Define dirty. The question says potable. The stuff that will hurt you requires very fine filtration that I don't think is feasible using natural elements. Filters for protozoan cysts and bacteria typically filter to 0.2 microns
Yes you can filter larger particulate matter (e.g. dirt) but that is not (typically) the dangerous stuff.
Is Water Treatment Even Necessary?
Q: Why do I need to treat water in wilderness areas?
A: Regardless of how pure water may look, any water source on the planet could be tainted with microscopic waterborne pathogens—disease-causing pests that, if ingested, could cause severe diarrhea, cramps, vomiting and fever.
Recent research, though, suggests that wilderness water at higher elevations is much cleaner than previously believed. Some experts argue that the blame for intestinal infections is more often traceable to preexisting conditions and lax sanitation, particularly unwashed hands.
Q: What's in wilderness water that can affect me?
A: Three groups of waterborne critters are most commonly linked to water-related illness:
- Protozoan cysts (Cryptosporidium parvum, Giardia lamblia). Tiny (1 to
300 microns; 1 micron = one-millionth of a meter).
- Bacteria (Escherichia coli, or E. coli, Salmonella, Campylobacter
jejuni, Yersinia entercolitica, Leptospira interrogans and many
others). Very tiny (0.1 to 10 microns).
- Viruses (hepatitis A, rotavirus, enterovirus, norovirus, Norwalk
virus). Exceptionally tiny (0.005 to 0.1 micron). Viruses are rarely
found in North American wilderness waters. Only purifiers (not
filters) eliminate viruses.