LiDAR analyses#

Classification (crates/ohl-lidar, classify), vegetation and as-built wires (crates/ohl-clearance, vegetation and wirefit). All three are a base to build on: the methods are simple, published and documented with their limits.

Classification#

Ground — a progressive morphological filter (K. Zhang, S.-C. Chen, D. Whitman, M.-L. Shyu, J. Yan, C. Zhang, "A progressive morphological filter for removing nonground measurements from airborne LIDAR data", IEEE Transactions on Geoscience and Remote Sensing 41(4), 2003):

1. A grid of cell size \( c \) (1 m) holds the lowest point of each cell.
2. The grid surface is opened — eroded (minimum filter) then dilated (maximum filter) — with square windows of \( w_k = 2r_k + 1 \) cells, \( r_k = 1, 2, 4, 8, \dots \) while \( w_k c \) is within the largest window (20 m). Opening removes objects smaller than the window.
3. A cell whose lowest point stands above the opened surface by more than \( \Delta h_k = s\,(w_k - w_{k-1})\,c + \Delta h_0 \) (at most \( \Delta h_{max} \); \( \Delta h_0 \) for the first window; slope \( s \) = 0.3, \( \Delta h_0 \) = 0.3 m, \( \Delta h_{max} \) = 3 m) holds no ground; the opened surface is the input of the next window.
4. The lowest points of the remaining cells make a TIN (no triangle sides above 50 m).

The minimum and maximum filters are separable (rows, then columns), so a window costs \( 2(2r+1) \) comparisons per cell. Empty cells are skipped.

Heights above the ground — every point to classify (by default classes 0 and 1; points already classified are kept) is measured above the TIN: within \( \pm \) 0.3 m it is ground (2); above, the first height band it falls in — ASPRS LAS 1.4 classes 3 (low vegetation, up to 0.5 m), 4 (medium, up to 2 m) and 5 (high, up to 60 m), all parameters — and above the last band high noise (18); below the ground, low noise (7). Buildings, poles and wires above the ground get vegetation classes too: telling them apart (planarity, linearity, returns) is not part of this base.

Throughput (release build, one core): 4 million points over \( 10^6 \) cells in 1.2 s, about 3 million points a second. Points are held in memory (32 bytes each; --max-points, default 200 million); grids over \( 10^8 \) cells are refused. Exit clauses: a time limit (30 minutes in ohl lidar classify), checked per grid row and every \( 2^{20} \) points.

The result is written as LAS/LAZ 1.4 (point format 6) or stored as a new point cloud in the project file.

Vegetation#

For each wire (a span of a section) of a weather case — a hot case for sag, a windy one for blowout — every point of the vegetation classes (by role):

• Encroaching: its radial distance to the wire (the nearest subconductor) is below the required clearance plus the growth until the next cut, \( g = \text{growth per year} \times \text{years} \) — the tree grown by \( g \) in every direction.
• Fall-in: with the ground below it, a point of height \( h \) above the ground (at least 2 m) is taken as a tree top on a trunk straight below; falling about its base \( P_b \), the top sweeps a sphere of radius \( h + g \). The tree can reach the wire when \( \min_{q \in \text{wire}} |q - P_b| < h + g \); the margin is the difference. Leaning trees and crown widths are not modelled.

Per wire: the counts, the nearest point, the tree with the smallest fall-in margin, and the worst points by margin (the smaller of the clearance margin and the fall-in margin). Wire segments are indexed on a 10 m plan grid with the search reach (clearance + growth + 5 m, or the tallest tree looked for), so each point is compared with the few segments near it — about 1 µs a point.

As-built wires#

Points of a span: points of the wire classes (roles conductor, shield_wire) within a plan distance (1 m) of the vertical plane through the modelled span's chord, within a height window (3 m) of the modelled wire (still air, a chosen temperature), away from the ends by 5 % of the span. Each becomes \( (s, z) \): horizontal distance along the chord from the first clamp, and elevation. Bundle subconductors are fitted together.

Fit: the catenary in the plane,

\[ z(s) = z_0 + C\left[\cosh\frac{s - s_0}{C} - 1\right], \]

with low point \( (s_0, z_0) \) and catenary constant \( C = H/w \), by least squares on the vertical residuals. The start is a least-squares parabola \( z = a u^2 + b u + e \) (\( u \) centred on the mean \( s \)): \( C = 1/2a \), \( s_0 = \bar s - b/2a \), \( z_0 = e - b^2/4a \). Gauss–Newton steps with Levenberg–Marquardt damping (D. W. Marquardt, J. SIAM 11(2), 1963) on the Jacobian

\[ \frac{\partial z}{\partial C} = \cosh u - 1 - u\sinh u, \quad \frac{\partial z}{\partial s_0} = -\sinh u, \quad \frac{\partial z}{\partial z_0} = 1, \qquad u = \frac{s - s_0}{C} \]

until the step is below \( 10^{-10} C \) (about ten iterations; exit clause 500). Points further than three RMS (at least 5 cm) from the first fit are dropped (birds, other wires inside the window) and the fit repeated once.

Sag of the fitted wire: the largest vertical distance below the chord from \( s = 0 \) to \( s = L \) — the model's definition. Where the curve is parallel to the chord, \( \sinh((s^* - s_0)/C) = m \) with \( m \) the chord's slope, so \( s^* = s_0 + C \operatorname{asinh} m \). Compared with the model: the sag difference, the height difference at mid-span and at the low point.

Temperature at the survey: the section is solved in still air at trial temperatures — with its stringing, creep condition and clamping as modelled — until the span's sag equals the fitted sag: regula falsi (Illinois variant) on \( \text{sag}(T) - D_{fit} \), increasing in \( T \), between −30 and 120 °C, to 0.01 °C or 0.1 mm of sag (about six solves; exit clause 60). Outside that range no temperature is given. The temperature is that of the model's sag–temperature relation: errors of the model (stringing tension, creep) show up as temperature.

Tests: a synthetic survey of the example line at 35 °C — points every 0.5 m along each span with ±3 cm noise, birds above the wires — gives every span's sag within 2 cm and its temperature within 2 °C (in practice ±0.5 °C). Validation: wirefit-001…002 (exact points of known catenaries; sag against a brute-force evaluation).