Spanner LAI Example

Spanner LAI Example

This example script demonstrates the calculation of Leaf Area Index (LAI) using gap fraction profiles, Leaf Area Density (LAD), and related methods in R. It utilizes the lidR and terra libraries.

Import Libraries

library(lidR)
library(terra)

Gap Fraction Profile Calculation

This function calculates the gap fraction profile of a canopy given a vertical distribution of points.

gap_fraction_profile <- function(z, dz = 1, z0 = 2) {
  zrange = range(z)
  if (z0 < zrange[1])
    z0 = floor((zrange[1] - z0)/dz) * dz + z0
  if (z0 >= zrange[2])
    return(data.frame(z = numeric(0), gf = numeric(0)))
  bk <- seq(z0, ceiling((zrange[2] - z0)/dz) * dz + z0, dz)
  histogram <- graphics::hist(z, breaks = c(-Inf, bk), plot = FALSE)
  h <- histogram$mids
  p <- histogram$counts
  cs <- cumsum(p)
  i <- cs[1:(length(cs) - 1)]/cs[2:length(cs)]
  i[is.na(i)] = 0
  z = h[-1]
  return(data.frame(z = z, gf = i))
}

Leaf Area Density (LAD) Calculation

This function calculates the Leaf Area Density (LAD) using the gap fraction profile.

LAD <- function(z, dz = 1, k = 0.5, z0 = 2) {
  ld <- gap_fraction_profile(z, dz, z0)
  if (nrow(ld) == 0)
    return(data.frame(z = numeric(0), lad = numeric(0)))
  lad <- ld$gf
  lad <- -log(lad)/(k * dz)
  lad[is.infinite(lad)] = NA
  return(data.frame(z = ld$z, lad = lad))
}

Formula Explanation

The Leaf Area Density (LAD) is calculated using the formula:

$$ lad = -\frac{\log(gf)}{k \times dz} $$

Where:

• ( gf ) is the gap fraction,
• ( k ) is the extinction coefficient, and
• ( dz ) is the vertical resolution.

Leaf Area Index (LAI) Calculation

This function computes the Leaf Area Index (LAI) by integrating the LAD over the height.

calculate_lai <- function(z_values, dz = 1, z0 = 2) {
  # Compute the Leaf Area Density (LAD)
  lad <- LAD(z_values, dz = dz, z0 = z0)

  # Integrate the LAD to compute LAI
  lai <- sum(lad$lad, na.rm = TRUE) * dz
  
  return(lai)
}

The Leaf Area Index (LAI) is the integral of LAD over height ( z ):

$$ LAI = \int LAD(z) dz $$

Where:

• ( LAD(z) ) is the leaf area density at height ( z ),
• ( dz ) is the height step used in the numerical integration.

Example Usage

You can use the calculate_lai function with a set of vertical height values to compute the LAI for a given canopy.

dz = 1
z0 = 2
z_values <- c(1, 3, 5, 7, 9, 11) # Example height values
lai <- calculate_lai(z_values, dz = dz, z0 = z0)
print(lai)

This will print the computed LAI for the provided z_values.