Foliage cover and radiation interception

Field Crops Research, 13 (1986) 75--87

75

Elsevier Science Publishers B.V., Amsterdam -- Printed in The Netherlands

FOLIAGE COVER AND RADIATION INTERCEPTION

M.D. STEVEN ~,3, P.V. BISCOE :,4, K.W. JAGGARD: and J. PARUNTU I

Environmental Physics Unit, University of Nottingham School of Agriculture, Sutton Bonington, Loughborough LE12 5RD (Great Britain) 2Broom's Barn Experimental Station, Higham, Bury St. Edmunds, Suffolk IP28 6PN (Great Britain) 3 Present address: Department of Geography, University of Nottingham, Nottingham NG7 2RD, Great Britain * Present address: ICI Agricultural Division, Fertiliser Sales Department, P.O. Box 1, Billingham, Cleveland TS23 1LB, Great Britain (Accepted 20 August 1985)

ABSTRACT Steven, M.D., Biscoe, P.V., Jaggard, K.W. and Paruntu, J., 1986. Foliage cover and radiation interception. Field Crops Res., 13: 75--87. A new method is described to measure foliage cover -- the fraction of the ground covered by foliage -- from photographs taken vertically above a crop. The relationship between cover and the interception of light by the crop is described by simple theory based on Beer's law and is shown to depend on the ratio of two attenuation coefficients. This ratio is calculated for the case of black leaves using various hypothetical leaf-angle distributions with angular distributions of solar radiation for clear and overcast skies. For black leaves radiation interception is always greater than cover but empirically, foliage cover and interception are almost equal in sugar beet and field beans, and cover is always greater than interception in barley. The value of the ratio may be determined from the measured relationship and used to estimate interception from photographic measurements. Extensions to the technique to measure the effect of environmental stresses in crops are described and the application of the technique to large-scale monitoring of crops is discussed.

INTRODUCTION I t has b e e n r e c o g n i s e d f o r s o m e t i m e t h a t o n e o f t h e p r i n c i p a l v a r i a b l e s i n t h e g r o w t h o f c r o p s is t h e f r a c t i o n o f t h e a v a i l a b l e s o l a r r a d i a t i o n i n t e r c e p t e d b y f o l i a g e . T h e p r o d u c t i v i t y o f c r o p s m a y b e a n a l y s e d as t h e p r o d u c t of the solar energy i n t e r c e p t e d over a season and the efficiency with which t h a t e n e r g y is c o n v e r t e d t o b i o m a s s ( M o n t e i t h , 1 9 7 7 ) . M e a s u r e m e n t s o f t h e r a d i a t i o n i n t e r c e p t e d are n o w w i d e l y a p p l i e d w i t h t h i s f o r m o f a n a l y s i s to investigate the effects of e n v i r o n m e n t a l or c u l t u r a l variables o n crop productivity. M e a s u r e m e n t s o f t h e r a d i a t i o n i n t e r c e p t e d are c o m m o n l y m a d e b y m o u n t -

0378-4290/86/$03.50

© 1986 Elsevier Science Publishers B.V.

76 ing tube solarimeters underneath the crop canopy. The transmitted irradiance is referred to a measurement of the incident irradiance above the canopy, made with a similar instrument (Szeicz et al., 1964). Biomass production is measured as the difference between successive harvests and the analysis of efficiency by this technique refers to periods of at least a week. For this reason, measurements of interception are averaged or integrated over a day, or several days, to eliminate any diurnal variation. While the individual instruments for the measurement of intercepted radiation are reasonably priced, the requirements of replication and the need to integrate the o u t p u t of the solarimeters make the technique too expensive for use in large-scale agronomic experiments. Also the tubes are quite delicate and should be in position t h r o u g h o u t the life of the canopy, and this reduces the flexibility of husbandry operations, particularly in row crops. A useful surrogate measurement for the fraction of radiation intercepted throughout a whole day is described here. Foliage cover -- the fraction of the ground covered by foliage -- may be measured on photographs taken vertically above the crop. In addition to being cheaper than interception measurements, the technique has the advantage that photographs sample a relatively large area, may be taken rapidly in the field, are easily reproducible and provide a permanent record of the state of the crop. This paper describes the measurement of foliage cover and its relationship with radiation interception. THEORY Monsi and Saeki (1953) showed that the mean transmittance T of radiation through a canopy of randomly distributed leaves could be represented by analogy with Beer's law as T = exp(--KL)

