Given a DCEL, how do you identify the unbounded face

I have constructed a DCEL using the procedure described in How do I construct a doubly connected edge list given a set of line segments?.

This correctly identifies all faces, however I'm struggling to come up with a way to identify the unbounded face surrounding my graph.

So far my only idea is that by building a polygonal representation of every face, I could find the face polygon which 'contains' all the others, but this seems kind of messy.