Phi-index of an 8-hour storm

Method

The $\phi$-index is the constant rate of loss such that the rainfall in excess of it equals the direct runoff. Losses = $15.0 - 8.7 = 6.3$ cm over 8 h.

Trial

Assume all hours exceed $\phi$: $15 - 8\phi = 8.7 \Rightarrow \phi = 0.7875$ cm/h. But the 1st hour (0.6) is below $\phi$, so exclude it (loss in that hour is 0.6 cm). Try 7 hours: $(15 - 0.6) - 7\phi = 8.7$, so $\phi = 0.8143$ cm/h. The 8th hour (0.75) is also below $\phi$, so exclude it too. Try 6 hours:

$$(15 - 0.6 - 0.75) - 6\phi = 8.7 \Rightarrow 13.65 - 6\phi = 8.7 \Rightarrow \phi = 0.825\ \text{cm/h}$$

Check

The excess equals the 8.7 cm runoff, and the 6 hours with excess have rainfall > 0.825 cm/h, so the trial is valid.

Answer: $\phi$-index = 0.825 cm/h.