It kind of looks like F Lydian. The Lydian scale starts on the note that is four steps higher than the tonic (in this case, Bb). But the F Lydian scale would not include Eb (Bb, C, D, E, F, G, A Bb). Including the Eb and starting the scale on Bb makes it Bb Major, or Bb Ionian.
It's my understanding that the key signature (the number of sharps or flats) will give you the tonic. From there, the Mode is determined by what note the scale starts on. Dorian starts on the next note up from the tonic. Phrygian starts one major third above the tonic. Lydian starts one perfect fourth higher. Mixolydian starts on the perfect fifth higher. Aeolian starts one major sixth higher (natural minor scale), and finally Locrian, which starts one major seventh higher than the tonic.
Regardless of which note the scale starts on, the key signature remains the same. So if it's in F, you'd only have one flat (Bb), regardless of what note you start on.