Tree inventory measurements

A tree inventory turns a segmented forest plot into a tree list: one row per tree, holding the measurements a forester records in the field (stem position, DBH, height, crown base, crown size), plus the quality evidence behind each number. This page is the method specification. Every measurement below names the definition it follows and the published method it uses. The defaults are ours, tuned on synthetic stems of known size. See Run a tree inventory for the steps.

Each tree is measured from the points that tree segmentation gave the same tree_instance id. Sky/miss points and deleted points are never used. All coordinates are world coordinates, so any global shift applied at import has already been added back.

Ground under the tree

Every height on this page is measured from the terrain surface (DTM), not from the tree's lowest point. On a slope, the lowest point of a stem is on its downhill side, so measuring from it would bias every height. The terrain elevation at any point under a tree comes from the first source that has a value there:

1. Height above ground. If Generate DEM has written the height_above_ground column, each point already carries the terrain elevation beneath it (z − height_above_ground). The ground elevation at the stem comes from a least-squares plane fitted through that value for the tree's points within 0.5 m (horizontally) of the stem and below 1 m, evaluated at the stem. A median is not used, because on a slope the stem points around the base sit unevenly up- and downslope of it, which biases a median.
2. Ground-labeled points. If only ground segmentation has run, the ground points are averaged on a 0.5 m grid. The elevation at the stem is that grid, inverse-distance weighted over the occupied cells within 2 m.
3. The tree's own lowest point. This is the last resort. The row is flagged ground_from_tree_min, because the value is biased on slopes.

The stem base elevation (the terrain at the stem center) is the reference for tree height and for the stem curve's heights. Breast height has its own reference, the uphill side of the stem (see DBH).

The sources are tried in this order for every lookup. A DEM that covers only part of the plot therefore hands the trees it missed to the ground labels, not to the last resort. The row's ground_source column records which source gave the stem base.

Stem axis, stem base and lean

Stem curve. A stem curve is the list of stem diameters, and their center positions, at a series of heights up the stem. Liang et al. (2014) measure it from TLS by fitting circles to thin cross-sections of the stem. We do the same in two passes.

1. Horizontal pass (to find the axis). Take 0.10 m thick horizontal slices every 0.25 m, from 0.5 m to 2.5 m above ground. Fit a circle to each slice (see Circle fitting). The median center of the successful fits is the seed. Then track the stem upward: each slice is searched only within 1.5·r + 0.10 m of the previous center, and a fit is accepted only if its radius stays within 0.5–1.5× the previous one and it is not flagged high_residual. Tracking stops after two failed slices in a row.
2. Stem axis. Fit a straight line through the accepted slice centers up to 4 m above ground, by least squares of center x and y on height. With fewer than three accepted centers, the axis is vertical through the seed.
3. Perpendicular pass (the reported stem curve). Cut slices perpendicular to the axis every 0.5 m of distance along it, starting at 0.5 m. A horizontal cut through a leaning cylinder is an ellipse, not a circle, and a circle fitted to it overstates the diameter by 1/cos(lean). A cut perpendicular to the axis is a true circle. The search window follows the previous center, so a gently curved stem is still tracked.

Stem base position. The stem base is where the stem axis meets the terrain. It is solved by intersecting the axis with the ground elevation at the axis's own x, y, by three fixed-point iterations. Its x, y is the tree's position in the tree list. Stem-mapping from the fitted stem, rather than from the point centroid, follows Liang et al. (2016) and Olofsson et al. (2014).

Lean is the angle between the stem axis and the vertical: lean = atan(√(bx² + by²)), where bx, by are the axis slopes (meters of horizontal offset per meter of height). The lean azimuth is the compass direction the stem leans toward: degrees clockwise from +y (grid north). It is left blank when the lean is below 0.5°.

Diameter at breast height (DBH)

Breast height is 1.3 m above the ground, the convention in countries using SI units. The US convention of 4.5 ft (1.37 m) is a setting (Kershaw et al., 2016; West, 2009). The textbook rules for where to measure are followed:

• On a slope, breast height is measured from the ground on the uphill side of the tree (Kershaw et al., 2016), that is, from the highest ground level at its base (West, 2009). The uphill ground is the highest DTM elevation around the stem's perimeter, sampled at 16 points on a circle of the stem's radius around the stem base. It is reported as breast_height_ref_z.
• On a leaning tree, breast height is measured parallel to the lean, and the diameter perpendicular to the stem's axis (Kershaw et al., 2016). The DBH slice is therefore centered on the stem axis, 1.3 m along the axis from the axis point level with the uphill ground, and cut perpendicular to the axis.

On flat ground with a vertical stem, all of this reduces to a horizontal slice 1.3 m above the stem base.

The breast-height slice is 0.10 m thick. If it holds fewer than 20 points, it is widened to 0.20 m and then to 0.30 m, and the thickness used is reported.

