Thiessen-Weighted Watershed Rainfall (Three Gauges)

A rain gauge measures one point, but a runoff calculation wants one rainfall depth for the whole watershed. The Thiessen method, as the Natural Resources Conservation Service describes it in the National Engineering Handbook (Part 630, chapter 4), gets there by giving each gauge its own share of the ground. Draw the perpendicular bisectors between neighbouring gauges and the watershed divides into polygons, so that every point lies in the polygon of its nearest gauge. Each gauge is then taken to represent its whole polygon. A gauge's weight is its polygon area divided by the watershed area, the weights add to 1, and the watershed depth $P_w$ is the sum of each weight times its gauge's depth, which is the area-weighted mean written in the formula.

That assumption is the method's real content. The arithmetic is exact once the polygons are drawn, but how good the answer is depends on how well each gauge stands for its polygon. If a gauge is added or removed, the polygons and weights must be redrawn. For more than three gauges, combine two first: the area-weighted mean of two gauges can stand as a single gauge with the sum of their areas, and the result is the same. The handbook does not use the Thiessen method for mountainous watersheds, where elevation strongly controls the rainfall and a gauge is not typical of the ground around it; its isohyetal method is the alternative.

The handbook's Example 4-2 has polygons of 81, 31 and 87 acres in a 199 acre watershed, so weights of 0.407, 0.156 and 0.437. Gauge depths of 1.40, 1.54 and 1.94 inches give weighted depths of 0.570, 0.240 and 0.848 inches, which sum to 1.658 inches; the handbook rounds this to 1.66 inches. The page gives 1.65789, against the printed 1.658. The areas may be in any unit as long as all three match. The page's own example uses three polygons of 1 ha with 10, 20 and 30 mm: equal areas mean equal weights, so the answer is the simple mean, 20 mm.

The calculation also runs backwards. If the watershed depth is known, the page can find a missing gauge depth or polygon area. With the printed average of 1.658 inches, the third gauge comes out at 1.9403 inches, against the printed 1.94. The watershed depth is the input the curve number method wants.