The Method of Manufactured Universes is presented as a validation framework for uncertainty quantification (UQ) methodologies and as a tool for exploring the effects of statistical and modeling assumptions embedded in these methods. The framework calls for a manufactured reality from which "experimental" data are created (possibly with experimental error), an imperfect model (with uncertain inputs) from which simulation results are created (possibly with numerical error), the application of a system for quantifying uncertainties in model predictions, and an assessment of how accurately those uncertainties are quantified. The application presented for this research manufactures a particle-transport "universe," models it using diffusion theory with uncertain material parameters, and applies both Gaussian process and Bayesian MARS algorithms to make quantitative predictions about new "experiments" within the manufactured reality. To test further the responses of these UQ methods, we conduct exercises with "experimental" replicates, "measurement" error, and choices of physical inputs that reduce the accuracy of the diffusion model's approximation of our manufactured laws. Our first application of MMU was rich in areas for exploration and highly informative. In the case of the Gaussian process code, we found that the fundamental statistical formulation was not appropriate for our functional data, but that the code allows a knowledgable user to vary parameters within this formulation to tailor its behavior for a specific problem. The Bayesian MARS formulation was a more natural emulator given our manufactured laws, and we used the MMU framework to develop further a calibration method and to characterize the diffusion model discrepancy. Overall, we conclude that an MMU exercise with a properly designed universe (that is, one that is an adequate representation of some real-world problem) will provide the modeler with an added understanding of the interaction between a given UQ method and his/her more complex problem of interest. The modeler can then apply this added understanding and make more informed predictive statements.

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This thesis examines the sensitivity of the marine controlled source electromagnetic (CSEM) method to sub-seafloor resistivity structure, with a focus on gas hydrate and free gas occurrences. Different analysis techniques are applied with progressive sophistication to a series of studies based on simulated and measured data sets. CSEM data are modelled in time domain for one-dimensional models with gas hydrate, free gas and/or permafrost occurrences. Linearized and non-linear inversion methods are considered to infer subsurface models from CSEM data. One study applies forward modelling and singular value decomposition to estimate uncertainties for permafrost models of the Beaufort Sea. This simulation study analyzes the resolution of the CSEM data for shallow water depth which is a challenging case because the electromagnetic signature of the air-water boundary may mask the sub-seafloor response. The results reveal a blind window as a function of water depth in which the CSEM data are insensitive to the sub-seafloor structure. However, the CSEM data are sensitive to the top and the bottom of the permafrost with increasing uncertainties with depth. The next study applies non-linear Bayesian inversion to CSEM data acquired in 2005/2006 on the Northern Cascadia margin to investigate sub-seafloor resistivity structure related to gas hydrate deposits and cold vents. Bayesian inversion provides a rigorous approach to estimate model parameters and uncertainties by probabilistically sampling of the parameter space. The resulting probability density function is interpreted here in terms of posterior median models, marginal and joint marginal probability densities for model parameters and credibility intervals. The Bayesian information criterion is applied to determine the amount of structure (number of layers) that can be resolved by the data. The parameter space is sampled with the Metropolis-Hastings algorithm in principal-component space. Non-linear, probabilistic inversion allows the analysis of unknown acquisition parameters such as time delays between receiver and transmitter clocks or unknown source amplitude. The estimated posterior median models and credibility intervals from Bayesian CSEM inversion are compared to reflection seismic data to provide a more complete geological interpretation. The CSEM data on the Northern Cascadia margin generally reveal a 1 to 3 layer sediment structure. Inversion results at the landward edge of the gas hydrate stability zone indicate a sediment unconformity as well as several potential cold vents which were previously unknown. The resistivities generally increase upslope due to sediment erosion along the slope. Inversion results on the middle slope infer several vent systems close to well-known Bullseye vent in agreement with ongoing interdisciplinary observations. Finally, a trans-dimensional (trans-D) Bayesian inversion is applied to CSEM data acquired in 2012 in the German North Sea to investigate possible free gas occurrences. Trans-D inversion treats the number of layers as an additional unknown sampled probabilistically in the inversion. %over the parameter space by evaluating probabilistically the transition to a higher or lower number of interfaces. Parallel tempering is applied to increase sampling efficiency and completeness. Inversion results for the German North Sea yield resistivities at the seafloor which are typical for marine deposits, while resistivities at greater depth increase slightly and can be correlated with a transition from fine-grained marine deposits (Holocene age) to coarse-grained, glacial sediments (Pleistocene age), which is observed in a sediment core. The depths of layer interfaces estimated from CSEM inversion match the seismic reflector related to the contrast between the two depositional environments. The CSEM survey targeted a strong, phase-reversed, inclined seismic reflector within the glacial sediments, potentially indicating free gas. While interface-depth estimates from CSEM inversion do not correlate closely with this reflector, resistivities are generally elevated above the strong seismic amplitudes and the thickness of the resistive layer follows the trend of the inclined reflector. However, the uncertainties of deeper interface depth estimates increase significantly and overlap with the targeted reflector at some of the measurement sites. Relatively low resistivities of a third layer correlate with sediments of late-Miocene origin with a high gamma-ray count indicating an increased amount of fine-grained sediments with organic material. The interface at the bottom of the third layer has wide uncertainties which relates to the penetration limit of the CSEM array. / Graduate

