Why do we use white noise to estimate echoes?
The mathematical justification would be that the Cramer Rao Bound on the estimate of time delay is inversely proportional to the time bandwidth product of the pulse, so in cases where there are many distinct echoes, the shorter the pulse is better to resolve each arrival, so one would increase the bandwidth to minimize the bound on time delay.

In the case where the signal is long, as illustrated by the matlab code below for a single delay:


  clear all
  x=randn(1,16*32768);
  x1=[x zeros(1,.5*128)];
  x2=[zeros(1,.5*128) x];
  figure(1)
  pwelch(x1+x2)  




The psd has periodic nulls, and you can get an estimate from the null spacing, and the wider the band (white noise), the more nulls that can be resolved.

$$
H(\omega)=1+\alpha e^{\jmath \omega \tau}
$$

For 3 delays it will be a bit more of a bother, but doable.  Actually if you don't know the waveform, like a passive SONAR problem, this is an essential approach.