The man, believed to be around 30, was discovered on the roof of Edinburgh’s Waverly Station beneath North Bridge at around 10am yesterday (Sun).
He was last seen outside the Bank Hotel on Edinburgh’s Royal Mile at around 4.15am.
Police insiders said that they were not looking for anyone in connection with the death.
Formal identification of the man has still to take place.
He is described as white, around 5ft 10ins tall, medium build, with short dark hair, and glasses.
He was wearing a dark blue and black striped shirt, black trousers and black trainers.
Police are appealing to anyone who may have seen him in the area afterwards to get in touch.
A police spokesman said: “We are appealing to anyone who thinks they saw the man between 4am and 4.30am today to contact us, in order that we can build a full picture of his final movements. Anyone else who has any information that can help our inquiries should also get in touch.”
Anyone with any information can contact Lothian and Borders Police on 0131 311 3131, or Crimestoppers in confidence and complete anonymity on 0800 555 111.