From your oscillscope traces, it looks like the region of interest is the about 50us wide, starting 90us after the A.out rise.
Since analogRead() takes about 100us per analogRead() - Arduino Reference so you could get a value in the window, and with the target at 1 foot out it's pretty much back to baseline by t+200us.
About the best you can do (without low-level, fast ADC trickery) is start one single analogRead() after or in sync with the pulse and use that value.