Instead of bowing the roof you might want to put a peak in it like most conventional above ground houses, though the peak need not be so pronounced. The greater the pitch the better the drainage. On the other hand the greater the pitch the harder it is to keep earth up there. It may tend to slide or wash off if the pitch is too great. A steep incline also makes it more difficult to utilize the roof—as a lawn area, for example. The pitch can be minimized somewhat by mounding more earth towards the lower part, though you will always want some surface pitch to encourage surface runoff.
A pitched roof offers the possibility of coming above ground if that is desired. It also offers excellent clerestory and gabling possibilities. Again, the roof should drain into French drains and solid earth while the open ends should face out onto sunken patios and/or greenhouses.
On all of these houses you will want to construct a sufficient roof overhang on the southern exposure to block the high summer sun but little enough to allow the lower angled winter sun to enter. Planting deciduous trees on the south helps there too. The leaves shade the house in the summer while the naked branches allow the winter sun to warm the structure.
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