Total rainfall and phi-index, 100 km2 basin

Each intensity acts for 30 min ($\Delta t = 0.5$ h).

Total storm rainfall

$$P = (4 + 2 + 6 + 5)\times0.5 = 8.5\ \text{cm}$$

$\phi$-index

Net storm runoff $R = 3$ cm, so the total loss $= 8.5 - 3 = 5.5$ cm.

Trial 1: assume all four intervals exceed $\phi$: $8.5 - 4(0.5\phi) = 3 \Rightarrow \phi = 2.75$ cm/h. But $2 < 2.75$, so the 2 cm/h interval gives no runoff.

Trial 2: only the intervals of 4, 6 and 5 cm/h contribute:

$$\left[(4 - \phi) + (6 - \phi) + (5 - \phi)\right]\times0.5 = 3 \Rightarrow 15 - 3\phi = 6 \Rightarrow \phi = 3.0\ \text{cm/h}$$

Check: $4, 6, 5 > 3.0$ and $2 < 3.0$. Excess depth $= (1 + 3 + 2)\times0.5 = 3.0$ cm. Valid.

Answer: Total storm rainfall = 8.5 cm; $\phi$-index = 3.0 cm/h.