if I were to suffer your "ordeal," I would cheat like mad - taking an accurate compass bearing, relating to some sort of generally visible landmark in the distance ( mountain, tall tree, grain elevator- you name it), and then marching off, probably counting paces to handle the distance covered.

I know of no desert, at least here in North America, that fortunately isn't replete with all kinds of terrain features - at a very minimum, distant mountain ranges..