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.

Two summaries fall directly out of a computed grid, and Phytograph reports both in the Profile & LAI window of a LAD result:

• The vertical profile — mean LAD per horizontal level — is how the canopy's foliage is distributed with height. It is what distinguishes a canopy that carries its leaf area in a dense upper crown from one that spreads the same total over its whole depth.
• The bulk LAI — total leaf area over the grid's ground footprint — is the single canopy-scale number, and the one most often quoted.

Neither counts occluded voxels. A voxel the beams never adequately probed is unmeasured, not empty, and treating its density as zero is the most common way both figures come out biased low. See estimating LAD.

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:

1. 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.)
2. Trace every beam through the voxel grid and measure the path length dr it travels inside each voxel.
3. Count returns to estimate the gap probability P (the fraction of beams that pass through the voxel without a return).
4. 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 traveling 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:

1. Constant value — e.g. G(θ) = 0.42.
2. 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.
3. 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.

You don't choose the algorithm, and there is no setting for it: it follows directly from whether those three per-pulse columns are present in the data. The Scan Parameters dialog reports the verdict under Detected from point data, and the LAD dialog echoes it read-only. See Return type is detected, not declared.

Keep the later echoes if the instrument recorded them

If your instrument records multiple echoes per pulse, import in a way that preserves them. Using only the first echo of each pulse — the "first-hits" method — biases LAD in a known direction: a beam that is partially intercepted before and after a voxel counts as zero transmission, so transmission is underestimated and LAD comes out too high.

Kent and Bailey (2024) quantified this against simulated scans with known leaf area. First-hits differed most from the reference of every method tested, consistently underestimating transmission and overestimating LAD — overestimating total leaf area by 149–266% in their most demanding case, against 80–122% for the equal-weighting method Phytograph uses for multi-return scans. (Their headline result is a separate question: intensity-based weighting of echoes did not reliably beat equal weighting, so equal weighting is "an acceptable choice" — which is why the multi-return path above weights each echo equally.)

The practical consequence is about file formats, since a format that stores one point per beam leaves no later echoes to weight. Formats that keep them:

• RIEGL .riproject / .PROJ — carries per-pulse target index and count directly.
• Structured .e57 — keeps every echo for a cell.
• Any format retaining timestamp, target_index, and target_count as columns (LAS/LAZ, PLY, or the text formats). In LAS/LAZ these are the standard return_number / number_of_returns fields, and they are carried even when every point holds the same value — a cropped single-tree extract in which every pulse declares the same return count still supports the inference below. Only a genuinely single-return file (number_of_returns of 1 throughout) is treated as carrying nothing.

.ptx cannot — it is a complete raster of one line per grid cell, so it collapses each pulse to a single echo and has no schema slot for those three columns. A multi-return scan re-imported from PTX is not a multi-return scan that quietly falls back; it is single-return, its return type field agrees, and no warning fires because nothing is inconsistent. The echoes are gone with nothing downstream able to tell you. PTX remains a perfectly good LAD input for genuinely single-return instruments, where it retains misses well (see Sky/miss points).

Cropped and segmented clouds

The equal-weighting inversion counts, for each pulse, the returns inside and beyond every voxel the beam pierces. Those counts come from the returns present in the cloud. So if a pulse hit a needle inside a voxel and then the ground behind it, and the ground return is gone (a crop to the tree, a ground filter, a leaf/wood classification that keeps only leaf points), the surviving needle return becomes the whole beam: a pulse that was half transmitted reads as fully intercepted, and LAD comes out too high. Worse, a pulse that crossed the voxel and returned only beyond it, which is the transmission signal itself, disappears altogether. The removed energy does not become a sky miss; it simply vanishes.

How well Phytograph recovers from that depends on where the crop happened.

Cropped in Phytograph (crop box, erase, filter). A deletion keeps the point's coordinates, and LAD feeds every deleted point that lies outside the voxel grid back to the inversion, because for the inversion a deleted return beyond the grid is a transmitted beam exactly as before and one before the grid contributes nothing either way. So cropping to the grid, or deleting noise outside it, changes nothing: the result is identical to the uncropped scan, and a miss buffer computed before the crop stays valid. The one deletion the inversion cannot repair is a point deleted inside the grid, and LAD warns with the count when it finds one.

