# 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. --- # Agent Instructions: Querying This Documentation If you need additional information that is not directly available in this page, you can query the documentation dynamically by asking a question. Perform an HTTP GET request on the current page URL with the `ask` query parameter: ``` GET https://swatplus.gitbook.io/io-docs/~/changes/GiVD5ID0WVyRIAumaMy2/weather-generator/precipitation/maximum-half-hour-rainfall/normalized-intensity-distribution.md?ask= ``` The question should be specific, self-contained, and written in natural language. The response will contain a direct answer to the question and relevant excerpts and sources from the documentation. Use this mechanism when the answer is not explicitly present in the current page, you need clarification or additional context, or you want to retrieve related documentation sections.