# How to derive a multi-reflection echo from a given single-reflection “trace”

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.

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## migrated from dsp.stackexchange.comMar 2 '12 at 19:30

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There has been a suggestion that this might be well served by migration to physics.se. I'm a mod there and would like some clarification. Is your problem here one of understanding the behavior of the backward propagating waves or simple one of not knowing how to do the math. In particular are you concerned about the reflection of return waves or the interference of multiple reflections? –  dmckee Mar 2 '12 at 18:38
@dmckee : I'm interested in deriving the multi-reflection trace from the single-reflection trace in terms of math (in terms of convolution if available). This is the part of the more complicated (inverse) problem: derive a reverse de-convolution process that would provide a single-reflection trace from the observed multi-reflection events. I don't object this question to be copied to physics, but I'd also like to find answers here or in math. –  mbaitoff Mar 2 '12 at 18:47
I'm not sure that you have answered my question. Do the return signals interact with the features, and are the signals coherent waves for which we need to worry about interference effects? These kinds of features would make the question one well suited to physics, without them the problem seems to be fairly simple one of performing a sum (or integration in the limit of continuous feature distribution). –  dmckee Mar 2 '12 at 18:51
To begin with, let's assume that both forward and backward waves are infinitely short simple wavelets that interact with the features. As the first step, the interference of the waves is not taken into account (only their arrival times matter), but moving later on, the problem will be to deal with the interference of the arrived wavelets. I don't see this problem so simple so that the integration would do. Moving on from discrete features' positions to features' distributions is even harder question. –  mbaitoff Mar 2 '12 at 19:01
Ah. Good. I recommend editing the question to make it clear that there is reinteraction of the reflected signal (and what that interaction is--it's not guaranteed to be the same at the outgoing interaction) and migration to physics. Mind you, the first thing a physicist will do is ask if the interaction is weak enough that they can neglect higher orders and reduce it to the the no-reinteraction case (which really is just a convolution). –  dmckee Mar 2 '12 at 19:22