Volume of line bundles via higher rank valuation vectors

I will explain a not too widely known approach to intersection numbers (going back to Parshin), based on higher rank valuation vectors. When computing the volume of an ample line bundle with it, one gets formulas which look a lot like the volume of some mystery polytope. One can also compute the volume via the Okounkov body. Both approaches produce enticingly similar formulae, but they appear to be of a genuinely different nature.