Hydration number: Carbonic acid in water
You may wish to refer to the publication of Saykally et al., The hydration structure of aqueous carbonic acid from X-ray
absorption spectroscopy

Basically, to determine hydration number, X-ray spectroscopy was used. This relates to the short existence of carbonic acid as, since it is so short lived, it is necessary to analyze it in such a time frame. As such, this was "the first measurement of X-ray absorption spectra for aqueous carbonic acid, enabled by the use of rapid-flow liquid microjet technology" which allowed it to be captured.

Then, by analyzing the spectra, it is possible to calculate the approximate hydration number, as "the increasing H-bond strength is proportional to the hydration
number (measured for distances up to $2.5\mathrm{\overset{\circ}A}$): $\ce{Na2CO3}$ (5.55), $\ce{HCO3}$
(4.26), $\ce{H2CO3}$ (3.17)".