(1)

where L is the leaf area index and K is an attenuation coefficient. The foliage cover is, in principle, defined by the vertical projection of canopy elements onto the ground surface and is equal to 1 -- Tv where Tv is the direct transmittance of light reflected from the soil along a vertical path between the canopy elements. The fraction of solar radiation intercepted by the canopy is equal to 1 -- Ti where Ti is the mean transmittance of radiation into the crop canopy along all possible paths, weighted according to the geometric distribution of direct and diffuse radiation. Thus, foliage cover and the fraction of radiation intercepted are determined by the same process, governed principally by the angular distribution of leaves. Differences arise because the angles of light penetration through the canopy are different and because the measurement of cover does not allow for the transmission of light by individual leaves. These two factors work in opposite directions. The transmission of radiation through leaves, especially in the near-infrared band, reduces the fraction measured as intercepted by a given foliage cover. But

77 radiation penetrating the canopy at low angles of incidence is more likely to be intercepted by vertical elements such as stems which do not register strongly in the measurement of cover. The theoretical relationship between cover and fractional radiation interception may be calculated from eqn. (1). The fractional foliage cover f is given by (2)

f = 1 -- exp(--gfL)

and fractional radiation interception i by (3)

i = 1 -- exp(--KiL)

Combining eqns. (2) and (3) gives the relationship i = I -- (i --f)KJKr

(4)

shown in Fig. 1 for various values of the ratio K i / K f. With horizontal black leaves, foliage cover should correspond exactly to fractional radiation interception because, with this arrangement, the penetration of radiation into the canopy is independent of angle and K i and K f are equal. The argument may be extended to conical distributions of leaves, in which all leaves are inclined at the same angle to the horizontal and distributed randomly in azimuth, but the theoretical correspondence is then limited to angles of penetration greater than the inclination of the leaves (Anderson, 1966). 100

8O

~= 60

co

40

2O

20

I 40

I 60

I 80

I 100

Foliage cover ( % )

Fig. 1. Theoretical relationships between radiation interception and foliage cover for values of Ki/K f of 1.4, 1.2, 1.0, 0.8 and 0.6 respectively (from top to bottom).

A simple model was used to calculate values of K i and K f for overcast and clear skies in summer. To represent clear skies, 4 days between March and June 1979 were selected from solar radiation records at the University of Nottingham School of Agriculture at Sutton Bonington (51.8°N, 1.2°W).

78 D a y s w e r e selected w i t h high global irradiance and l o w d i f f u s e irradiance, indicating t h a t t h e r e was little o r no cloud. T h e daily global i r r a d i a t i o n o n t h e s e d a y s was r e p a r t i t i o n e d a c c o r d i n g to t h e c a l c u l a t e d f r a c t i o n e m a n a t ing f r o m nine circular s k y zones, s y m m e t r i c a b o u t t h e zenith, each z o n e c o r r e s p o n d i n g to a 10 ° r a n g e o f e l e v a t i o n angles. H o u r l y values o f d i r e c t solar irradiance w e r e c a l c u l a t e d f r o m t h e d i f f e r e n c e b e t w e e n global and diffuse irradiance records. T h e solar e l e v a t i o n was c a l c u l a t e d f r o m t h e locat i o n , d a t e a n d t i m e (Sellers, 1965} a n d t h e direct solar irradiance a l l o c a t e d t o the c o r r e s p o n d i n g sky zone. T h e diffuse irradiance was divided into a circumsolar component, allocated with the direct beam, and a background c o m p o n e n t w h i c h was divided b e t w e e n all t h e sky z o n e s a c c o r d i n g to t h e s k y r a d i a n c e m o d e l o f Steven a n d U n s w o r t h ( 1 9 7 9 ) . T h e z o n a l c o m p o n e n t s o f r a d i a t i o n f o r each h o u r w e r e s u m m e d t o give t h e d i s t r i b u t i o n o f global radiation with sky elevation for the whole day. The distribution for overcast skies was c a l c u l a t e d f r o m t h e m e a n r a d i a n c e d i s t r i b u t i o n o f S t e v e n and U n s w o r t h (1980). T h e s e d i s t r i b u t i o n s f o r clear a n d o v e r c a s t skies are summ a r i s e d in T a b l e 1. TABLE 1 Integrated daily distributions of solar radiation with sky elevation showing the fraction of radiation in each angular range Sky

