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Numerical realization of the Bayesian inversion accelerated using surrogate models
Bérešová, Simona
The Bayesian inversion is a natural approach to the solution of inverse problems based on uncertain observed data. The result of such an inverse problem is the posterior distribution of unknown parameters. This paper deals with the numerical realization of the Bayesian inversion focusing on problems governed by computationally expensive forward models such as numerical solutions of partial differential equations. Samples from the posterior distribution are generated using the Markov chain Monte Carlo (MCMC) methods accelerated with surrogate models. A surrogate model is understood as an approximation of the forward model which should be computationally much cheaper. The target distribution is not fully replaced by its approximation. Therefore, samples from the exact posterior distribution are provided. In addition, non-intrusive surrogate models can be updated during the sampling process resulting in an adaptive MCMC method. The use of the surrogate models significantly reduces the number of evaluations of the forward model needed for a reliable description of the posterior distribution. Described sampling procedures are implemented in the form of a Python package.
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Three-dimensional ambient noise tomography of the Bohemian Massif
Valentová, Ľubica ; Gallovič, František (advisor) ; Burjánek, Jan (referee) ; Kristek, Jozef (referee)
We have performed 3D ambient noise tomography of the Bohemian Massif. We invert adopted inter-station dispersion curves of both Love and Rayleigh waves in periods 4-20 s, which were extracted from ambient noise cross-correlations, using a two-step approach. In the first step, the inter-station dispersion curves are localized for each period into the so-called dispersion maps. To account for finite-frequency effects, gradient method employing Fréchet kernels is used. Assuming membrane wave approximation of the surface wave propagation at each period, the kernels were calculated using the adjoint method. To reduce the effect of data noise, the kernels were regularized by Gaussian smoothing. The proper level of regularization is assessed on synthetic tests. In the second step, the phase-velocity dispersion maps are inverted into a 3D S-wave velocity model using the Bayesian approach. The posterior probability density function describing the solution is sampled by more than one million models obtained by Monte-Carlo approach (parallel tempering). The calculated variance of the model shows that the well resolved part corresponds to the upper crust (i.e., upper 20 km). The mean velocity model contains mainly large scale structures that show good correlation with the main geologic domains of the Bohemian...
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Influence of velocity model uncertainty in earthquake source inversions
Halló, Miroslav ; Gallovič, František (advisor) ; Duputel, Zacharie (referee) ; Vavryčuk, Václav (referee)
Title: Influence of velocity model uncertainty in earthquake source inversions Author: Miroslav Halló Department: Department of Geophysics Supervisor: doc. RNDr. František Gallovič, Ph.D., Department of Geophysics Abstract: Earthquake ground motions originate from rupture processes on faults in Earth. Constraints on earthquake source models are important for better un- derstanding of earthquake physics and for assessment of seismic hazard. The source models are inferred from observed waveforms by inverse modeling, which is subject to uncertainty. For large tectonic earthquakes the major source of un- certainty is an imprecise knowledge of crustal velocity model. The research topic of this Thesis is the influence of the velocity model uncertainty on the inferred source models. We perform Monte-Carlo simulations of Green's functions (GFs) in randomly perturbed velocity models to reveal the effects of the imprecise veloc- ity model on the synthetic waveforms. Based on the knowledge gained, we derive closed-form formulas for approximate covariance functions to obtain fast and effective characterization of the GFs' uncertainty. We demonstrate that approxi- mate covariances capture correctly the GF variability as obtained by the Monte- Carlo simulations. The proposed approximate covariance functions are...
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Three-dimensional ambient noise tomography of the Bohemian Massif
Valentová, Ľubica ; Gallovič, František (advisor) ; Burjánek, Jan (referee) ; Kristek, Jozef (referee)
We have performed 3D ambient noise tomography of the Bohemian Massif. We invert adopted inter-station dispersion curves of both Love and Rayleigh waves in periods 4-20 s, which were extracted from ambient noise cross-correlations, using a two-step approach. In the first step, the inter-station dispersion curves are localized for each period into the so-called dispersion maps. To account for finite-frequency effects, gradient method employing Fréchet kernels is used. Assuming membrane wave approximation of the surface wave propagation at each period, the kernels were calculated using the adjoint method. To reduce the effect of data noise, the kernels were regularized by Gaussian smoothing. The proper level of regularization is assessed on synthetic tests. In the second step, the phase-velocity dispersion maps are inverted into a 3D S-wave velocity model using the Bayesian approach. The posterior probability density function describing the solution is sampled by more than one million models obtained by Monte-Carlo approach (parallel tempering). The calculated variance of the model shows that the well resolved part corresponds to the upper crust (i.e., upper 20 km). The mean velocity model contains mainly large scale structures that show good correlation with the main geologic domains of the Bohemian...
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