Amplitude Variation with Oset (AVO) inversion is one of the techniques that is being used to estimate subsurface physical parameters such as P-wave velocity, S-wave velocity, and density or their attributes. AVO inversion is an ill-conditioned problem which has to be regularized in order to obtain a stable and unique solution. In this thesis, a Bayesian procedure that uses a Multivariate Cauchy distribution as a prior probability distribution is introduced. The prior includes a scale matrix that imposes correlation among the AVO attributes and induces a regularization that provokes solutions that are sparse and stable in the presence of noise. The performance of this regularization is demonstrated by both synthetic and real data examples using linearized approximations to the Zoeppritz equations. / Geophysics

This thesis presents Bayesian geoacoustic inversion of seabed reflection-coefficient data as part of the U.S. Office of Naval Research Seabed Characterization Experiment 2017 at the New England Mud Patch. First, a linearized, ray-based Bayesian inversion of acoustic arrival times is carried out for high-precision estimation of experiment geometry and uncertainties, representing an important first step to inferring seabed properties using geoacoustic inversion methods such as reflection inversion. The high-precision estimates for source-receiver ranges, source depths, receiver depths, and water depths at reflection points along the survey track are used to calculate grazing angles, with angle uncertainties computed using Monte Carlo methods. The experiment geometry uncertainties are obtained using analytic linearized estimates, and verified with nonlinear analysis. Second, a trans-dimensional (trans-D) Bayesian inversion of reflection-coefficient data is carried out for geoacoustic parameters and uncertainties of fine-grained/cohesive sediments. The trans-D inversion samples probabilistically over an unknown number of seabed interfaces and the parameters of a zeroth- or first-order autoregressive error model. The numerical method of parallel tempering reversible jump Markov-chain Monte Carlo sampling is employed. Spherical-wave reflection coefficient modelling is applied using plane-wave decomposition in the Sommerfeld integral. The inversion provides marginal posterior probability profiles for Buckingham's viscous grain-shearing parameters: porosity, grain-to-grain compressional modulus, material exponent, and compressional viscoelastic time constant as a function of depth in the sediment. These parameters are used to compute dispersion relationships for each layer in the model, providing marginal posterior probability profiles for compressional-wave velocity and attenuation at different frequencies, as well as density. The geoacoustic inversion results are compared to independent measurements of sediment properties. / Graduate

Projecting the ice sheets' contribution to sea-level rise is difficult because of the complexity of accurately modeling ice sheet dynamics for the full polar ice sheets, because of the uncertainty in key, unobservable parameters governing those dynamics, and because quantifying the uncertainty in projections is necessary when determining the confidence to place in them. This work presents the formulation and solution of the Bayesian inverse problem of inferring, from observations, a probability distribution for the basal sliding parameter field beneath the Antarctic ice sheet. The basal sliding parameter is used within a high-fidelity nonlinear Stokes model of ice sheet dynamics. This model maps the parameters "forward" onto a velocity field that is compared against observations. Due to the continental-scale of the model, both the parameter field and the state variables of the forward problem have a large number of degrees of freedom: we consider discretizations in which the parameter has more than 1 million degrees of freedom. The Bayesian inverse problem is thus to characterize an implicitly defined distribution in a high-dimensional space. This is a computationally demanding problem that requires scalable and efficient numerical methods be used throughout: in discretizing the forward model; in solving the resulting nonlinear equations; in solving the Bayesian inverse problem; and in propagating the uncertainty encoded in the posterior distribution of the inverse problem forward onto important quantities of interest. To address discretization, a hybrid parallel adaptive mesh refinement format is designed and implemented for ice sheets that is suited to the large width-to-height aspect ratios of the polar ice sheets. An efficient solver for the nonlinear Stokes equations is designed for high-order, stable, mixed finite-element discretizations on these adaptively refined meshes. A Gaussian approximation of the posterior distribution of parameters is defined, whose mean and covariance can be efficiently and scalably computed using adjoint-based methods from PDE-constrained optimization. Using a low-rank approximation of the covariance of this distribution, the covariance of the parameter is pushed forward onto quantities of interest.

