Deriving the Gaussian dispersion equation requires the assumption of constant conditions for the entire plume travel distance from the emission source point to the downwind ground-level receptor. Yet we cannot say with any reasonable certainty that the windspeed at the plume centerline height and the atmospheric stability class are known exactly or that they are constant for the entire plume travel distance. Whether such homogeneity actually occurs is a matter of pure chance, particularly for large distances. There are few knowledgeable dispersion modelers who would dispute that Pasquill's coefficients could be in error by plus or minus 25 percent, especially when used for non-level, complex terrain and for large distances ranging up to 50 kilometers or more. 1-hour average concentrations.
Atmospheric turbulence is also constant throughout the plume travel distance.
The dispersion pattern is probabilistic and can be described exactly by Gaussian distribution. Terrain conditions can be accomodated by using one set of dispersion coefficients for rural terrain and another set for urban terrain. The basic Gaussian dispersion equation is not intended to handle terrain regimes such as valleys, mountains or shorelines. In short, the Gaussian models assume an ideal steady-state of constant meteorological conditions over long distances, idealized plume geometry, uniform flat terrain, complete conservation of mass, and exact Gaussian distribution. Such ideal conditions rarely occur.