Our house is completely underground as far as the roof and sides of the house go. We've thought about covering the atrium, but covering it like a bomb shelter would block any light coming into two bedrooms, one of which we use for our office, and block ventilation coming in from the front. It would have to be completely sealed, including keeping the outside door closed to keep the water from pouring down from street level. But then we would have to install French drains and re-route the water around the house. Right now at least the drains in the atrium do work except for sudden deluges.
The cover option would be more feasible if the first owner installed a couple of skylights in the roof. It is too late to do that now because of the rebar frame. Also, the atrium is the only convenient access to our house. The long winding driveway keeps us from coming in through the garage. (I've slipped and fallen on my back trying to navigate the driveway on foot, and as I've stated in the article, we can't drive our cars out of the garage and up it.)
We've also thought about building a shell of a building on top of the whole house, but that would be about 3,000 sq. ft. of roof alone if we covered the atrium, too. It is just a poor design, poor location, and shoddy construction that doesn't deserve to keep on existing. We were much younger when we bought it and just didn't think it through. Several friends warned us that underground houses were impractical in our area and would leak. We should have listened to them.