This holds for as long as the deleted points are kept, which is until you choose Permanently apply deletions in the Erase panel. The display refresh that follows a crop, erase or filter keeps them. Applying deletions removes the points for good, so LAD can no longer restore the ones outside the grid, and it warns that the result may read high. The same happens to a cloud made by split, extract, duplicate or merge: it holds only the points it took, so the rest of the scan's returns are gone from it. Either crop without applying deletions, or re-run Backfill Misses on the new or applied cloud. Backfilling against the points that remain recovers the lost pulses as misses, which brings LAD back in line and clears the warning. This recovery applies only once the points are gone — after Permanently apply deletions, a split, or an extract. While a crop is still just a mask, Backfill deliberately restores the deleted hits and reconstructs the scan as measured, so re-running it after an ordinary crop changes nothing (and is not needed: LAD already restores the cropped points outside the grid). Likewise, after you move or rotate a cloud, LAD warns until you re-run Backfill Misses, because the misses' beam directions predate the move.

Cropped before import (a segmented tree file). The removed returns were never in the session, so the inversion works from what the file kept. If the surviving returns still carry the per-pulse target_index and target_count the scanner wrote (LAS return_number / number_of_returns, or the same columns in a text file), a removed return can be placed from the surviving indices alone, since a pulse's returns are ordered by range and a beam crosses the grid in one contiguous segment:

• an index below the smallest surviving index was before the grid and changes nothing;
• an index above the largest surviving index was beyond the grid and is counted as transmitted through every voxel the beam pierces;
• an index between two surviving returns cannot be placed. It is left out, and the result reports how many.

That covers pulses that kept a return inside the grid. Pulses that lost all their returns have nothing to carry a count, so run Backfill Misses on the cropped cloud: the gapfill synthesizes a miss for every pulse with no return, which restores the beams that crossed the grid and returned only beyond it. The limit of that recovery is occlusion: a pulse whose only return was on something between the scanner and the grid (a neighboring crown, a trunk outside the grid) comes back as a miss too, and a miss counts as transmitted, so those voxels read low. Nothing in the cropped file can tell a blocked beam from a transmitted one. For a per-tree profile in a closed stand, crop in Phytograph rather than before import, or keep the returns between the scanner and the tree in the file.

Two warnings catch the cases the inversion cannot repair: LAD warns when a session has deleted points inside the voxel box, and when the inversion found returns it could not place. Separating wood from leaf area is a correction applied after the inversion, not a point filter before it — which is exactly how Phytograph does it; see Wood area below.

Wood area

Branches and trunks intercept beams too, so an inversion that treats every return as foliage reports plant area density under the name LAD. When the cloud carries a wood/leaf classification (run Segment Wood / Leaf first), Phytograph splits each voxel's result into a leaf part and a wood part, and reports both.

The split happens after the inversion, never by filtering points before it. Every return and every sky miss stays in the cloud, so the beam bookkeeping is untouched; what the classification changes is only how each voxel's measured interception is attributed. Deleting wood points instead would turn a beam stopped by a branch into a transmitted one and bias the leaf density — the failure described under cropped and segmented clouds.

The two numbers use different, deliberate conventions, and both mean "the area that intercepts light":

	Reported as	Projection coefficient
LAD	one-sided leaf area per m³	G ≈ 0.5 for a spherical leaf-angle distribution
WAD	total woody surface area per m³	G = 0.25 for randomly-oriented cylinders

A leaf is flat, so its "two-sided" area is twice its one-sided area. A branch has no such doubling — its two-sided area is its surface area. Using each convention's own coefficient makes the two add up: LAI + WAI = PAI, and the summary export reports all three.

The wood projection coefficient is measured from the triangulation, the same mesh the leaf side uses. The result says which coefficient it applied and whether it was measured or assumed.

This works for a reason worth stating, because it is counter-intuitive: the triangulation is an orientation measurement, not an area one. A branch is a cylinder, and every point on a cylinder's surface has a normal perpendicular to its axis — so the axis is simply the direction those normals avoid. A terrestrial scan only ever triangulates the scanner-facing half of a branch, and that costs nothing here: a direction is not an area, and the hidden half is redundant by symmetry. Measured on a synthetic trunk, the visible half alone recovers the axis to a small fraction of a degree.

Where no mesh exists (a supplied-G(θ) run, a moving platform) or the branch axis is not confident, Phytograph falls back to the randomly-oriented-cylinder value (0.25) and labels it assumed. That fallback is sound rather than desperate: because a cylinder is symmetric about its axis, the coefficient spans only about ±13% across every achievable branch-angle distribution, and on a realistically mixed canopy it lands within a few percent.

Both LAD and WAD are absolute areas, not just a ratio. A discrete-return scan records the first thing each beam hits, so leaf and wood compete as independent risks along the path — and for that process the share of returns that are wood is exactly the share of interception that wood is responsible for. Multiplying that share by the total interception the inversion measured gives each medium's own area density.

