Phi-index of storm, 20 cm rain, 12 cm runoff

Total rainfall $= 0.7+2.2+3.0+4.6+3.6+3.2+2.0+0.7 = 20.0$ cm; runoff $R = 12$ cm, so total loss $= 8$ cm over 8 h.

Trial 1: if all 8 hours exceed $\phi$: $20 - 8\phi = 12 \Rightarrow \phi = 1.0$ cm/h. But the 1st and last hours (0.7 cm) are below 1.0, so they do not produce runoff.

Trial 2: exclude the two 0.7 cm hours (their full 1.4 cm is lost) and use 6 hours:

$$(20 - 1.4) - 6\phi = 12 \Rightarrow 18.6 - 6\phi = 12 \Rightarrow \phi = 1.1\ \text{cm/h}$$

Check: hourly rain 2.2, 3.0, 4.6, 3.6, 3.2 and 2.0 are all above 1.1, and 0.7 is below. Excess $= (2.2+3.0+4.6+3.6+3.2+2.0) - 6\times1.1 = 18.6 - 6.6 = 12.0$ cm. Valid.

Answer: $\phi$-index = 1.1 cm/h.