The work engaged by the authors was motivated by radar applications. But, although the actual radar systems dwell on electromagnetic waves, and targets cause to interrelate the field components, which gives rise to vector wave equations, the authors consider only one scalar functions \(u(t,x)\) satisfying the equation \[ \Delta u- \{1/c^*(t, x)\}^2 \partial^2_t u= 0, \] where \(c^*\) stands for the speed of the wave in the space including the targets. The usual aim of the problem consists in finding the region in which the target scattering density defined by \(V(t,x)= 1/c^2- \{1/c^*(t,x)\}^2\) is different from zero. Here \(c\) denotes the speed of the wave in vacuum. The attention of the authors is restricted to extracting structure-relevant information directly from measured data without needing recourse to constructing first an image of the target. Their approach relates the singular structure of the target (edges, for examples) to the singular structure of the data set. The paper includes many new definitions and theorems.