May I suggest getting help from an architect who has experience with earth-sheltered buildings in your region? So many important details: Roof weight, water retention and drainage, infiltration, insulation (earth is a poor insulator though effective for heat storage), ventilation, earth to air temperature comparisons, soil types, plantings, weather variability…
More thoughts…
Raúl asked a very important question about his project: “How long will it last?” In my opinion, all one’s thinking about buildings should be informed by the idea that the life of a good building ought to be measured in the hundreds of years. Too often green roof structures seem to aim for much, much less, sometimes mere years, or at best, decades. Wood and billboard tarps obviously fall into the much much less category, and while the appeal of lower initial cost is difficult to shake, it’s wise I think to consider long-term costs, including maintenance, repairs, rebuilds, and potentially even redesigns. (Long term calculations are full of pitfalls, but it’s more than reasonable to give it a good go in the planning stage.) This is not to say that a wood and tarp roof is never a good idea, only that it should, like all building ideas, be subjected to cold, hard analysis.
Last but not least is site consideration. The best place to build an earth-sheltered building is on wasteland, which will be improved by a living roof, as opposed to already biologically active ground, which is almost always diminished by a building, even a successful underground building.
I’ve designed a couple of earth sheltered structures for temperate climates, and for the record, am a fan of insulated concrete forms (ICFs for short) which can be engineered for walls and roofs. Also, notably, though a design professional myself, I was happy for help from more experienced colleagues.
All of this is of course stated out of context, and should be thoughtfully regarded in light of personal circumstance.