Canopy Architecture Models

PhyTorch implements models for characterizing canopy architecture, including leaf angle distribution which describes how leaves are oriented within a plant canopy.

Leaf Angle Distribution

Leaf angle distribution (LAD) describes the angular distribution of leaf inclination angles within a canopy. This is a critical parameter for understanding light interception, photosynthesis, and canopy radiative transfer.

Beta Distribution Model

The leaf angle distribution follows a beta distribution parameterized by shape parameters μ and ν:

$$f(\theta) = \frac{\sin^{\mu-1}(\theta) \cdot \cos^{\nu-1}(\theta)}{B(\mu, \nu) \cdot 90}$$

where:

• $\theta$ = Leaf inclination angle from horizontal (0-90°)
• $\mu$ = First shape parameter (controls horizontal tendency)
• $\nu$ = Second shape parameter (controls vertical tendency)
• $B(\mu, \nu)$ = Beta function

Canonical Distribution Types

Following de Wit (1965), leaf angle distributions are classified into six canonical types based on their fitted μ and ν parameters:

Type	Description	μ	ν	Example Species
Planophile	Mostly horizontal leaves	2.770	1.172	Oak, broadleaf crops
Erectophile	Mostly vertical leaves	1.172	2.770	Grasses, willow
Plagiophile	Mostly oblique leaves	3.326	3.326	Many conifers
Extremophile	Both horizontal and vertical	0.433	0.433	Some shrubs
Uniform	Equal distribution	1.000	1.000	Theoretical
Spherical	Spherical distribution	1.101	1.930	Many tree species

Usage

from phytorch import fit
from phytorch.models.canopy import LeafAngleDistribution
import numpy as np

# Leaf angle data (degrees from horizontal)
# Can be binned frequency data or individual measurements
data = {
    'theta': np.array([5, 15, 25, 35, 45, 55, 65, 75, 85]),  # Angle bins
    'frequency': np.array([0.05, 0.12, 0.18, 0.22, 0.20, 0.13, 0.07, 0.02, 0.01])  # Relative frequency
}

# Fit the model
model = LeafAngleDistribution()
result = fit(model, data)

# View fitted parameters
print(f"μ = {result.parameters['mu']:.3f}")
print(f"ν = {result.parameters['nu']:.3f}")
print(f"R² = {result.r_squared:.4f}")

# Classify into canonical type
classification = model.classify(result.parameters)
print(f"\nCanopy type: {classification['type']}")
print(f"Distance from canonical: {classification['distance']:.3f}")
print(f"Canonical μ: {classification['canonical_mu']:.3f}")
print(f"Canonical ν: {classification['canonical_nu']:.3f}")

# Plot the distribution
result.plot()

Example: Planophile Canopy

from phytorch import fit
from phytorch.models.canopy import LeafAngleDistribution
import numpy as np

# Simulate planophile canopy (mostly horizontal leaves)
data = {
    'theta': np.array([10, 20, 30, 40, 50, 60, 70, 80]),
    'frequency': np.array([0.25, 0.30, 0.20, 0.12, 0.07, 0.04, 0.01, 0.01])
}

model = LeafAngleDistribution()
result = fit(model, data)

classification = model.classify(result.parameters)
print(f"Canopy type: {classification['type']}")  # Expected: planophile

Example: Erectophile Canopy

from phytorch import fit
from phytorch.models.canopy import LeafAngleDistribution
import numpy as np

# Simulate erectophile canopy (mostly vertical leaves)
data = {
    'theta': np.array([10, 20, 30, 40, 50, 60, 70, 80]),
    'frequency': np.array([0.01, 0.02, 0.05, 0.10, 0.18, 0.25, 0.28, 0.11])
}

model = LeafAngleDistribution()
result = fit(model, data)

classification = model.classify(result.parameters)
print(f"Canopy type: {classification['type']}")  # Expected: erectophile

Parameters

Parameter	Description	Typical Range	Units	Default
mu	First shape parameter	0.4-3.5	-	1.5
nu	Second shape parameter	0.4-3.5	-	1.5

Data Requirements

Leaf angle distribution data can be provided in two formats:

1. Binned frequency data (recommended):

   • theta: Center of angle bins (degrees, 0-90)
   • frequency: Relative frequency in each bin (normalized to sum to 1)

2. Individual measurements:

   • theta: Individual leaf angle measurements
   • frequency: Can be omitted (assumes equal weight)

Measuring Leaf Angles

Leaf inclination angles are typically measured using:

• Protractors or inclinometers for direct measurement
• Hemispherical photography analysis
• LiDAR-based canopy scanning
• Manual sampling of representative leaves

Best practices:

• Sample 30-50 leaves per canopy for robust estimates
• Stratify sampling across canopy layers if relevant
• Measure from horizontal (0°) to vertical (90°)
• Record multiple canopies per species/treatment for variability

Interpretation

Shape Parameter Relationships:

• $\mu > \nu$: Tendency toward horizontal leaves (planophile)
• $\mu < \nu$: Tendency toward vertical leaves (erectophile)
• $\mu \approx \nu$: Symmetrical distribution (plagiophile, uniform, or spherical)
• Low $\mu$ and $\nu$ ($< 1$): Bimodal distribution (extremophile)

Ecological Significance:

• Planophile canopies maximize light interception in low-light environments
• Erectophile canopies reduce light saturation and overheating in high-light environments
• Spherical distributions are common in mature forest canopies
• LAD affects canopy photosynthesis, water use efficiency, and microclimate

Classification Confidence

The distance value in classification results indicates how close the fitted distribution is to the canonical type:

• Distance < 0.5: Strong match to canonical type
• Distance 0.5-1.0: Moderate match
• Distance > 1.0: Weak match, intermediate between types

Applications

Leaf angle distribution is used in:

• Radiative transfer models: Light penetration and absorption
• Photosynthesis models: Canopy-scale carbon assimilation
• Remote sensing: Vegetation indices and LAI retrieval
• Crop modeling: Yield prediction and optimization
• Climate models: Surface energy balance

Custom Parameter Bounds

Constrain parameters based on expected canopy structure:

from phytorch import fit, FitOptions

# Constrain to planophile-like distributions
options = FitOptions(
    bounds={
        'mu': (2.0, 4.0),  # Higher mu favors horizontal
        'nu': (0.5, 2.0)   # Lower nu
    }
)

result = fit(LeafAngleDistribution(), data, options)

Model Comparison

Compare leaf angle distributions across species or treatments:

# Fit multiple canopies
canopies = {
    'Oak': oak_data,
    'Grass': grass_data,
    'Pine': pine_data
}

model = LeafAngleDistribution()
results = {}

for name, data in canopies.items():
    result = fit(model, data)
    classification = model.classify(result.parameters)
    results[name] = classification
    print(f"{name}: {classification['type']} (μ={classification['mu']:.2f}, ν={classification['nu']:.2f})")

References

• de Wit, C. T. (1965). Photosynthesis of Leaf Canopies. Agricultural Research Reports No. 663, Pudoc, Wageningen.
• Campbell, G. S. (1986). Extinction coefficients for radiation in plant canopies calculated using an ellipsoidal inclination angle distribution. Agricultural and Forest Meteorology, 36(4), 317-321.
• Wang, W. M., Li, Z. L., & Su, H. B. (2007). Comparison of leaf angle distribution functions: Effects on extinction coefficient and fraction of sunlit foliage. Agricultural and Forest Meteorology, 143(1-2), 106-122.