Elevation

In depth

The terms are used loosely, so the definitions matter. A digital elevation model is the general term for a gridded representation of height. A digital surface model records the top of whatever is there, including canopy and roofs, and a digital terrain model records the bare ground [1]. Their difference is the height of objects above the ground.

where $\mathrm{nDSM}$ is the normalised digital surface model, the height of buildings, trees and other objects above the ground; $\mathrm{DSM}$ is the digital surface model; and $\mathrm{DTM}$ is the digital terrain model; all three are grids of height in $\mathrm{m}$ on the same datum.

Figure 1 draws both surfaces in panel (a). Panel (b) shows the geometry of the first measurement method. Two views from height $H$ with separation $B$ project a point of height $h$ to different positions on the datum, separated by $\Delta$. Similar triangles give the height.

where $h$ is the height of the point above the datum, in $\mathrm{m}$; $H$ is the height of the two viewpoints above the datum, in $\mathrm{m}$; $B$ is the base, the distance between the two viewpoints, in $\mathrm{m}$; and $\Delta$ is the parallax, the separation of the point's two projections on the datum, in $\mathrm{m}$.

A larger base-to-height ratio $B/H$ gives more parallax per metre of height and so better precision, at the price of more difficult image matching. Finding $\Delta$ for every pixel is the matching problem, solved today by dense methods such as semi-global matching [2, 3].

The second method is laser ranging. A laser scanner carried on an aircraft measures the time of flight of each pulse, so the range follows equation 1 on the radar imaging page, and combined with the aircraft's position and the beam direction it gives the height. Part of each pulse passes between leaves, so one pulse can return from both the canopy and the ground, and this is why laser scanning can produce both models at once [4, 5].

The third is interferometric radar. From a repeat-pass pair the topographic phase of equation 3 on the radar interferometry page is the signal, and it is easier to think of in terms of the height that causes one full cycle of phase, the height of ambiguity. With $p = 1$ when two receivers share one transmitted pulse and $p = 2$ when each pass transmits its own,

where $h_a$ is the height of ambiguity, the change in height that adds one full cycle of $2\pi$ to the interferometric phase, in $\mathrm{m}$; $\lambda$ is the radar wavelength, in $\mathrm{m}$; $R$ is the slant range from the radar to the point, in $\mathrm{m}$; $\theta$ is the incidence angle; $p$ is 1 or 2 as above, without unit; and $B_\perp$ is the perpendicular baseline, the component of the separation between the two antennas at right angles to the line of sight, in $\mathrm{m}$.

A large perpendicular baseline gives fine height resolution but makes phase unwrapping hard, and a small one the opposite. Radar interferometry for topography was demonstrated in the 1980s [6]. The TanDEM-X mission flew two satellites in close formation to obtain single-pass interferograms free of temporal change, and produced a global elevation model [7, 8]. The theory of the design trade-offs is set out in [9]. Whatever the method, a model has a vertical accuracy that varies with terrain and cover, so the stated accuracy and the surface it refers to (surface or terrain) are the first things to read in a product description.