Paper presented at the 2nd International Conference on Tropical Climatology, Meteorology and Hydrology TCMH-2001, Brussels, Belgium; 12-14 December 2001. This paper is a copyright © by C.B.S. Teh, 2001. Note: Table and Figures are at the bottom of this page.



Modelling the partitioning of solar radiation capture and evapotranspiration in intercropping systems


Christopher B.S. TEH, Lester P. SIMMONDS, Tim R. WHEELER

KEYWORDS. Radiation; Evapotranspiration; Shuttleworth-Wallace; Maize; Sunflower; Intercrop

SUMMARY. The primary purpose of this study was to model the partitioning of radiation capture and evapotranspiration in a two-crop (maize and sunflower) intercropping system. Two field experiments were conducted in 1998 and 1999. Detailed canopy architecture was measured, and transpiration and soil evaporation were measured using sap flow gauges and lysimeters, respectively. One- (1D) and two-dimensional (2D) models were developed for modelling radiation attenuation in one dimension (vertical) and two dimensions (vertical and horizontal), respectively. The simpler 1D model was slightly more accurate than the more complex 2D model, where their mean errors (95% error range in brackets) for estimating the fractional radiation interception were 0.01 (-0.09 – 0.11) and -0.04 (-0.13 – 0.06), respectively. Nevertheless, the hourly simulations by the 2D model followed the measured diurnal trend of total radiation capture more closely than those by the 1D model. The Shuttleworth-Wallace evapotranspiration equation was extended and applied to intercropping systems. Its mean prediction error for transpiration was near zero (-0.01 mm h-1), and its accuracy was not affected by plant growth stages, but simulated transpiration during high measured transpiration rates was underestimated. There were also larger errors in predictions by both models for daily soil evaporation than for plant transpiration.



Factors affecting the plant-radiation regime are the amount and quality of incident radiation, canopy architecture, and the optical properties of leaves and soil (Sinoquet and Caldwell, 1995). The simplest way to model how irradiance varies within canopies is to assume that irradiance varies only vertically; that is, in one dimension. However, this assumption is only valid in homogenous canopies, achieved when canopies are closed and randomly mixed. Consequently, some workers (e.g., Sinoquet and Bonhomme, 1992; De Castro and Fetcher, 1998) have extended the modelling of radiation attenuation in two or three dimensions in attempts to produce more representative and accurate models.

Compared to radiation modelling, evapotranspiration modelling is usually more difficult because it involves a simultaneous mathematical analysis of above and below ground spaces (Tournebize et al., 1996). Evaporation from the soil or transpiration from the plants is often estimated using the Penman-Monteith (PM) equation (Monteith, 1965), but the PM equation can only be used to estimate either soil evaporation or plant transpiration, but not both simultaneously. A recent, important extension of the PM equation is the Shuttleworth-Wallace (SW) equation (Shuttleworth and Wallace, 1985) because it specifies explicitly the energy exchanges at the soil and canopy, thereby making it possible to distinguish the fraction of water transpired and that evaporated from soil (Farahani and Ahuja, 1996). The SW equation is still relatively new and not extensively used (Farahani and Ahuja, 1996), but field tests of this equation have been promising (Lafleur and Rouse, 1990; Wallace et al., 1990; Farahani and Bausch, 1995).

Thus, the main objectives of this study were: 1) to model the partitioning of captured radiation and evapotranspiration in a maize-sunflower intercropping system in partial- and fully-grown canopies, and 2) to determine the degree of complexity that is required for radiation modelling.

Development of models

Two kinds of models were developed: a one-dimensional (1D), and a two-dimensional (2D) model. The 1D model was so-called because it modelled the radiation regime in one dimension, where irradiance varies only in a vertical direction. This means at a given canopy height, the irradiance at any point within the canopy is equal. The 2D model, however, modelled the radiation regime in two dimensions, where irradiance varies both vertically and horizontally. Both these two models, however, modelled evapotranspiration in the same way by applying the extended SW equation to intercropping systems.

Partitioning of captured radiation

Both 1D and 2D models use the same weighting factor as given by Spitters (1989) to partition the captured radiation for a given crop species i. The share of intercepted radiation for each species depends on its leaf area index and extinction coefficient; that is, for species i, its share of captured radiation is weighted by

where Ac,i is the amount of radiation captured by crop species i; A is the total amount of radiation captured by all crops in the system; and w i is the weighting factor, a value between 0 and 1, for crop species i:


where ki and Li are the extinction coefficient and leaf area index of crop species i, respectively; n is the total number of crops in the system; and s i is the leaf scattering coefficient of radiation for crops species i (Tournebize and Sinoquet, 1995) which is 0.14 and 0.10 for sunflower and maize, respectively (Masoni et al., 1994).