Wood accuracy depends on voxel size

The one assumption is that leaf and wood are mixed within a voxel. A trunk thick enough to block a beam by itself takes nearly every return in its footprint, so a voxel large enough to hold both a trunk and the foliage beside it will misattribute between them. Holding the true areas fixed and varying only how the wood is distributed:

Wood in the voxel	Wood area error	Leaf area error
Fine twigs, well mixed	−0.2%	±0.0%
Mostly twigs, small trunk	−6%	−0.2%
Half twigs, half trunk	−21%	−0.9%
Mostly trunk	−53%	−0.6%

Leaf area stays accurate throughout — wood is the minority component, so misattributing it barely moves the leaf number. The split makes LAD better, never worse.

For accurate wood area the voxel has to be comparable to the trunk, so that the trunk fills its footprint rather than competing with foliage inside the same cell. Simulated for a 20 cm trunk in a canopy of LAD 2:

Voxel side	Trunk fills	Wood area error
2.0 m	10%	−20%
1.0 m	20%	−23%
0.5 m	40%	−20%
0.25 m	80%	−8%
0.15 m	100%	±0%

Fine woody material in the crown — twigs and small branches — is mixed through the foliage and is accurate at any resolution. It is specifically trunks and thick scaffold branches that need the fine grid.

The reason a finer grid is the only remedy is that a trunk saturates the beams that meet it: with a trunk in 5% of a voxel's footprint, 95% of the beams never meet wood at all and the few that do are stopped almost every time. A saturated return is a lower bound, not a measurement, so no reweighting of the beam data can recover what was never sampled.

If you need woody area for a trunk and cannot use a fine grid, measure it structurally instead: a QSM fits cylinders to the woody skeleton and reports their surface area directly, which is the right tool for trunks and the wrong one for foliage.

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:

1. 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 color. 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.
2. 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.
3. 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 — which is screened separately, below.
• If the data fall outside the method's validity envelope, no interval is reported rather than a misleading one.

Occlusion

Foliage intercepts beams, so a voxel behind dense canopy is reached by few beams traveling only a short way through it. Its Beer's-law inversion then goes wrong in a specific direction: LAD is overestimated, increasingly so as the probed path shortens.

Phytograph screens for this with the total probed beam path through each voxel — the sum, over all beams, of the chord each cut through it. Below a threshold the voxel is reported as occluded rather than measured: excluded from the leaf-area total and from the group-scale interval, written as NoData in every export, and drawn in its own color in the viewer. The default threshold is 100 × the voxel side length, the form the criterion takes in Soma, Pimont & Dupuy (2021).

Occlusion means beams went in and were stopped short — some path through the voxel, but not enough. A voxel that no beam entered at all is a different thing: it lies outside what the scan swept, which is ordinary for the headroom and margin around a canopy, and it keeps its honest LAD of zero rather than being counted as occluded. Folding those in would swamp the occlusion figure with geometry: on a 3 m box drawn around a 1 m canopy, doing so flags 69 % of the grid, of which only a handful is real occlusion.

Why not use the confidence interval instead?

It is tempting to screen on the Pimont interval — it is already computed, and a badly-sampled voxel ought to have a wide one. It does not work, for a reason worth knowing:

The sampling variance carries a factor I(1 − I)/N, where N is the beam count and I the fraction intercepted. Deep occlusion lowers N, which widens the interval — but it also drives I toward 1, which shrinks that numerator. The two effects largely cancel. Pushed far enough the interval gets narrower as occlusion worsens: a voxel reached by a handful of nearly-all-intercepted beams can report a confident-looking interval around a wildly inflated value.

Underneath that is a simpler point: occlusion produces bias, and a confidence interval describes variance. An interval is not obliged to cover a systematic offset, and this one does not. The interval remains the right tool for what it measures, which is why it is used to weight neighbors during filling — just not to detect occlusion in the first place.

Filling

Occluded voxels can optionally be estimated from the reliable ones by LAD-kriging (Soma et al. 2020), a kriging variant that treats each donor voxel's own sampling variance as a known measurement error, so better-measured neighbors count for more. Filled voxels stay marked as interpolations and their leaf area is reported apart from the measured total — treating a fill as data is the very bias the screening exists to remove.

References

• Pimont, F. et al. (2018). Estimators and confidence intervals for plant area density at voxel scale with T-LiDAR. Remote Sensing of Environment 215, 343–370.
• Soma, M. et al. (2020). Mitigating occlusion effects in leaf area density estimates from terrestrial LiDAR through a specific kriging method. Remote Sensing of Environment 245, 111836.
• Soma, M., Pimont, F. & Dupuy, J.-L. (2021). Sensitivity of voxel-based estimations of leaf area density with terrestrial LiDAR to vegetation structure and sampling limitations. Remote Sensing of Environment 257, 112354.

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