Sensitivity Kernel for the Weighted Norm of the Frequency-Dependent Phase Correlation (original) (raw)
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Geophysical Journal International, 2011
We propose a new approach to computing the sensitivity kernels used in seismic tomography based on a Green's function database. For any perturbation in the Earth's structural model, the waveform Fréchet derivative can be expressed in terms of strain Green's tensors, which are themselves functions of the reference Earth model only. The Fréchet derivative of any seismic observable can then be obtained from waveform Fréchet derivative. Given a reference model, a strain Green's tensor database can be established, thus eliminating the need for repetitive wavefield evaluations in all subsequent synthetic and kernel calculations, and reducing the CPU time. For a spherically symmetric reference Earth model, the strain Green's tensor database can be constructed on a (r, ) grid by normal-mode summation. The stored strain Green's tensors can then be used to quickly evaluate the wavefield between any source and receiver. The generality of the strain Green's tensors makes it possible to compute the Fréchet kernels for any phase on a seismogram (P, S, Pdiff , surface waves, etc.), for any type of data (traveltime, amplitude, splitting intensity, waveform, etc.), and for any parameter (isotropic, anisotropic, attenuation, etc.). The kernel calculation at each point in the medium is reduced to the convolution of two sets of strain Green's tensors extracted from the database, which makes the approach extremely efficient. Efficient Fréchet Kernel calculation-I 923 For regional tomography based upon phase measurements of long period surface waves (a Rayleigh wave of 100-s period has a wavelength of about 300 km), this effect can limit the resolution potential significantly. Efforts have been made more recently to develop efficient numerical methods to compute full-wave sensitivity kernels for seismic tomography based on 1-D reference models (e.g. ). Using the approach of Zhao et al. (2000) and based upon normal-mode coupling to calculate complete Fréchet kernels for the phase of Rayleigh waves, have shown that regional surface wave tomography can indeed resolve structures as small as 100 km, smaller than the size of the first Fresnel zone. However, the application of the normal-mode approach has so far been rather limited owing to a low numerical efficiency.
Geophysical Journal International, 2011
Seismic tomography has been developed largely on the basis of the Born approximation, in which any waveform-derived seismic observable, such as a perturbation in traveltime, is linearly related to the local perturbation of a model parameter, such as the speed of a seismic wave. This relation defines a Fréchet kernel which can be expressed as the convolution of two strain Green's tensors from the source and receiver to the perturbation location. We develop a new approach for computing the Fréchet kernels using pre-calculated databases of strain Green's tensors. After deriving the general expressions of Fréchet kernels in terms of the strain Green's tensors, we obtain specific expressions for the sensitivities of traveltime and amplitude to both isotropic and anisotropic perturbations of the elastic moduli. We also derive the Fréchet kernels of the SKS-splitting intensity for anisotropic model parameters. Numerical examples of Fréchet kernels are presented for a variety of seismic phases to demonstrate the efficiency and flexibility of this new approach and its potential for both regional and global finite-frequency tomography applications.
Journal of Geophysical Research: Solid Earth, 2014
We have successfully applied full-3-D tomography (F3DT) based on a combination of the scattering-integral method (SI-F3DT) and the adjoint-wavefield method (AW-F3DT) to iteratively improve a 3-D starting model, the Southern California Earthquake Center (SCEC) Community Velocity Model version 4.0 (CVM-S4). In F3DT, the sensitivity (Fréchet) kernels are computed using numerical solutions of the 3-D elastodynamic equation and the nonlinearity of the structural inversion problem is accounted for through an iterative tomographic navigation process. More than half-a-million misfit measurements made on about 38,000 earthquake seismograms and 12,000 ambient-noise correlagrams have been assimilated into our inversion. After 26 F3DT iterations, synthetic seismograms computed using our latest model, CVM-S4.26, show substantially better fit to observed seismograms at frequencies below 0.2 Hz than those computed using our 3-D starting model CVM-S4 and the other SCEC CVM, CVM-H11.9, which was improved through 16 iterations of AW-F3DT. CVM-S4.26 has revealed strong crustal heterogeneities throughout Southern California, some of which are completely missing in CVM-S4 and CVM-H11.9 but exist in models obtained from previous crustal-scale 2-D active-source refraction tomography models. At shallow depths, our model shows strong correlation with sedimentary basins and reveals velocity contrasts across major mapped strike-slip and dip-slip faults. At middle to lower crustal depths, structural features in our model may provide new insights into regional tectonics. When combined with physics-based seismic hazard analysis tools, we expect our model to provide more accurate estimates of seismic hazards in Southern California.
