Generalized seismological data functionals

SUMMARY We have formulated a new waveform‐analysis procedure to recover phase and amplitude information from individual seismograms that makes use of the ability to compute complete seismograms from realistic earth models. The basic tool is the isolation filter, a composite waveform constructed to select data from a desirable portion of the seismogram. When the cross‐correlation between this synthetic waveform and an observed seismogram is localized in the time domain by windowing and in the frequency domain by narrow‐band filtering, the resulting cross‐correlagram can be approximated by a five‐parameter Gaussian wavelet. One of these five parameters is the bandwidth of the correlagram, specified by the narrow‐band filter; the other four define a set of time‐like, frequency‐dependent quantities {δ t x : x = q, p, a, g}, which are functionals of earth structure. δ t p is the differential phase delay and δ t g is the differential group delay of the observed waveform relative to the synthetic, and δ t q and δ t a are the corresponding frequency‐dependent amplitude parameters. We have developed a procedure for measuring the four generalized seismological data functionals by fitting a Gaussian wavelet to the windowed, filtered cross‐correlagram. To relate the GSDFs to earth structure, we apply corrections to the differential times for the effects of windowing and filtering. Solving a linear system of four equations in four unknowns yields a set of differential dispersion parameters {δ x : x = q, p, a, g}. Formulae expressing the perturbations of the GSDFs in terms of the perturbations to the dispersion parameters for the individual component waveforms, including all interference effects, have been derived. Under a set of approximations valid for a large class of isolation filters, these can be simplified to yield easily computed expressions for the Fréchet kernels of the δ x 's. The calculation of these Fréchet kernels requires no high‐frequency approximations, and it can be extended to the investigation three‐dimensional earth structure.