total field, scatt field, incident field....

fisico30

in scattering problems, there is this formulation

total field= incident+ scattered

But once a scatterer is hit by an incident field, it re-emits, re-radiates in different directions.
Why should the field at a particular place always have to be the sum of the scattered and the incident? maybe the incident is not there?

We are like assuming that the incident field moves across the scatterer as if the object was not there. Then the scattered field modifies the incident field to get the total field.....

Behind the scatterer, I would, for instance just see scattered field....