Three-dimensional Frechet differential kernels for seismicdelay times

Abstract

Summary Seismic traveltimes are the most widely exploited data in seismology. Their Fréchet or sensitivity kernels are important tools in tomographic inversions based on the Born or single‐scattering approximation. The current study is motivated by a paradox posed by two seemingly irreconcilable observations in the numerical calculations for the sensitivity kernels of the traveltime perturbations. Calculations of kernels for 2‐D media by the normal‐mode approach indicate that traveltimes are most sensitive to the structure on and around the geometrical ray paths corresponding to the seismic arrivals, whereas calculations for 3‐D media by geometrical ray theory predict exactly zero sensitivities on the ray paths. In the current work, we employ these two completely different wave‐propagation approaches, the more efficient geometrical ray theory and the more accurate normal‐mode theory, to investigate the 3‐D sensitivities of the delay times to shear‐wave speed variations. Expressions for the delay‐time Fréchet kernels are presented for both methods, and extensive numerical experiments are conducted for various types of seismic phases as well as for different reference earth models. The results show that the contradictory observations are but two examples of a wide range of behaviours in the delay‐time sensitivity. For most of the seismic phases in realistic reference models with multiple discontinuities, wave‐speed gradients and low‐velocity zones, the wavefields are highly complicated and ray theory, which describes the response by the contributions of a few geometrical rays between the source and receiver, produces qualitatively different delay‐time kernels from those obtained by the normal‐mode theory, which includes essentially all contributions.

Finite-Frequency Kernels Based on Adjoint Methods

“…The new approach incorporates travel-time effects associated with wavefront healing and recognizes the inherent frequency dependence of the body-wave travel time or surface-wave phase. For layercake or spherically symmetric Earth models, sensitivity or Fréchet kernels may be calculated based on surface-wave Green's functions (Marquering et al, 1999), normal modes (Zhao et al, 2000), or asymptotic, ray-based methods Hung et al, 2000;Zhou et al, 2004). Simple 3D travel-time kernels for phases like P and S are shaped like bananas with a doughnutlike cross section, and thus the kernels are commonly referred to as "banana-doughnut" kernels.…”

Wave-equation-based travel-time seismic tomography – Part 1: Method

“…Considering that the derivations of the travel-time sensitivity kernel in Eq. (11) and previous finite-frequency sensitivity kernels (e.g., Dahlen et al, 2000;Zhao et al, 2000;Tromp et al, 2005;Fichtner et al, 2006) are mainly based on the Born approximation, which requires that the reference velocity model c(x) for synthetic seismogram s(t) is very close to the real model c(x) + δc(x) for real data d(t), i.e., |δc(x)| c(x). Since |δc(x)| c(x), it is straightforward to get that |s(t) − d(t)| |s(t)| if both s(t) and d(t) satisfy the same wave equation.…”

“…Ray theory is actually only valid when the scale length of the variation of material properties is much larger than the seismic wavelength (Rawlinson et al, 2010). To take into account the sensitivity to off-ray structures, finite-frequency tomography methods that construct 2-D or 3-D travel-time and P. Tong et al: Part 1: Method amplitude sensitivity kernels are proposed, including those based on the paraxial approximation and dynamic ray tracing (e.g., Marquering et al, 1999;Dahlen et al, 2000;Tian et al, 2007;Tong et al, 2011) and those based on the normal mode theory (e.g., Zhao et al, 2000;Zhao and Jordan, 2006;To and Romanowicz, 2009). Tomographic models with improved resolutions were reported by recent finite-frequency tomographic studies (e.g., Montelli et al, 2004;Hung et al, 2004Hung et al, , 2011Gautier et al, 2008), although comparison to ray-based tomography remains controversial (de Hoop and van der Hilst, 2005a;Dahlen and Nolet, 2005;de Hoop and van der Hilst, 2005b).…”

An objective rationale for the choice of regularisation parameter with application to global multiple-frequency <i>S</i>-wave tomography

“…Nolet, 2008), alternatives are available (e.g. Zhao et al, 2000;Tromp et al, 2005;Nissen-Meyer et al, 2007;Zhao and Chevrot, 2011a, b). The most promising ones are based on numerical techniques (e.g.…”