Satellites such as GOCE and GRACE measure the geoid with unprecedented accuracy. Altimeters can determine local surface elevation with millimetre-precision. This makes me wonder: to what precision and accuracy can we measure (or calculate) the volume of the Earth?
I'm a lay when it comes to solid Earth, and I find it surprisingly hard to find even the order of magnitude of an answer ($10^0$ metre? $10^3$ metre? $10^{-3}$ metre?). Wikipedia is extremely limited on the question and links to a surprisingly poor NASA factsheet that doesn't include any error estimates, cites no sources, and is imprecise in providing context or defining what it's talking about.
One can calculate the volume of Earth using semi-major and minor axes obtained from reference ellipsoids, but I have no idea how large an error is introduced by this approximation.
What are the precision and accuracy do we know the shape of the Earth?