Foliage distribution and plant-radiation regime for the 1D model

The foliage distribution of a crop is characterised mathematically by the 1D model using the G-function defined by Ross and Nilson (1965) as

where gL(rL) is the leaf normal distribution function which expresses the probability that a leaf normal is around direction rL, where direction rL is described by a very narrow normal inclination range q L+ q L and normal azimuth range f L+ f L (Lemeur, 1973a, b); cos rLr is the cosine angle between the leaf normal direction rL and sun direction r, and is calculated by

and is the projection of the part of the leaf area with normals within the solid angle dW L around the sun direction r. Consequently, G(r) is the regarded as the average projection per unit foliage area in the sun direction r, where the sun direction is described by inclination q and azimuth f . The function g(rL) or g(q L, f L) is simplified by assuming that the leaf normal inclination q L is independent of the leaf normal azimuth f L, so then , where g(q L) is the distribution function for leaf normal inclination, and g(f L) is the function describing leaf normal azimuth distribution (Lemeur, 1973a).

G(q , f ) is corrected to account for two situations: 1) radiation scattering by leaves, and 2) when leaves are not randomly distributed but clumped as found in row crops:

where is the corrected G-function; and W (q ) is the clumping factor (Tournebize and Sinoquet, 1995; Campbell and Norman, 1998; Kustas and Norman, 1999a, b). W (q ) is determined empirically by


and e is the ratio of plant height to width; and Lc is the localised leaf area index determined by

where fc is the fractional canopy cover which is the fraction per unit ground area occupied by canopy cover and can be approximated by taking the ratio of canopy width to row spacing (Campbell and Norman, 1998). Extinction coefficient k is related to the G-function by

(Ross and Nilson, 1965; Lemeur, 1973a, b).

Direct radiation within mixed canopies is calculated by

where I0,dr is the amount of direct radiation above canopy; n is total number of crops; and p is the mean canopy reflection coefficient calculated by

(Goudriaan, 1977, 1988). Diffuse radiation within mixed canopies is calculated by

and integrated numerically using the 5-point Gaussian method (Goudriaan, 1988). The B(q , f ) is the brightness function or the amount of diffuse radiation component coming from sky direction q inclination and f azimuth (Charles-Edwards et al., 1986).

The total amount of direct radiation Adr and diffuse radiation Adf intercepted by all crops are calculated by

The amount of radiation captured by crop species i is then calculated by


where Adr,c,i and Adf,c,i are the amount of direct and diffuse radiation captured by crop species i, respectively; and w i is determined from Eq. (1).

Foliage distribution and plant-radiation regime for the 2D model

Unlike the 1D model, the 2D model divided the canopy space, as described by Sinoquet and Bonhomme (1992), into a set of contiguous rectangular cells, forming a two-dimensional grid network that is perpendicular to the planting row direction (Fig. 1). The aerial space from the soil surface to the canopy top is divided into Nz horizontal layers of thickness Ez, and Nx vertical sections of thickness Ex. The horizontal cell thickness Ez need not be equal to the vertical cell thickness Ex.

The probability (Pk) of total radiation intercepted within the k-th cell visited by a single beam is calculated by


where the multiplicative series c=1 to (k-1) represents every cell visited sequentially by the beam in reaching the target cell k; Gjc(r) is the G-function for the j-th crop in the c-th cell; r f,jc is the leaf area density for the j-th crop in the c-th cell; sc is the beam path length in the c-th cell; and n is the total number of crops (Tournebize and Sinoquet, 1995). Consequently, the fraction of total radiation captured by crop species i in k-th cell Fki is determined by

where w ki is determined from Eq. (1). The height and width of cells were fixed at 0.2 m in this study (Ex = Ez = 0.2 m).

As shown in Eq. (3), three properties must be determined for each cell in the grid network to determine the irradiance within the canopies: the horizontal and vertical distributions of 1) G-function G(r), 2) leaf area density r f, and 3) distance of beam travel s. Unlike the 1D model, the 2D model used a simpler form of the G-function:

where Lki,j is area of the j-th leaf of crop species i in the k-th cell; and N is total number of leaves in k-th cell from crop species i (Thanisawanyangkura et al., 1997). Information about the leaf area density in each cell are obtained from the plant profile method as described by Stewart and Dwyer (1993). Calculations for the distance of beam travel are based on simple geometry as described by Allen (1974), Gijzen and Goudriaan (1989), and Sinoquet and Bonhomme (1992).

