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When inverting large matrices, iterative techniques are necessary because of their speed and low memory requirements, as opposed to singular value decomposition (SVD). Recently, there have been attempts to obtain information on the quality of the solutions calculated using conjugate gradient (CG) methods such as LSQR. The purpose of this note is to comment on the paper titled "Estimation of resolution and covariance for large matrix inversions' by Zhang & McMechan (1995), who extend Paige and Saunders' LSQR algorithm to obtain an orthonormal basis used to approximate resolution and covariance. We show that for larger problems, where the number of orthogonal vectors is several orders of magnitude smaller than the number of model parameters, the vectors obtained do not adequately span the range of the model space. We use a synthetic borehole experiment to illustrate the differences between the singular value spectrum obtained through the more complete method of SVD and the Ritz value spectrum that results from a simple extension of LSQR, We also present a trivial numerical example to illustrate the differences between Zhang & McMechan's approximate resolution matrix and the true resolution. 相似文献
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Optimal parametrization of tomographic models 总被引：1，自引：0，他引：1
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Seismic tomography shows that upwelling beneath Iceland is confined to the upper mantle 总被引：3，自引：0，他引：3
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Nullspace shuttles 总被引：1，自引：0，他引：1
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Three-dimensional waveform sensitivity kernels 总被引：2，自引：0，他引：2
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Large scale seismic anisotropy in the Earth's mantle is likely dynamically supported by the mantle's deformation; therefore, tomographic imaging of 3-D anisotropic mantle seismic velocity structure is an important tool to understand the dynamics of the mantle. While many previous studies have focused on special cases of symmetry of the elastic properties, it would be desirable for evaluation of dynamic models to allow more general axis orientation. In this study, we derive 3-D finite-frequency surface wave sensitivity kernels based on the Born approximation using a general expression for a hexagonal medium with an arbitrarily oriented symmetry axis. This results in kernels for two isotropic elastic coefficients, three coefficients that define the strength of anisotropy, and two angles that define the symmetry axis. The particular parametrization is chosen to allow for a physically meaningful method for reducing the number of parameters considered in an inversion, while allowing for straightforward integration with existing approaches for modelling body wave splitting intensity measurements. Example kernels calculated with this method reveal physical interpretations of how surface waveforms are affected by 3-D velocity perturbations, while also demonstrating the non-linearity of the problem as a function of symmetry axis orientation. The expressions are numerically validated using the spectral element method. While challenges remain in determining the best inversion scheme to appropriately handle the non-linearity, the approach derived here has great promise in allowing large scale models with resolution of both the strength and orientation of anisotropy. 相似文献
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Rays propagating through strongly laterally varying media exhibit chaotic behaviour. This means that initially close rays diverge exponentially, rather than according to a power law. This chaotic behaviour is especially pronounced if the medium contains laterally varying interfaces. By studying simple 2-D and 3-D versions of models with laterally varying interfaces, the importance of chaotic ray behaviour is determined. A model of the Moho below Germany produces sharp variations with epicentral distance of the number of arrivals. In addition, the number of caustics grows dramatically: up to 1200 caustics are present between a distance of 0 and 800 km. Using the theory of Hamiltonian systems, a more in-depth study of the chaotic character of the ray equations is obtained. It is found that for realistic heterogeneous models most of the relevant rays will exhibit chaotic behaviour. The degree of chaos is quantified in terms of predictability horizons. Beyond the predictability horizons ray tracing cannot be carried out accurately. For the models under consideration, the length from the source to the predictability horizon has an order of magnitude of 1000 km. The chaotic behaviour of the rays makes it necessary to use extensions of asymptotic ray theory, such as Maslov theory, to compute seismic waveforms. It is shown that pseudo-caustics, an important obstacle in computing Maslov synthetics, are a generic feature of the 2-D laterally varying models that are studied. Eventually, the use of asymptotic methods is restricted because of the inaccuracy in the computation of the ray paths. 相似文献