Overcast Clear Clear Clear Clear

Date

23 18 19 22

March April May June

Radiation (MJ/m 2)

Elevation

Global

Diffuse

0--30 °

30--60 °

60--90 °

15.3 15.4 24.3 27.3

-

0.19 0.51 0.17 0.27 0.18

0.51 0.47 0.80 0.71 0.80

0.30 0.02 0.03 0.02 0.02

4.2 7.8 10.1 8.0

T h e values o f K w e r e c a l c u l a t e d f o r a range o f h y p o t h e t i c a l d i s t r i b u t i o n s o f black leaves. F o l l o w i n g M o n t e i t h ( 1 9 7 3 ) , t h e value o f K f o r a conical d i s t r i b u t i o n o f b l a c k leaves was c a l c u l a t e d b y K = cos(a)

(5)

w h e n t h e angle o f e l e v a t i o n o f t h e i n c o m i n g r a d i a t i o n ~ was g r e a t e r t h a n t h e angle a o f t h e leaves t o t h e h o r i z o n t a l , a n d b y K = n-1 [(n - - 20 0)cos a + 2sin a cotl3 sin 00]

(6)

w h e r e 0 = cos-~(tan/~ c o t a) w h e n / 3 was less t h a n a. F o r a spherical or rand o m d i s t r i b u t i o n o f leaves, in w h i c h all leaf angles are e q u a l l y r e p r e s e n t e d , K was c a l c u l a t e d b y K = 0.5 cosec(/~)

(7)

79

To calculate Kf, the value of/~ was always 90 ° so t h a t Kf does n o t depend on the radiation distribution or t he day used. Values of Ki were calculated initially for each sky zone, taking/3 f r om the mid-point of the 10 ° range of zenith angles. Radiation transmission for a series o f values of L was calculated with these values o f Ki and multiplied by the p r o p o r t i o n o f radiation in each sky zone. The transmitted fractions for each zone were summed to derive th e total transmission o f radiation, T. The effective value of Ki for the whole sky and the whole day was then calculated for each value o f L by inverting eqn. (1). Beer's law does not apply strictly when different a t t e n u a t i o n coefficients are com bi ne d in this way and the derived values of Ki do depend to a small e x t e n t on L. Table 2, however, shows t hat the main source o f variation in the ratio Ki/K f is neither the leaf area index nor the state o f t he sky, but t he angular distribution of the leaves. For black l e a v e s , K i is always greater than K f but the difference is n o t large for leaf angles up to a b o u t 40 ° f r om the horizontal. With higher leaf angles, differences in th e values be c om e m or e apparent and show m ore d e p e n d e n c e on L and o n sky conditions. T he values o f Ki/Kf were slightly higher for the clear day in March than f or o t h e r days because a larger p r o p o r t i o n o f the radiation came f r o m lower sky elevations, but generally the differences between clear and overcast skies are small because most of the incident solar radiation in summer {direct plus diffuse) comes from angles o f elevation greater than 30 ° . The model described here is n o t an a t t e m p t to simulate the canopy completely, b u t serves to establish the physical link bet w een the interception TABLE 2 C o m p u t e d values o f

Ki/K f for

black leaves w i t h L = 1 (a) a n d L = 3 (b)

Conical leaf d i s t r i b u t i o n leaf angle (degrees)

Spherical leaf distribution

0

20

40

60

80

KZ:

1

0.94

0.77

0.50

0.17

0.50

Overcast 23 March 18 April 19 May 22 J u n e

1 1 1 1 1

1.02 1.05 1.02 1.03 1.02

1.12 1.33 1.11 1.17 1.11

1.45 2.26 1.59 1.61 1.50

3.61 6.89 4.60 4.52 4.18

1.53 2.25 1.67 1.71 1.60

1 1 1 1 1

1.01 1.03 1.01 1.02 1.01

1.07 1.23 1.07 1.11 1.07

1.30 1.98 1.47 1.40 1.35

2.88 5.81 4.12 3.72 3.62

1.39 2.01 1.57 1.53 1.48

(a)

(b) Overcast 23 March 18 April 19 May 22 J u n e

80 of radiation and foliage cover. The model only accounts for the geometry of ideal leaf canopies and does not allow for transmission or reflection by leaves. These factors do not affect Kf because the measurement of foliage cover is only concerned with the presence or absence of foliage, whereas transmission reduces Ki. The canopies of real crops have various angular distributions of leaves, but measurements of radiation interception in field crops show that on clear days K i only depends strongly on solar elevation near dawn and dusk, when ~ is less than 30 ° (Monteith, 1973; Szeicz, 1974). Even in temperate latitudes the proportion of daily radiation which comes from so low in the sky is small and the model indicates that the relationship of fractional radiation interception with foliage cover would, therefore, usually be consistent. DETERMINATION OF COVER Measurements of foliage cover may be obtained from vertical photographs taken over the crop. In this study, photographs of barley and beans were taken with a 35-mm camera with a standard 50-mm lens, giving a maximum field of view of 46 ° , or -+23° from the vertical. This departure from the vertical projection is not serious as the attenuation coefficient K will only vary within this range of angles for those canopy elements that are oriented within 23 ° of the vertical. Photographs of sugar beet were taken with a 28mm lens, but the measurement of cover was confined to a central area covering 2/3 o f the long axis on the film frame. The height of the camera above the canopy is not important in itself, but should be great enough to allow sufficient depth of field to keep all levels of the canopy in reasonable focus and for the photograph to extend over several rows of the crop. It is also sometimes useful to slightly overexpose the photographs so that canopy elements in deep shadow can be distinguished. Photography under diffuse light, either on cloudy days or using a large diffuser to shade the sun, helps to reduce the contrasts between light and shade and makes the photographs easier to analyse. The photographs may be analysed manually by superposition of a grid and counting the numbers of intersections falling on soil, leaves, or other components of the canopy. The fractional cover is then estimated from the proportion of points falling on leaves or other foliage elements. It is best to use a randomised grid (Appendix A) for this purpose to avoid the possibility of an interaction between the grid spacing and the row spacing of the crop; this could lead to systematic error. The size of the points at the grid intersections should be as small as possible to ensure maximum precision in determining their position on the photograph. With these precautions against bias, the accuracy of the technique is governed by the binomial probability distribution: about 1000 points are required to give 99% confidence of accuracy to -+5% cover (Neave, 1977). To speed the process (and to limit the tedium), this number of points may conveniently be spread

81

across photographs of replicate areas of the crop, without t o o much loss o f accuracy. Automatic analysis is also possible using digital image analysers such as the Quantimet (Cambridge Instruments, Cambridge}. These devices rely on high photographic contrast between leaves and soil, which may be obtained using Kodak high speed infrared film with a Wratten 88A filter. Photography under diffuse light is essential. The accuracy of this m e t h o d is limited by the ability to tune the analyser to separate leaves from soil and this can be a small subjective source of error. Estimates of this error in barley, made by varying the tuning factor to either side of the selected optimum, suggest that with careful analysis and high-contrast photography the error can be limited to + 3% or less. However, the resolution of presentday image analysers is less than can be achieved by the human eye and automatic analysis works best on large-leaved crops. MEASUREMENTS AND RESULTS

The relationship between foliage cover and fractional radiation interception was measured in a number of crops of sugar beet (Beta vulgaris L.) grown in t w o seasons (1979 and 1981) and at two sites in East Anglia {Fig. 2). The data represent different rates of application of nitrogen fertiliser (nil or 125 kg N/ha) and both irrigated and unirrigated treatments of two varieties, N o m o and Bush Mono G. Radiation interception was measured using t w o replicate sets of three solarimeters beneath the canopy, each set traversing six rows of the crop. The signals from these solarimeters and those above the crops were measured over periods of 24 h by using integrators (Delta-T Devices, Cambridge). Foliage cover was measured on infrared photographs analysed automatically by the Quantimet image 100

/

8O

60

.~_ 40

20

0

i 20

J 40 Foliage

I 60 cover

810

I 100

(%)

Fig. 2. Empirical relationship between interception and cover in sugar beet.

