From a 3D-Var point of view,
we
want to find a Ta
that minimizes the cost function J:
Toy temperature data assimilation, variational
approach
We have a forecast Tb and a radiance obs
Innovation:
This analysis temperature Ta is closest to both the forecast Tb and the observation yo and maximizes the likelihood of Ta~Ttruth given the information we have.
It is easier to find the analysis increment Ta-Tb that minimizes the cost
function J