Deriving tree height from shadow geometry

Vegetation management on a transmission corridor comes down to one question asked thousands of times: is this tree tall enough to matter? Clearance is a vertical measurement, and a satellite image is a plan view. It shows you the crown from directly above. It tells you nothing, directly, about how far that crown is from the ground.

There are two ways to get the missing dimension. One of them is much better than the other, and most systems use the worse one because it is easier.

The easy way: allometry

Allometry infers height from crown diameter using a power relationship — height equals some coefficient times diameter raised to some exponent. Measure the crown from above, look up the coefficients for the species, get a height.

It is genuinely useful, and it is genuinely a statistical answer. The equation describes how trees of that type tend to grow. It does not describe the tree in front of you, which may have been topped by a previous cutting cycle, may be leaning, may be growing in a gully, or may simply be an outlier. On a corridor the outliers are the entire point: the average tree is not the one that causes the outage.

The failure mode is worse than it looks, because allometric coefficients are usually quoted for a region and a species and then applied far outside both. A table fitted in one forest type, used in another, produces confident numbers that are wrong in a consistent direction.

The better way: measure the shadow

A tree standing in sunlight casts a shadow whose length is a function of exactly two things — the height of the tree and the elevation of the sun. Both are knowable. The sun's position for a given latitude, longitude, date and time is deterministic; it is in the scene metadata. So if you can measure the shadow in the image, you can solve for the height.

This is a measurement of that tree, not an estimate of trees like it. A topped tree casts a short shadow. A leaning tree casts a shadow that reflects where its crown stands. The outliers stay outliers instead of being averaged into the population.

Why it is not the default

Because shadows are inconvenient. They fall across other objects. They merge when trees are close together. They vanish under cloud, at low sun angles, and in dense canopy where every shadow lands on another crown. A shadow-based method that only works in ideal conditions is not a system, it is a demonstration.

So the practical design is shadow-primary with allometry as the fallback, and — this is the part that matters — a record of which one produced each number. A height derived from a clean shadow and a height inferred from a growth curve are not the same kind of fact, and a system that presents them identically has thrown away the most important thing it knew.

• The shadow bearing is derived from the scene's own sun azimuth, not assumed. Get this wrong by 180° and every measurement is confidently wrong.
• The raster's orientation is asserted at startup rather than trusted. A north-up assumption that silently fails produces plausible numbers in the wrong direction.
• Allometric coefficients are treated as starting points to be calibrated against the scene, never as constants. A fitted pair hardcoded as if it were physics is a bug that never throws.

What this changes operationally

Clearance is a threshold decision — cut or do not cut — and the cost of the two errors is wildly asymmetric. Cutting a tree that was never going to reach the conductor wastes a crew day. Not cutting one that was going to reach it is an outage, and in the wrong season a fire.

A method that is right on average is the wrong tool for a threshold decision on outliers. That is the whole argument for measuring rather than inferring.

Clearance is a threshold decision taken on outliers. A method whose accuracy derives from averaging is characterising the wrong population.