It will depend on:
1) Where your project is at and what the frost depths are there.
2) How frost susceptible your soil is.
3) Whether your can rely on your tunnel to be continuously heated.
4) Whether or not shallow frost protection strategies are employed (not that this is shallow).
Up where I'm at in Canada, the unheated frost depth might be in the 10'-12' range. So I'd want my tunnel foundations lower then that if the tunnel were unheated. And you'll pretty close to that by the sound of it.
You profile indicates a Georgia address. If your project is within a state or two of Georgia, I would expect you to be okay for frost.