We consider the necessity of half-off-shell transition-matrix elements (near-field) information on iterative approaches to the inverse acoustic scattering problem. We then show that, because of the manner in which half-off-shell effects enter, the recently introduced Volterra inverse series correctly predicts the first two moments of the interaction, while the Fredholm inverse series is correct only for the first moment. Finally, we demonstrate that the Volterra approach provides a method for exactly obtaining interactions that can be written as a sum of delta functions.