Published online 2 October 2006
Published in Crop Sci 46:2343-2347 (2006)
© 2006 Crop Science Society of America
677 S. Segoe Rd., Madison, WI 53711 USA
PERSPECTIVES
A Reminder of the Limitations in Using Beer's Law to Estimate Daily Radiation Interception by Vegetation
Thomas R. Sinclair*
Agronomy Physiology Lab., Univ. of Florida, P.O. Box 110965, Gainesville, FL, 32611-0965
* Corresponding author (trsincl{at}ifas.ufl.edu)
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ABSTRACT
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Radiation extinction coefficients for leaf canopies are often calculated using Beer's Law based on midday measurements of radiation interception. However, the assumption of Beer's Law is not appropriate for leaf canopies and this empirical approach needs to be used with caution. Alternatively, classic derivations of radiation interception by leaf canopies have resulted in a similar exponential form defined as a function of time of day, day of year, and latitude. A common experimental approach of determining the extinction coefficient from midday measurements results in a minimum coefficient that underpredicts the total daily radiation interception. Two approaches are explored to improve the estimates of daily radiation interception under a clear sky. The first approach compared extinction coefficients calculated for midday against ones calculated as representative of radiation interception over the entire day. A linear correlation was found between the two extinction coefficients so that the midday extinction coefficient could be corrected to obtain a coefficient representative of the entire day under a clear sky. The second approach estimated the time of day when measurements could be made to obtain directly an extinction coefficient representative of the entire day. While such times in the day could be identified, this approach is impractical because the periods for taking these measurements are short and dependent on leaf angle.
Abbreviations: G, leaf shadow projection • ki, instantaneous extinction coefficient • kd, daily extinction coefficient • L, leaf area index • PAR, photosynthetically active radiation • Rb, irradiance transmitted through the leaf canopy • Rbo, irradiance incident on the leaf canopy • 
, leaf angle • ß, solar angle
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INTRODUCTION
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THE AMOUNT of solar radiation intercepted by vegetation has long been recognized as a key variable in estimating C assimilation rate and growth of plants. In recent years, growth of vegetation over fixed time intervals is often expressed directly as a function of intercepted radiation through the calculation of radiation use efficiency, which is the ratio of canopy growth divided by intercepted solar radiation (Sinclair and Muchow, 1999). Consequently, there is renewed interested in determining the daily amount of solar radiation intercepted by various leaf canopies.
Direct measurement of the daily intercepted radiation is a challenge because instruments need to be placed under leaf canopies and left there for the entire period for which radiation use efficiency is to be determined. Such an experiment involving several treatments and several replications would require permanent installation of a number of radiation sensors requiring a large investment in equipment and in time required to maintain sensors. One solution frequently used to overcome the equipment limitation is to use only one or a few sensors to make spot measurements under leaf canopies at one time of day, usually midday, as representative of radiation interception for the entire day. Such radiation interception data are often interpreted by deriving a radiation extinction coefficient in an exponential radiation-interception equation analogous to the Beer–Bouguer–Lambert Law.
 | [1] |
where: - Rb = irradiance transmitted through the leaf canopy,
- Rbo = irradiance incident on the leaf canopy,
- k = radiation extinction coefficient, and
- L = leaf area index.
Monsi and Saeki (1953) originally suggested that the exponential equation as a function of the leaf area in horizontal layers in leaf canopies was a useful expression of vegetation radiation interception.
A key assumption in Beer's Law, however, is that radiation passes through a medium containing small absorbing and scattering particles distributed uniformly throughout the medium. The basic assumption of Eq. [1], however, is not met in natural vegetation because the canopy is composed of discrete, contiguous leaf surfaces with characteristic angles of display. Further, the angle of the incident direct beam radiation for leaf canopies changes through the day as solar elevation changes. Measurements made near midday, for example, are taken at maximum solar elevation resulting in the greatest penetration of radiation through the leaf canopy. The bias of the midday measurements toward maximum radiation penetration, and hence maximum radiation use efficiency, is dependent on a number of factors, including latitude, time of year, and time of day (Goudriaan, 1982), and on the angles of the leaf segments. Muchow (1985) pointed out that small values for fraction of intercepted solar radiation measured at midday resulted in the greatest deviations from the midday value at other times of the day. Charles-Edwards and Lawn (1984) presented a correction factor to apply to midday measures to obtain a daily integral of light interception but the correction was developed empirically. Their experimental results were obtained for crops sown in rows and, hence, the results should not necessarily be expected to be stable across differing row widths or among species with differing canopy structures.
