Normalized Intensity Distribution

The rainfall intensity distribution given in equation 1:3.3.1 can be normalized to eliminate units. To do this, all time values are divided, or normalized, by the storm duration and all intensity values are normalized by the average storm intensity. For example,

$\hat i =\frac{i}{i_{ave}}$ 1:3.3.2

$\hat t=\frac{T}{T_{dur}}$ 1:3.3.3

where $\hat i$ the normalized rainfall intensity at time $\hat t$, $i$ is the rainfall intensity at time T (mm/hr), $i_{ave}$ is the average storm rainfall intensity (mm/hr),$\hat t$ is the time during the storm expressed as a fraction of the total storm duration (0.0-1.0), $T$ is the time since the beginning of the storm (hr), and $T_{dur}$ is the duration of the storm (hr).

The normalized storm intensity distribution is:

$\hat i(\hat t)={\hat i_{mx}*exp[\frac{\hat t - \hat t_{peak}}{d_1}] , \hat i_{mx}*exp[\frac{\hat t_{peak}-\hat t}{d_2}]}$ 1:3.3.4

$0 \le \hat t \le \hat t_{peak}$$\hat t_{peak} < \hat t< 1.0$

where $\hat i$ the normalized rainfall intensity at time $\hat t$, $\hat i_{mx}$ is the normalized maximum or peak rainfall intensity during the storm,$\hat t$ is the time during the storm expressed as a fraction of the total storm duration (0.0-1.0), $\hat t_{peak}$ is the time from the beginning of the storm till the peak intensity expressed as a fraction of the total storm duration (0.0-1.0), $d_1$ and $d_2$ are equation coefficients.

The relationship between the original equation coefficients and the normalized equation coefficients is:

$\delta_1=d_1*T_{dur}$ 1:3.3.5

$\delta_2=d_2*T_{dur}$ 1:3.3.6

where $\delta_1$ is the equation coefficient for rainfall intensity before peak intensity is reached (hr), $d_1$is the normalized equation coefficient for rainfall intensity before peak intensity is reached, $\delta_2$ is the equation coefficient for rainfall intensity after peak intensity is reached (hr), $d_2$ is the normalized equation coefficient for rainfall intensity after peak intensity is reached, and $T_{dur}$ is the storm duration (hr).

Values for the equation coefficients, $d_1$ and $d_2$, can be determined by isolating the coefficients in equation 1:3.3.4. At $\hat t$ = 0.0 and at $\hat t$= 1.0,

$\frac{\hat i}{\hat i_{mx}} \approx 0.01$

$d_1=\frac{\hat t-\hat t_{peak}}{ln(\frac{\hat i}{\hat i_{mx}})}=\frac{0-\hat t_{peak}}{ln(0.01)}=\frac{\hat t_{peak}}{4.605}$ 1:3.3.7

$d_2=\frac{\hat t_{peak}-\hat t}{ln(\frac{\hat i}{\hat i_{mx}})}=\frac{\hat t_{peak}-1}{ln(0.01)}=\frac{1.0-\hat t_{peak}}{4.605}$ 1:3.3.8

where $d_1$ is the normalized equation coefficient for rainfall intensity before peak intensity is reached, $d_2$ is the normalized equation coefficient for rainfall intensity after peak intensity is reached, $\hat t$ is the time during the storm expressed as a fraction of the total storm duration (0.0-1.0), $\hat t_{peak}$ is the time from the beginning of the storm till the peak intensity expressed as a fraction of the total storm duration (0.0-1.0), $\hat i$ is the normalized rainfall intensity at time $\hat t$ , and $\hat i_{mx}$is the normalized maximum or peak rainfall intensity during the storm.