Making rain or surface water potable requires extensive filtration and biological action. Two ways of providing a biologically active area to clean the water is with a slow sand filter, or a fluidized bed filter. Slow sand filters are often used in third world nations to purify water. The top of the sand layer builds up a biologically active layer that traps and metabolizes organic matter.
The fluidized bed filter is also known as a quicksand filter. It pumps water through a bed of sand, turning the sand liquid. This captures foreign matter, and has a lot of surface area for bacteria to work on organic matter. Mostly used in aquarium systems, I don’t believe it’s as good as a slow sand filter for potable water production. It also requires energy to operate, unlike the slow sand filter.