S-hydrograph from rectangular 1-hr UH

The S-hydrograph (S-curve) is the hydrograph due to a continuous rain of 1 cm/h indefinitely. It is obtained by adding 1-h UHs, each lagged by 1 h:

$$S(t) = U(t) + U(t-1) + U(t-2) + \dots$$

The 1-h UH is a rectangle: height 0.25 per hour, base 4 h (area = 0.25 × 4 = 1 cm ✓). Between 0 and 1 h it has ordinate 0.25, between 1 and 2 h 0.25, and so on up to 4 h.

Construction (ordinates over each hour)

Since each UH rectangle lasts only 4 h, the maximum number of overlapping UHs is 4, so the S-curve reaches its equilibrium value 1.00 at t = 4 h (= base time of the UH) and stays there.

 S (per h)
 1.00 |           ___________________
      |        /
 0.75 |      /
 0.50 |    /
 0.25 |  /
    0 +--+--+--+--+--+--+--+-- t (h)
      0  1  2  3  4  5  6

Answer: S(t) = 0.25 t for 0 ≤ t ≤ 4 h (0.25, 0.50, 0.75, 1.00 at the ends of hours 1 to 4), then constant at 1.00 per hour. The equilibrium value equals 1 cm/h, the intensity of the continuous rain, as it should.