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Inversion and Depth Range of Dipole Electromagnetic Induction Measurements in Geophysics Fernando César Moura de Andrade Institute of Hydrogeology, Engineering Geology and Applied Geophysics Faculty of Science, Charles University Electromagnetic induction geophysical methods are, basically, composed by a transmitter which produces a magnetic field and a set of receivers which measure the primary magnetic field, from the transmitter, superimposed by secondary magnetic fields inducted in the subsurface. Equipment operating at, relatively, low frequencies and with short distances between the transmitter and the receivers are usually called conductivity meters and operate at low inductions numbers. The depth of investigation, in such kind of equipment, depends mainly on the transmitter-receiver distance, on the orientations of the magnetic dipoles and the height of the instrument from the ground, in order that a depth sounding can be done changing these parameters in a single measurement location. Making a series of these multi-configuration measurements, two-dimensional, or even three-dimensional surveys, can be performed and, subsequently, inverted in order to produce an image of the subsurface of the earth. Forward modelling and inversion of multi-configuration electromagnetic induction data can be made...

Seismic hazard analysis is concerned with estimating risk to human populations due to earthquakes and the other natural disasters that they cause. In many parts of the world, earthquake-generated tsunamis are especially dangerous. Assessing the risk for seismic disasters relies on historical data that indicate which fault zones are capable of supporting significant earthquakes. Due to the nature of geologic time scales, the era of seismological data collection with modern instruments has captured only a part of the Earth's seismic hot zones. However, non-instrumental records, such as anecdotal accounts in newspapers, personal journals, or oral tradition, provide limited information on earthquakes that occurred before the modern era. Here, we introduce a method for reconstructing the source earthquakes of historical tsunamis based on anecdotal accounts. We frame the reconstruction task as a Bayesian inference problem by making a probabilistic interpretation of the anecdotal records. Utilizing robust models for simulating earthquakes and tsunamis provided by the software package GeoClaw, we implement a Metropolis-Hastings sampler for the posterior distribution on source earthquake parameters. In this work, we present our analysis of the 1852 Banda Arc earthquake and tsunami as a case study for the method. Our method is implemented as a Python package, which we call tsunamibayes. It is available, open-source, on GitHub: https://github.com/jwp37/tsunamibayes.

<div>Structural health monitoring (SHM) has immense potential to improve the safety of aerospace, mechanical, and civil structures because it allows for continuous, real-time damage prognostication. However, conventional SHM methodologies are limited by factors such as the need for extensive external sensor arrays, inadequate sensitivity to small-sized damage, and poor spatial damage localization. As such, widespread implementation of SHM in engineering structures has been severely restricted. These limitations can be overcome through the use of multi-functional materials with intrinsic self-sensing capabilities. In this area, composite materials with nanofiller-modified polymer matrices have received considerable research interest. The electrical conductivity of these materials is affected by mechanical stimuli such as strain and damage. This is known as the piezoresistive effect and it has been leveraged extensively for SHM in self-sensing materials. However, prevailing conductivity-based SHM modalities suffer from two critical limitations. The first limitation is that the mechanical state of the structure must be indirectly inferred from conductivity changes. Since conductivity is not a structurally relevant property, it would be much more beneficial to know the displacements, strains, and stresses as these can be used to predict the onset of damage and failure. The second limitation is that the precise shape and size of damage cannot be accurately determined from conductivity changes. From a SHM point of view, knowing the precise shape and size of damage would greatly aid in-service inspection and nondestructive evaluation (NDE) of safety-critical structures. The underlying cause of these limitations is that recovering precise mechanics from conductivity presents an under determined and multi-modal inverse problem. Therefore, commonly used inversion schemes such as gradient-based optimization methods fail to produce physically meaningful solutions. Instead, metaheuristic search algorithms must be used in conjunction with physics-based damage models and realistic constraints on the solution search space. To that end, the overarching goal of this research is to address the limitations of conductivity-based SHM by developing metaheuristic algorithm-enabled methodologies for recovering precise mechanics from conductivity changes in self-sensing composites.</div><div><div><br></div><div>Three major scholarly contributions are made in this thesis. First, a piezoresistive inversion methodology is developed for recovering displacements, strains, and stresses in an elastically deformed self-sensing composite based on observed conductivity changes. For this, a genetic algorithm (GA) is integrated with an analytical piezoresistivity model and physics-based constraints on the search space. Using a simple stress based failure criterion, it is demonstrated that this approach can be used to accurately predict material failure. Second, the feasibility of using other widely used metaheuristic algorithms for piezoresistive inversion is explored. Specifically, simulated annealing (SA) and particle swarm optimization (PSO) are used and their performances are compared to the performance of the GA. It is concluded that while SA and PSO can certainly be used to solve the piezoresistive inversion problem, the GA is the best algorithm based on solution accuracy, consistency, and efficiency. Third, a novel methodology is developed for precisely determining damage shape and size from observed conductivity changes in self-sensing composites. For this, a GA is integrated with physics-based geometric models for damage and suitable constraints on the search space. By considering two specific damage modes —through-holes and delaminations —it is shown that this method can be used to precisely reconstruct the shape and size of damage. </div><div><br></div><div>In achieving these goals, this thesis advances the state of the art by addressing critical limitations of conductivity-based SHM. The methodologies developed herein can enable unprecedented NDE capabilities by providing real-time information about the precise mechanical state (displacements, strains, and stresses) and damage shape in self-sensing composites. This has incredible potential to improve the safety of structures in a myriad of engineering venues.</div></div>

