Fréchet kernels for finite-frequency traveltimes-I. Theory

Abstract

Summary We use body wave ray theory in conjunction with the Born approximation tocompute 3‐D Fréchet kernels for finite‐frequency seismic traveltimes, measured by cross‐correlation of a broad‐band waveform with a spherical earth synthetic seismogram. Destructive interference among adjacent frequencies in the broad‐band pulse renders a cross‐correlation traveltime measurement sensitive only to the wave speed in a hollow banana‐shaped region surrounding the unperturbed geometrical ray. The Fréchet kernel expressing this sensitivity is expressed as a double sum over all forward‐propagating body waves from the source and backward‐propagating body waves from the receiver to every single scatterer in the vicinity of this central ray. The kernel for a differential traveltime, measured by cross‐correlation of two phases at the same receiver, is simply the difference of the respective single‐phase kernels. In the paraxial approximation, an absolute or differential traveltime kernel can be computed extremely economically by implementing a single kinematic and dynamic ray trace along each source‐to‐receiver ray.

Towards a quantitative interpretation of global seismic tomography

“…Refining the models including anisotropy and anelasticity will become essential to increasing the seismologic information. Careful Earth parameterization, the use of finite frequency kernels [Dahlen et al, 2000], wavefield scattering and multiresolution analysis [e.g., De Hoop and van der Hilst, 2005], and adjoint methods [Tromp et al, 2005] will no doubt prove crucial in this ambitious but exciting endeavour.…”

“…The uneven distribution of earthquakes and stations and the fact that surface waves provide relatively weak constraints on compressional wave speed means that high resolution is mainly restricted to seismically active regions, such as plate boundaries, or continental regions with many seismograph stations, whereas large areas beneath oceans remain without effective sampling. In modern applications, the uneven data coverage is partly balanced by the use of adaptive grids [Abers and Roecker, 1991;Fukao et al, 1992;Widiyantoro and van der Hilst, 1996;Sambridge and Gudmundson, 1998;Bijwaard et al, 1998;Kárason and van der Hilst, 2001;Montelli et al, 2004] and, in part, remedied by the use of different data types along with 3D sensitivity kernels to account for frequency differences [e.g., Dahlen et al, 2000;Kárason and van der Hilst, 2001;Montelli et al, 2004;van der Hilst et al, in preparation].…”

“…Since these are often measured at different frequencies, it is becoming important to consider finite frequency effects, which is a topical subject of theoretical seismology [e.g., Dahlen et al, 2000;De Hoop and van der Hilst, 2005]. The biggest challenge is to be able to perform a complete waveform inversion; we are now able to calculate exact seismograms in a full 3D Earth, but the computational resources are still lacking to apply these exciting techniques to a realistic inverse problem [Tromp et al, 2005].…”

Seismic wave propagation in fully anisotropic axisymmetric media

“…In a series of papers, Nissen-Meyer et al (2007a,b, 2008) developed a novel approach to global seismic wave propagation that is based on a decomposition of the 3-D wave equation into a series of uncoupled 2-D equations that is valid for axisymmetric models. The axisymmetric approach has three major advantages over full 3-D methods: (1) it enables the storage of the wavefields that provide the basis for computing Fréchet sensitivity kernels (Dahlen et al 2000), which is not feasible with full 3-D methods due to disk space requirements; (2) it allows the inclusion 2.5-D lateral heterogeneities that are effectively modelled as ringlike structures around the symmetry axis giving rise to various applications in a high-frequency approximation that are not tractable with 1-D methods and (3) it is computationally several orders of magnitude less expensive than full 3-D methods and hence allows the simulation of higher frequencies. Axisymmetric approaches have been presented earlier using finite difference (Alterman & Karal 1968;Igel & Weber 1995, 1996Chaljub & Tarantola 1997;Thomas et al 2000;Takenaka et al 2003;Toyokuni et al 2005) or pseudospectral methods (Furumura et al 1998), but most of these studies assume azimuthally symmetric sources (monopoles) and hence cannot model arbitrary earthquake sources, but rather resemble explosive sources or a certain geometry for strike slip events (Jahnke et al 2008).…”

“…Stähler et al (2012) use this method to compute finite frequency sensitivity kernels for triplicated P waves, which is inaccurate with other methods such as the one by Dahlen et al (2000) due to the strong influence of the upper-mantle discontinuities in comparison to their sensitivity to mantle heterogeneity. Colombi et al (2012Colombi et al ( , 2013) compute boundary topography kernels, analyse the sensitivity of different phases in comparison to there sensitivity to mantel heterogeneity and invert for CMB topography.…”