Meaning of "Objectively analyzed"
In my field of oceanography, objective analysis (OA) is often taken as a synonym of optimal interpolation (OI). (e.g. Emery &amp; Thomson, Section 4.2; and in many lectures). 

The same method is also referred to as objective mapping or Gauss-Markov smoothing. In this case OA is a least squared interpolation where the estimate is unbiased and the expected error is a minimum in the least squares sense.  

In many other cases, objective analysis is taken to be a specific type of optimal interpolation. A good example is the group from University of Liege. In this case OI includes other methods like kriging, successive corrections or the Cressman method. 

In my case, I tend to follow the nomenclature and suggestions by Ide et al. (1997).