<p>Applications in materials and biological imaging are currently limited by the ability to collect high-resolution data over large areas in practical amounts of time. One possible solution to this problem is to collect low-resolution data and apply a super-resolution interpolation algorithm to produce a high-resolution image. However, state-of-the-art super-resolution algorithms are typically designed for natural images, require aligned pairing of high and low-resolution training data for optimal performance, and do not directly incorporate a data-fidelity mechanism.</p><p><br></p><p>We present a Multi-Resolution Data Fusion (MDF) algorithm for accurate interpolation of low-resolution SEM and TEM data by factors of 4x and 8x. This MDF interpolation algorithm achieves these high rates of interpolation by first learning an accurate prior model denoiser for the TEM sample from small quantities of unpaired high-resolution data and then balancing this learned denoiser with a novel mismatched proximal map that maintains fidelity to measured data. The method is based on Multi-Agent Consensus Equilibrium (MACE), a generalization of the Plug-and-Play method, and allows for interpolation at arbitrary resolutions without retraining. We present electron microscopy results at 4x and 8x super resolution that exhibit reduced artifacts relative to existing methods while maintaining fidelity to acquired data and accurately resolving sub-pixel-scale features.</p>

In this note we explain the use of the Akiake Information Criterion and its related model comparison indices (usually derived for maximum likelihood estimator inverse problem formulations) in the context of least squares (ordinary, weighted, iterative weighted or “generalized”, etc.) based inverse problem formulations. The ideas are illustrated with several examples of interest in biology.

Seismic hazard analysis is concerned with estimating risk to human populations due to earthquakes and the other natural disasters that they cause. In many parts of the world, earthquake-generated tsunamis are especially dangerous. Assessing the risk for seismic disasters relies on historical data that indicate which fault zones are capable of supporting significant earthquakes. Due to the nature of geologic time scales, the era of seismological data collection with modern instruments has captured only a part of the Earth's seismic hot zones. However, non-instrumental records, such as anecdotal accounts in newspapers, personal journals, or oral tradition, provide limited information on earthquakes that occurred before the modern era. Here, we introduce a method for reconstructing the source earthquakes of historical tsunamis based on anecdotal accounts. We frame the reconstruction task as a Bayesian inference problem by making a probabilistic interpretation of the anecdotal records. Utilizing robust models for simulating earthquakes and tsunamis provided by the software package GeoClaw, we implement a Metropolis-Hastings sampler for the posterior distribution on source earthquake parameters. In this work, we present our analysis of the 1852 Banda Arc earthquake and tsunami as a case study for the method. Our method is implemented as a Python package, which we call tsunamibayes. It is available, open-source, on GitHub: https://github.com/jwp37/tsunamibayes.

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Iterative Learning Control (ILC) considers systems which perform the given desired trajectory repetitively. The command for the upcoming iteration is updated after every iteration based on the previous recorded error, aiming to converge to zero error in the real-world. Iterative Learning Control can be considered as an inverse problem, solving for the needed input that produces the desired output. However, digital control systems need to convert differential equations to digital form. For a majority of real world systems this introduces one or more zeros of the system z-transfer function outside the unit circle making the inverse system unstable. The resulting control input that produces zero error at the sample times following the desired trajectory is unstable, growing exponentially in magnitude each time step. The tracking error between time steps is also growing exponentially defeating the intended objective of zero tracking error. One way to address the instability in the inverse of non-minimum phase systems is to use basis functions. Besides addressing the unstable inverse issue, using basis functions also has several other advantages. First, it significantly reduces the computation burden in solving for the input command, as the number of basis functions chosen is usually much smaller than the number of time steps in one iteration. Second, it allows the designer to choose the frequency to cut off the learning process, which provides stability robustness to unmodelled high frequency dynamics eliminating the need to otherwise include a low-pass filter. In addition, choosing basis functions intelligently can lead to fast convergence of the learning process. All these benefits come at the expense of no longer asking for zero tracking error, but only aiming to correct the tracking error in the span of the chosen basis functions. Two kinds of matched basis functions are presented in this dissertation, frequency-response based basis functions and singular vector basis functions, respectively. In addition, basis functions are developed to directly capture the system transients that result from initial conditions and hence are not associated with forcing functions. The newly developed transient basis functions are particularly helpful in reducing the level of tracking error and constraining the magnitude of input control when the desired trajectory does not have a smooth start-up, corresponding to a smooth transition from the system state before the initial time, and the system state immediately after time zero on the desired trajectory. Another topic that has been investigated is the error accumulation in the unaddressed part of the output space, the part not covered by the span of the output basis functions, under different model conditions. It has been both proved mathematically and validated by numerical experiments that the error in the unaddressed space will remain constant when using an error-free model, and the unaddressed error will demonstrate a process of accumulation and finally converge to a constant level in the presence of model error. The same phenomenon is shown to apply when using unmatched basis functions. There will be unaddressed error accumulation even in the absence of model error, suggesting that matched basis functions should be used whenever possible. Another way to address the often unstable nature of the inverse of non-minimum phase systems is to use the in-house developed stable inverse theory Longman JiLLL, which can also be incorporated into other control algorithms including One-Step Ahead Control and Indirect Adaptive Control in addition to Iterative Learning Control. Using this stable inverse theory, One-Step Ahead Control has been generalized to apply to systems whose discrete-time inverses are unstable. The generalized one-step ahead control can be viewed as a Model Predictive Control that achieves zero tracking error with a control input bounded by the actuator constraints. In situations where one feels not confident about the system model, adaptive control can be applied to update the model parameters while achieving zero tracking error.

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