# Development and Evaluation of a Fluctuating Plume Model for Odor Impact Assessment
## 2 Model Development
### Gaussian Fluctuating Plume Model for Statistical Moments (Marco's Model)  
Following [PERSON] et al. (2015), when the model deals with situations where the hypotheses underlying a Gaussian Plume Model are valid and the meteorology and turbulence are nearly steady-state and horizontally homogeneous, the horizontal meandering of the plume centroid can be described using a Gaussian PDF:  
\[p_{ym}(x,y_{m})\ =\ \frac{1}{\sqrt{2\pi}\sigma_{ym}}exp\left(-\frac{\left(y_{m}-y_{ s}\right)^{2}}{2\sigma_{ym}^{2}}\right) \tag{8}\]  
Indeed, in the vertical direction this homogeneity cannot exist, due to the action of the soil as a lower bound to dispersion and, in convective conditions, for the action of the PBL top \(z_{i}\) as upper bound. However, the simplest PDF that describes a realistic vertical meandering motion of the centroid of the instantaneous plume (principally in near neutral conditions) considers only the interaction with the ground, and therefore:  
\[p_{zm}(x,z_{m})\ =\ \frac{1}{\sqrt{2\pi}\sigma_{zm}}\left\{exp\left(-\frac{ \left(z_{m}-z_{s}\right)^{2}}{2\sigma_{zm}^{2}}\right)+exp\left(-\frac{\left(z _{m}+z_{s}\right)^{2}}{2\sigma_{zm}^{2}}\right)\right\} \tag{9}\]  
where \(y_{s}\) and \(z_{s}\) are the emission point coordinates. In these equations, \(\sigma_{ym}\) and \(\sigma_{zm}\) are the meandering dispersion parameters in horizontal and vertical direction and will be described in Section 2.6. For a generic instantaneous plume with the centroid in position (\(y_{m}\), \(z_{m}\)), described by PDF (8) and (9), the relative average concentration, \(\overline{C_{r}}\), of the instant plume is  
\[\overline{C_{r}}\ =\ \frac{Q}{U_{m}}p_{yr}(x,y,y_{m})p_{zr}(x,z,z_{m}) \tag{10}\]  
where \(U_{m}\) is the mean wind velocity at the height \(z_{m}\) and \(Q\) is the constant emission rate. For the quantification of \(\overline{C_{r}}\), the explicit forms of the PDFs, statistically describing the relative transverse and vertical dispersion, are needed. The relative dispersion is fully represented by the following two relationships:  
\[p_{yr}(x,y_{m})\ =\ \frac{1}{\sqrt{2\pi}\sigma_{yr}}exp\left(-\frac{\left(y_{m}- y\right)^{2}}{2\sigma_{yr}^{2}}\right) \tag{11}\]  
\[p_{zr}(x,z_{m})\ =\ \frac{1}{\sqrt{2\pi}\sigma_{zr}}\left\{exp\left(-\frac{ \left(z_{m}-z\right)^{2}}{2\sigma_{zr}^{2}}\right)+exp\left(-\frac{\left(z_{ m}+z\right)^{2}}{2\sigma_{zr}^{2}}\right)\right\} \tag{12}\]  
Further, in this case, a completely Gaussian relative transversal dispersion has been assumed, given the lack, in the hypotheses considered, of lateral barriers to the plume. Regarding the vertical relative dispersion, only the soil-bound has been considered. In order to make these PDFs effectively calculable, the standard deviations \(\sigma_{yr}\) and \(\sigma_{zr}\) must be known. Detailed explanation will be given in Section 2.6. Taking all this into account and assuming an instantaneous concentration fluctuating according to a Gamma distribution, the \(n^{\text{-th}}\) moment of the concentration is obtained by placing the (1), (6), (7), (8), (9), and (10) in (2). The result is the complex relation (13), which cannot be solved analytically due to the presence of some integrals, which quantifies both the average value and any \(n^{\text{-th}}\) moment of the concentration. In order to effectively calculate the average concentration and the variance, which are indispensable parameters for the quantification of olfactory annoyance, it is first necessary to know the analytical relations that express the different variances present (those relevant to the meandering movement and those relevant to the relative dispersion) and the parameter \(i^{2}_{cr}\) necessary for the complete definition of the Gamma distribution (see Equation (6)). \[