What are the methods/tools for generating realistic skies with clouds and atmospheric shading?
FOSS alternatives and spherical projections get extra points.
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I've seen someone generate impressive skymaps by modeling a scene in Blender, complete with elaborate particles, shaders, post processing, etc., and then use six 90-degree cameras to render out the skybox (which you could then convert to spherical, if you wanted. Might even be able to render directly to spherical.) This is of course most useful if you've got some kind of fantasy or surreal environment that'd be difficult to capture with photography.
If the 3D package you are using supports render-to-texture functionality you could always unwrap a sphere/hemisphere and bake an environment (your skybox in this case) to the unwrapped sphere texture. You can do this with a 6-sided box as well. I've played around with this in 3DS Max but I think Blender can do it too.