Separate static echos from acoustic signal (ICA?)
This probably should be a comment, but I didn't earn that option yet. You can try measuring instrument impulse response. If I understand correctly, your system is essentially one-dimensional. Then if you use thin plate or plate of known thickness instead of thick object/material, your output signal will be an approximation of impulse response. Using it as deconvolution kernel will allow you to measure response specific to material, or object.