Agronomy Journal 93:333-338 (2001)
© 2001 American Society of Agronomy
AGROCLIMATOLOGY
Inclusion of the Fractal Dimension of Leafless Plant Structure in the Beer-Lambert Law
Kayhan Foroutan-pour,
Pierre Dutilleul and
Donald L. Smith
Department of Plant Science, McGill University, Macdonald Campus, 21,111 Lakeshore, Ste-Anne-de-Bellevue, Quebec, Canada H9X 3V9
Corresponding author (cydp{at}musica.mcgill.ca)
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ABSTRACT
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The Beer-Lambert law is commonly used to describe the relationship between the proportion of light penetrating a plant canopy and the leaf area index (LAI). Although the geometric distribution of leaf area has a potential effect on the ability of a plant to intercept light, the equation contains no term to account for it. In this study, the geometric distribution of leaf area was quantified by the fractal dimension of leafless plant structure (FD). The objective was to evaluate the contribution of plant structure complexity to the Beer-Lambert law, by including FD in the equation. The crop was soybean [Glycine max. (L.) Merr.]. Data were collected according to a block design with four blocks and five weekly repeated measures. The analyzed variables were LAI and light penetration (% per plant), and FD, estimated using leafless plants photographed from the side that allowed the maximum appearance of branches and petioles. Statistical analyses were performed week by week, on weekly means and on block means. When LAI and FD were significantly correlated (i.e., at the end of canopy development and on weekly means), inclusion of either variable as regressor in the equation provided similar goodness-of-fit. In other instances, inclusion of FD as a multiplicative factor of LAI increased the r2 value up to 0.31. In all instances, the correlation between light penetration and FD was stronger than between light penetration and LAI. In summary, the application of the Beer-Lambert law for light penetration into the canopy is improved by inclusion of FD.
Abbreviations: FD, fractal dimension • LAI, leaf area index
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INTRODUCTION
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BECAUSE they are photosynthetic, higher plants must array their foliage in space to optimize light interception and gas exchange. The relationship between light penetration and leaf area has been modeled with the Beer-Lambert law for light penetration into translucent media (e.g., Monsi and Saeki, 1953; Anderson, 1966; Brown, 1984; Lang, 1987; Brown and Parker, 1994; Maass et al., 1995; Vose et al., 1995):
 | (1) |
where I is the irradiance under the crop canopy, I0 is the irradiance above the crop canopy, LAI is the leaf area index, and k is the extinction coefficient or fraction of light intercepted per LAI unit. Related formulae that account for the clumping of canopy tissues (Oker-Blom and Kellomäki, 1983) or the effect of sun angle (Smith et al., 1991) have also been used.
Equation [1] describes the relationship between proportional light penetration and LAI. It has no term characterizing the geometric structure of the canopy, although this can have a large effect on the ability of a plant to intercept light (Williams et al., 1968; Tetio-Kagho and Gardner, 1988a, 1988b; Aries et al., 1993). For example, two soybean plants with similar leaf area may have different foliage distributions in space and hence, different patterns of light interception (Fig. 1). In fact, the architecture of soybean vegetation is determined by the branching pattern through which the petioles array the leaves in space. Although the leaves intercept sunlight, they are not a major determinant of the plant structure, and the geometric distribution of leaf area is determined by the petioles and the branches that support them. Thus, the leafless structure, including the petioles, is of interest in soybean, and combining LAI with an appropriate quantification of the leafless structure might improve the description of light penetration into the canopy in such a case.
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Fig. 1. Images of two soybean plants (a, c) with their corresponding leafless structure (b, d). The plants are from canopies with LAIs of 4.18 (a) and 4.14 (c). Light penetrations (% per plant) are 11.0 (a) and 17.8 (c), and fractal dimensions of leafless structure are 1.45 (b) and 1.37 (d), respectively
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Fractal analysis has provided a novel approach for quantifying geometric structures such as the plant canopy structure (Morse et al., 1985; Corbit and Garbary, 1995; Kübler and Dudgeon, 1996; Critten, 1997; Zeide, 1998; Foroutan-pour et al., 2000). Specifically, the fractal dimension of leafless plant structure has been shown to measure the complexity of the vegetative structure of soybean plants (Foroutan-pour et al., 1999a).
In this study, the geometric distribution of leaf area of soybean plants at different stages of development is quantified by the fractal dimension of leafless plant structure (FD). The objective is to evaluate the contribution of plant structure complexity to the Beer-Lambert law, by including FD in the equation. The collected data are analyzed in a number of approaches to find those that emphasize instances like that illustrated in Fig. 1.
