# Development and Evaluation of a Fluctuating Plume Model for Odor Impact Assessment
## 4 Results and Discussions
### Peak-to-Mean Factor, \(R_{90}\)  
The \(R_{90}\) was calculated considering both the Gamma and the Modified Weibull [83] as representative PDF for the instantaneous ambient concentration. In Figure 6, the results for the peak-to-mean factor are reported. The orange results refer to the Gamma PDF and the blue ones to the Modified Weibull. It is possible to see the simulated values (\(y\)-axis) of \(R_{90}\) plotted in ordinate towards the field data (\(x\)-axis). On Figure 6a the scatter plot is reported. In addition to the bisector, the 1:0.5 and 1:2 lines have been reported. 90% of the results obtained by the Gamma PDF are within the range of _FAC2_. Instead, 80% of the results obtained with the Modified Weibull are within the _FAC2_ range. Focusing on the numerical indexes, in the case of the Gamma PDF, \(MB=0.35\), \(NMB=0.15\) and \(NMSE=0.28\) were obtained. Considering the Modified Weibull, \(MB=0.87\), \(NMB=0.37\) and \(NMSE=0.79\) are calculated. All the calculated correlation indexes are reported in Table 5. Because of the presence of the factor 1.5 (see Equation (21)), present in the Modified Weibull, the results for this PDF show higher values. On Figure 6b a q-q plot is reported. The results do not lay strictly on the bisector, but very similar trends can be seen for both PDFs: the obtained data seem to be diverse only for a static translation, probably due to the presence of the \"safety factor\" 1.5 [97]. In Figure 7, the predicted \(R_{90}\) contour plots calculated testing the Gamma PDF are shown for all considered trials, while in Figure 8 the predicted \(R_{90}\) contour plots calculated testing the Modified Weibull. Even in these graphs, it is clear that the values calculated using the Modified Weibull are higher than those obtained with the Gamma PDF as a reference. Figures 7 and 8 provide further information with respect to the scatter plot and q-q plot. It is possible to see that the values of \(R_{90}\) vary along the plumes. Not precisely at the emission source (located in the domain center) but in the strict proximity, there are higher \(R_{90}\) values (yellow-orange areas). This result depicts a different trend with respect to the one proposed by empirical peak-to-mean literature [17; 22]. They proposed a monotonic trend for the peak-to-mean value, maximum at the emission and decreasing with the downwind distance. Here-obtained results depict a non-monotonous \(R_{90}\) trend, with a peak near the source but not at the emission, then decreasing with the downwind distance, as proposed by [17; 22]. This model shows a greater complexity, due to the non-unique dependence on the distance. Another observation evident from Figures 7 and 8 is that \(R_{90}\) increases with distance from the plume centerline. This evidence is in accordance with the theory of [PERSON] [98]. Referring to the cited paper, the dependency of the peak-to-mean factor from \(y\) is exponential and \(R_{90}\) increases by getting closer to the extremes of the plume, according also to experimental evidences [44; 77]. Reaching the edges of the domain, the colors of the figures indicate decreasing values with distance. A characteristic shape, in certain experiments V-shape, in others Y-shape, is formed because of these dependences on \(x\) and \(y\) (lighter yellow). All these considerations are valid for both the Gamma and the Modified Weibull PDFs. It is important to note that the obtained values fall between 1 and 4, in particular 1-3 for the Gamma PDF and 1.5-4 for the Modified Weibull. More in detail, the dispersion of the points in the scatter plot for both the PDFs is evident. This can be explained by considering that the physics of the model is not completely satisfactory: there is not a monotonic trend between the simulated values of \(R_{90}\) and the experimental ones. Similar tendency of dispersion is evident also in [99], where the authors tested a new approach: the use of Lagrangian Particle Model LASTAT coupled with the implementation of the concentration-variance transport and the Modified Weibull [83].