Author
Alejandro Morales Sierra

Centre for Crop Systems Analysis - Wageningen University

Published

March 29, 2023

In this example we extend the forest growth model to include PAR interception and a radiation use efficiency to compute the daily growth rate.

The initial setup is as follows:

``````using VPL
using Sky
using Plots
using Distributions
import Random
Random.seed!(123456789)

## Model definition

### Node types

The data types needed to simulate the trees are given in the following module. The difference with respec to the previous model is that Internodes and Leaves have optical properties needed for ray tracing (they are defined as Lambertian surfaces).

``````# Data types
module TreeTypes
using VPL
using Distributions
# Meristem
Base.@kwdef mutable struct Meristem <: VPL.Node
age::Int64 = 0   # Age of the meristem
end
# Bud
struct Bud <: VPL.Node end
# Node
struct Node <: VPL.Node end
# BudNode
struct BudNode <: VPL.Node end
# Internode (needs to be mutable to allow for changes over time)
Base.@kwdef mutable struct Internode <: VPL.Node
age::Int64 = 0         # Age of the internode
biomass::Float64 = 0.0 # Initial biomass
length::Float64 = 0.0  # Internodes
width::Float64  = 0.0  # Internodes
sink::Exponential{Float64} = Exponential(5)
material::Lambertian{1} = Lambertian(τ = 0.1, ρ = 0.05) # Leaf material
end
# Leaf
Base.@kwdef mutable struct Leaf <: VPL.Node
age::Int64 = 0         # Age of the leaf
biomass::Float64 = 0.0 # Initial biomass
length::Float64 = 0.0  # Leaves
width::Float64 = 0.0   # Leaves
sink::Beta{Float64} = Beta(2,5)
material::Lambertian{1} = Lambertian(τ = 0.1, ρ = 0.05) # Leaf material
end
# Graph-level variables -> mutable because we need to modify them during growth
Base.@kwdef mutable struct treeparams
# Variables
PAR::Float64 = 0.0   # Total PAR absorbed by the leaves on the tree (MJ)
biomass::Float64 = 2e-3 # Current total biomass (g)
# Parameters
RUE::Float64 = 5.0   # Radiation use efficiency (g/MJ) -> unrealistic to speed up sim
IB0::Float64 = 1e-3  # Initial biomass of an internode (g)
SIW::Float64 = 0.1e6   # Specific internode weight (g/m3)
IS::Float64  = 15.0  # Internode shape parameter (length/width)
LB0::Float64 = 1e-3  # Initial biomass of a leaf
SLW::Float64 = 100.0 # Specific leaf weight (g/m2)
LS::Float64  = 3.0   # Leaf shape parameter (length/width)
budbreak::Float64 = 1/0.5 # Bud break probability coefficient (in 1/m)
plastochron::Int64 = 5 # Number of days between phytomer production
leaf_expansion::Float64 = 15.0 # Number of days that a leaf expands
phyllotaxis::Float64 = 140.0
leaf_angle::Float64 = 30.0
branch_angle::Float64 = 45.0
end
end

import .TreeTypes``````

### Geometry

The methods for creating the geometry and color of the tree are the same as in the previous example but include the materials for the ray tracer.

``````# Create geometry + color for the internodes
function VPL.feed!(turtle::Turtle, i::TreeTypes.Internode, vars)
# Rotate turtle around the head to implement elliptical phyllotaxis
rh!(turtle, vars.phyllotaxis)
HollowCylinder!(turtle, length = i.length, height = i.width, width = i.width,
move = true, color = RGB(0.5,0.4,0.0), material = i.material)
return nothing
end

# Create geometry + color for the leaves
function VPL.feed!(turtle::Turtle, l::TreeTypes.Leaf, vars)
# Rotate turtle around the arm for insertion angle
ra!(turtle, -vars.leaf_angle)
# Generate the leaf
Ellipse!(turtle, length = l.length, width = l.width, move = false,
color = RGB(0.2,0.6,0.2), material = l.material)
# Rotate turtle back to original direction
ra!(turtle, vars.leaf_angle)
return nothing
end

# Insertion angle for the bud nodes
function VPL.feed!(turtle::Turtle, b::TreeTypes.BudNode, vars)
# Rotate turtle around the arm for insertion angle
ra!(turtle, -vars.branch_angle)
end``````

### Development

The meristem rule is now parameterized by the initial states of the leaves and internodes and will only be triggered every X days where X is the plastochron.

