Consider a one-dimensional semi-bounded "ray" (or thin "cylinder") of stratum (or just any homogeneous substance) which has several "features" along it's length (say, thin films). Consider a wave that starts at time $T=0$ from the bounded end and propagates along the stratum. When the wave hits the "feature", it's split into two - one is partially reflected back and another is partially passes through. Back-traveling wave(s) arrival time(s) and amplitude(s) are registered at the bounded end when they reach it.
Consider that the one-reflection "profile" of the stratum tube is known. That is, there is a "trace" - a function in time that is zero everywhere except at times when the first wave hits the feature, and at those points the function is equal to the reflection coefficient. Also consider that the bounded end has the reflection coefficient close to 1, so once the wave reaches the bounded end, it's weakened "ghost" goes forward again. This is illustrated below (blue - substance, light blue - features, red - single-reflection function):
Once the single-reflection trace is known, how can I derive the double-reflection trace, triple-reflection trace, and infinite-reflection trace? I've heard something about the auto-convolution, but don't know how to apply it here.