Photo Scale

Photo scale is the classical statement of the same similar triangles that give the ground sample distance, written the way a photogrammetrist working with film would state it: as a representative fraction. $S = f/(H-h)$, and a result of 1:20 000 means one unit on the photograph corresponds to twenty thousand of the same units on the ground. The fraction is unit-free, which is its charm — it reads the same in millimetres, in inches and in chains, and it survives being photocopied at a different size only if you also copy the scale bar, which is why prints carried one.

The two heights are the piece that catches people. $H$ is the flying height above the DATUM — mean sea level, or whatever vertical reference the project uses — and $h$ is the elevation of the ground at the point you care about, above that same datum. Their difference is the height of the camera above that particular piece of ground. Entering a height above take-off for $H$ and an elevation above sea level for $h$ mixes two datums and produces nonsense, and it is the usual reason this equation returns something absurd.

A photograph has no single scale. This is the most important thing on the page, and the equation states it plainly once you notice that $h$ is in it. Every point at a different elevation sits at a different $H - h$, so a hilltop images at a larger scale than the valley beside it. The scale printed on an aerial photograph is the AVERAGE scale, evaluated at the mean terrain elevation, and it is exactly correct nowhere on the frame. This is the same geometry as relief displacement — the reason buildings appear to lean outward from the centre of a frame, and the reason a tall smokestack shows its side rather than just its top. You cannot reliably scale a distance off a raw photograph across varying terrain, and a photomosaic assembled from raw frames is not a map. Removing that variation, point by point, using a terrain model, is the whole job of orthorectification, and an orthophoto is the product you are allowed to measure on.

Focal length choice used to be a real design decision because of exactly this. The six-inch (152 mm) wide-angle lens was the workhorse of aerial mapping for decades; 88 mm super-wide-angle covered more ground per frame and gave a strong base-to-height ratio and therefore good heights; 305 mm narrow-angle gave less relief displacement, which mattered enormously over cities where leaning buildings hide the streets beside them. The trade has not changed with digital sensors — wide angle means good geometry and bad occlusion, narrow angle the reverse — only the focal lengths have.

Two smaller notes. Scale is a ratio here, so 1:20 000 is entered as 0.00005 rather than as 20 000; if you have the denominator, divide one by it. And the historical distinction between "large scale" and "small scale" trips everyone at first: 1:1 000 is a LARGE scale because the fraction is larger, and it shows a small area in great detail. 1:100 000 is a small scale covering a large area. It is the fraction that is being described, not the ground.

Digital work has largely replaced this relation with the ground sample distance, which says the same thing in a unit you can act on — centimetres per pixel rather than a dimensionless fraction. The two are the same geometry: $\mathrm{GSD} = p/S$. Scale survives because it is what the archives are labelled with, because it is how a photogrammetric camera's coverage is still specified, and because it makes the terrain dependence visible in a way that a single GSD figure quietly hides.