``````# Create right side of the growth rule (parameterized by the initial states
# of the leaves and internodes)
function create_meristem_rule(vleaf, vint)
meristem_rule = Rule(TreeTypes.Meristem,
lhs = mer -> mod(data(mer).age, vars(mer).plastochron) == 0,
rhs = mer -> TreeTypes.Node() +
(TreeTypes.Bud(),
TreeTypes.Leaf(biomass = vleaf.biomass,
length  = vleaf.length,
width   = vleaf.width)) +
TreeTypes.Internode(biomass = vint.biomass,
length  = vint.length,
width   = vint.width) +
TreeTypes.Meristem())
end``````

The bud break probability is now a function of distance to the apical meristem rather than the number of internodes. An adhoc traversal is used to compute this length of the main branch a bud belongs to (ignoring the lateral branches).

``````# Compute the probability that a bud breaks as function of distance to the meristem
function prob_break(bud)
# We move to parent node in the branch where the bud was created
node =  parent(bud)
# Extract the first internode
child = filter(x -> data(x) isa TreeTypes.Internode, children(node))[1]
data_child = data(child)
# We measure the length of the branch until we find the meristem
distance = 0.0
while !isa(data_child, TreeTypes.Meristem)
# If we encounter an internode, store the length and move to the next node
if data_child isa TreeTypes.Internode
distance += data_child.length
child = children(child)[1]
data_child = data(child)
# If we encounter a node, extract the next internode
elseif data_child isa TreeTypes.Node
child = filter(x -> data(x) isa TreeTypes.Internode, children(child))[1]
data_child = data(child)
else
error("Should be Internode, Node or Meristem")
end
end
# Compute the probability of bud break as function of distance and
# make stochastic decision
prob =  min(1.0, distance*vars(bud).budbreak)
return rand() < prob
end

# Branch rule parameterized by initial states of internodes
function create_branch_rule(vint)
branch_rule = Rule(TreeTypes.Bud,
lhs = prob_break,
rhs = bud -> TreeTypes.BudNode() +
TreeTypes.Internode(biomass = vint.biomass,
length  = vint.length,
width   = vint.width) +
TreeTypes.Meristem())
end``````

### Light interception

As growth is now dependent on intercepted PAR via RUE, we now need to simulate light interception by the trees. We will use a ray-tracing approach to do so. The first step is to create a scene with the trees and the light sources. As for rendering, the scene can be created from the `forest` object by simply calling `Scene(forest)` that will generate the 3D meshes and connect them to their optical properties.

However, we also want to add the soil surface as this will affect the light distribution within the scene due to reflection from the soil surface. This is similar to the customized scene that we created before for rendering, but now for the light simulation.

``````function create_soil()
soil = Rectangle(length = 21.0, width = 21.0)
rotatey!(soil, π/2) # To put it in the XY plane
VPL.translate!(soil, Vec(0.0, 10.5, 0.0)) # Corner at (0,0,0)
return soil
end
function create_scene(forest)
# These are the trees
scene = Scene(vec(forest))
soil = create_soil()
soil_material = Lambertian(τ = 0.0, ρ = 0.21)
add!(scene, mesh = soil, material = soil_material)
# Return the scene
return scene
end``````

