# Development and Evaluation of a Fluctuating Plume Model for Odor Impact Assessment
## 2 Model Development
### New Operational Gaussian Fluctuating Plume Model  
The estimation of the olfactory annoyance requires to quantify the odorants peak concentration defined as an appropriate percentile \(C_{p}\) of the instantaneous concentrations collected within one hour: this can be achieved from a PDF able to describe the instantaneous concentration. If this PDF is a two-parameter distribution (such as a Log-Normal, a Weibull, or a Gamma), then the knowledge of the hourly spatial distribution of the mean concentration and the variance of the concentrations are sufficient to obtain the peak. Our objective is, firstly, to obtain, from relation (13), for a generic \(n^{th}\) moment (uncentered), the appropriate relations for the average concentration \(C\) and for the variance of concentrations \(\sigma^{2}_{C}\) and, secondly, to simplify these relations by making them suitable for screening and regulatory use. Equation (13), obtained by [PERSON] et al. (2015), is totally general if the assumptions of applicability of a plume model are valid. Its practical applicability, however, is not straightforward, since it contains some integrals requiring a numerical solution. However, assuming the simple hypothesis that \(i_{\sigma}\) depends only on the downwind distance \(x\)[54], it is possible to analytically solve the integrals in (13) and finally obtain a totally algebraic relationship, even if rather complex, for the generic \(n^{th}\) moment (uncentered) of the instantaneous concentration:  
\[ \begin{array}{rl}\overline{c^{n}}\ =\ \left(\frac{Q}{2\pi\upsilon_{ym}c_{zx} \Gamma_{m}}\right)^{n}\frac{\upsilon_{yr}}{\left(\upsilon_{ym}^{2}+\upsilon_{ yy}^{2}\right)^{n}}\ \frac{\upsilon_{yr}}{\left(\upsilon_{ym}^{2}+\upsilon_{ yy}^{2}\right)^{n}}\cdot\frac{1}{\lambda^{n}}\ \frac{\Gamma(n+\lambda)}{\Gamma(\lambda)}\cdot\\ \cdot exp\left[-\frac{n\left(y-y_{iz}\right)^{2}}{2\left(\upsilon_{ym}^{2}+ \upsilon_{yy}^{2}\right)}\right]\cdot\sum\limits_{k=0}^{n}\left\{\left(\begin{array} []{c}n\\ k\end{array}\right)\cdot exp\left[-\frac{\left(n-k\right)\cdot(z-z_{k})^{2}}{2 \left(\upsilon_{ym}^{2}+\upsilon_{yy}^{2}\right)}\right]\cdot\\ \cdot exp\left[-\frac{k\left(z+z_{k}\right)^{2}}{2\left(\upsilon_{ym}^{2}+ \upsilon_{yy}^{2}\right)}\right]\cdot exp\left[-\frac{\left(2z\right)^{2}k(n-k )}{2\left(\upsilon_{ym}^{2}+\upsilon_{yy}^{2}\right)}\right]\right\}\end{array}  \tag{14}\]  
where \(\left( \begin{array}{c}n\\ k\end{array} \right)\) is the binomial coefficient \(\frac{n!}{k!(n-1)!}\). Considering _(n = 1)_, we obtain the Gaussian relationship for the spatial distribution of the mean concentration for a Gaussian plume:  
\[\begin{array}{rl}C\ =\ \overline{c}\ =\ \left(\frac{Q}{2\pi\upsilon_{ym}c_{zx} \Gamma_{m}}\right)\frac{\upsilon_{yr}}{\left(\upsilon_{ym}^{2}+\upsilon_{ yy}^{2}\right)^{n}}\frac{\upsilon_{yr}}{\left(\upsilon_{ym}^{2}+\upsilon_{yy}^{2} \right)^{n}}\frac{1}{\lambda}\ \frac{\Gamma(1+\lambda)}{\Gamma(\lambda)}\cdot\\ \cdot exp\left[-\frac{\left(y-y_{iz}\right)^{2}}{2\left(\upsilon_{ym}^{2}+ \upsilon_{yy}^{2}\right)}\right]\cdot\sum\limits_{k=0}^{n}\left\{\left( \begin{array}{c}1\\ k\end{array} \right)\cdot exp\left[-\frac{\left(1-k\right)\cdot(z-z_{k})^{2}}{2 \left(\upsilon_{ym}^{2}+\upsilon_{yy}^{2}\right)}\right]\cdot\\ \cdot exp\left[-\frac{k\left(z+z_{k}\right)^{2}}{2\left(\upsilon_{ym}^{2}+ \upsilon_{yy}^{2}\right)}\right]\cdot exp\left[-\frac{\left(2z\right)^{2}k(1-k )}{2\left(\upsilon_{ym}^{2}+\upsilon_{yy}^{2}\right)}\frac{\upsilon_{yy}^{2}}{ \upsilon_{yy}^{2 Remembering the property of Euler's Gamma function \(\Gamma(1+\lambda)\ =\ \lambda\cdot\Gamma(\lambda)\) and that \(\left( \begin{array}{c}1\\ k\end{array}