Leaf area density¶

Leaf area density (LAD) is the one-sided leaf area per unit volume, in m²/m³, estimated per voxel of a 3-D grid you place over a scanned canopy. It is the spatial building block of leaf area index (LAI) — sum LAD over a column of voxels and multiply by their height and you get the LAI of that column.

Phytograph computes LAD with the PyHelios LiDAR plugin from one or more terrestrial scans that carry scanner parameters.

LAD is not a sum of leaf areas¶

A common misconception is that LAD is the triangulated leaf surface area divided by the voxel volume. It is not. You cannot see every leaf from a scanner — leaves occlude each other — so simply meshing the hit points and summing triangle areas systematically undercounts the foliage deeper in the canopy.

Instead, LAD is recovered by inverting Beer's law on the laser beams that pass through each voxel:

Triangulate the hit points. The mesh is not the answer — it is used only to estimate the per-voxel G-function, G(θ): the mean projection of the leaves onto the plane perpendicular to the beam. For a random (spherical) leaf-angle distribution G(θ) ≈ 0.5; for erect (vertical) leaves it is lower. (You can also skip this step and prescribe G(θ) directly from a known leaf-angle distribution.)
Trace every beam through the voxel grid and measure the path length dr it travels inside each voxel.
Count returns to estimate the gap probability P (the fraction of beams that pass through the voxel without a return).
Invert P = exp(−G(θ) · LAD · dr) for LAD, per voxel.

Because it works from transmission (gaps) rather than from visible surface, the inversion accounts for occluded foliage — which is the whole point.

Leaf-angle distributions and G(θ)¶

G(θ) is the mean leaf-projection coefficient: the fraction of leaf area projected onto the plane perpendicular to a beam travelling at zenith angle θ, averaged over the leaves' orientations. It depends on the canopy's leaf-angle distribution g_L(θ_L) (how leaf inclinations are spread between horizontal and vertical) and on the beam zenith. For a given distribution,

G(θ) = ∫ A(θ, θ_L) · g_L(θ_L) dθ_L

where A is the standard azimuthally-averaged projection kernel (Warren-Wilson / Lemeur). Three common cases:

Spherical (random): leaf normals point in every direction equally; G(θ) ≈ 0.5 at all zeniths. This is the usual default.
Planophile (mostly horizontal): G is high near nadir and falls toward the horizon.
Erectophile (mostly vertical): the opposite — G is low near nadir and rises toward the horizon.

Phytograph normally estimates G(θ) per voxel from a triangulation of the hit points (the surface normals give the leaf orientations). But you can also prescribe it directly — see Override G(θ) directly. Three ways to specify it:

Constant value — e.g. G(θ) = 0.42.
de Wit classical distribution — pick a named family (spherical, planophile, erectophile, plagiophile, extremophile, uniform); G(θ) is derived by integrating the kernel over the actual beam zenith angles in your scan.
Beta (μ, ν) — Goel–Strebel parameters (the same convention used by Adjust leaf angles and the leaf-angle de Wit fitting on the triangulation side); G(θ) is derived as for de Wit.

Each can be applied uniformly across the grid or as a vertical profile (one method, with the parameter varying by z-level) when leaf inclination changes with canopy height.

The voxel grid is required¶

For triangulation a grid is optional. For LAD it is mandatory: LAD is defined per voxel, so the grid is the calculation. You supply it as a voxel box created in the viewer, with its position, size, and per-axis cell counts (Nx × Ny × Nz) setting the grid. A 1×1×1 box with 1×1×1 cells yields a single canopy-wide LAD value; subdividing it gives a 3-D density field.

Single- vs multi-return scans¶

The inversion differs by return type, and Phytograph detects it from each scan's parameters:

Single-return (discrete): one return per pulse. The gap fraction is estimated directly from hit/miss classification, so the scan still needs miss information — either imported miss points or a timestamp column to gapfill from (see Sky/miss points below).
Multi-return (full-waveform): several returns per pulse. The beams are grouped by pulse and weighted equally, and sky/miss rays are gap-filled before the inversion. This needs per-return metadata — timestamp, target_index, and target_count — preserved from the source file. When a scan is marked multi-return but those columns are missing, Phytograph falls back to the single-return algorithm and warns you.

You don't choose the algorithm directly — it follows each scan's return type (set in its scan parameters).

Sky/miss points and gapfilling¶

The inversion measures gaps — beams that passed through a voxel without a return — so it needs to know about the beams that hit the sky and returned nothing at all (the "misses"). Phytograph handles misses three ways, in order of preference:

Imported misses. Scans from an E57 or structured PLY carry real miss points (flagged is_miss). The inversion uses them directly. Toggle Show misses on the scan row to verify they're present — they draw in a distinct colour. Until the scan has a scanner origin, the misses show at their true (typically far-field) coordinates; once you give the scan a scanner origin, they're drawn on a sphere just beyond the farthest hit so they stay visible against the cloud.
Gapfilled misses. A scan with no miss points but a timestamp column has its misses recovered automatically at compute time: Phytograph reconstructs the scan grid from the pulse timestamps and fills the gaps. It reports how many misses it recovered.
No miss information. A scan with neither miss points nor a timestamp can't account for the gaps, so the inversion cannot run — it would have no valid denominator and would only produce a biased number. Phytograph stops with a clear message asking you to re-import the scan from a format that carries misses (E57 / structured PLY) or one with a timestamp column so misses can be gapfilled.

Reading the result¶

The result is a grid of translucent voxel cells colored by LAD through the shared colormap, with a colorbar in m²/m³. Hover a cell to read its exact LAD, G(θ), and hit count. Empty voxels (no returns) are hidden by default.

How certain is the estimate?¶

Every inversion also reports a sampling-uncertainty interval following Pimont et al. (2018). It depends on the element width you set in the LAD dialog — the characteristic width of a leaf or needle (broadleaf ≈ 0.05 m, conifer ≈ 0.002 m) — which scales the variance without changing the LAD point estimate.

When you select the result, the panel shows a group-scale confidence interval aggregated over all solved voxels (e.g. Mean LAD 1.23 [1.15–1.31] m²/m³, 95% CI). This group-scale interval is the one to report: per-voxel intervals are routinely ±50–100% and only trustworthy in narrow regimes, whereas the aggregate is much tighter (typically ±5–10%).

Two caveats worth keeping in mind:

The interval is conditional on the beams that entered the voxels. It quantifies sampling noise, not occlusion bias — foliage that no beam ever reached is invisible to both the estimate and its interval.
If the data fall outside the method's validity envelope, no interval is reported rather than a misleading one.

See Estimate leaf area density for the step-by-step workflow.