At 10.33am on Tuesday 3 May, officers were called to Hampton Wick station following reports of a casualty on the tracks.
Paramedics also attended and a man was taken to hospital for treatment to serious injuries, where he remains.
The incident is not being treated as suspicious, but despite extensive enquiries officers have been unable to identify the man and his family.
The man is described as white, in his 40s, with short dark shaven hair and facial hair. He also has a number of large and distinctive tattoos on his chest and arms.
Anyone with information is asked to contact BTP by texting 61016 or calling 0800 40 50 40 quoting reference 200 of 03/05/22.
