Perturbation kernels for generalized seismological data functionals (GSDF)

Abstract

S U M M A R YIn seismic waveform analysis and inversion, data functionals can be used to quantify the misfit between observed and model-predicted (synthetic) seismograms. The generalized seismological data functionals (GSDF) of Gee & Jordan quantify waveform differences using frequency-dependent phase-delay times and amplitude-reduction times measured on timelocalized arrivals and have been successfully applied to tomographic inversions at different geographic scales as well as to inversions for earthquake source parameters. The seismogram perturbation kernel is defined as the Fréchet kernel of the data functional with respect to the seismic waveform from which the data functional is derived. The data sensitivity kernel, which is the Fréchet kernel of the data functional with respect to structural model parameters, can be obtained by composing the seismogram perturbation kernel with the Born kernel through the chain rule. In this paper, we extend GSDF analysis to broad-band waveforms by removing constraints on two control parameters defined in Gee & Jordan and derive the seismogram perturbation kernels for the modified GSDF analysis. The modifications given in this paper are consistent with the original GSDF theory in Gee & Jordan around the centre frequency and improve the stability of GSDF analysis towards the edges of the passband. We also present numerical examples of perturbation kernels for the modified GSDF analysis and their data sensitivity kernels using a homogenous half-space structure model and a complex 3-D structure model.Advances in parallel computing technology and numerical methods have made large-scale, 3-D numerical simulations of seismic wavefields much more affordable and they open up the possibility of 'full 3-D tomography' (F3DT), in which the starting model as well as the derived model perturbation is 3-D in space and the Fréchet kernel is computed using the full physics of 3-D wave propagation. Two physically equivalent but computationally distinct approaches to F3DT (Chen et al. 2007a) have been developed, the scattering-integral (SI) method, which sets up the inverse problem by calculating and storing the Fréchet kernels for individual misfit measurements ) and the adjoint-wavefield method, which constructs the gradient of the objective function through correlating the forward wavefield from the source and the adjoint-wavefield from the receivers (Tarantola 1986;Tromp et al. 2005). The first successful application of F3DT using real data from natural earthquakes was conducted in Chen et al. (2007b) to improve the 3-D crustal structure in the Los Angeles Basin region using the SI method. Recently, Tape et al. (2009) has adapted the adjoint-wavefield method to image the crustal structure in Southern California using waveform data from local earthquakes and Fichtner et al. (2009) has adapted the adjoint-wavefield method to continental-scale tomography and inverted for upper-mantle structure in the Australasian region. In these successful F3DT applications, time-and frequencydependent phas...

Full‐3‐D tomography for crustal structure in Southern California based on the scattering‐integral and the adjoint‐wavefield methods

“…For all ambient-noise correlagrams used in our inversion, the total amount of disk space needed to store the strain fields for about 200 virtual sources was about 39 Tb. Once all the strain fields were obtained, the data sensitivity kernels for all misfits measured on ambient-noise correlagrams were computed using the scattering-integral method [Zhao et al, 2005[Zhao et al, , 2006Chen et al, 2010]. The strain fields were all deleted from the disk after the data sensitivity kernels for all ambient-noise correlagrams were obtained.…”

“…For each waveform, we also measured the peak of the unfiltered cross-correlation function, which gives the "broadband" delay time ΔT c , i.e., the phase delay of the observed waveform relative to the synthetic averaged across the full bandwidth of the waveform [Sipkin and Jordan, 1980;Luo and Schuster, 1991;Tanimoto, 1995;Zhao and Jordan, 1998;Dahlen et al, 2000]. We calculated the Fréchet kernels for different measurement types using the appropriate seismogram perturbation kernels [Chen et al, 2007a[Chen et al, , 2007b[Chen et al, , 2010, as described in Appendix D.…”

“…Our basin structure is therefore mostly constrained by ambient-noise correlagrams for stations pairs crossing the SSJB. The Fréchet kernels of the ambient-noise correlagram between two stations were computed by assuming a spatially Dirac virtual source located at one of the stations [Chen et al, 2010;Xu et al, 2013]. For uneven distributions of noise sources, the virtual source at one of the stations is usually smeared in both space and time and the algorithm that accounts for this smearing in the Fréchet kernels is given in Tromp et al [2010].…”

“…For the broadband cross-correlation traveltime delay, the SPK is the negative of the energy-normalized synthetic particlevelocity seismogram within the specified time window [Luo and Schuster, 1991;Tanimoto, 1995;Zhao and Jordan, 1998;Dahlen et al, 2000;Chen et al, 2007a]. For the frequency-dependent GSDF phase delay, the SPK, J p (t, ω), where ω is the angular frequency at which the narrowband phase-delay misfit was measured, was derived in Chen et al [2010]. The SPK for the frequency-dependent GSDF group delay J g (t, ω) can be approximated by numerically differentiating ωJ p (t, ω) with respect to ω using…”

“…The starting model is 3-D, and the Fréchet (sensitivity) kernels are calculated using the full physics of 3-D wave propagation [Zhao et al, 2005[Zhao et al, , 2006Tromp et al, 2005;Liu and Tromp, 2006]. F3DT can employ any functional of the seismogram as observable, including frequency-dependent phase and group delays of earthquake waveforms [Gee and Jordan, 1992] and two-station correlagrams of the ambient seismic field Chen et al, 2010;Xu et al, 2013]. F3DT accounts for the nonlinearity of the structural inverse problem through iterated cycles of forward simulation, data measurement, kernel calculation, and inversion.…”