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MATERIALS AND METHODS
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Field Experimentation
An experiment was carried out at the Emile A. Lods Agronomy Research Center, Macdonald Campus of McGill University, Ste-Anne-de-Bellevue, QC, Canada. The experiment was arranged as a block design with four blocks (plots) and five temporal (weekly) repeated measures. It was conducted on an Endoaquept soil. The soybean variety used was `Maple Glen'. This variety is commonly grown in eastern Canada. In the years preceding this experiment, the site had been used to produce corn (Zea mays L.). Soybean seeds were inoculated with Bradyrhizobium japonicum before sowing. Soybean was planted in 2.25 by 5 m plots (6 rows spaced 37.5 cm apart) on 15 May 1997. The plant population density was 450000 plants ha-1. To avoid border effects, two rows of soybean with the same plant population density as the plants in the plot were seeded on either side of the plot and, additionally, a 3 m spacing was left between adjacent plots. Plants received no irrigation, and were fertilized as recommended (CPVQ, 1994) with 20 kg N ha-1, 32 kg P ha-1, and 62 kg K ha-1 applied during field preparation, as recommended by a soil test.
Measurements and Data Collection
The LAI and canopy light penetration were measured weekly, 48, 55, 62, 69, and 76 d after planting. Measurements were made at random locations along a diagonal transect through each plot. The average value for the plot was retained for further calculation. The LAI was measured with a LAI-2000 Plant Canopy Analyzer (LI-COR, Lincoln, NE), and canopy light penetration (µmol s-1 m-2) with a Line Quantum Sensor (LI-COR, Lincoln, NE). We rigorously followed the operating instructions for measuring devices. In particular, the LAI measurements were made early in the day. Light penetration (%) of plant canopy was calculated as (I/I0) x 100, where I0 is the irradiance above a plant canopy and I is the irradiance under the same plant canopy. To have measurements on an individual plant basis, light penetration (% per plant) was calculated using the plant population density in question. Note that light penetration hereafter refers to an individual plant, except when the context indicates otherwise.
For fractal analysis, one soybean plant was randomly sampled in each plot at each time of measurement of LAI and canopy light penetration. The sampled plants were cut at the base of the stem. The leaf blades were detached immediately, leaving petioles, and the structure of the leafless plant was photographed from the side that allowed the maximum appearance of branches and petioles. The images of leafless soybean plants collected during the 5 weeks of measurement represent the development of the soybean canopy. Therefore, all the estimates of FD in this study were obtained from images of leafless soybean plants, following Foroutan-pour et al. (1999a)(1999b).
Procedure of Fractal Dimension Estimation
Mandelbrot (1983) discusses a number of techniques that can be used to estimate the fractal dimension. One of these techniques is the box counting method, which has been shown to be appropriate for estimating FD from two-dimensional images (Foroutan-pour et al., 1999b). In this method, each image is covered by a sequence of grids made of squares of descending sizes. For each grid, two values are recorded: the number of squares intersected by the image, N(s), and the side length of squares, s. The regression slope (D) of the straight line formed by plotting log[N(s)] against log(1/s) indicates the degree of complexity, or FD, which ranges between 1 and 2 (1 
D 2) (Mandelbrot, 1983). An image for which FD is equal to 1 or 2 will be considered as completely differentiable or very rough and irregular, respectively. For example, a straight line has an FD of 1, as the number of boxes intersecting it increases exponentially from to 2 to 4, 8, ... , 2n; this object of classical geometry is not the only image with an FD of 1 in fractal analysis (Mandelbrot, 1983). We used the following linear regression equation to estimate FD:
 | (2) |
where log denotes the natural logarithm, C is a constant and N(s) is proportional to (1/s)D (Mandelbrot, 1983).
In this study, the box counting method of FD estimation was performed using the Fractal Dimension Calculator software for the Apple Macintosh computer, written by Paul Bourke of the School of Architecture, University of Auckland, New Zealand (available at paul@bourke.gen.nz.). We followed the procedure presented in Foroutan-pour et al. (1999b).
Inclusion of Fractal Dimension in the Beer-Lambert Law
To be advantageous, the inclusion of FD in the Beer-Lambert law must provide an improved explanation of light penetration, over the use of LAI alone. In particular, the inclusion of an FD value in the equation must be consistent with the pattern of light penetration. If FD was included as a divisor of LAI, the pattern of decreasing light penetration into the canopy with increasing leaf area would be eclipsed by any increase of the plant structural complexity. The LAI and FD cannot be added to each other or subtracted from each other, because they are not of the same nature. Among the four elementary arithmetic operations, the remaining one consists of multiplying LAI by FD, which is in accordance with the negative relationship to be expected between proportional light penetration and plant structural complexity.
