Among wave-propagation theorists, the removal of all multiples is called inversion. It seems that for inversion theory to be applicable to the real problem, the theory must include a way to deal with an unknown, spectrally incomplete, shot waveform. Routine work today typically ignores inversion theory and presumes the mathematical limit within which multiples may be handled as a shot waveform. Despite many further theoretical developments and the continuing active interest of many practical workers, routine deconvolution is little changed. I will state in simple terms what I believe to be the essence of deconvolution.
Every seismogram has a spectrum.
The spectrum is a product of many causes. Some causes are of fundamental interest. Others are extraneous. It is annoying when a seismogram is resonant just because of some near-surface phenomena. Deconvolution is basically a process in which strong resonances are measured, and then a filter is designed to suppress them.