Two particular inverse scattering problems are of special interest. The first concerns the discovery of a perturbation in the speed of sound by analyzing the return signal from a blast wave set off above it. The second concerns the determination of a potential from the back scattering of plane waves. In one dimensional problems the two cases are very closely related. In higher dimensions the situation becomes much more complicated. We present here a new approach to four such classical higher dimensional inverse problems for determining the coefficients of a partial differential equation, including the two mentioned. The idea stems from the work of Deift and Trubowitz [1], in one space dimension. No conclusive theorem is found but the approach provides an algorithm for iteration which might lead to an existence theorem and which will be explored numerically elsewhere.