Hi, beekie,
I highly recommend a video by Ceaser Millan, a Professional Dog Behaviorist. He rehabilitates dogs, and trains people.
He has his own TV program, the Dog Whisperer, on the NGEO (National Geographic) channel, on Friday nights mainly, but also comes on some other days. This video is available from:
http://www.dogpsychologycenter.com/
I also ordered his new book, "Ceaser's Way", from Walmart online, around $15 US, and it's very good!
There are other good videos available on the web, but this one is really worth your attention.
There are many episodes in Ceaser's episodes, dealing with dogs chasing critters while walking, while in the back yard, and even as your issue, digging out from under a fence to chase them!
It can be solved, with training. Once your dog gets the idea that she is NOT the boss; that you are, she will stop.
Training a dog changes the dog's behavior, and requires the owner to be knowledgeable on how to do it. You have to be boss, and the dog has to know it.
Take her outside, and when you see her start to dig, IMMEDIATELY firmly push her down with your fingers, say "NO". Start rubbing your hands over her body, giving her "massage" pleasure, getting her mind off the digging. This has to be consistent, and persistent, every time, not stopping. It might a coupld of weeks, for her to get the idea. It will work.
You have to stick with it, doing it every time you see her start to dig. She is doing it now, cause she knows you won't stop her, and keep stopping her. Training her not to do it, requires time. I do wish you the best.