# Generated Time to Peak Intensity

The normalized time to peak intensity is calculated by SWAT+ using a triangular distribution. The triangular distribution used to generate the normalized time to peak intensity requires four inputs: average time to peak intensity expressed as a fraction of total storm duration $$(\hat t_{peakM})$$, maximum time to peak intensity expressed as a fraction of total storm duration $$(\hat t_{peakU})$$, minimum time to peak intensity expressed as a fraction of total storm duration $$(\hat t_{peakL})$$and a random number between 0.0 and 1.0.

&#x20;                  The maximum time to peak intensity, or upper limit of the triangular distribution, is set at 0.95. The minimum time to peak intensity, or lower limit of the triangular distribution is set at 0.05. The mean time to peak intensity is set at 0.25.&#x20;

&#x20;                  The triangular distribution uses one of two sets of equations to generate a normalized peak intensity for the day. If $$rnd_1\le[\frac{\hat t_{peakM} - \hat t_{peakL}}{\hat t_{peakU} - \hat t_{peakL}}]$$ then

$$\hat t_{peak}=\hat t_{peakM} * \frac{\hat t_{peakL}+[rnd_1*(\hat t_{peakU}-\hat t_{peakL})*(\hat t_{peakM}-\hat t_{peakL})]^{0.5}}{\hat t_{peak,mean}}$$            1:3.3.9

If  $$rnd_1>[\frac{\hat t_{peakM} - \hat t_{peakL}}{\hat t_{peakU}-\hat t_{peakL}}]$$  then

$$\hat t_{peak}=\hat t_{peakM}*\frac{\hat t_{peakU}-(\hat t_{peakU}-\hat t_{peakM})*[\frac{\hat t_{peakU}(1-rnd_1)-\hat t_{peakL}(1-rnd_1)}{\hat t_{peakU}-\hat t_{peakM}}]^{0.5}}{\hat t_{peak,mean}}$$     1:3.3.10

where $$\hat t_{peak}$$ 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),$$\hat t_{peakM}$$is the average time to peak intensity expressed as a fraction of storm duration, $$rnd_1$$ is a random number generated by the model each day, $$\hat t_{peakL}$$ is the minimum time to peak intensity that can be generated,$$\hat t_{peakU}$$is the maximum time to peak intensity that can be generated, and $$\hat t_{peak,mean}$$ is the mean of $$\hat t_{peakL} , \hat t_{peakM}$$ and $$\hat t_{peakU}$$ .

