To make sure the energy distributions of the emission and absorption peak overlap, we can set the source on a train that moves towards the scatterer. Then the emitted photons will get a blue shift and the distributions can overlap when the train has the correct velocity.
Here lays some problems about realizing the set-up: the train needs to have a large constant velocity, which is difficult to realize (but impossible of course).
The second problem is that there needs to be a gigantic track where this train can ride on. This needs to be gigantic because of this large velocity.
A last problem is the fact that we can only measure for a short period, because we can only measure as long as the train has not passed the scatterer.