How do we measure (or empirically calculate) the amount of entropy in the atmosphere and oceans?
There is a convention for moist air entropy built from dry air entropy being 0 at T = 273.15K and P = 101325Pa, and liquid water entropy being 0 at its triple point. See the page numbered 47 of this NIST document:

http://www.nist.gov/calibrations/upload/5241.pdf

Entropy is not measured directly.  

Entropy $S$ is defined as $dS \equiv dQ_{rev}/T$

Therefore, entropy can measured by measuring heat transfer and temperature. 

See Calculation of Entropy Change in Some Basic Processes for more information.