Police had been called to Cottingley Road in Allerton on Monday afternoon after a man's body was found in a beck.
Tonight (Tuesday) the family of James Huby, 54, said they had to accept "we have lost our brother James in tragic circumstances".
In a statement issued on behalf of the family they said: "James will be sorely missed by his family and friends. James has been suffering with long term, complex, medical issues for many years but he is at peace now.
"We would like to thank James' friends who supplied information to the police and to all family members who contributed with information and local leafleting towards finding James.
"We would also like to express our gratitude and thanks to the officers and staff of West Yorkshire Police who have committed considerable expertise and resources to searching for and eventually finding James. It is through their dedication and care that he was found so quickly."
They added that they request their privacy be respected as they support his wife Veronica and close family members.