Browsing by Subject "Full waveform inversion"
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Item Bayesian Model Selection for the Solution of Spatial Inverse Problems with Geophysical, Geotechnical and Thermodynamical Applications(2014-08-14) Seyedpour Esmailzadeh, SabaBayesian inference is based on three evidence components: experimental observations, model predictions and expert?s beliefs. Integrating experimental evidence into the calibration or selection of a model, either empirically of physically based, is of great significance in almost every area of science and engineering because it maps the response of the process of interest into a set of parameters, which aim at explaining the process? governing characteristics. This work introduces the use of the Bayesian paradigm to construct full probabilistic description of parameters of spatial processes. The influence of uncertainty is first discussed on the calibration of an empirical relationship between remolded undrained shear strength Su?r and liquidity index IL, as a potential predictor of the soil strength. Two site-specific datasets are considered in the analysis. The key emphasis of the study is to construct a unified regression model reflecting the characteristics of the both contributing data sets, while the site dependency of the data is properly accounted for. We question the regular Bayesian updating process, since a test of statistics proves that the two data sets belong to different populations. Application of ?Disjunction? probability operator is proposed as an alternative to arrive at a more conclusive Su?r?IL model. Next, the study is extended to a functional inverse problem where the object of inference constructs a spatial random field. We introduce a methodology to infer the spatial variation of the elastic characteristics of a heterogeneous earth model via Bayesian approach, given the probed medium?s response to interrogating waves measured at the surface. A reduced dimension, self regularized treatment of the inverse problem using partition modeling is introduced, where the velocity field is discretized by a variable number of disjoint regions defined by Voronoi tessellations. The number of partitions, their geometry and weights dynamically vary during the inversion, in order to recover the subsurface image. The idea of treating the number of tessellation (number of parameters) as a parameter itself is closely associated with probabilistic model selection. A reversible jump Markov chain Monte Carlo (RJMCMC) scheme is applied to sample the posterior distribution of varying dimension. Lastly, direct treatment of a Bayesian model selection through the definition of the Bayes factor (BF) is developed for linear models, where it is employed to define the most likely order of the virial Equation of State (EOS). Virial equation of state is a constitutive model describing the thermodynamic behavior of low-density fluids in terms of the molar density, pressure and temperature. Bayesian model selection has successfully determined the best EOS that describes four sets of isotherms, where approximate (BIC) method either failed to select a model or fevered an overly-flexible model, which specifically perform poorly in terms of prediction.Item A mixed unsplit-field PML-based scheme for full waveform inversion in the time-domain using scalar waves(2010-05) Kang, Jun Won, 1975-; Kallivokas, Loukas F.; Stokoe, Kenneth H.; Tonon, Fulvio; Ghattas, Omar; Gonzalez, OscarWe discuss a full-waveform based material profile reconstruction in two-dimensional heterogeneous semi-infinite domains. In particular, we try to image the spatial variation of shear moduli/wave velocities, directly in the time-domain, from scant surficial measurements of the domain's response to prescribed dynamic excitation. In addition, in one-dimensional media, we try to image the spatial variability of elastic and attenuation properties simultaneously. To deal with the semi-infinite extent of the physical domains, we introduce truncation boundaries, and adopt perfectly-matched-layers (PMLs) as the boundary wave absorbers. Within this framework we develop a new mixed displacement-stress (or stress memory) finite element formulation based on unsplit-field PMLs for transient scalar wave simulations in heterogeneous semi-infinite domains. We use, as is typically done, complex-coordinate stretching transformations in the frequency-domain, and recover the governing PDEs in the time-domain through the inverse Fourier transform. Upon spatial discretization, the resulting equations lead to a mixed semi-discrete form, where both displacements and stresses (or stress histories/memories) are treated as independent unknowns. We propose approximant pairs, which numerically, are shown to be stable. The resulting mixed finite element scheme is relatively simple and straightforward to implement, when compared against split-field PML techniques. It also bypasses the need for complicated time integration schemes that arise when recent displacement-based formulations are used. We report numerical results for 1D and 2D scalar wave propagation in semi-infinite domains truncated by PMLs. We also conduct parametric studies and report on the effect the various PML parameter choices have on the simulation error. To tackle the inversion, we adopt a PDE-constrained optimization approach, that formally leads to a classic KKT (Karush-Kuhn-Tucker) system comprising an initial-value state, a final-value adjoint, and a time-invariant control problem. We iteratively update the velocity profile by solving the KKT system via a reduced space approach. To narrow the feasibility space and alleviate the inherent solution multiplicity of the inverse problem, Tikhonov and Total Variation (TV) regularization schemes are used, endowed with a regularization factor continuation algorithm. We use a source frequency continuation scheme to make successive iterates remain within the basin of attraction of the global minimum. We also limit the total observation time to optimally account for the domain's heterogeneity during inversion iterations. We report on both one- and two-dimensional examples, including the Marmousi benchmark problem, that lead efficiently to the reconstruction of heterogeneous profiles involving both horizontal and inclined layers, as well as of inclusions within layered systems.