Circle fitting

Stem cross-sections are circles contaminated by branches, leaves, and the occluded back of the stem. The fit therefore has two stages: a robust search that finds the stem's points, then a precise fit to those points.

1. Robust search. One of two methods, chosen in the tool:

   • RANSAC (Fischler & Bolles, 1981), the default. Repeatedly fit a circle exactly through 3 random points and count the points within the inlier distance (default 0.02 m) of it. Keep the circle with the most inliers. The number of trials adapts to the best inlier ratio w found so far, N = log(1 − 0.99) / log(1 − w³), capped at 2000.
   • Randomized Hough transform (Xu, Oja & Kultanen, 1990). Map 3 random points to one point (a, b, r) in parameter space and vote for its cell in a sparse accumulator (cell 0.01 m). Take the cell with the most votes after 2000 samples as the candidate. Its inliers are the points within the inlier distance of it.

   Both reject candidate circles with radii outside 0.02–1.5 m. Both use a fixed random seed, so the same tree always gives the same answer.

2. Precise fit on the inliers.

   • Taubin's algebraic fit (Taubin, 1991) gives the starting circle. Algebraic fits have a closed form, but Kåsa's fit (Kåsa, 1976) underestimates the radius when only part of the circle is visible. Taubin's fit has a much smaller bias (Al-Sharadqah & Chernov, 2009), which matters for TLS stems, where the far side of the trunk is always occluded.
   • The geometric fit then minimizes the sum of squared orthogonal distances Σ(‖pᵢ − c‖ − r)² by Levenberg–Marquardt, starting from Taubin's circle (Chernov & Lesort, 2005; Chernov, 2010). This is the reported circle.
   • Inliers are recomputed against the geometric circle and the geometric fit is repeated once.

Quality evidence

Every circle (DBH and each stem-curve slice) reports:

Field	Meaning
rms_m	Root-mean-square orthogonal residual of the inliers, in meters.
arc_coverage	Fraction of 36 angular sectors (10° each) around the fitted center that hold at least one inlier. 1.0 = the whole circumference was seen.
max_gap_deg	Largest empty angular gap between inliers, in degrees.
n_points, n_inliers	Points in the slice, and inliers of the fit.

A circle fitted to a short arc is poorly constrained. As the arc shrinks, the bias and instability of every circle fit grow (Chernov & Lesort, 2005; Al-Sharadqah & Chernov, 2009), and the radius can trade off against the position of the center. Arc coverage is therefore reported for every fit. Arc coverage and the flag thresholds below are our own quality measures, tuned on synthetic stems; they are not taken from a published standard. DBH carries these flags:

Flag	Raised when
few_points	fewer than 20 inliers
partial_arc	arc coverage below 0.5 (less than half the circumference seen)
high_residual	rms_m above max(0.01 m, 0.1·r)
no_stem	no circle could be fitted at breast height; DBH is blank

A flagged DBH is still reported. The flag is the warning, and the value is not silently dropped.

A stem-curve row is ok = false when its fit has few_points or high_residual, or when its radius is outside 0.5–1.5× the previous ok slice's radius (a jump no stem makes in 0.5 m, usually a branch or a neighbor's stem). partial_arc does not make a slice not-ok. Seeing about half of each stem is the normal case for a single scan position, and the fits stay accurate there. Only ok slices steer the tracking up the stem, and only ok slices define the stem radius that separates stem from crown.

Height

Tree height is the highest point of the tree minus the ground elevation at the stem base. TLS tends to underestimate height, because the top of a tall tree is often occluded or only sparsely hit (Liang et al., 2018). The value is what the points show. It is not corrected.

Crown

The crown is the tree's points that are not stem. A point is stem if its perpendicular distance to the stem axis is within the stem radius at its height plus max(0.10 m, 0.5·r). The stem radius at a given height is interpolated from the stem curve, and the last radius is held above it. This test uses only geometry, so it works on leaf-off scans and does not need leaf/wood labels.

Crown base height. Definitions of the crown base vary. One convention in the mensuration literature is the first whorl of branches with live foliage in at least 3 of the 4 quadrants around the stem (Kershaw et al., 2016). LiDAR methods find the crown base from the vertical distribution of a tree's returns (Popescu & Zhao, 2008). We combine the two, with a decision rule of our own:

1. Bin the crown points by height (0.2 m bins). A bin is "occupied" if it holds at least 5% as many points as the fullest bin.
2. Walk down from the top occupied bin. Stop at the first empty stretch of 1.0 m or more. The occupied bins above that stretch are the continuous crown. Isolated low branches below such a gap are excluded.
3. Within the continuous crown, the base is the lowest bin whose points occupy at least 3 of the 4 quadrants around the stem axis. A quadrant counts when it holds at least 5% of that bin's points. This keeps a one-sided branch just below the crown from becoming its base.

