It is often necessary to restore the original ground motion from a seismogram. For this purpose, the seismogram must be filtered with the inverse response of the seismograph, a process known as deconvolution. It is rarely carried out in its exact mathematical form because the signal-to-noise ratio deteriorates outside the passband of the seismograph, and the deconvolution must be limited to a passband where the signal-to-noise ratio is still acceptable. Generally speaking, the ground motion is not entirely determined by the seismic record, and its reconstruction is a geophysical inverse problem whose solution must be constrained by a-priori information. Similar considerations apply to the case that records from seismographs with different responses must be homogenized. The PREPROC software package [Plesinger et al. 1996] contains different routines for a band-limited deconvolution and its manual offers a concise introduction into the problem.
A deconvolution can be realized in frequency or time domain. The frequency domain is convenient for the construction of approximate inverses with a view to the frequency-dependent signal-to-noise ratio of the data. Causal and acausal solutions are available; waveforms may be better preserved with an acausal inverse but the resulting precursory signals may give rise to misinterpretation. A division-by-zero problem exists in the low-frequency limit and prevents the construction of an exact inverse. In the time domain, both approximate and quasi-exact inverses can conveniently be realized with recursive (IIR) filters. They are always causal. Short time series can often be deconvolved with a quasi-exact inverse; no division-by-zero problem is encountered but a quadratic or cubic trend may be generated, which can be removed afterwards by polynomial fitting.