In this paper, simple calculations are presented to highlight the possible bias in the midday determination of an empirical k value for use in Beer's Law (Eq. [1]). These calculations were done assuming only direct beam solar radiation to highlight the possible discrepancies in the approaches. Similar computations can be made for diffuse sky radiation conditions once the angular distribution of the incident diffuse radiation has been defined (see Appendix).
Two possible approaches are considered to remove the "midday bias" in the extinction coefficient for use in estimating radiation use efficiency. The first approach is to derive a correction factor that can be applied to midday measurements to better represent daily, integrated radiation interception. The second approach is to estimate the appropriate time of day to make measurements that directly give estimates of daily interception of integrated radiation.
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REVIEW OF THEORY
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Instantaneous Radiation Interception Model
Empirical estimations of the radiation extinction coefficient in Eq. [1] continue to be made because the exponential model often fits the radiation interception data very well. Indeed, theoretical analyses have shown that an exponential model is appropriate (deWit, 1965; Duncan et al., 1967), although the assumptions of the derivation are different from that of Beer's Law. The derivation of Duncan et al. (1967) is straightforward starting with an assumed random distribution of leaf area segments within horizontal layers. From geometrical descriptions of the angle of the incident radiation above the horizon (ß) and the angle of the individual leaf elements above the horizon (), they derived the following equation.
 | [2] |
where: G = leaf shadow projection calculated from ß and . Duncan et al. (1967) presented a table of values for G computed for various combinations of ß and . Comparison of Eq. [1] and [2] shows that the empirical extinction coefficient for a leaf canopy (k) should not be expected to be a constant through the day or through the growing season. The value of k is actually a summary term for G/sin (ß), and hence, varies with both ß and .
Equation [2] describes the instantaneous radiation interception for a leaf canopy with a horizontally random distribution of leaf elements. This equation is dependent both on ß and . Of course, the value of ß for direct bean radiation depends on the latitude at which the data are collected, time of year, and the time of day. The influence of is expressed through a complex interaction with ß in the calculation of G, which was presented by Duncan et al. (1967). Their calculations of G were presented in tabular form but to facilitate the comparisons here, regressions of G against ß were done for assumed of 30°, 45°, and 60°. Of course, the regressions could be extended to other , but for many crop canopies the average are in the range of 30° to 60°.
Weighted Daily k Values
The instantaneous value in Eq. [2] of [G/sin (ß)], which will be referred to as ki, was calculated for midday and compared to estimates of a daily weighted value that could appropriately represent k in Eq. [1]. The values of daily k (kd) were based on integrated incident and intercepted radiation over the daily cycle calculated from instantaneous values obtained from Eq. [2].
The first step in these calculations was to define the incident radiation through the daily cycle. The exponential equation presented by Goudriaan (1982) for a humid sky was used to calculate the incident photosynthetically active radiation (PAR) at each time step based on ß.
 | [3] |
where: 640 = PAR irradiance above the atmosphere (J m–2 s–1). The value of G at each time step was obtained based on ß and using the regression equations presented in Fig. 1.
The second step was to calculate the amount of transmitted radiation (Rb) using Eq. [2] for each 10-min interval throughout the day. The instantaneous values of Rbo and Rb were summed over the day and used to calculate daily Rb/Rbo. The daily value of Rb/Rbo was then used to calculate kd by rearrangement of Eq. [1]:
 | [4] |
Values of kd were calculated for L values of 0.5, 1, 3, and 5. In addition, the response of kd to variations in (30, 45, 60°) and latitudes (20°, 30°, and 40° N) were calculated. These calculations were done for day of year of 180, 220, and 240 d, which are representative of times during the major crop growth period in the northern hemisphere.
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RESULTS AND DISCUSSION
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The value of G changed smoothly with changes in ß (Fig. 1). The least variation in G with changing ß occurred with the greatest . The results of Duncan et al. (1967) were readily represented for each by fourth-order polynomials, which had R2 values of 0.96 or greater.
The variation in ki through the daily cycle was calculated for the three assumed (Fig. 2). These results showed that during the midday period at 30° latitude, Day 180 and equal to 30°, the value of ki was stable for an extended period. These results were obtained because the decreases in sin (ß) that occurred on either side of solar noon were approximately matched by decreases in the value of G. These results indicated that determination of ki based on midday measurements of canopy radiation interception gives a stable result and there is likely not a need to consider the exact time at which the measurements are made at midday.
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Fig. 2. Instantaneous extinction coefficients (ki) throughout the day for canopies of various leaf angles.