The value is reported as height above the stem-base ground.

Crown projected area is the area of the convex hull of the crown points projected onto the horizontal plane. It is the laser-scanning counterpart of the field crown projection (Fleck et al., 2011), and convex hulls are an established way to compute it from point clouds (Lin et al., 2017).

Crown diameter. Two values are reported:

• crown_diameter_mean_m is the mean of the widest crown width and the width at right angles to it. This follows the field convention of measuring the crown at its widest point and again at right angles, and averaging the two (Kershaw et al., 2016). The two widths are reported as crown_max_width_m and crown_perp_width_m.
• crown_diameter_equiv_m is the diameter of the circle with the same area as the projection, 2·√(A/π).

Crown eccentricity. Two meanings are in use, and both are reported:

• crown_offset_m / crown_offset_azimuth_deg give the horizontal distance, and direction, from the stem base to the centroid of the crown projection. Crown asymmetry, in both direction and degree, is used as a measure of competition between trees (Seidel et al., 2011). This offset is our way of expressing it.
• crown_ellipse_eccentricity describes the shape of the projection. It is √(1 − λ₂/λ₁), where λ₁ ≥ λ₂ are the principal second moments of the hull polygon. 0 is a circle, and values near 1 are a long, narrow crown.

Voxel crown volume. Divide space into cubes (default 0.10 m) and count the cubes that hold at least one crown point. Multiply the count by the volume of one cube (Hosoi & Omasa, 2006). The result depends on the cube size: smaller cubes approach the volume actually occupied by returns, and larger cubes approach an envelope. The cube size is reported with the result.

Derived per-tree values

• Basal area g = π/4 · DBH² (m², DBH in meters). This is the cross-sectional area of the stem at breast height (Kershaw et al., 2016).
• Slenderness is the ratio of total height to DBH (Wang, Titus & LeMay, 1998), both in meters, so it has no units.

Both are left blank when DBH has the no_stem flag.

References

• Al-Sharadqah, A. & Chernov, N. (2009). Error analysis for circle fitting algorithms. Electronic Journal of Statistics 3:886–911.
• Chernov, N. (2010). Circular and Linear Regression: Fitting Circles and Lines by Least Squares. CRC Press.
• Chernov, N. & Lesort, C. (2005). Least squares fitting of circles. Journal of Mathematical Imaging and Vision 23(3):239–252.
• Fischler, M.A. & Bolles, R.C. (1981). Random sample consensus: a paradigm for model fitting with applications to image analysis and automated cartography. Communications of the ACM 24(6):381–395.
• Fleck, S. et al. (2011). Comparison of conventional eight-point crown projections with LIDAR-based virtual crown projections in a temperate old-growth forest. Annals of Forest Science 68(7):1173–1185.
• Hosoi, F. & Omasa, K. (2006). Voxel-based 3-D modeling of individual trees for estimating leaf area density using high-resolution portable scanning lidar. IEEE Transactions on Geoscience and Remote Sensing 44(12):3610–3618.
• Kåsa, I. (1976). A circle fitting procedure and its error analysis. IEEE Transactions on Instrumentation and Measurement IM-25(1):8–14.
• Kershaw, J.A., Ducey, M.J., Beers, T.W. & Husch, B. (2016). Forest Mensuration, 5th ed. Wiley-Blackwell.
• Lin, W., Meng, Y., Qiu, Z., Zhang, S. & Wu, J. (2017). Measurement and calculation of crown projection area and crown volume of individual trees based on 3D laser-scanned point-cloud data. International Journal of Remote Sensing 38(4):1083–1100.
• Liang, X. et al. (2014). Automated stem curve measurement using terrestrial laser scanning. IEEE Transactions on Geoscience and Remote Sensing 52(3):1739–1748.
• Liang, X. et al. (2016). Terrestrial laser scanning in forest inventories. ISPRS Journal of Photogrammetry and Remote Sensing 115:63–77.
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• Olofsson, K., Holmgren, J. & Olsson, H. (2014). Tree stem and height measurements using terrestrial laser scanning and the RANSAC algorithm. Remote Sensing 6(5):4323–4344.
• Popescu, S.C. & Zhao, K. (2008). A voxel-based lidar method for estimating crown base height for deciduous and pine trees. Remote Sensing of Environment 112(3):767–781.
• Seidel, D. et al. (2011). Crown plasticity in mixed forests — quantifying asymmetry as a measure of competition using terrestrial laser scanning. Forest Ecology and Management 261(11):2123–2132.
• Taubin, G. (1991). Estimation of planar curves, surfaces, and nonplanar space curves defined by implicit equations with applications to edge and range image segmentation. IEEE Transactions on Pattern Analysis and Machine Intelligence 13(11):1115–1138.
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