The voxel solver resolves radiation paths between cells and treats vegetation inside each cell as a spatially homogeneous mixture. This page explains which quantities are observed, which are assumptions, and how the resulting calculation conserves energy.
For a propagation direction \(\mathbf{d}\), component \(c\) in voxel \(v\) has equivalent area density \(a_{v,c}\) in \(\mathrm{m^2\,m^{-3}}\) and projection coefficient \(G_c(\mathbf{d})\). Its extinction coefficient is
The DDA traversal supplies the length \(\ell\) travelled along the ray inside the cell. Beer–Lambert transmission is
No additional division by solar elevation or \(\cos\theta\) belongs in this formula: obliquity is already represented by the traversed length. The standard spherical leaf-angle assumption gives \(G=0.5\), and we use that value for all components by default. A different constant \(G\) can be supplied per component.
PAD estimated by inverting lidar attenuation already depends on the inversion projection model, sampling, and any density cap. The AMAPVox PAD estimator documentation describes those dependencies. Using another projection coefficient in a simulation changes the effective extinction; so you should also consider updating the value for computing PAD.
The JSONL input supplies fractions of lidar points within each voxel that are associated to individual plants, and what proportion of these points are wood, leaf or unknown.
For an identified individual \(i\) with point fraction \(p_{v,i}\), let \(f_{v,i,o}\) be its fraction of points in organ class \(o\): leaf, wood, or unknown. In point_proxy mode, their product supplies an equivalent area fraction:
This assumes the identified, classified points represent the vegetation responsible for the cell's full PAD. Differential visibility, return probability, scanning geometry, and classification error can violate that assumption. The result is suitable for explicit proxy experiments; it is not a direct measurement of individual LAD or WAD.
If the model provides an independently justified identified-area fraction \(\gamma_v\) through identified_area_fraction, the allocation is \(PAD_v\gamma_v p_{v,i}f_{v,i,o}\); the remaining \(PAD_v(1-\gamma_v)\) must stay as unattributed vegetation. Such a coverage fraction cannot be recovered from the existing identified-only point ratios. Without that keyword, point_proxy uses the representativeness assumption above in cells containing identified points. A positive explicit fraction in a cell without identified points is rejected.
A positive-PAD cell without an individual list still intercepts radiation. It becomes an unattributed component of unknown organ. Its attenuation, absorption, and scattering remain in the scene budget rather than disappearing from the calculation.
An unknown organ of a known individual is different: that energy remains assigned to the individual and is reported as its unknown organ. It must not be reassigned silently to leaves or counted as APAR. The existing format does not retain organ counts for points without an individual ID, so the reader cannot infer a leaf/wood split for the unattributed component.
For every known-PAD cell, the component densities satisfy
PAD=0 remains transparent even if classified points are present. PAD=null is unknown, not observed empty space. The input workflow rejects unknown values by default; explicit zero imputation must remain visible in diagnostics.
For incoming energy \(E\), the energy intercepted by the cell is
Numerically, -expm1(-κ * ℓ) avoids cancellation for weak attenuation. If \(\kappa_v>0\), each component receives
If \(\kappa_v=0\), the ray passes through without interception. Applying a separate Beer–Lambert law to every individual using the same incoming energy would duplicate photons. Giving the individuals an arbitrary traversal order would introduce a spatial arrangement the file does not contain. Summed extinction and proportional allocation avoid both errors, as in the mixed turbid-medium treatment of RATP.
When all components share the same \(G\), the allocation reduces to equivalent area fractions and total extinction remains \(G\,PAD\). With different constant \(G_c\), both extinction and the interception fractions change.
Suppose a cell has \(PAD=2\), individual A represents 30% of its retained points, and 20% of A's points are classified as wood. The proxy density of A's wood is \(2\times0.30\times0.20=0.12\ \mathrm{m^2\,m^{-3}}\). The rest of A contributes \(0.48\) if all its remaining points are leaves. These are conditional fractions: 20% wood within A is 6% of the cell's total proxy area, rather than 20% of the cell.
With a common \(G=0.5\), a 1 m path and 100 J entering this cell, total interception is \(100(1-e^{-1})\simeq63.21\) J. A's wood receives 6% of that, about 3.79 J, before optical absorption and scattering are applied. It does not receive the entire beam.
Wood transmittance is zero. Once a ray encounters wood material, the intercepted energy is absorbed or reflected:
This says nothing about gaps around unresolved branches inside a voxel. Those gaps remain governed by wood density, projection, and Beer–Lambert transmission. Black wood, \(\rho=0\), is a possible simplifying assumption; real bark can reflect radiation. Choose reflectance per band when data exist. Use separate parameters for unknown organs and unattributed vegetation rather than silently applying physiological leaf properties to them.
Wood surface conventions also require care: a cylinder's outer surface and its projected silhouette are different areas. The effective PAD and point fractions used here do not reconstruct either geometric quantity. Comparisons with explicit branch meshes must state which area convention and projection are being matched.
For each component and band, absorptance is \(\alpha_{c,b}=1-\rho_{c,b}-\tau_{c,b}\). The intercepted share splits into
The isotropic solver can sum component emissions within a cell and reuse its geometric transport. Every subsequent intercepted order is allocated with the same extinction shares before absorption and re-emission. Individual identity creates output components; it does not require a separate set of rays for each tree.
The terminal budget includes all vegetation absorption, soil absorption, boundary escapes, and unresolved scattering energy. Cumulative interception over all scattering orders can exceed external incoming energy, because reflected or transmitted radiation can encounter vegetation again. This is because it counts repeated interactions. The budget is conservative in the end: the sum of all absorbed, scattered, and escaped energy equals the total incoming energy (there is a difference between intercepted and absorbed energy). Only absorbed PAR of the leaf components contributes to leaf APAR. Whole-plant PAR absorption also includes wood and unknown organs and is a different quantity.
Energy is reported in joules per step. Dividing by proxy area and step duration gives \(\mathrm{W\,m^{-2}}\) per equivalent plant surface; that denominator must not be presented as independently measured leaf area.
LightOptions.toricity is the common boundary reference. Periodic rays wrap horizontally; finite scenes receive light through the top and upwind sides, and lose rays at open sides. Independent terrain truncates a path at the nearest exact soil hit. Positive PAD below that hit receives no radiation. See Terrain And Soil for surface alignment and periodicity.
A homogeneous voxel mixture and a finite set of leaves need not give identical shadows: one assumes unresolved statistical gaps, the other resolves actual positions and orientations. Validate homogeneous limits analytically, refine ray sampling and voxel size, and compare converged aggregate absorption and soil illumination for geometrically comparable cases. Explicit solid trunks, few-leaf cells, strong clumping, and directional scattering require particular care. A good energy balance is necessary, but does not validate the proxy area fractions or leaf-angle assumptions.
The current model uses regular cells, constant projection coefficients, isotropic vegetation scattering, and Lambertian opaque soil. It assumes components are homogeneously mixed within each cell, without resolved within-cell clumping or organ positions. Independent measurements and sensitivity analysis remain necessary for uncertain PAD, point attribution, wood projection, and optical coefficients. Angular leaf distributions, anisotropic phase functions, adaptive grids, and solid-voxel reconstructions are separate future extensions.