols depict single estimates from the numeric model at r o w - - s u n angles of 0 ° (•) and 90 ° (o). Simple model estimates were based o n an ellipsoidal distribution of foliage elements. •
O
Influence of non-rectangularplanting patterns Light penetration to the soil surface of Crop A was strongly dependent on angle of sight and the light intercepting properties of the foliage. At low values of p, row-sun angle had little influence on ~ over a range of zenith angles, but ~b played a greater role at higher values of p (Fig. 4). The influence of b on estimates from the numeric model is demonstrated in Fig. 5, where effects of changes in ¢ at a zenith angle of 57 ° are shown. At p = 1 m - i , there was little change in ~ with increase in ~b, and the m a x i m u m standard error of five numeric estimates of ~ at each value of ~b was only 0.4%. However, Fig. 5 shows t h a t ~ first declined to a m i n i m u m of approximately 20% at ~b = 20 °, then rose to a m a x i m u m of 35% a t e = 90 ° f o r p = 16 m-1. Beyond the minimum, standard errors of the numeric estimates increased from 0.6% to a m a x i m u m of 5.6% at ¢ = 70 °. These data demonstrate the need for multiple estimates of ~ when using the numeric model with even a relatively minor random c o m p o n e n t in the structure of the canopy. The data a l s o demonstrate t h a t the simple model provided satisfactory estimates of the means.
310 188
--~
.
.
.
.
~
i08
80 - ~ --
--,
, ~
. . . . . .
-
88
78
~
60
CD
, "\
58
7[3
~
68
~
5B
2 28
28
',
"
18 i
0
i
28
i
i
48
i
i
68
]~
~
~
88
ZENITH RNGLE OF SIGHT [degrees)
I I
L
0
I
i
i
38
i
L
I I
I
i
J
68
i
I i
98
ROW-SUN RNGLE [degreesl
F i g . 4 . N u m e r i c e s t i m a t e s o f light p e n e t r a t i o n b e l o w Crop A w i t h change in z e n i t h angle
o f sight at r o w - - s u n angles o f 0 ° densities o f 1 m - 1 (©, e ) a n d 1 6 m row--sun angles o f 0 ° ( ) and based on an ellipsoidal distribution
( o p e n s y m b o l s ) and 9 0 ° (solid s y m b o l s ) at leaf area - 1 ( ~ , m). The curves depict simple m o d e l e s t i m a t e s at 9 0 ° ( . . . . ), r e s p e c t i v e l y . S i m p l e m o d e l e s t i m a t e s were
o f foliage elements.
F i g . 5. N u m e r i c e s t i m a t e s o f light p e n e t r a t i o n b e l o w Crop A w i t h change in r o w - - s u n angle at a z e n i t h angle o f 57 °. Leaf area densities in c o n s t i t u e n t plants were 1 m - 1 ( 0 ) and 1 6 m - 1 ( e ) . Standard errors o f five e s t i m a t e s at each r o w - - s u n angle are s h o w n for the latter. The curves d e p i c t simple m o d e l e s t i m a t e s based on an ellipsoidal distribution o f foliage elements.
DISCUSSION
The basic canopy structures considered in this study varied from the more-or-less isolated plants of Crop A to the continuous (Crop B) and hedgerow (Crop C) systems characteristic of many well-developed r o w crops (Table I). The profile data generated by varying ZL in these crops represent the equivalent of 15 crops comprised of plants with varying severities of truncation, and indirectly encompass an additional 12 crops when it is considered that truncation achieved by varying zL is equivalent to that achieved by inverting entire ellipsoids and varying Zv. The range of angles of sight and the effects of truncation on canopy structure were sufficient to vary ~mm over the range, 0--92% (Table III, Fig. 2). The diversity of crop-sun geometries considered here therefore provides adequate scope for appraising the flexibility and general application of the simple model. The simple model was appraised at two levels. The first concerned its ability to estimate ~ n and the second focused on its ability to estimate
311 1.o
:. . . .
. ....
!