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Since the value of ki was found to be stable under a range of fairly high sun angles, stability in midday ki was also found when changing latitude from 30° to 20° or to 40° (Fig. 2). Similarly, midday ki on Day 210 or 240 was very similar to those calculated on Day 180. These results indicated that midday ki is likely to be nearly constant through the growing season if leaf angle does not change.
However, stability in ki does not extend over the whole day. At the ends of the day with lower ß, G does not continue to decrease (Fig. 1) so the values of ki increase with further decreases in ß. Due to the fact that the changes in G with ß are different for each leaf angle, the time at which ki increases is different among the different . Stability in ki only exists for about 4 h around solar noon for of 60°, 6 h for of 45°, and 8 h for of 30°. Hence, the time restrictions on measuring light interception in canopies to obtain a "representative" midday value for ki are somewhat more limited as the values of increase.
The midday values of ki decrease as increases (Fig. 2). That is, more erect leaves allow greater radiation penetration deeper in the canopy than more horizontal leaves. However, this situation reverses at the end of the day in that ki increases at the ends of the day resulting in greater ki for the erect leaf crop than for the horizontal leaf crop. The time of crossover in ki among canopies of different in the example in Fig. 2 at latitude 30° and Day 180 was at about 4.5 h on either side of solar noon. At the ends of the day, therefore, the deeper penetration of radiation occurs in canopies with more horizontal leaves.
The values of ki increase substantially at the ends of the day due to low ß. That is, the canopy readily intercepts the radiation by top leaves when the incident angle is low. As a result, virtually all radiation incident to the canopy at the ends of the day is intercepted by the leaves (Fig. 3).
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Fig. 3. Incident and total intercepted radiation calculated for a crop on Day 180 at latitude 30° with leaf area index (L) of 3 and a leaf angle of 45°.
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Since ki increases so greatly at the ends of the day, the midday values of ki fail to account for the much higher radiation interception at low ß. Consequently, ki values estimated from midday measurements of radiation interception by the crop canopy are an underestimate of intercepted solar radiation on a daily basis. This, in turn, results in an overestimation of radiation use efficiency when calculated based on midday ki.
Equation [4] was used to calculate kd based on the amount of total radiation intercepted in the day relative to incident radiation. Calculation of kd, however, is not independent of L because the relative amount of radiation intercepted at various times through the day depends on L. That is, increased L results in proportionally more radiation being intercepted at midday than at the ends of the day so the estimate of kd is decreased. This result is indicated in Fig. 4 where the value of kd decreases to some extent as L increases. The largest change in kd with L occurred with the greatest . There was only small change in kd with changes in latitude and day of the year during the growing season.
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Fig. 4. Daily extinction coefficient (kd) calculated at various leaf area indices and for three leaf angles.
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Values of kd and midday ki were calculated for a range of latitudes and day numbers in addition to and L. Of course, in all cases, the value of kd was greater than midday ki (Fig. 5). That is, the interception of radiation was greater over the entire daily cycle than that estimated from midday ki. Based on these calculations for clear sky conditions, values of kd were linearly correlated (R2 = 0.98) with midday ki. The deviation between the two variables increased at lower values, however. These results are fully consistent with the experimental observation of Muchow (1985) that the deviation between a daily estimate of radiation interception and midday estimates was greater at lower levels of fraction intercepted radiation.
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Fig. 5. Daily extinction coefficient (kd) plotted against midday ki calculated for the same conditions. The dash line represents the line of equality for the two variables and the solid line is the regression between the two variables.
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Based on the regression between kd and midday ki (Fig. 5), kd for the case of equal to 60° was calculated to be about 20% greater than midday ki. However, there was considerable variation in the kd value at equal to 60° resulting from the sensitivity of kd to L. This variation in kd at low values of midday ki indicates that a correction simply based on the regression in Fig. 5 may not be adequate. This concern would be greatest at low L when calculation of radiation interception is especially sensitive to the value of the extinction coefficient (Eq. [1]). To fully make the correction in midday ki to estimate kd, it would likely be necessary to account for the influence of L on the daily extinction coefficient.
An alternative to correcting midday ki to obtain kd is to make measurements of radiation interception by the leaf canopy at times in the day when ki approximates the value of kd. Clearly, in this case the measurements would need to be made near the ends of the day when ki is greater than midday ki. The output ki for each 10 min though the day was scanned to find the time at which the ki matched the value of kd for each set of conditions. The appropriate time to determine ki as a direct estimate of kd are given in Fig. 6 for the morning period only because it is more likely in most climates that the sky will be more clear in the morning than in the afternoon. As already discussed, the change in ki through the daily cycle is dependent on . Increases in ki occur closer to midday in an erect leaf canopy than in a more horizontal leaf canopy. Hence, the time of day to measure radiation interception by a canopy to derive an estimate of kd is dependent on leaf angle. The calculated time to measure radiation interception to estimate kd (Fig. 6) for crops with various leaf angles was approximately 0735 to 0755 h for a 30° crop, 0815 to 0835 h for a 45° crop, and 0845 to 0905 h for a 60° crop.
