Species diversity can be calculated through available indices after going through precise data collection however , we do not have any universally accepted index which could be used to compute biodiversity in any system.
this is question of possibility to approximate data for all community (principle of integrative and representative ability of ecosystems) or how to explain the M. state & Metacommunity Processes. I'm attaching a file from Encyclopedia "Encyclopedia of Biodiversity, Volume 2", where the question is discussing. You have their not long but critical Ref. in the question, esp important is the paper of Caley MJ and Schluter D (1997) The relationship between local and regional diversity. Ecology78: 70.
Species diversity is merely an aspect of biodiversity as a whole. There may be multiple species in an area, but one might be dominant and extremely higher in biomass than the others. We still need to take into account the relative abundances.