A beam with the same inclination and azimuth can enter at any point on the cell, thus the number of beams "pushed" into the cell was pre-determined by several trial runs to obtain the minimum number of beams that can be used without sacrificing accuracy if more beams were used. This study used two beams to be pushed into a cell for a given sun azimuth and inclination. In several runs of radiation simulation, the error of using two beams instead of ten beams did not exceed 5%.

Each computed sc is then substituted into Eq. (3) to determine Pk so that the mean probability of intercepted radiation for a given beam direction can finally be calculated (). Direct radiation intercepted by all crops within cell k is calculated as

where I0,dr is amount of direct radiation above canopy; and p is mean crop reflection coefficient. Diffuse radiation intercepted by all crops within cell k is calculated as

where I0,df(W ) is the incident diffuse radiation coming from direction W . To determine I0,df(W ), the sky is divided into five equal inclination intervals (p /10) and eight equal azimuth intervals (p /4). Consequently, the amount of incident diffuse radiation I0,df(W ) from each of the 40 directions is determined using the brightness function B(q , f ).

The amount of direct and diffuse radiation captured by crop species i within k-th cell are determined similar to Eq. (2):

Note that the above equation is used to determine the amount of radiation captured by an individual crop species within a given cell. To determine the amount of radiation captured by the whole crop, the 2D model calculated the total amount of radiation that was captured while a beam travelled from the canopy top (represented by the uppermost cell row in the network) to the soil surface (represented by the most bottom cell row in the network) (Fig. 1).


The Shuttleworth-Wallace (SW) (1985) equation was extended to include the transpiration from two or more crops and evaporation from the soil (Wallace, 1997). The energy budget of the system is described in a series of equations which are the sum of the various latent heat, sensible heat and radiation fluxes (Fig. 2). With some algebraic manipulations, it can be shown that the total latent heat flux of the system with n crops is given by


where cp is the specific heat of water at constant pressure (4182 J kg-1 K-1); D is the vapour pressure deficit, or ; D is the mean rate of change of saturated vapour pressure with temperature, or ; g is the psychometric constant (0.658 mb K-1); A and As are the total energy available to the system and soil, respectively, and Ac,i is the amount of energy available to crop i so that:

where Fi is the fraction of radiation intercepted by crop species i. Fi can thus be regarded as the link between radiation and evapotranspiration models. The available energy available to the soil As is:

The partitioning of the various latent heat fluxes is determined from the total latent heat flux l E which is the sum of all latent heat fluxes in the intercropping system, or in a two-crop intercropping system:

where D0 is the vapour pressure deficit at the canopy source height, or

The SW model required several resistance components to be known: raa (resistance between mean canopy flow and reference height); rc,is (bulk stomatal resistance); rc,ia (bulk canopy boundary layer resistance); rsa (resistance between soil and mean canopy flow); and rss (soil surface resistance). These resistances are calculated from the equations given by Choudhury and Monteith (1988), and Shuttleworth and Gurney (1990).

Materials and methods

Maize (Zea mays L. cv. Hudson) and sunflower (Helianthus annuus L. cv. Sanluca) were sown on 22 May 1998 at Sonning Farm, Reading, UK (51° 27’ N and 0° 58’ W). Total field size was 0.13 ha, and planting rows were in a NE-SW direction. Inter-row distance was 0.6 m, but intra-row planting distance for maize was 0.3 m and sunflower was 0.6 m, so that the ratio of maize to sunflower was 2:1. The planting density of maize was 3 plants m-2, and 1.5 plants m-2 for sunflower.

