Air Quality Forecasting - also referred as 'Atmospheric Dispersion Modeling' (wikipedia) is the art of simulating how air pollutants (eg PM2.5 or Ozone) disperse is the ambient atmosphere. The result of the simulation gives the ambient concentration for each air pollutants, from which the Air Quality Index can be calculated.
There are dozens of dispersion models and many universities and institutions around the world are running one or several of them for different regions (eg Europe, Continental US, Asia ...). Below is a non-exhaustive and selected list of few models which are being used for providing Air Quality forecast on the "World Air Quality Index" project.
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