As the polar bear turns in their direction the adventurers, who are armed with a rifle, begin to shout and stamp their skis to scare off the bear who duly turns away from the small group.
The footage, filmed in in the Svalbard region of Norway in 2006, has emerged after a 17-year-old boy on the 2011 expedition was killed in a polar bear attack that left four others injured.
Svalbard is an archipelago of islands in the Arctic circle and is home to an estimated 1,500 polar bears from the Arctic's Barents Sea population.
The BSES, a youth development charity that organises scientific expeditions to remote areas and is based in Kensington, west London, has not yet named the 17-year-old boy.
It said the injured men were trip leaders Michael Reid and Andrew Ruck and trip members Patrick Flinders from Jersey, and Scott Smith.
They suffered head injuries but are now stable, Norwegian authorities said.
The father of Patrick Flinders, Terry, said he believed the polar bear had crossed a trip wire and into his son's tent.
"According to the doctor and the other people Patrick was trying to fend off the polar bear by hitting it on the nose - why, I don't know, but he did and... the polar bear attacked him with his right paw across his face and his head and his arm," he said.
BSES chairman Edward Watson said the society had been in touch with the family of the boy who died and had offered "our utmost sympathy".
He said the society's executive director was travelling to Svalbard, adding: "We are continuing to gather information on this tragedy."
A spokeswoman for the Svalbard governor's office said: "We got a call via satellite phone from a British group of campers that there had been a polar bear attack and that one person was dead and that others were injured and they needed assistance."
"There are no roads in the area where the incident happened so we scrambled a helicopter.