Measurements on canopy architecture followed the method by Lemeur (1973a, b) and Ross (1981), where leaf inclination and leaf azimuth were measured using a protractor, measuring tape and compass. Canopy architecture was measured for four to six periods in a day on three plants and all leaves of a plant were measured. Irradiance was measured using a sunfleck ceptometer (Decagon Devices Inc., Pullman, Washington, USA; Model SF-80). Sap flow was measured using customised sap flow gauges based on the concept of stem heat balance (Kucera et al., 1977). The sap-flow gauges were fitted to two maize and two sunflower plants. Data was collected every 10 s and averaged at 10-min intervals using a Campbell CR10 (Campbell Scientific Inc., Shepshed, UK) data logger. PVC lysimeters were used to measure soil evaporation (six replicates). The lysimeters measured 50 mm in diameter and 120 mm in depth. The lysimeters were placed in the middle of two planting rows, and the openings of the lysimeters were placed level with the soil surface. The soil in the lysimeters were changed every seven days or after each raining period, and the lysimeters were weighed every one to two days. Daily and half hourly weather data (air temperature, total incoming radiation, wind speed and vapour pressure) were obtained from the automatic weather station at Sonning Farm.

An exact field experiment as in 1998 was conducted again in 28 May 1999 to obtain the early crop growth periods.

Results and discussion

Radiation capture

For the 1D model, there was a close clustering of points along the 1:1 ratio line, and there was no trend of estimation error (Fig. 3). The mean error (simulated minus measured) was nearly zero (0.01) and 95% of these errors were limited to a narrow range (-0.09 to 0.11). The more complex 2D model, however, tended to underestimate when the fractional radiation interception was around 0.80-0.90 (Fig. 3). Mean error was -0.04 which indicated an overall tendency to underestimate, but compared to the 1D model, 95% of the prediction errors from the 2D model were limited to a narrower range (-0.13 to 0.06).

Though the 1D model generally simulated the radiation interception more accurately than the 2D model, the latter model depicted the diurnal trend or pattern of radiation interception more accurately (Fig. 4). Radiation interception depended on the solar position, whereby radiation interception decreased gradually as the sun begun to align in parallel to the row direction (NE-SW). This gradual decline, however, decreased abruptly and sharply at about 14:30 hours when the sun was parallel to the row direction. The radiation intercepted was at the lowest at 15:30 hours, and after this hour radiation interception begun to increase. The existence of a planting row structure has been shown to affect the diurnal pattern of radiation interception (e.g., McCaughey and Davis, 1974, Wallace et al., 1990).

Plant transpiration

Both the 1D and 2D models predicted transpiration of the intercrop components with reasonable accuracy (Table 1 and Fig. 5). For each model, there was a close clustering of points along the 1:1 line of equality, and their mean errors were zero or near zero which indicated little bias in estimation errors. Moreover, 95% of these estimation errors occurred in a narrow range within -0.09 to 0.07 mm h-1. The 2D model had a slightly broader error range than the 1D model, and the mean error by the 2D model was –0.01 compared to 0.00 by the 1D model. This revealed that the 2D model was slightly less accurate than the 1D model. Nevertheless, both models tended to underestimate transpiration slightly for intercrop maize and intercrop sunflower when measured transpiration exceeded 0.15 and 0.40 mm h-1, respectively (Fig. 5). And this underestimation was slightly larger for the 2D model than the 1D model. Prediction by the 2D model was less accurate than the 1D model because, as shown in Fig 3, the 2D model predicted radiation interception in the intercrop slightly less accurately than the 1D model. It followed that because the 2D model tended to underestimate radiation when measured intercepted radiation exceeded 0.80, this would also lead to an underestimation of transpiration especially during high transpiration rates. This is because intercepted radiation is one of the major driving forces of transpiration (Wallace, 1995).

Fig. 6 and 7 show the diurnal transpiration rates for the intercrop components. As expected, transpiration rates usually increased from morning, reaching a maximum at about 12:00 to 14:00 hours, then decreasing towards late evening. This observed diurnal trend of transpiration generally followed the measured diurnal trend of temperature, solar irradiance and vapour pressure deficit (Ozier-Lafontaine et al., 1997).

Overall, both 1D and 2D models predicted the partitioning of transpiration between maize and sunflower in the intercropping system with reasonable accuracy. The accuracy of both models were quite robust to different plant growing stages. Nevertheless, accuracy could be improved by differentiating between sunlit and shaded leaves. In this study, total leaf area of maize and sunflower was assumed to intercept all radiation, and this may lead to a considerable amount of underestimation of canopy resistance which would increase with canopy development (Ozier-Lafontaine et al., 1997). Furthermore, both models in this study did not model soil water movement but assumed that the soil was always at field capacity, having a constant volumetric water content of 0.20 m3 m-3. This could be why both models tended to slightly underestimate transpiration during high measured transpiration rates. Another possible source of error was the uncertainties in characterising the micrometeorological conditions of each crop species such as the convective transfer and wind speed profile within the heterogeneous and clumped canopies. Finally, the factor of plant hydraulic capacitance was not considered in this study. Plant hydraulic capacitance, which is the storage of water in the plant, can cause discrepancies between the diurnal pattern of sap flow and transpiration especially for plants that are water-stressed and have large stems (Caspari et al., 1993; Zhang et al., 1997).

