LAD: an R package to estimate leaf angle distribution from measured leaf inclination angles


Leaf angle distribution (LAD) is an important factor for characterizing the optical features of vegetation canopies (Ross 1981). It influences several processes such as photosynthesis, evapotranspiration, spectral reflectance and absorptance (Vicari et al. 2019).  LAD influence on radiation transmission is also described by the leaf projection function (also known G-function), which is the projection coefficient function of unit foliage area on a plane perpendicular to the viewing direction (Ross 1981). Both parameters are therefore crucial to indirectly estimate Leaf Area Index (LAI) from gap fraction or radiation trasmission measurements.
Despite its importance, LAD is one of the most poorly characterized parameters due to the difficulty of directly measure leaf inclination angles, particularly in tall trees . As alternative to direct manual measurements, Ryu et al. (2010) proposed a robust and simple leveled photographic method to measure leaf inclination angles, which was proven comparable to manual clinometer measurements (Pisek et al. (2011)).
Using this leveled photographic method, some dataset of leaf inclination angle distributions have been compiled by some authors (Pisek et al. 2013; Raabe et al. 2015; Chianucci et al. 2018; Pisek and Adamson 2020).

In this article we presented “LAD”, an R package to calculate the Leaf angle distribution (LAD) function and the G-function from measured leaf inclination angles.

Basic theory

The method proposed by Ryu et al. (2010) consists of acquiring leveled images of the canopy using a digital camera. As some species may display phototropism, leaves shall be measured in all the azimuth directions and along the vertical profile of the surrounding canopy. Crowns of trees can be observed using towers, extendable poles, ladder, nearby tall buildings, or unmanned aerial vehicles (McNeil et al. 2016). The measurement of leaf inclination angle requires the identification of the leaf plane, from which the leaf normal is measured (Fig. 1). For this reason, the leaves must be selected from those oriented approximately parallel to the viewing direction of the camera (i.e., the leaves shown as a line in the image, Fig. 1), avoiding bended leaves to be measured.

alt text

Figure 1. Example of leaf inclination angle measurements from leveled photography. The measure of leaf inclination angle requires the identification of a hypothetical leaf plane xy, from which the leaf surface normal \(\hat{N}\) is calculated with respect to the zenith (z). In a 2-D space, such as a digital image, the only measurable leaf normal is that oriented perpendicular to the camera viewing direction, since the leaf inclination plane is parallel to the leaves (the red dashed line on the left side of the figure). From Chianucci et al. 2018, modified

Once a reliable set of leaf inclination angles measurements \(\theta_L\) are taken (a minimum of 75 measurements per species are recommended by Pisek et al. (2013)), two parameters \(\mu,\nu\) are derived for fitting a Beta distribution:



where \(\sigma_0^2\) and \(\sigma_t^2\) are the maximum standard deviation and variance of t, respectively, and are calculated following Wang et al.(2007):



with \(t=2\theta_L/\pi\), and \(\theta_L\) expressed as radians.

The two \(\mu,\nu\) parameters were then used to fit a Beta distribution function to represent the LAD of the considered canopy Goel and Strebel (1984):


Following de Wit (1965), LAD functions can be described using six common functions based on empirical evidence of the natural variation of leaf normal distributions and mathematical considerations: 

In spherical canopies, the relative frequency of leaf inclination angle is the same as for a sphere; planophile canopies are dominated by horizontally oriented leaves; plagiophile canopies are dominated by inclined leaves; erectophile canopies are dominated by vertically oriented leaves; extremophile canopies are characterized by both horizontally and vertically oriented leaves; uniform canopies are characterized by equal proportion of leaf inclination angles for any angle.

The fitted LAD function can then be compared with these six distributions, to determine the actual distribution type. The distribution type is classified using a leaf inclination index (Ross (1975)):


The LAD function can also be used to calculate the G-function (Ross (1981)), which corresponds to the fraction of foliage area on a plane perpendicular to the view angle \(\theta\)



\(A(\theta,\theta_L) = \begin{cases} cos\theta\times{cos\theta_L} &\text{if } \vert{cot\theta \times{cot\theta_L}}\vert >1 \\ cos\theta\times{ cos\theta_L} [1+(2/\pi)(tan\psi-\psi)] &\text{if } \vert{cot\theta \times{cot\theta_L}}\vert \le{1} \end{cases}\)


\(\psi=cos^{-1}(cot\theta\times{ cot\theta_L})\)


You can install the development version of LAD from GitLab using devtools (Wickham et al. 2021):

# install.packages("devtools")

The LAD package

The package have two key functions to calculate LAD, G-function, summary leaf inclination angle statistics:

fitLAD(): fit a LAD distribution

fitLAD calculates the LAD function and G-function from two-parameters \(\mu,\nu\) of Beta distribution.

