The comparison of seismograms plays a central role in seismology in diverse ways such as relative time-shifts, propagation effects between stations for a common source, and inversion for source or structural studies. Different measures for comparison have been used in the various situations, but all can be linked by the use of the concept of a transfer operator between a reference seismogram and a comparator trace. Transfer operators are implicit in various methods of phase velocity estimation, receiver functions and anisotropy studies, and measures for estimating arrival times and amplitude variations.
Such transfer operators have a number of important roles; first they allow a visual assessment of the similarities of seismograms, secondly they provide a useful description of propagation effects for a common source in terms of the evolution from a reference station, and thirdly the transfer operator provides a means of representing seismogram differences in inversion without dominance by the largest amplitude arrivals.
Whereas many time-domain measures of the degree of fit between an observed seismogram and the corresponding synthetic seismogram depend on the difference between the traces, which can be readily disturbed by minor misalignment, the transfer operator can readily represent a time offset while retaining a suitable measure of the similarities between the traces.
The transfer operator concept can be applied with weighting or windowing of seismograms, and can be expressed in the time and frequency domains, or even in frequency time. This approach provides a means of representing and quantifying differences in the character of two seismograms that are visually apparent, in the time or frequency domain, but which get suppressed in any single measure of fit.
We show how transfer operators can be usefully employed in many aspects of seismology with emphasis on frequency-domain representations at low frequency, and the time domain for higher frequency applications. We can express the general goal of inversion as the reduction of the transfer operator between observed and synthetic seismograms to the identity, thereby avoiding dominance by the largest arrivals and enhancing the influence of the full range of propagation processes. Broad classes of measures for comparison of times of arrival and amplitudes with quasi-linear properties can be constructed from the transfer operators through the use of a simple weighting function. This versatility highlights the unifying character of the transfer operator; and greatly simplifies the design of measurements targeted at specific aspects of the Earth’s structure.