Daily soil evaporation

Simulations from both 1D and 2D models followed the observed trend of soil evaporation quite closely (Fig. 8). In the 1998 experiment, for example, daily soil evaporation increased from DAS 70 to 75, and decreased from DAS 78 to 84. Simulations by both models also showed the same increasing and decreasing patterns for DAS 70 to 75 and DAS 78 to 84, respectively. Of the two models, the 1D model was more accurate in estimating daily soil evaporation. Its mean error was closer to zero (0.02 mm day-1) as compared to the mean error by the 2D model (-0.08 mm day-1), showing that the 2D model tended to underestimate the daily soil evaporation. Nevertheless, the 95% error range for the 1D model was larger (-0.80 to 0.85 mm day-1) than the 2D model (-0.84 to 0.69 mm day-1). This showed that though the 1D model was, overall, more accurate than the 2D model, 95% of the estimation errors by the former model occurred in a slightly wider range than by the latter model.


This study successfully developed two models (the 1D and 2D models) that were shown to be reasonably accurate in modelling: 1) the total captured radiation, 2) the partitioning of total captured radiation between two crops of comparable heights: maize and sunflower, and 3) the partitioning of transpiration between these two crops. The accuracy of these two models were also robust to the different crop growth stages and canopy covers. However, compared to the 2D model, the 1D model was overall slightly more accurate. That the simpler 1D model was slightly more accurate than the more complex 2D model already indicated a huge simplification step in the modelling process. Computations by the 1D model were simpler, less data-demanding and much faster than the 2D model.


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Table 1. Simulation error of transpiration (mm h-1)


Intercrop Component

Mean Error (simulated – measured)

Mean – 2SD

Mean + 2SD

1D model











2D model










SD - standard deviation



Fig. 1. The canopy space is divided into a network of cells in the 2D model




Fig. 2. Schematic diagram of the various energy fluxes, temperatures, vapour pressures and resistances in a two-crop intercropping system. Key: l E, l Ec,1, l E c,2 and l Es are latent heat fluxes from the system, first crop, second crop and soil, respectively; H, Hc,1, Hc,2 and Hs are sensible heat fluxes from the system, first crop, second crop and soil, respectively; Rn and Rsn are net radiation fluxes into the system and to the soil, respectively; G is heat conduction into the soil; Tr, Tf,1, Tf,2, To and Ts are temperatures for the reference height, first crop, second crop, mean canopy flow (canopy source height) and soil, respectively; er and eo are vapour pressure at the reference height and mean canopy flow, respectively; ew(T) is saturated vapour pressure at temperature T; raa is aerodynamic resistance between the mean canopy flow and reference height; rc,1s and rc,2s are bulk stomatal resistance for the first crop and second crop, respectively; rc,1a and rc,2a are bulk boundary layer resistance of the canopy for the first crop and second crop, respectively; rsa is aerodynamic resistance between the soil and mean canopy flow; and rss is soil surface resistance.





Fig. 3. Comparisons between simulated and measured fraction of total incident radiation intercepted, where simulation was by the: (a) 1D model, and (c) 2D model



Fig. 4. Diurnal faction of total incident radiation intercepted in the 1998 experiment on DAS 78





simulation by 1D model

simulation by 2D model









Fig. 5. Comparisons between simulated and measured transpiration for the intercrop components in the 1998 and 1999 experiments









Fig. 6. Comparisons between simulated and measured diurnal transpiration for the intercrop maize on: (a) DAS 71, 1998; (b) DAS 78, 1998; (c) DAS 95, 1998; (d) DAS 44, 1999; (e) DAS 48, 1999; and (f) DAS 53, 1999









Fig. 7. Comparisons between simulated and measured diurnal transpiration for the intercrop sunflower on: (a) DAS 71, 1998; (b) DAS 78, 1998; (c) DAS 95, 1998; (d) DAS 44, 1999; (e) DAS 48, 1999; and (f) DAS 53, 1999






Fig. 8. Comparisons between simulated and measured daily soil evaporation for the: (a) 1998 experiment, and (b) 1999 experiment