Z E (AA
z
0.8
Z
u
0.6
F--
~
O-
0.4
~
0.2
CO Z (.3
8.0
,_ i
L 2
L 4
,
,
,
,
,
5
6
7
8
9
10
LERF RRER INDEX OVER SHRDED REGION
Fig. 6. Change in ~ per u n i t change in ~min as a f u n c t i o n of leaf area index from u n i f o r m ( ) and ellipsoidal ( . . . . ) distributions of foliage elements over the potentially shaded region, R*.
light penetration given varying light intercepting properties of the foliage. Figure 6 demonstrates the general sensitivity of ~ to estimates of ~m~ for all model crops. It clearly shows the need for accurate estimates of ~min (and hence f*, see eq. 7) in the region where the average projected leaf area index over R* is greater than approximately 1.5. For example, ~ is increased by approximately 0.6% for a 1% overestimate of ~min with a leaf area index of 2 over R*. Figure 6 also demonstrates that estimates of e are essentially determined by the estimate of emm when the leaf area index over R* is greater than approximately 4.5. Appraisal of several alternatives to eqs. 8 showed that estimates of emm were generally more sensitive to the estimate of shadow extent across rows than that within rows (data not presented). The simple model provided reliable estimates of emm (Fig. 2). When individual plants were assumed to exhibit uniform light intercepting characteristics, the simple model provided satisfactory estimates of e even when a uniform distribution of leaf elements over R* was assumed (Table IV). Mann et al. (1979) have demonstrated that light penetration is underestimated if uniform distributions of foliage elements are used to approximate non-uniform distributions in single plants confined to eUipsoidal envelopes. This was evident here, where the assumption of a uniform distribution of foliage elements over R* was c o m m o n l y associated with low estimates of e. The ellipsoidal spatial distribution partly removed this bias, and the simple model was found inadequate only with extreme
312 variations in the vertical distribution of foliage (Table V). The use of an average value for F* where relatively high values of p are confined to a small volume of the individual plants (combinations 4, 6, 7 and 10, Tables II and V) is clearly inappropriate. Otherwise, the simple model incorporating the ellipsoidal distribution of foliage elements over R* provided reliable estimates of ~ over a range of canopies. The model provides a simple framework for analysing the efficiency of light interception b y row crops in terms of three parameters: light interception by isolated or spaced members of the crop, the fraction of light wasted through failing to intercept the canopy volume and the extent of interplant competition for the light intercepted by the canopy. The analysis derives from eq. 11, which may be rewritten as ~c = ( l - - f * )
+ f* exp
(--F*/cosO + F'/cosO) exp ( - - F ' / c o s 0)
Here, ~c is the fraction of light transmitted by the canopy and F' is the projected leaf area index of the isolated plant over Amax, the area of shadow cast b y a solid representation of the plant. Isolated plants transmit the fraction, ~s (= exp [-- F'/cos 0 ] ), of the incoming light to the soil surface within the extent of the shadow and intercept the fraction, 1 - ~s. The proportion of light wasted by failing to impinge on the foliage volume is l - f*. Thus, the extent of interplant competition or interference over the potentially shaded region can be taken as C = 1 - - exp
(--F*/cosO + F'/cosO).
