Vertical Precision from Base-to-Height Ratio

This is the page that should be printed on the back of every drone mapping quotation. Height in photogrammetry comes from the intersection of rays from two or more camera stations, and how sharply those rays cut depends on how far apart the stations are relative to how far away the ground is. That ratio, $B/H$, is the base-to-height ratio, and the height precision is the image measurement precision magnified twice: once by the image scale $H/f$, and once by the inverse of the base-to-height ratio. $\sigma_Z = (H/B)(H/f)\sigma_p$.

The first thing to take from it is how much worse height is than the pixel. Compare it with the ground sample distance, $\mathrm{GSD} = Hp/f$, and you find $\sigma_Z/\mathrm{GSD} = H/B$ when $\sigma_p$ is one pixel. On a typical drone flight at 80% endlap the base-to-height ratio is around 0.1 to 0.15, so vertical precision is seven to ten times the GSD. A flight planned at 1.4 cm GSD is delivering roughly 11 cm of height precision from its geometry alone — and the two numbers are routinely quoted as if they were the same. They are not, they are not close, and the ratio is knowable in advance from one division.

The second thing is that more overlap makes height worse. This is counterintuitive and it is important. Pushing the endlap up shrinks the base, and $\sigma_Z$ rises in exact proportion. Overlap is bought for reliable matching and redundancy, and those are real goods — but past about 80% you are trading height accuracy for them, not buying it. The operator who pushes to 90% "to be safe" has made the elevation model measurably worse while doubling the image count. If the deliverable is a surface model, that is the wrong trade.

The third thing is the one nobody says out loud: vertical precision degrades with the SQUARE of flying height while GSD degrades only linearly. The $H^2$ in the numerator is not decoration. At a fixed base, doubling the flying height doubles the GSD and QUADRUPLES the height error. Plan a flight on GSD alone and the penalty for going higher looks like a factor of two; the elevation model takes a factor of four. That asymmetry is the strongest argument there is for flying low, and it almost never appears in a flight plan because the planner only prints the GSD. (The honest qualification: if the base scales down with the height, as it does when the overlap PERCENTAGE is held constant rather than the base, then $\sigma_Z$ falls only linearly with $H$. Both cases occur, they give very different answers, and it is worth knowing which one your planner is doing before quoting either figure.)

Two words about $\sigma_p$, the image measurement precision. For automated matching on ordinary textured ground, something like half a pixel to one pixel is realistic; on strong texture in good light a third of a pixel is achievable; sub-tenth-pixel figures belong to laboratory targets and not to a field survey. Run the equation backwards on a claimed accuracy and see what $\sigma_p$ it would demand — if the answer is a small fraction of your pixel pitch, that accuracy is not coming from this geometry, and it is worth finding out where it is supposed to be coming from before it goes on a report.

And the largest caveat of all: this is a PRECISION, not an accuracy. It describes the random spread the geometry permits under ideal conditions. It says nothing about systematic error — a poor camera calibration, an unmodelled rolling shutter, an atmospheric refraction correction that was skipped, or a block tied to too few control points in the wrong places. The infamous vertical "bowl" of an uncontrolled drone block, where the middle of the model sits a metre below the edges, is entirely systematic; this equation cannot see it and never will. Good geometry is necessary and not sufficient. Ground control that is independently surveyed, well distributed including the edges and the middle, and partly held back as CHECK points rather than all used in the adjustment, is what turns a precise block into an accurate one — and check points are the only thing that ever tells you the truth about a deliverable.

One last route out of the trap. The base-to-height ratio is a proxy for the range of angles from which the ground is seen, and there is another way to widen that range: point some cameras sideways. Oblique imagery sees the same ground from very different directions, which is a long base by another name, and it is why oblique rigs reconstruct vertical structure — building façades, riverbanks, stockpile flanks — so much better than a nadir-only block at the same GSD.