Given the scene, we can create the light sources that can approximate the solar irradiance on a given day, location and time of the day using the functions from the Sky package (see package documentation for details). Given the latitude, day of year and fraction of the day (`f = 0` being sunrise and `f = 1` being sunset), the function `clear_sky()` computes the direct and diffuse solar radiation assuming a clear sky. These values may be converted to different wavebands and units using `waveband_conversion()`. Finally, the collection of light sources approximating the solar irradiance distribution over the sky hemisphere is constructed with the function `sky()` (this last step requires the 3D scene as input in order to place the light sources adequately).

``````function create_sky(;scene, lat = 52.0*π/180.0, DOY = 182)
# Fraction of the day and day length
fs = collect(0.1:0.1:0.9)
dec = declination(DOY)
DL = day_length(lat, dec)*3600
temp = [clear_sky(lat = lat, DOY = DOY, f = f) for f in fs] # W/m2
Ig   = getindex.(temp, 1)
Idir = getindex.(temp, 2)
Idif = getindex.(temp, 3)
# Conversion factors to PAR for direct and diffuse irradiance
f_dir = waveband_conversion(Itype = :direct,  waveband = :PAR, mode = :power)
f_dif = waveband_conversion(Itype = :diffuse, waveband = :PAR, mode = :power)
Idir_PAR = f_dir.*Idir
Idif_PAR = f_dif.*Idif
# Create the dome of diffuse light
dome = sky(scene,
Idir = 0.0, # No direct solar radiation
Idif = sum(Idir_PAR)/10*DL, # Daily Diffuse solar radiation
nrays_dif = 1_000_000, # Total number of rays for diffuse solar radiation
sky_model = StandardSky, # Angular distribution of solar radiation
dome_method = equal_solid_angles, # Discretization of the sky dome
ntheta = 9, # Number of discretization steps in the zenith angle
nphi = 12) # Number of discretization steps in the azimuth angle
# Add direct sources for different times of the day
for I in Idir_PAR
push!(dome, sky(scene, Idir = I/10*DL, nrays_dir = 100_000, Idif = 0.0)[1])
end
return dome
end``````

The 3D scene and the light sources are then combined into a `RayTracer` object, together with general settings for the ray tracing simulation chosen via `RTSettings()`. The most important settings refer to the Russian roulette system and the grid cloner (see section on Ray Tracing for details). The settings for the Russian roulette system include the number of times a ray will be traced deterministically (`maxiter`) and the probability that a ray that exceeds `maxiter` is terminated (`pkill`). The grid cloner is used to approximate an infinite canopy by replicating the scene in the different directions (`nx` and `ny` being the number of replicates in each direction along the x and y axes, respectively). It is also possible to turn on parallelization of the ray tracing simulation by setting `parallel = true` (currently this uses Julia’s builtin multithreading capabilities).

In addition `RTSettings()`, an acceleration structure and a splitting rule can be defined when creating the `RayTracer` object (see ray tracing documentation for details). The acceleration structure allows speeding up the ray tracing by avoiding testing all rays against all objects in the scene.

``````function create_raytracer(scene, sources)
settings = RTSettings(pkill = 0.9, maxiter = 4, nx = 5, ny = 5, dx = 20.0,
dy = 20.0, parallel = true)
RayTracer(scene, sources, settings = settings, acceleration = BVH,
rule = SAH{3}(5, 10));
end``````

The actual ray tracing simulation is performed by calling the `trace!()` method on the ray tracing object. This will trace all rays from all light sources and update the radiant power absorbed by the different surfaces in the scene inside the `Material` objects (see `feed!()` above):

``````function run_raytracer!(forest; DOY = 182)
scene   = create_scene(forest)
sources = create_sky(scene = scene, DOY = DOY)
rtobj   = create_raytracer(scene, sources)
trace!(rtobj)
return nothing
end``````

The total PAR absorbed for each tree is calculated from the material objects of the different internodes (using `power()` on the `Material` object). Note that the `power()` function returns three different values, one for each waveband, but they are added together as RUE is defined for total PAR.