This has a minimum of 0 with no interference or competition between plants (i.e., F* =F', f*AR =Amax), and a maximum of 1. Now, ~s, ~c and f* are directly available from the model. An indirect measure of interplant competition or interference over R* is therefore C -- [ ( l - - f * )
q-f*~s--~c]/f*~s
and the simple equation, ~c = ( 1 - - f * ) ÷ f * ( 1 - - C ) ~ s describes canopy interception (1 -- ~c ) as a function of wastage (1 -- f* ), the ability of the isolated, single plant to intercept light ( 1 - ~s), and interference, C. Combined with its flexibility, the results show that the simple model applies to many crops which would otherwise require extensive mathematical treatment. Furthermore, the m e t h o d of estimating the area of the potentially shaded region subtending the canopy and superimposing an appropriate distribution of foliage applies generally. The approach used in this paper should therefore apply with other distributions of plants on the soil surface and alternatives to the ellipsoidal approximation of single plant extent. Appraisal of similar models over a more diverse range of canopy structures warrants attention, as the present analysis indicates that direct
313 application of Beer's Law is only appropriate where the area of the shadow envelope cast by an individual plant is commensurate with the ground area per plant, and where the distribution of foliage elements over the shadow envelope can be considered as uniform. LIST OF MAJOR SYMBOLS a, b, c, h dx dy ex
parameters describing extent o f ellipsoidal foliage envelope distance between rows distance between plants within rows extent of the shadow cast by a solid representation of an individual plant across the rows ey extent of the shadow cast by a solid representation of an individual plant along the rows f* potentially shaded fraction of the plane subtending the canopy Ox, Oy, Oz axes of Cartesian coordinate system ZL, ZU heights of truncating the ellipsoidal foliage envelope Amax area of shadow cast by a solid representation of an individual plant AR ground area per plant F projected leaf area index F* projected leaf area index over the potentially shaded region R horizontal plane subtending part or all of canopy R* potentially shaded region of a plane subtending the canopy b random offset preventing a rectangular distribution of plants on the soil surface 0 zenith angle of a beam ¢ row-sun angle p projected leaf area density fraction of light penetrating the canopy e* beam-lit fraction of the potentially shaded region ~mm minimum light penetration to the plane subtending the canopy ~E, ~V model estimates of light penetration using ellipsoidal and uniform distributions of projected foliage elements, resp. APPENDIX
Derivation o f eq. 12 The foliage elements subtended by R* were assumed to be confined to an ellipsoidal envelope (semiaxes ~, ~ and 7) at a constant leaf area density, p'. With beams from the zenith, the normalised density function o f projected leaf elements on the plane subtending the ellipsoid is given by g(z) = 2p'z/(47r~fJTp'/3 ) = 3z/27A*
(A.1)
314
within the elliptical shadow envelope. Here, A* = ~r~. Beyond the ellipse, g(z)
=
0
Equation A.1 describes an ellipsoidal spatial distribution of foliage elements over the ellipse with g(x, y) = 3(1 -- x2/c~ 2 -- y 2 / ~ 2 ) l / 2 / 2 " y A * Now A, the area of a subellipse concentric with the elliptical shadow envelope (0 ~
= A*(1--z2/'y
2)
Rearranging and substituting for z in eq. A1, g(A) is given by g(A) = 3 ( 1 - - A / A * ) l n / 2 A * Applying eq. 1, 5" is given by A*
5*
J exp [ - - g ( A ) ( F ' A * ) ] d A / A * 0
where F is the mean projected leaf area index over the elliptical shadow envelope, and is given by F = 47p'/3A*
Thus A*
5* =
j exp [-- (3F/2) ( 1 - - A / A * ) I / 2 ] d A / A * 0
Integration and substitution of F*/cos 0 for F yields eq. 12. The same expression is generated when foliage elements are assumed to be confined to an hemiellipsoidal foliage envelope. REFERENCES Acock, B., Thornley, J.H.M. and Warren Wilson, J., 1970. Spatial variation of light in the canopy. In: Prediction and Measurement of Photosynthetic Productivity. Pudoc, Wageningen, pp. 91--102. Allen Jr., L., 1974. Model of light penetration into a wide-row crop. Agron. J., 66: 41-47. Charles-Edwards, D.A. 1981. The Mathematics of Photosynthesis and Productivity. Academic Press, London, 127 pp. Charles-Edwards, D.A., 1982. Physiological Determinants of Crop Growth. Academic Press, London, 161 pp. Charles-Edwards, D.A. and Thornley, J.H.M., 1973. Light interception by an isolated plant -- a simple model. Ann. Bot., 37 : 919--928. Charles-Edwards, D.A. and Thorpe, M.R., 1976. Interception of diffuse and direct-beam radiation by a hedgerow apple orchard. Ann. Bot., 40: 603--613.
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