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Fig. 6. The time in the morning at which radiation interception by a canopy should be measured to obtain directly a ki value that is equivalent to the kd value for the conditions of differing leaf area indices and leaf angle.
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The results of this analysis are discouraging in regards to using the approach of measuring radiation interception under clear sky at alternate times of the day to estimate directly kd. The difficulty in this alternate experimental approach is that ki changes rapidly at the ends of the day (Fig. 2). Therefore, the time window for making the "kd measurement" is narrow and only a limited number of measurements are possible within the short time frame. Further, "kd measurement" is sensitive both to and L, so that a substantial amount of information is needed about the leaf canopy to define the appropriate timeframe to make measurements. Therefore, it seems likely that the better approach to estimating kd would be to determine midday ki and make the appropriate correction, including an accounting for L if such data are available.
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APPENDIX: DIFFUSE RADIATION
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Calculation of an extinction radiation for diffuse radiation is complicated because of the range in possible distributions of radiation originating from various sky segments. A number of complicating atmospheric conditions as well as the angle of the sun gives a wide number of possibilities for the angular distribution of incident diffuse radiation.
The simplest assumption for diffuse radiation is to assume a fully overcast condition with radiation originating uniformly from all segments of the sky. Therefore, the diffuse radiation can be assumed to originate from a hemisphere and the calculation of k is independent of time of day, time of year, or latitude. The fraction of radiation (F) received from each sky segment between angles 1 and 2 can be calculated using the following equation.
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The extinction coefficient considering the entire sky was obtained by calculating the intercepted radiation originating from each 10° segment of the sky. Summing the intercepted radiation from each sky segment gives total intercepted radiation, which was then used in Eq. [3] to obtain a kd for diffuse radiation.
Figure 7 presents the estimates of kd under a uniformly diffuse sky for various and L. This figure is similar to Fig. 4 in that both increasing and increasing L resulted in decreased kd. The influence of L on kd is greater under diffuse radiation than direct radiation because of the proportion of total radiation coming from comparatively low angles with diffuse radiation. Therefore, at low L much of the intercepted radiation is from low sky angles resulting in a large value for kd. Increasing L increases the interception of the radiation originating from higher sky angles and causes kd to decrease. For the assumed uniform source of diffuse radiation, these calculations demonstrate that L of the vegetation, especially with higher leaf angles, may have substantial impact on the value of kd. Therefore, it is not possible to assume a constant kd even under diffuse radiation when there are changes in and/or L through the growing season.
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Fig. 7. Extinction coefficient under uniformly diffuse incident radiation plotted for canopies of various leaf area indices and leaf angles.
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NOTES
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Florida Agricultural Experiment Station, Journal Series No. R-11004.
Received for publication January 23, 2006.
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REFERENCES
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- Charles-Edwards, D.A., and R.J. Lawn. 1984. Light interception by grain legume row crops. Plant Cell Environ. 7:247–251.
- deWit, C.T. 1965. Photosynthesis and leaf canopies. Agr. Res. Rep. no. 663, Agr. Publ. Doc., Wageningen.
- Duncan, W.G., R.S. Loomis, W.A. Williams, and R. Hanau. 1967. A model for simulating photosynthesis in plant communities. Hilgardia 38:181–205.[Web of Science]
- Goudriaan, J. 1982. Potential production processes. p. 98–113. In F.W.T. Penning de Vries and H.H. van Laar (ed.) Simulation of plant growth and crop production. Pudoc, Wageningen, The Netherlands.
- Monsi, M., and T. Saeki. 1953. Uber den Lichtfaktor in den Pflanzengesellschaften. (In German.) Jpn. J. Botany 14:22–52.
- Muchow, R.C. 1985. An analysis of the effects of water deficits on grain legumes grown in a semi-arid tropical environment in terms of radiation interception and its efficiency of use. Field Crops Res. 11:309–323.[CrossRef]
- Sinclair, T.R., and R.C. Muchow. 1999. Radiation use efficiency. Adv. Agron. 65:215–263.
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ABSTRACT
INTRODUCTION