``````# Run the ray tracer, calculate PAR absorbed per tree and add it to the daily
# total using general weighted quadrature formula
function calculate_PAR!(forest;  DOY = 182)
# Reset PAR absorbed by the tree (at the start of a new day)
reset_PAR!(forest)
# Run the ray tracer to compute daily PAR absorption
run_raytracer!(forest, DOY = DOY)
# Add up PAR absorbed by each leaf within each tree
for l in get_leaves(tree)
vars(tree).PAR += power(l.material)[1]
end
end
return nothing
end

# Reset PAR absorbed by the tree (at the start of a new day)
function reset_PAR!(forest)
for tree in forest
vars(tree).PAR = 0.0
end
return nothing
end``````

### Growth

We need some functions to compute the length and width of a leaf or internode from its biomass

``````function leaf_dims(biomass, vars)
leaf_biomass = biomass
leaf_area    = biomass/vars.SLW
leaf_length  = sqrt(leaf_area*4*vars.LS/pi)
leaf_width   = leaf_length/vars.LS
return leaf_length, leaf_width
end

function int_dims(biomass, vars)
int_biomass = biomass
int_volume  = biomass/vars.SIW
int_length  = cbrt(int_volume*4*vars.IS^2/pi)
int_width   = int_length/vars.IS
return int_length, int_width
end``````

Each day, the total biomass of the tree is updated using a simple RUE formula and the increment of biomass is distributed across the organs proportionally to their relative sink strength (of leaves or internodes).

The sink strength of leaves is modelled with a beta distribution scaled to the `leaf_expansion` argument that determines the duration of leaf growth, whereas for the internodes it follows a negative exponential distribution. The `pdf` function computes the probability density of each distribution which is taken as proportional to the sink strength (the model is actually source-limited since we imposed a particular growth rate).

``````sink_strength(leaf, vars) = leaf.age > vars.leaf_expansion ? 0.0 :
pdf(leaf.sink, leaf.age/vars.leaf_expansion)/100.0
plot(0:1:50, x -> sink_strength(TreeTypes.Leaf(age = x), TreeTypes.treeparams()),
xlabel = "Age", ylabel = "Sink strength", label = "Leaf")``````
``````sink_strength(int) = pdf(int.sink, int.age)
plot!(0:1:50, x -> sink_strength(TreeTypes.Internode(age = x)), label = "Internode")``````

Now we need a function that updates the biomass of the tree, allocates it to the different organs and updates the dimensions of said organs. For simplicity, we create the functions `leaves()` and `internodes()` that will apply the queries to the tree required to extract said nodes:

``````get_leaves(tree) = apply(tree, Query(TreeTypes.Leaf))
get_internodes(tree) = apply(tree, Query(TreeTypes.Internode))``````

The age of the different organs is updated every time step:

``````function age!(all_leaves, all_internodes, all_meristems)
for leaf in all_leaves
leaf.age += 1
end
for int in all_internodes
int.age += 1
end
for mer in all_meristems
mer.age += 1
end
return nothing
end``````

The daily growth is allocated to different organs proportional to their sink strength.

``````function grow!(tree, all_leaves, all_internodes)
# Compute total biomass increment
tvars = vars(tree)
ΔB    = max(0.5, tvars.RUE*tvars.PAR/1e6) # Trick to emulate reserves in seedling
tvars.biomass += ΔB
# Total sink strength
total_sink = 0.0
for leaf in all_leaves
total_sink += sink_strength(leaf, tvars)
end
for int in all_internodes
total_sink += sink_strength(int)
end
# Allocate biomass to leaves and internodes
for leaf in all_leaves
leaf.biomass += ΔB*sink_strength(leaf, tvars)/total_sink
end
for int in all_internodes
int.biomass += ΔB*sink_strength(int)/total_sink
end
return nothing
end``````

Finally, we need to update the dimensions of the organs. The leaf dimensions are

``````function size_leaves!(all_leaves, tvars)
for leaf in all_leaves
leaf.length, leaf.width = leaf_dims(leaf.biomass, tvars)
end
return nothing
end
function size_internodes!(all_internodes, tvars)
for int in all_internodes
int.length, int.width = int_dims(int.biomass, tvars)
end
return nothing
end``````

### Daily step

All the growth and developmental functions are combined together into a daily step function that updates the forest by iterating over the different trees in parallel.

