The main dome ring is fabricated from two layers of 12 mm plywood with a skirt of 3 mm plywood.
In the absence of any suitable level surface (apart from the top of the walls), I built the dome on pegs that were driven into the lawn.
Cutting the plywood into curved sections was the most laborious part of the job. As well as the horizontal rings, a considerable number of sections are required for the ribs that hold the skin of the dome.
To make life a bit easier, I built this jig for cutting curved sections using a standard router. I used a 6 mm cutter, which just survived the whole job, taking 6 mm deep cuts on each pass.
Hopefully the picture says it all, but it is basically a long arm made from shelf bracket fixing channel with a sliding wooden extension that can be clamped at different radiuses.