The Earth Sciences Institute was never mentioned in dialogue. The name comes from an assignment patch worn by the A-6 excavation team.
The Science Council may be associated in some way with the Institute, since the A-6 team checked in with them. The Star Trek Encyclopedia (4th ed., vol. 1, p. 227) stated the ESI was a part of the Science Council and, additionally, the institute operated a research station near the Arctic Circle.