Terrain-Surface-Area Cover from LiDAR: Inverse-Probability-Weighted Estimation and a Global Cramér-von Mises Lack-of-Fit Test

Abstract

Canopy cover is normally reported as a fraction of a horizontal map plane. When the domain of inference is the land surface, the appropriate target is the fraction of terrain surface area under canopy. We define this estimand on a C1 graph surface and show that it differs from vertical-projection cover by the planimetric covariance of cover with the area density, divided by the mean area density; the rumple index is a different quantity. A LiDAR pulse is recorded as a canopy return only when detection succeeds, so the naive return fraction estimates cover multiplied by the canopy detection probability. We study an inverse-probability-weighted estimator that uses that probability as the inclusion probability, and a weighted corner-averaged Cramér-von Mises lack-of-fit statistic for constant mean surface-area cover under independent canopy returns, calibrated by a parametric bootstrap. Synthetic validation on the flat plane and the Gaussian hill gives a naive bias near -0.40 and a weighted bias near 0.00; at 5000 pulses the test holds size at 0.02-0.06 across the simulated null cells and reaches power 0.79 under surface-uniform and 0.85 under plane-uniform sampling against broad slope-dependent cover, but only 0.12-0.13 against a compact hotspot. The definitions and their exact reductions are machine-checked in Lean 4. No field data yet.