The only time I've noticed where iced played a roll on how deer moved across a frozen river, was when the ice was still developing. It's as if the deer know, and I think by trial and error, that they can break through the ice. It's when it gets thick enough to walk on, they resume back to their crossing habits.
Ever see a deer cross ice? It's funny to watch. You can tell the ones who have already taken a dive because they walk stiff legged taking small steps. LOL!
iSnipe