Wavefield inversion has been considered with much attention by R. E. Kleinman. Some of his investigations have been led with the authors and their colleagues. Of particular examination here are two solution algorithms (complete family and binary-specialized modified gradient) for the retrieval of scatterers buried in a layered embedding. But the main purpose of this contribution is to illustrate the lasting impact of his work in this demanding field both at theoretical and numerical levels, and in so doing to sketch some challenging issues to be addressed if one wishes, as R. E. Kleinman worked so much for, to "accelerate the transition from mathematical model to practical numerical solution".