Mr Wigram, 51, who was originally from Wellington in New Zealand, sparked a major search involving local lifeboat crews and a Coastguard rescue helicopter on Friday when he was reported missing following a trip in his distinctive red kayak.
He had last been seen at about 11am last Thursday when he parked his car in Bigton, on the west of the mainland, before entering the water with his kayak.
Mr Wigram is understood to have been wearing a wetsuit when he vanished after going into the sea at the St Ninian's Isle beach.
A major search began for Mr Wigram, described as an experienced kayaker, when he was officially reported missing on Friday.
The alarm was raised by a member of the public who noticed he had still not returned to his car.
A rescue helicopter and two lifeboats searching the area were stood down at lunchtime on Saturday as the search was called off.
A Shetland Coastguard spokesman said the kayaker could not have survived in the water for long because of the low temperature of the sea.
He said: "The survivability is now long gone. He would only have had one hour or so survival time if he had been in the water."
Mr Wigram had been in Shetland for more than ten years.
A chemical engineer by profession, he had worked in the oil industry and in forestry in eight countries across the globe before arriving in Shetland to work at the Sullom Voe oil terminal.
Five years ago he briefly worked as a probationary maths teacher at Brae High School on Shetland before returning to work as a chemical engineer, working as an independent consultant, mostly in Norway.
A close friend, who did not wish to be named, said Mr Wigram would be sadly missed by the many people on Shetland who knew him.
She said: "When he came to Shetland he made himself very familiar with the place very quickly and got to know an awful lot of people very quickly. He was very warmly regarded by a lot of people.
"He was a very free spirit and very much his own person – and he will be very deeply missed by family and friends."
The friend explained: "He tended not to stay in any one job for long periods of time. I think he liked being independent.
"But he loved kayaking and was very experienced. It was certainly his main hobby Shetland.
"He would go out in his kayak on a regular basis and he kept himself very fit.
"He loved the natural environment and he loved challenging himself within the natural environment. He probably did go out where others might have hesitated."
A spokesman for Northern Constabulary said: "Police in Shetland are continuing their inquiries into the missing man in Bigton, Shetland, who can now be named as Kester Wigram, 51, originally of Wellington in New Zealand."