模擬波與波線走時：有限頻寬與波線層析成像的極限

Abstract

Up-to-date seismic tomographic models essentially rely on infinite-frequency ray theory or finite-frequency Born-Frech&eacute;t kernel theory (BKT) which translates observed travel time shifts of seismic waves into 3-D aspherical velocity variations within the earth. From a ray-theoretical point of view, a seismic wave only 'feels' the structure right on an infinite-thin, least-time geometrical ray path. Linearized ray theory (LRT) assumes the travel time shift is unchanged to first order for infinitesimally small variation in the ray path, and thus expressed as a line integral of seismic slowness perturbations along the unperturbed ray in a radially-symmetric earth. Nonlinear travel time tomography uses general ray theory (GRT) to reevaluate the 'exact' ray-theoretical travel times along the changing path trajectories in different starting 3-D models prior to each iterative inversion. Both theories are strictly valid for infinite-frequency waves. In reality, the wavefront of a finite-frequency wave naturally undergoes a diffractive healing process. Destructive interference of waves scattered off property heterogeneity among different frequencies renders the region of strong sensitivity of a travel time shift confined to the vicinity of the first Fresnel zone. Recent development in Born-Frech&eacute;t kernel theory has gone beyond high-frequency ray approximation; Born single scattering theory is employed to account for the effects of wavefront healing and off-path scattering upon the travel time shift measured by cross-correlation of an observed seismic pulse with its spherical-earth synthetic. The 3-D Born-Frech&eacute;t kernel expressing such travel time sensitivity is identically zero everywhere along the unperturbed ray path; rather, the maximum sensitivity lies within the fringe of its tubular geometry surrounding the ray. We conduct a validation study of these three fundamental theories for seismic tomography by forward modeling finite-frequency travel times of scalar wave propagation in heterogeneous random media. We obtain 'ground-truth' travel time shifts from cross correlation of numerically-computed pressure-response seismograms with the corresponding pulses in the homogenous medium, and compare the measured data with those predicted by GRT, LRT, and BKT. Both ray theories suffer from poor travel time approximations whenever the scale length of medium heterogeneity is shorter than half of the first Fresnel zone width. Born-Frech&eacute;t kernel theory, on the other hand, provides accurate predictions for various scale lengths, but only for weak heterogeneity strength (e < 4%). With the increasing e, the high-order change in travel times due to the detoured wave paths in heterogeneous media no longer is negligible, as assumed in LRT and BKT. General ray theory computes the travel times along the fastest paths with a ray-bending method and prominently improves modeling of high-frequency travel time fluctuations in strongly heterogeneous media (e > 3%) with intermediate to large scale lengths. Our forward-modeling experiments suggest that a nonlinear, finite-frequency theory for computing the 3-D sensitivity kernels of seismic travel times in 3-D earth models is necessarily required to resolve the mantle heterogeneity on various scale lengths and full strength spectra.