``````get_meristems(tree) = apply(tree, Query(TreeTypes.Meristem))
function daily_step!(forest, DOY)
# Compute PAR absorbed by each tree
calculate_PAR!(forest, DOY = DOY)
# Grow the trees
# Retrieve all the relevant organs
all_leaves = get_leaves(tree)
all_internodes = get_internodes(tree)
all_meristems = get_meristems(tree)
# Update the age of the organs
age!(all_leaves, all_internodes, all_meristems)
# Grow the tree
grow!(tree, all_leaves, all_internodes)
tvars = vars(tree)
size_leaves!(all_leaves, tvars)
size_internodes!(all_internodes, tvars)
# Developmental rules
rewrite!(tree)
end
end``````

### Initialization

The trees are initialized on a regular grid with random values for the initial orientation and RUE:

``````RUEs = rand(Normal(1.5,0.2), 10, 10)
histogram(vec(RUEs))``````
``````orientations = [rand()*360.0 for i = 1:2.0:20.0, j = 1:2.0:20.0]
histogram(vec(orientations))``````
``origins = [Vec(i,j,0) for i = 1:2.0:20.0, j = 1:2.0:20.0];``

The following initalizes a tree based on the origin, orientation and RUE:

``````function create_tree(origin, orientation, RUE)
# Initial state and parameters of the tree
vars = TreeTypes.treeparams(RUE = RUE)
# Initial states of the leaves
leaf_length, leaf_width = leaf_dims(vars.LB0, vars)
vleaf = (biomass = vars.LB0, length = leaf_length, width = leaf_width)
# Initial states of the internodes
int_length, int_width = int_dims(vars.LB0, vars)
vint = (biomass = vars.IB0, length = int_length, width = int_width)
# Growth rules
meristem_rule = create_meristem_rule(vleaf, vint)
branch_rule   = create_branch_rule(vint)
axiom = T(origin) + RH(orientation) +
TreeTypes.Internode(biomass = vint.biomass,
length  = vint.length,
width   = vint.width) +
TreeTypes.Meristem()
tree = Graph(axiom = axiom, rules = (meristem_rule, branch_rule),
vars = vars)
return tree
end``````

## Visualization

As in the previous example, it makes sense to visualize the forest with a soil tile beneath it. Unlike in the previous example, we will construct the soil tile using a dedicated graph and generate a `Scene` object which can later be merged with the rest of scene generated in daily step:

``````Base.@kwdef struct Soil <: VPL.Node
length::Float64
width::Float64
end
function VPL.feed!(turtle::Turtle, s::Soil, vars)
Rectangle!(turtle, length = s.length, width = s.width, color = RGB(255/255, 236/255, 179/255))
end
soil_graph = RA(-90.0) + T(Vec(0.0, 10.0, 0.0)) + # Moves into position
Soil(length = 20.0, width = 20.0) # Draws the soil tile
soil = Scene(Graph(axiom = soil_graph));
render(soil, axes = false)``````

And the following function renders the entire scene (notice that we need to use `display()` to force the rendering of the scene when called within a loop or a function):

``````function render_forest(forest, soil)
scene = Scene(vec(forest)) # create scene from forest
scene = Scene([scene, soil]) # merges the two scenes
display(render(scene))
end``````

## Simulation

We can now create a forest of trees on a regular grid:

``````forest = create_tree.(origins, orientations, RUEs);
render_forest(forest, soil)
start = 180
for i in 1:50
println("Day \$i")
daily_step!(forest, i + start)
if mod(i, 5) == 0
render_forest(forest, soil)
end
end``````