Roy Gross, chief of the Suffolk County Society for the Prevention of Cruelty to Animals, said the bird was still able to fly, eat and walk, despite the arrow sticking straight through its body.
"It appears to be a target arrow," Gross said, adding that hunting arrows have razor tips that are more lethal. "That's probably why it's still alive."
Gross said investigators still have not been able to get the bird, which was first seen near Sag Harbor Cove across from Long Beach Road and Fosters Memorial Beach.
"If it is not captured and given veterinary care, it will die a long, slow death," Gross said. He said the SPCA is searching for help from any group that has specialized equipment to capture birds.
The agency has also offered a $1,000 reward for information leading to the arrest and conviction of the person who shot the goose.
Canada geese, which are migratory birds, are under federal protection, Gross said. It is also illegal to shoot them with a bow and arrow, Gross said.
"It has to be suffering," he said. "It's just a horrible act of animal cruelty."
Anyone with information about this case is asked to call the SPCA at 631-382-7722.
The shooting is the second in recent months on Long Island. In October, a domestic white Pekin duck was found with a target arrow through its neck in North Patchogue. The SPCA was also searching for the shooter in that case.