The overall goal of this thesis is to develop non-linear Bayesian methods for three-dimensional tracking of a moving acoustic source in shallow water despite environmental uncertainty, with application to data from a horizontal line array (HLA) of hydrophones. As a precursor, Bayesian geoacoustic inversion is applied to estimate seabed model parameters and their uncertainties. A simulation study examines the effect of source and array factors on geoacoustic information content in matched-field inversion of HLA data, as quantified in terms of model parameter uncertainties. Bayesian geoacoustic inversion is applied to both controlled-source and ship-noise data from a HLA deployed on the seafloor in a shallow-water experiment conducted in the Barents Sea. A new approach is introduced to account for data error reduction due to averaging data over time-series subsegments (snapshots), based on empirically apportioning measurement and theory error, with effects on inversion results compared to those of existing approaches. It is further demonstrated that combining data from multiple, independent time-series segments (for a moving source) in the inversion can significantly reduce geoacoustic parameter uncertainties. Geoacoustic uncertainties are also shown to depend on ship range and orientation, with lowest uncertainties for short ranges and for the ship stern/propeller oriented toward the array. Sediment sound-speed profile and density estimates from controlled-source and ship-noise data inversions are found to be in good agreement with values from geophysical measurements. Two non-linear Bayesian matched-field inversion approaches are developed for three-dimensional source tracking despite environmental uncertainty. Focalization-tracking maximizes the posterior probability density (PPD) over track and environmental parameters. Synthetic test cases show that the algorithm substantially outperforms tracking with poor environmental estimates and generally obtains results close to those achieved with exact environmental knowledge. Marginalization-tracking integrates the PPD over environmental parameters to obtain joint marginal distributions over source coordinates, from which track uncertainty estimates and the most probable track are extracted. Both approaches are applied to data from the Barents Sea experiment. Focalization-tracking successfully estimates the tracks of the towed source and a surface ship in cases where simpler tracking algorithms fail. Marginalization-tracking generally outperforms focalization-tracking and gives uncertainty estimates that encompass the true tracks.

This dissertation is a comprehensive summary of four papers on the development and application of new strategies for locating tremor and relocating events in earthquake catalogs. In the first paper, two new strategies for relocating events in a catalog are introduced. The seismicity pattern of an earthquake catalog is often used to delineate seismically active faults. However, the delineation is often hindered by the diffuseness of earthquake locations in the catalog. To reduce the diffuseness and simplify the seismicity pattern, a relocation and a collapsing method are developed and applied. The relocation method uses the catalog event density as an a priori constraint for relocations in a Bayesian inversion. The catalog event density is expressed in terms of the combined probability distribution of all events in the catalog. The collapsing method uses the same catalog density as an attractor for focusing the seismicity in an iterative scheme. These two strategies are applied to an aftershock sequence after a pair of earthquakes which occurred in southwest Iceland, 2008. The seismicity pattern is simplified by application of the methods and the faults of the mainshocks are delineated by the reworked catalog. In the second paper, the spatial distribution of seismicity of the Hengill region, southwest Iceland is analyzed. The relocation and collapsing methods developed in the first paper and a non-linear relocation strategy using empirical traveltime tables are used to process a catalog collected by the Icelandic Meteorological Office. The reworked catalog reproduces details of the spatial distribution of seismicity that independently emerges from relative relocations of a small subset of the catalog events. The processed catalog is then used to estimate the depth to the brittle-ductile transition. The estimates show that in general the northern part of the area, dominated by volcanic processes, has a shallower depth than the southern part, where tectonic deformation predominates. In the third and the fourth papers, two back-projection methods using inter-station cross correlations are proposed for locating tremor sources. For the first method, double correlations, defined as the cross correlations of correlations from two station pairs sharing a common reference station, are back projected. For the second method, the products of correlation envelopes from a group of stations sharing a common reference station are back projected. Back projecting these combinations of correlations, instead of single correlations, suppresses random noise and reduces the strong geometrical signature caused by the station configuration. These two methods are tested with volcanic tremor at Katla volcano, Iceland. The inferred source locations agree with surface observations related to volcanic events which occurred during the tremor period.

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...

In this thesis, acoustic inversion methods are employed to estimate array element locations and the geoacoustic properties of the seabed using measured acoustic data consisting of noise from a surface ship in the Gulf of Mexico. The array element localization utilizes relative travel-time information obtained by cross-correlating the recorded time series of ship noise received at spatially separated hydrophones. The relative travel-time data are used in an inversion, based on the regularized least-squares method and the acoustic ray tracing equations, to obtain improved estimates of the receiver and source positions and their uncertainties. Optimization and Bayesian matched-field inversion methods are employed to estimate seabed geoacoustic properties and their uncertainties in the vicinity of a bottom-moored vertical line array using the recorded surface ship noise. This study is used to test the feasibility of matched-field methods to detect temporal changes in the geoacoustic properties of the seabed near a known gas hydrate mound in the Gulf of Mexico. Finally, a synthetic study is performed that demonstrates how ignoring environmental range dependence of seabed sound speed and water depth in matched-field inversion can lead to biases in the estimated geoacoustic parameters. The study considers the distributions of optimal parameter estimates obtained from a large number of range-independent inversions of synthetic data generated for random range-dependent environments. Range-independent Bayesian inversions are also performed on selected data sets and the marginal parameter distributions are examined. Both hard- and soft-bottom environments are examined at a number of scales of variability in sound speed and water depth.