Direct observation of a p-τ curve in a slant stacked wave field

Abstract

The well known Wiechert-Herglotz technique for computing a velocity-depth profile from refraction seismograms uses data in the form of ray parameter as a function of intercept time (a p-τ curve). The technique of slant stacking used by reflection seismologists automatically produces a p-τ curve from a common shot profile, thereby bypassing arbitrary travel-time picking and several intermediate calculations of previous methods. Successful preliminary results are presented for both synthetic and real data. This method deteriorates for widely separated geophones and where a number of apparent sources with low inter-source coherency coexist in a single profile, but is capable of completely sorting out triplications in data with less extreme contrasts without human intervention.

Short-pulse inversion of inhomogeneous media: a time-domain diffraction tomography

“…where, anticipating extension to the time-domain, we have scaled the spectral parameter ξ = (ξ 1 , ξ 2 ) by the frequency parameter k. Consequently, ξ has a pure frequencyindependent interpretation in terms of the cosines of the plane-wave angles with respect to the (x 1 , x 2 ) axes (see (12)). Notationally, we use a tilde to define plane-wave spectra.…”

“…Equation ( 14) describes the field as a superposition of plane waves. In the 'propagating spectrum' range (|ξ| < 1) they propagate in the directions (12)). In the 'evanescent spectrum' range |ξ| > 1, ζ = i|ζ | (for ω > 0) and the plane waves decay in the positive z direction.…”

“…The 0266-5611/96/060977+17$19.50 c 1996 IOP Publishing Ltd time-dependent plane-wave spectrum is obtained from the data through the slant-stack transform (or τ -p transform, see e.g. [11,12]), which constitutes a Radon transform in the spacetime coordinates. Thus, by formulating the time-domain scattering and inverse scattering problems in the Radon space we shall obtain a direct diffraction-tomography relation between the time-dependent plane-wave spectrum of the scattered field and the Radon transform of the medium.…”