Phi-index and W-index

$\phi$-index

The $\phi$-index is the constant rate of rainfall (cm/h or mm/h) above which the rainfall volume equals the direct runoff volume. Rain with an intensity below $\phi$ is assumed to be lost completely. When the intensity is above $\phi$, the loss is assumed to occur at the constant rate $\phi$.

$$\phi = \frac{P - R}{t_e}$$

where $P$ is the total rainfall, $R$ is the direct runoff depth and $t_e$ is the duration of rainfall in which intensity exceeds $\phi$. It is found by trial from the hyetograph so that the area above the $\phi$ line equals $R$. It is simple, but it is only an average and ignores the fact that infiltration decreases with time.

 i
 |        oooo
 |        oooo        oooo
 |  oooo  oooo        oooo
 |- - - - - - - - - - - - - - - - -  phi line
 |  ####  ####        ####
 |  ####  ####  ####  ####
 |  ####  ####  ####  ####  ####
 +--------------------------------- t
   oooo = rainfall excess (runoff R), #### = loss

W-index

The W-index is the average infiltration rate during the time that rainfall intensity exceeds the infiltration capacity. It corrects the $\phi$-index for depression storage and initial losses:

$$W = \frac{P - R - S}{t_r}$$

where $S$ is the initial loss plus depression storage (interception) and $t_r$ is the time in which the rainfall intensity is actually larger than the infiltration capacity. Because it removes the storage, $W$ is slightly less than $\phi$.

Both indices are used to estimate rainfall excess when direct infiltration data are missing.