Geophysical Journal International, 2009
Although the use of the first-order Born approximation for the computation of seismic observables and sensitivity kernels in 3-D earth models shows promise for improving tomographic modelling, more work is necessary to systematically determine how well such methods forward model realistic seismic data compared with more standard asymptotic methods. Most work so far has been focused on the analysis of secondary data, such as phase velocity, rather than time domain waveforms. We here compare synthetic waveforms obtained for simple models using standard asymptotic approximations that collapse the sensitivity to 3-D structure on the great circle plane and those obtained using the 3-D linear Born approximation, with accurate numerical 3-D synthetics. We find, not surprisingly, that 3-D Born more accurately models the perturbation effects of velocity anomalies that are comparable in wavelength to or are smaller than the first Fresnel zone. However, larger wavelength and amplitude anomalies can easily produce large phase delays that cause the first-order (linear) Born approximation to break down, whereas asymptotic methods that incorporate the effect of heterogeneity in the phase rather than in the amplitude of the waveform are more robust. Including a path average phase delay to the Born calculated waveforms significantly improves their accuracy in the case of long-wavelength structure, while still retaining the ability to correctly model the effect of shorter-wavelength structure. Tests in random models with structural wavelengths consistent with existing global seismic models indicate that the linear Born approximation frequently breaks down in realistic earth models, with worse misfit for first and second orbit Rayleigh and higher mode surface waveforms than the great-circle based approximations at all distances tested (>20 •). For fundamental modes, the average misfit for the waveforms calculated with the linear Born formalism is quite poor, particularly for distances larger than 60 •. The modified Born formalism consistently improves the fit relative to the linear Born waveforms, but only outperforms the great-circle based approximations for the higher mode surface waveforms. We note, however, that phase delay kernels for multitaper measurements of fundamental mode Rayleigh wave phase velocities developed from the Born approximation do not demonstrate the problems associated with the linear waveform kernels. There is general agreement with measurements and moderate improvement relative to phase delays predicted by the path-average approximation.
Full-3d waveform tomography for southern california
2011
We have successfully applied full-3-D tomography (F3DT) based on a combination of the scattering-integral method (SI-F3DT) and the adjoint-wavefield method (AW-F3DT) to iteratively improve a 3-D starting model, the Southern California Earthquake Center (SCEC) Community Velocity Model version 4.0 (CVM-S4). In F3DT, the sensitivity (Fréchet) kernels are computed using numerical solutions of the 3-D elastodynamic equation and the nonlinearity of the structural inversion problem is accounted for through an iterative tomographic navigation process. More than half-a-million misfit measurements made on about 38,000 earthquake seismograms and 12,000 ambient-noise correlagrams have been assimilated into our inversion. After 26 F3DT iterations, synthetic seismograms computed using our latest model, CVM-S4.26, show substantially better fit to observed seismograms at frequencies below 0.2 Hz than those computed using our 3-D starting model CVM-S4 and the other SCEC CVM, CVM-H11.9, which was improved through 16 iterations of AW-F3DT. CVM-S4.26 has revealed strong crustal heterogeneities throughout Southern California, some of which are completely missing in CVM-S4 and CVM-H11.9 but exist in models obtained from previous crustal-scale 2-D active-source refraction tomography models. At shallow depths, our model shows strong correlation with sedimentary basins and reveals velocity contrasts across major mapped strike-slip and dip-slip faults. At middle to lower crustal depths, structural features in our model may provide new insights into regional tectonics. When combined with physics-based seismic hazard analysis tools, we expect our model to provide more accurate estimates of seismic hazards in Southern California.
Exploration Geophysics, 2004
Frequency-space domain full-wave tomography is a promising technique for delineating detailed subsurface structure with high resolution. However, this method requires criteria for the selection of a set of optimal temporal frequency components, to achieve stability in the sequence of inversion processes together with computational efficiency. We propose a method of selecting optimal temporal frequencies, based on wavenumber continuity. The proposed method is tested numerically and is shown to be able to select an optimal set of frequency components that are sufficient to image the anomalies.