To assess whether this type of inclusion of FD in the equation improves the description of proportional light penetration, a regression model was fitted after log-transformation, once with the original equation:
 | (3) |
and once by including FD as a multiplicative factor of LAI in the equation:
 | (4) |
Note that the coefficients (slopes) k and k' in Eq. [3] and [4] are equal if FD = 1.
We have used the two equations above to fit regression models to our data in three approaches. In the first one (i.e., the week-by-week approach), each week of data was analyzed separately. In the second approach, weekly means were calculated across blocks, so that each point in the regressions originated from plants with similar development stages. In the third approach, block means were calculated across weeks and each point was an average over plants at different stages of development. The motivation in following three approaches was to find instances in which the ranges of FD and LAI values were, respectively, wide and narrow. These approaches are typical of the analysis of repeated measures (see, e.g., Dutilleul et al., 1998). They were also followed here in the correlation analyses (see below).
Statistical Analyses
Two types of statistical analyses were performed on the light penetration, LAI and FD data: (i) preliminary correlation analyses using Pearson's r linear coefficient and (ii) major axis regressions without intercept, including tests of difference in the major axis slope between Eq. [3] and [4]. All analyses were performed in each of the three approaches described above.
The central part of these analyses is the fitting of Eq. [3] and [4] and the comparison of their goodness-of-fit. Major axis regression had to be preferred over classical linear regression because of the random nature of the regressors, LAI and (LAI x FD), neither of these being fixed by the experimenter. We used the squared linear correlation coefficient r2 to measure the goodness-of-fit. On the other hand, the difference between major axis slopes provides a different type of information by measuring the discrepancy of FD from 1.
The major axis slopes k and k' in Eq. [3] and [4] were estimated following Sokal and Rohlf (1995)(p. 544). Differences between major axis slopes were tested according to Steel and Torrie (1980)(p. 258), following Sokal and Rohlf's recommendation that the standard error of a major axis slope can be approximated by the standard error of the linear regression slope (i.e., when the regressor is not random). The absence of intercept in Eq. [3] and [4] was taken into account in all our regression analyses. The correlation analyses were carried out with SAS procedure CORR (SAS Inst., 1997).
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RESULTS
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Correlation Analyses
Results are summarized in Table 1. In all three approaches, correlations between light penetration and LAI and between light penetration and FD are negative, whereas those between LAI and FD are positive. The strongest correlations are observed between light penetration and LAI and between light penetration and FD in Week 5 and on weekly means, and between LAI and FD in Weeks 3 and 5 and on weekly means. Remarkably, all correlations between light penetration and FD are >0.9 in absolute value. Accordingly, all these correlations are either significant or almost significant at the 0.05 level.
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Table 1. Pearson's r linear correlation coefficients with the associated probabilities of significance (below, in parentheses), for each pair of variables and by approach
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The correlation between light penetration and LAI is significant (Week 5 and on weekly means) or its significance is third in ascending order (Week 3) if LAI and FD are significantly correlated. Otherwise, the correlation between light penetration and LAI is weaker, especially at the earlier stages of canopy development (Weeks 1 and 2) and on block means. Thus, the amount of leaf area, as measured by LAI, is correlated with light penetration when the range of values of both variables is wider, that is, when both are studied during the whole canopy development or at the end of it. The complexity of the geometric distribution of leaf area, as quantified by FD, does not seem to follow the same pattern. Recall that the statistical concept of correlation is independent of the scale of observation, since it is a standardized covariance (Sokal and Rohlf, 1995, p. 555–608). This said, one must bear in mind that the FD data analyzed here were directly estimated from the sampled plants, whereas the LAI and light penetration data were both calculated on an individual plant basis from average values for the plot. Notwithstanding, the results above imply that in a situation where LAI and FD change together in the same direction, either of them or both can be used to predict light penetration into the canopy.
By working with block means, the temporal increase in LAI vanished and the mean LAI values did not fluctuate much. Working with block means or with weekly data collected at earlier stages of canopy development is similar, to some extent, to working with plants having similar LAI but distinct geometric structures quantified by FD. Such plants are characterized by different light penetrations into the canopy (Fig. 1). In these instances, both variables may need to be included in the Beer-Lambert equation to describe light penetration more completely.
Inclusion of Fractal Dimension in the Beer-Lambert Law
In this part of our results, we report on the fitting of equations that correspond to the original version of the Beer-Lambert law (Eq. [3]) and its modification by the inclusion of FD (Eq. [4]). Results are presented in Fig. 2 and 3. The r2 values reported below for the regression of light penetration on FD were obtained as the square of the r values for log[(I/I0) x 100] and FD in Table 1.