82 analyser. The results indicate very close correspondence, in spite of the effects of treatment, variety and other factors, such as disease. Fig. 3 shows the same relationship for field beans (Vicia faba L., cvs. Minica and Alfred) in an experiment testing three water regimes: droughted, ensured by covering the ground with plastic; a well-irrigated crop; and a control treatment, subject to natural conditions. Light interception was measured by three replicate solarimeters in each crop, angled to traverse exactly one row each. The measurements were derived from integrated values of the radiometric signals over the main part of the day, from about 09.00 to 16.00 GMT. Foliage cover was measured on Ektachrome photographs by the manual techniques described earlier. The relationship is again a near-perfect correspondence for most of the data, irrespective of treatment. The four circled points represent the last measurement of the season in the droughted and control crops when the leaves had dropped off, leaving near-vertical bare stalks. These stalks continued to intercept radiation and shaded the ground but contributed little to the measurement of cover. 100

v

8O

": :..:;.

~ 60 ®

.~ 4(1

®®

•

20

I

20

410 Foliage

610 cover

I

80

J

100

(%)

Fig. 3. E m p i r i c a l r e l a t i o n s h i p b e t w e e n i n t e r c e p t i o n a n d cover in field beans.

A less-direct correspondence was found in an experiment on barley (Hordeurn vulgare L., cv. Midas) where four rates of nitrogen fertiliser application were compared (Fig. 4). Light interception was measured using four replicate solarimeters, each one crossing five rows of the crop. Other details of measurement were as for the bean crop. The foliage cover was always greater than the radiation interception, by as much as 10% in mid-season, indicating that in the case of barley, Ki and Kf differ considerably. Eqn. (4) was applied to derive the value of Ki/Kf as the slope of the relationship between log(1 -- i) and log(1 -- f) in Fig. 5. The value depended to some extent on the rate of nitrogen application (Table 3), but the combined value

83

100

v

8O

¢.) 6 0

E ._~ 4 0

2O

0

-0"

~

20

I

40

6

0

Foliage cover

810

I

100

(%)

Fig. 4. Empirical relationship between interception and cover in spring barley. Log

-2

( 1-1

cover

)

'

o

%o o O ' ~ / d ~ ' ~ ° -

a

Ae • m

g

3

I

-2

Fig. 5. A logarithmic transformation of the relationship in Fig. 4. The slope of the regression is equal to Ki/K f. Nitrogen application rates: (o) 0; (A) 40; (o) 80; (e) 160 kg/ha.

for all treatments was 0.74 and only the crop with no nitrogen fertiliser departed significantly from this (at a confidence level of 98%). In addition to the measurements o f cover and interception described above, measurements were made in the barley crops of the leaf area index and of the interception of photosynthetic quanta using a Li-Cor quantum meter (Lincoln, NE) mounted on a track underneath the canopy (Paruntu, 1984). Values of Ki for radiation interception and Kq for quantum interception were derived, and their ratios are shown in Table 3. The quantum

84 TABLE 3 A t t e n u a t i o n ratios in barley Applied N (kg/ha)

Ki/K f Kq/K i Kq/Kf

Combined

0

40

80

160

0.55 1.26 0.69

0.70 1.23 0.86

0.74 1.31 0.97

0.76 1.51 1.14

0.74 1.38 1.03

interception values could not be compared directly with the cover data because the measurements were made on different days. Instead, the radiation interception values were converted to q u a n t u m interception q, using the relation, (1 --

q) = (1 --

i)Kq/Ki

(8)

which is equivalent to (1 -- q) = (I

--

f)Kq/Kf

(9)

The derived relationship between foliage cover and quantum interception is shown in Fig. 6 with the corresponding values of Kq/Kf, found by linear regression of log(1 -- q) on log(1 -- f), shown in Table 3. The conversion from radiometric units to photosynthetic quanta brings most of the data back into an almost direct correspondence but the difference between nitrogen treatments is increased, indicating that, for a given cover, the unfertilised crop intercepted a smaller fraction of the photosynthetic quanta.