The functions returns a list with two elements:

Example of graphical output:


calcLAD(): calculate LAD statistics

The calcLAD(dataframe,angles,type,...) function allows to calculate either summary leaf inclination angle statistics or the LAD function and G-function from leaf inclination angle measurements. Three important considerations must be drawn when using this function:

  1. The function considers a dataframe as input; it doesn’t work on single vector(s) of measurements;
  2. The function needs one or more grouping column(s) to calculate the attributes;
  3. The function yields a summarized list of attributes derived from measured leaf inclination angles or an extended calculation of LAD and G-function depending of the function argument type.

Package usage

We will illustrate the function usage with example data from Chianucci et al. (2018), which represents the largest leaf inclination angle dataset for 138 temperate and boreal woody species.

##   N ICP_Code     Family           Genus       Species         Author_citation
## 1 1     <NA> Araliaceae Eleutherococcus sessiliflorus (Rupr. & Maxim.) S.Y.Hu
## 2 2     <NA> Araliaceae Eleutherococcus sessiliflorus (Rupr. & Maxim.) S.Y.Hu
## 3 3     <NA> Araliaceae Eleutherococcus sessiliflorus (Rupr. & Maxim.) S.Y.Hu
## 4 4     <NA> Araliaceae Eleutherococcus sessiliflorus (Rupr. & Maxim.) S.Y.Hu
## 5 5     <NA> Araliaceae Eleutherococcus sessiliflorus (Rupr. & Maxim.) S.Y.Hu
## 6 6     <NA> Araliaceae Eleutherococcus sessiliflorus (Rupr. & Maxim.) S.Y.Hu
##   Canopy_sector Angle_degree     Date
## 1  whole canopy       20.973 11-10-11
## 2  whole canopy       34.472 11-10-11
## 3  whole canopy        7.787 11-10-11
## 4  whole canopy        1.868 11-10-11
## 5  whole canopy       52.667 11-10-11
## 6  whole canopy       65.433 11-10-11

The calcLAD() function requires selecting the columns with the leaf inclination angle measurements data (in degrees), the type of calculations to be performed (‘summary’ or ‘extended’) and the grouping column(s).

By selecting type='extended' the function calculate LAD function (pdf) and G-function (G) for each group:

## # A tibble: 12,282 × 5
##    Genus  Species  theta    pdf     G
##    <chr>  <chr>    <int>  <dbl> <dbl>
##  1 Acacia dealbata     1 0.0116 0.688
##  2 Acacia dealbata     2 0.012  0.688
##  3 Acacia dealbata     3 0.0123 0.688
##  4 Acacia dealbata     4 0.0125 0.687
##  5 Acacia dealbata     5 0.0126 0.686
##  6 Acacia dealbata     6 0.0127 0.685
##  7 Acacia dealbata     7 0.0128 0.684
##  8 Acacia dealbata     8 0.0128 0.683
##  9 Acacia dealbata     9 0.0129 0.682
## 10 Acacia dealbata    10 0.0129 0.68 
## # ℹ 12,272 more rows

By selecting type='summary' the function calculate mean statistics (mean leaf inclination angle (MTA); standard deviation (SD), frequency (NR), Beta parameters (mu,nu) and distribution type):

## Joining with `by = join_by(Genus, Species)`
## Joining with `by = join_by(Genus, Species)`

## # A tibble: 138 × 8
##    Genus  Species          MTA    SD     N    mu    nu distribution
##    <chr>  <chr>          <dbl> <dbl> <int> <dbl> <dbl> <chr>       
##  1 Acacia dealbata        40.9  24.5   283  1.27  1.06 uniform     
##  2 Acer   burgerianum     63.7  13.7   115  2.33  5.64 erectophile 
##  3 Acer   campestre       50.1  15.0    75  3.51  4.4  plagiophile 
##  4 Acer   griseum         33.6  20.9    80  2.09  1.24 planophile  
##  5 Acer   miyabei         44.9  15.5    80  3.72  3.71 plagiophile 
##  6 Acer   monspessulanum  20.1  14.8   105  4.21  1.21 planophile  
##  7 Acer   negundo         59.4  17.5   114  1.68  3.26 erectophile 
##  8 Acer   platanoides     26.8  16.7  1254  3.56  1.51 planophile  
##  9 Acer   pseudoplatanus  33.8  17.1   102  3.44  2.07 planophile  
## 10 Acer   rubrum          30.3  16.1  1001  3.98  2.03 planophile  
## # ℹ 128 more rows


The package was carried out within the Agritech National Research Center and received funding from the European Union Next-GenerationEU (National Recovery and Resilience Plan (NRRP) – MISSION 4 COMPONENT 2, INVESTMENT 1.4 – D.D. 1032 17/06/2022, CN00000022). This manuscript reflects only the authors’ views and opinions, neither the European Union nor the European Commission can be considered responsible for them.


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