  The purpose of this paper is to solve the inverse scattering problem of
nonlinear Alfv\'en waves governed by the three dimensional ideal incompressible
MHD system. Bridging together geometric methods and weighted energy estimates,
we establish a couple of scattering isomorphisms to substantially strengthen
our previous rigidity results. This answer is consistent with the physical
intuition that Alfv\'en waves behave exactly in the same manner as their
scattering fields detected by the faraway observers. The novelty of the present
work is twofold: for one thing, the relationship between Alfv\'en waves
emanating from the plasma and their scattering fields at infinities is explored
to the best; for another thing, the null structure inherent in MHD equations is
thoroughly examined, especially when we estimate the pressure term.
