Hi-

The warp field is made in the base/master dset grid. A value sits at each grid point, and tells you where to pull data from in the input/source dset. Through an interpolation kernel (NN, wsinc5, cubic splines, etc.), a value is mapped via the warp dset to that new location.

The Jacobian is a vector field related to the gradient of the warp. Most people use the scalar-valued determinant of the Jacobian det(J) (or, even more specifically, log(det(J))) to measure the properties of the warp. The idea is that the magnitude of det(J) tells you about expansions (>1) or contractions (<1).

The warps calculated by AFNI are diffeomorphic and invertible. So, I believe that the whether you calculate det(J) on the warp or its inverse, the information should be equivalent (just inverted: if warp shrinks by a factor, then the inverse warp expands by that same factor), according to this:
[en.wikipedia.org]

So, whether you calculate the Jacobian on a warp or its inverse, shouldn't deeply matter-- it is probably more a question of which grid you want your information on. And which way you want to think about "expansion".

Of course, if you want a veeery good answer, we should get Bob's opinion...

--pt