100 oo

80

~® 60 E

i ,o 0

20

0

~

20

40

60

Foliage

cover (%)

80

I 100

Fig. 6. The relationship b e t w e e n interception of p h o t o s y n t h e t i c quanta and cover in spring barley.

85 DISCUSSION The attenuation coefficients, K, for interception or for cover depend b o t h on the angular distribution of leaves and on their optical properties. The close correspondence between K i and Kf in some crops appears to be a result of compensation between these two factors as Ki/Kf is always greater than 1 for black leaves {Table 2). In healthy green crops the conversion to quantum photosynthetic fluxes is roughly equivalent to considering the case of black leaves because almost all the intercepted photosynthetic radiation is adsorbed b y chlorophyll, and Kq/Ki in many crops is almost invariant at a value of a b o u t 1.4 {Marshall and Willey, 1983). Our results indicate a lower value in barley crops starved of nitrogen, and the spread of values of Kq/Kf is larger than the spread o f Ki/Kf suggesting that the absorption of photosynthetic radiation by chlorophyll in these crops is less efficient. However, in spite of these disparities b e t w e e n nitrogen treatments in barley, the application of a c o m m o n conversion factor Ki/K f would allow radiation interception to be estimated with reasonable accuracy from photographic measurements of cover. The measurement of cover by photographic techniques has further potential applications. The partition of intercepted radiation between different canopy elements can also be estimated, using the same analogy that allows radiation interception to be calculated from cover. To a first approximation, the interception by canopy elements is in proportion to the relative fractions o f the photograph covered by those elements. Colour photographs have been used by the authors to estimate light interception by chlorotic leaves in diseased and stressed crops. This estimation cannot easily be done b y leaf sampling because such techniques take no account of the relative spatial distributions of green and chlorotic tissue. However, vertical overhead photographs automatically give greater weight to those leaves at the t o p of the canopy that intercept the greater part of the radiation. The partition of radiation between canopy elements also determines the spectral composition of radiation reflected from the canopy -- the spectral signature used in remote sensing. A similar development of the technique uses photographs to differentiate between canopy elements in direct sunlight and canopy elements in shade. Variations in spectral signature may then be analysed in terms of variations in the partition of radiation between canopy elements (Steven and Rollin, 1985). A further extension of the technique is to use photographs taken at an inclined angle, preferably with a long focal-length lens. In principle this is similar to an inclined point quadrat measurement (Warren-Wilson, 1960, 1965) except that only the first intersection between the quadrat (in this case the measurement point on the photograph) and the canopy is recorded. Because the penetration of radiation through the canopy depends on angle, inclined photographs can, in principle, be used to define the angular distribution of K and by inversion, the angular distribution of the leaves.

86

The results of this study show that in many crops the relationship between radiation interception and foliage cover is sufficiently close for the latter to be used as a substitute for more elaborate measurements of light interception. Less direct relationships may be found in other crops, but nevertheless these will be consistent with simple theory and the conversion requires only a single coefficient, Ki/K f. Once this coefficient is established for a particular crop, by the slope of the relationship between log(1 -- i) and log(1 -- f), photographic measurements of cover may be used as a surrogate for solarimeter data in the analysis of radiation interception and growth. The advantage is that photographs are cheap and easy to take and allow this analysis to be applied more widely; to large scale field trials, to commercial agricultural applications and to more complex canopies such as intercrops and natural vegetation. They are also useful when cover information over a wide area is required at a particular time, and these techniques have been applied as part of the " G r o u n d - T r u t h " exercise in experiments designed to estimate cover by remote sensing from aircraft or satelites (Steven et al., 1983). ACKNOWLEDGEMENTS

