I believe its the Elizabeth from Columbia, but the Booker your playing is native to the rapture universe. When you play Elizabeth in the second Episode you will probably return to Columbia near the end, connecting the two.
The AD on the poster threw me a little, but maybe this Booker has also lost a child...
Being a detective in the rapture universe may have given Noir Booker access to the Bathysphere's (maybe you become part of Ryan's inner circle during the episode...) explaining why Columbia Booker was also able to use them at the end of the game.
Originally Posted by Jack Frost
Well, Booker was able to operate the bathysphere which was only suppose to function for those who had Ryan's DNA, so...
No, the Bathysphere's were only supposed to function for those in Ryan's "Inner Circle" but they were faulty and allowed everyone who was related to anyone in the "inner circle" as is mentioned in the following audio diary:
Originally Posted by audio diary
"We're putting all the bathyspheres in lockdown until further notice. Ryan had us install some kinda genetic device into the things so only Ryan and his inner circle will be able to use 'em without dispensation. But the boys tell me the keys are pretty unreliable. Sisters, cousins-anybody in the ballpark, genetically, will be able to come and go as they see fit."
Sander Cohan for example could use the Bathysphere's (how else did he get to Olympic Heights from Fort Frolic if you didn't kill him) and I don't believe he was related to Ryan in any way.