Odd question, but what data would be needed to to calculate how far winds will carry a strong odor (such as the smells of a restaurant, pizza parlor or burning wood/building?
There is wind direction and wind speed. But how does one measure the strength of an odor and predict how far it will travel or how far the winds will carry it before it disappears?
The first problem to tackle is "what is an odor"? An odor is a chemical aerosol or gas, which are small molecules suspended in the atmosphere. To track the odors of pizza or burning wood you would first need to identify the molecules associated with the scent. One example is α-pinene, which is the molecule that gives pine trees their scent. Food cooking typically gives off other volatile organic compounds (VOCs) and the particular matter associated with it. VOCs physically and chemically combine to form more types of aerosols and there can also be increases in ozone.
Meteorology, emissions sources, and atmospheric photochemistry are combined in chemical transport models (CTMs) so that the air quality can be predicted. One example is WRF-Chem, which is used for large regions and many sources. Another example is CALPUFF, which is better suited for near-source impacts and capturing the effects of complex terrain. The reason you want a model with chemistry is that molecules can be reactive and will change as they interact with other molecules turning them into things that no longer smell like pizza or wood burning. The reason you need a meteorological model is so that pollutants will be advected with the wind and interact with the moisture budget, solar radiation, terrain, etc.
For large-scale plumes you can simulate your molecules as being emitted at a constant rate from a point source and see how this plume evolves with time. For smaller scale flows like a pizzeria, you may need use a model that can accommodate small scale features like buildings that will have an effect on atmospheric flows. Lastly, you'll also need to account for other emission sources as the perceived strength of the smell is also going to depend on what else you are smelling (e.g. car exhaust) and these other aerosols may be reactive or combined with the odors you are interested in tracking.
Once you have done this you will have a time varying plume of "smell" as output from your model. This will likely be output as a number concentration of aerosol molecules per unit volume per grid box. To turn this into "odor strength" is probably a harder problem and one I am not familiar with. As a first order approximation though, wherever the plume is located, so is the smell; and the higher the concentration of molecules, the stronger the odor.
Ofc what casey said is true, but I think the basic physics involved can be understood pretty easily by the mention of the advection-diffusion-equation:
This is the easiest model possible for this process, covering transport of the chemical compound with concentration $c(\vec x, t)$ by the wind (first term) and diffusion in adjacent air parcells (second term).
The operator $\vec v \cdot \vec \nabla$ will transport anything that you put behind it, in real space. I.e. perhaps you know this from the Navier-Stokes-equation where the famous "self-transport"-term $(\vec v \cdot \vec \nabla)\vec v$ causes alot of complications.
Then $\mu \Delta$ is the molecular diffusivity times the Laplace-operator which appears also in any random-walk / diffusion modell.
So this could give you the means to calculate the question you posed even by hand. Ofc the "disapperaing" of an odor, is just c falling below some threshhold value, where it is still noticable for human biology.
One interesting thing to note here, is that in a stratified atmosphere you obviously have a preferred direction in 3D-space. This will make the diffusivity $\mu$ more efficient in the horizontal plane, than vertically. You could take this into account by splitting your Laplacian into $\mu_{vertical}\partial_{zz} + \mu_{horizontal}(\partial_{xx} + \partial_{yy})$.