The authors are grateful to C.J.A. Clark and Mrs. M. Green who were responsible for a large part of the fieldwork and interpretation of photographs. Assistance was also given by Mrs. C. Daniels and Miss S. Harrison. MDS also expresses his thanks to the Agricultural Research Council for their financial support. REFERENCES Anderson, M., 1966. Stand structure and light penetration. II. A theoretical analysis. J. Appl. Ecol., 3: 41--54. Marshall, B. and Willey, R., 1983. Radiation interception and growth in an intercrop of pearl millet/groundnut. Field Crops Res., 7: 141--160. Monsi, M. and Saeki, T., 1953. Uber den Lichtfaktor in den Pflanzengesellschaften und seine Bedeutung fiir die Stoffproduktion. Jpn. J. Bot., 14: 22--52. Monteith, J., 1973. Principles of Environmental Physics. Edward Arnold, London. Monteith, J., 1977. Climate and the efficiency of crop production in Britain. Philos. Trans. R. Soc., London Ser. B, 281 : 277--294. Neave, H., 1978. Statistics Tables. George Allen and Unwin, London. Paruntu, J., 1984. Growth and interception of light by barley in relation to stress. Ph.D. Thesis, University of Nottingham, Nottingham. Sellers, W., 1965. Physical Climatology. University of Chicago Press, Chicago, IL. Steven, M. and Rollin, E., 1985. The dependence of crop spectral signatures on viewing angle -- a SPOT simulation study. Rep. Dep. of Trade and Industry, Department of Geography, University of Nottingham, Nottingham. Steven, M. and Unsworth, M., 1979. The diffuse solar irradiance of slopes under cloudless skies. Q. J. R. Meteorol. Soc., 105: 593--602. Steven, M. and Unsworth, M., 1980. The angular distribution and interception of diffuse solar radiation below overcast skies. Q. J. R. Meteorol. Soc., 106: 57--61.

87 Steven, M., Biscoe, P. and Jaggard, K., 1983. E s t i m a t i o n of sugar b e e t p r o d u c t i v i t y f r o m r e f l e c t i o n in the red and infrared spectral bands. Int. J. R e m o t e Sensing, 4: 325--334. Szeicz, G., 1974. Solar radiation in crop canopies. J. Appl. Ecol., 11: 1 1 1 7 - - 1 1 5 6 . Szeicz, G., M o n t e i t h , J. and Dos Santos, J., 1964. T u b e solarimeter to measure radiation a m o n g plants. J. Appl. Ecol., 1: 169--174. Warren-Wilson, J., 1960. Inclined p o i n t quadrats. N e w Phytol., 59: 1--8. Warren-Wilson, J., 1965. Stand structure and light penetration. I. Analysis by point quadrats. J. Appl. Ecol., 2: 383--390. APPENDIX A Construction of a r a n d o m i s e d grid: (1) Measure the dimensions x and y of the p h o t o g r a p h area on which you intend to work. If using a projector, measure x and y on the p r o j e c t e d image. (2) Choose the a p p r o x i m a t e n u m b e r o f points N you require for the measurement. A b o u t 1000 gives accuracy to 5%, but fewer points can be used if there is replication. (3) With a regular square grid the spacing o f points along each axis would be x y / N = s. With a r a n d o m grid the mean spacing should be a p p r o x i m a t e l y the same. (4) Select a string of r a n d o m single digits f r o m tables o r f r o m a r a n d o m n u m b e r generator. Treat 0 as 10. The m e a n value of a long string of such digits should approach 5.5. Take s' = s ] 5 . 5 as a scaling factor. (5) Taking each r a n d o m digit d in turn, mark o f f points along the X axis at r a n d o m intervals of d x s' until the X dimension is filled. Construct lines parallel to the Y axis through each point. (6) R e p e a t (5) with a new set of digits for the Y axis, constructing the lines at r a n d o m intervals parallel to the X axis. (7) The intersections of the grid should n o w be spaced at r a n d o m integer intervals f r o m one to ten times s' along b o t h axes. These intersections are the m e a s u r e m e n t points. (8) Variations on this m e t h o d will give equally good results. It is not strictly necessary to m a k e the average spacing on the X and Y axes the same and it can be helpful to choose the spacings and N so that s' is in c o n v e n i e n t units o f measurement.