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Fig. 2. Major axis regressions fitted to each week of data separately, using Eq. [3] (a–e) and Eq. [4] (f–j). Two points overlap one another in panel e. Note that the regressor is random in a major axis regression, whereas it is fixed in a classical regression. See text (Materials and Methods section) for details about the estimation of the slope in a major axis regression without intercept. Light penetration [(I/I0) x 100] is expressed as a percentage per plant
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Fig. 3. Major axis regressions fitted to the weekly means calculated across blocks, using Eq. [3] (a) and Eq. [4] (b), and to the block means calculated across weeks, using Eq. [3] (c) and Eq. [4] (d). See text (Materials and Methods section) for details
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In all instances, the goodness-of-fit was better for Eq. [4] than for Eq. [3]. The largest differences in r2 were observed in Weeks 1, 2, and 4 and on block means, with 
r2 = 0.24, 0.14, 0.31, and 0.16, respectively. The smallest differences in r2 were observed in Week 5 and on weekly means (r2 < 0.03). By comparison, the r2 varied from 0.81 to 0.99 in the regression of light penetration on FD alone. These results confirm those reported for the correlation analyses and our previous interpretation applies here.
The major axis slopes, with LAI vs. (LAI x FD) as regressor, were significantly different (p < 0.05) in Weeks 3 and 5 and on weekly means. Without reaching the 0.05 level, the difference between major axis slopes in Eq. [3] and [4] was close to significance in Week 4 (p = 0.1109) and on block means (p = 0.0799).
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DISCUSSION
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In this study, the fractal dimension of leafless plant structure was shown to correlate negatively with light penetration and provide an improved fitting when included in the Beer-Lambert equation as a multiplicative factor of LAI. This result may seem contradictory or unexpected because the amount of light that penetrates the canopy is light that is not intercepted by leaves, whereas FD is estimated on plants from which leaves are detached while leaving petioles. However, FD quantifies the geometric distribution of leaf area by measuring the complexity of the branching pattern through which the petioles array the leaves in space. Similarly, the lack of significant difference between major axis slopes does not contradict the better goodness-of-fit of Eq. [4] compared with Eq. [3] in Weeks 1 and 2, since the comparison of major axis slopes is actually an assessment of the discrepancy of FD from 1. This is in accordance with the linear branching pattern of soybean seedlings at the early stages of canopy development (Foroutan-pour et al., 1999a). Despite small sample sizes (4 or 5), our results were statistically significant, with one-third of the correlations being significant (p < 0.05) and 40% almost significant (0.05 p < 0.10). In particular, 100% of the correlations between light penetration and FD fell in either category. Accordingly, light penetration could be regressed solely on FD, but this could not be called a modification of the Beer-Lambert law. Thus, it appears that the comprehensible approximate relationships announced by Critten (1993)(p. 25) for light penetration are now available, either with FD as multiplicative factor of LAI in the equation or with FD as regressor.
More generally, we have investigated a three-dimensional phenomenon (i.e., light penetration into the plant canopy) by applying a novel technique (i.e., fractal analysis) in two dimensions. Some previous work (Foroutan-pour et al., 1999a) supports this approach based on photographs of leafless plants from the side that allows the maximum appearance of details on branches and petioles. Differences with viewing angle can be substantial, though, as Foroutan-pour et al. (2000) observed for corn for which leaves were not detached and photographs were taken along and perpendicular to the row as well as from the side that allowed the maximum appearance of details. The very strong correlations between light penetration and FD that were obtained in this study demonstrate that the side that one of us (KFP) had chosen for taking photographs was correct. It is true that photographs may also be taken from the top of the plant, but our experience does not support this approach because of shading, overlapping, and hiding effects. Recently acquired equipment will allow us to expand our research work on this subject to three dimensions.
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CLOSING REMARKS
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On the basis of our results, Eq. [4] is recommended in situations where LAI is almost constant among plants and differences in light penetration into the canopy are dependent on the distribution of leaf area in space. This is the case at given points in time (i.e., early stages of plant canopy development) and on average across measurement times (i.e., on block means here). At later stages of development and on weekly means, using LAI as regressor is sufficient. In all cases, including FD as a multiplicative factor of LAI in the Beer-Lambert law does not impair its application to light penetration into soybean plant canopies. We encourage the study of this feature in other crops.
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ACKNOWLEDGMENTS
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The senior author acknowledges a scholarship from the Iranian government during a part of this work. The second and third authors acknowledge NSERC individual operating grants. The comments of three anonymous reviewers and the associate editor, Dennis Timlin, were highly appreciated.
Received for publication May 13, 1999.
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ABSTRACT
INTRODUCTION
