This is sort of Astronomical archaeology. We're looking back in time twice - once for the light travel time from the observed area to the telescope (The standard sort of Astronomical Time Machine) and then again for the bounce. If you miss observing something in real time, you can always look at it's echo later.

I'm sure there are other applications of this concept - echoes of the Big Bang, anyone? *

* I'm well aware that this wouldn't actually work, since the universe appears to have been opaque for the first little while when it was a glowing ball of plasma. Only later, when electrons and protons recombined to form matter did the Universe become transparent. (Atomic bombs have a similar double-flash history for similar reasons, and I'm sure the detailed light curve from a Supernova has all sorts of fascinating and tell-tale wiggles.)

How about echoes of GRB's? If these are really collimated beams, then yu wouldn't expect an echo from one you actually saw, but you might get an echo from one you DIDN'T see?