The Danube spends its height before it finds its water

#How was this measured?

The terrain is the Copernicus digital elevation model at 90-metre resolution, made from radar images taken by the TanDEM-X satellites, averaged here to cells of about 1.7 km.

The rivers come from HydroRIVERS, a global river network derived from satellite elevation data. Its 42,994 stretches in the Danube basin each carry a long-term mean flow from the WaterGAP hydrological model; the map shows the 14,248 with at least 1 m³/s. Checked against gauges, the model is close: about 1,940 m³/s at Vienna against a measured 1,920, about 2,310 at Budapest against 2,350, about 5,480 at Baziaș against 5,523. At the mouth it runs about 4% high.

The river's height is the lowest point of the elevation model within about 700 metres of the river line, smoothed so the river never flows uphill. Satellite elevation models flatten water surfaces, so these heights are a snapshot of the river's level when the images were taken. Bulgarian river agency gauges check them: their zero marks sit at 26.75 m at Novo Selo, 22.65 m at Lom, 14.89 m at Svishtov and 11.80 m at Ruse, and the satellite surface lies 0.1 to 2.2 m above each. In Vienna, though, the satellite reads the water up to eight metres too low (153 m at the Reichsbrücke, where the official mean water level is 161.42 m), so between Korneuburg and the Freudenau dam the profile uses the mean water levels published by via donau, Austria's waterway agency.

The power is gross hydraulic power: the density of water times gravity times the mean flow times the drop, summed segment by segment. It is the rate at which the falling river releases energy, before any turbine, and it is computed from mean flows, so floods and droughts average out.

The raindrop follows HydroRIVERS from the nearest mapped river stretch downstream to the sea. Its distances are measured along the mapped network, which runs 7 to 12% longer than the official river kilometres. The height of the starting point is the average ground in a cell of about 1.7 km, so on a mountain it reads lower than the summit. The countries come from Natural Earth's borders; Moldova's short stretch of Danube bank at Giurgiulești, too short for those cells, is added for every path that passes it. The river names come from Natural Earth's river lines, matched where they lie within three kilometres of a mapped river; two wrongly matched names were corrected (the Mur on the lower Drava, the Someșul Cald on the Someșul Mic and the Someș), and names that cover less than eight kilometres of a path are left out.

Two figures rest on weaker evidence. The kilometre of Donaueschingen, about 2,779, is inferred from the official German stationing; the Danube Commission gives the river's length from there as 2,845 km, so the shares of length here could be a point or two lower. The 27-metre head of the Iron Gates I dam comes from a Romanian press interview, and the Serbian side's 1,140 MW from a press report quoting the operator.