On the Generalization of Seismic Tomography Algorithms
American Journal of Computational Mathematics, 2014
The seismic tomography problem often leads to underdetermined and inconsistent system of equations. Solving these problems, care must be taken to keep the propagation of data errors under control. Especially, the non-Gaussian nature of the noise distribution (for example outliers in the data sets) can cause appreciable distortions in the tomographic imaging. In order to reduce the sensitivity to outlier, some generalized tomography algorithms are proposed in the paper. The weighted version of the Conjugate Gradient method is combined with the Iteratively Reweighted Least Squares (IRLS) procedure leading to a robust tomography method (W-CGRAD). The generalized version of the SIRT method is introduced in which the (Cauchy-Steiner) weighted average of the ART corrections is used. The proposed tomography algorithms are tested for a small sized tomography example by using synthetic traveltime data. It is proved that-compared to their traditional versions-the outlier sensitivities of the generalized tomography methods are sufficiently reduced.
Time-Dependent Seismic Tomography
2008
Of methods for measuring temporal changes in seismic-wave speeds in the Earth, seismic tomography is among those that offer the highest spatial resolution. 3-D tomographic methods are commonly applied in this context by inverting seismic wave arrival time data sets from different epochs independently and assuming that differences in the derived structures represent real temporal variations. This assumption is dangerous because the results of independent inversions would differ even if the structure in the Earth did not change, due to observational errors and differences in the seismic ray distributions. The latter effect may be especially severe when data sets include earthquake swarms or aftershock sequences, and may produce the appearance of correlation between structural changes and seismicity when the wave speeds are actually temporally invariant. A better approach, which makes it possible to assess what changes are truly required by the data, is to invert multiple data sets simultaneously, minimizing the difference between models for different epochs as well as the rms arrival-time residuals. This problem leads, in the case of two epochs, to a system of normal equations whose order is twice as great as for a single epoch. The direct solution of this system would require twice as much memory and four times as much computational effort as would independent inversions. We present an algorithm, tomo4d, that takes advantage of the structure and sparseness of the system to obtain the solution with essentially no more effort than independent inversions require.
Model parametrization in seismic tomography: a choice of consequence for the solution quality
Physics of the Earth and Planetary Interiors, 2001
To better assess quality of three-dimensional (3-D) tomographic images and to better define possible improvements to tomographic inversion procedures, one must consider not only data quality and numerical precision of forward and inverse solvers but also appropriateness of model parametrization and display of results. The quality of the forward solution, in particular, strongly depends on parametrization of the velocity field and is of great importance both for calculation of travel times and partial derivatives that characterize the inverse problem. To achieve a quality in model parametrization appropriate to high-precision forward and inverse algorithms and to high-quality data, we propose a three-grid approach encompassing a seismic, a forward, and an inversion grid. The seismic grid is set up in such a way that it may appropriately account for the highest resolution capability (i.e. optimal data) in the data set and that the 3-D velocity structure is adequately represented to the smallest resolvable detail apriori known to exist in real earth structure. Generally, the seismic grid is of uneven grid spacing and it provides the basis for later display and interpretation. The numerical grid allows a numerically stable computation of travel times and partial derivatives. Its specifications are defined by the individual forward solver and it might vary for different numerical techniques. The inversion grid is based on the seismic grid but must be large enough to guarantee uniform and fair resolution in most areas. For optimal data sets the inversion grid may eventually equal the seismic grid but in reality, the spacing of this grid will depend on the illumination qualities of our data set (ray sampling) and on the maximum matrix size we can invert for. The use of the three-grid approach in seismic tomography allows to adequately and evenly account for characteristics of forward and inverse solution algorithms, apriori knowledge of earth's structure, and resolution capability of available data set. This results in possibly more accurate and certainly in more reliable tomographic images since the inversion process may be well-tuned to the particular application and since the three-grid approach allows better assessment of solution quality.
Physics of the Earth and Planetary Interiors, 2000
We investigate the impact of the theoretical limitations brought by asymptotic methods on upper-mantle tomographic Ž. models deduced from long-period surface wave data period) 80 s , by performing a synthetic test using a non-asymptotic formalism. This methodology incorporates the effects of back and multiple forward scattering on the wave field by summing normal modes computed to third order of perturbations directly in the 3D Earth, and models the sensitivity to scatterers away from the great-circle path. We first compare the methods we used for the forward problem, both theoretically and numerically. Then we present results from the computation of 7849 synthetic Love waveforms in an upper mantle model consisting of two heterogeneities with power up to spherical harmonic degree 12. The waveforms are subsequently inverted Ž. using a 0th order asymptotic formalism equivalent to a path-average approximation in the surface waves domain. We show that the main structures are retrieved, but that the theoretical noise on the output model is of the same order as the noise due to the path-coverage and a priori constraints.