Bayesian Methods for Gas-Phase Tomography

Gas-phase tomography refers to a set of techniques that determine the 2D or 3D distribution of a target species in a jet, plume, or flame using measurements of light, made around the boundary of a flow area. Reconstructed quantities may include the concentration of one or more species, temperature, pressure, and optical density, among others. Tomography is increasingly used to study fundamental aspects of turbulent combustion and monitor emissions for regulatory compliance. This thesis develops statistical methods to improve gas-phase tomography and reports two novel experimental applications. Tomography is an inverse problem, meaning that a forward model (calculating measurements of light for a known distribution of gas) is inverted to estimate the model parameters (transforming experimental data into a gas distribution). The measurement modality varies with the problem geometry and objective of the experiment. For instance, transmittance data from an array of laser beams that transect a jet may be inverted to recover 2D fields of concentration and temperature; and multiple high-resolution images of a flame, captured from different angles, are used to reconstruct wrinkling of the 3D reacting zone. Forward models for gas-phase tomography modalities share a common mathematical form, that of a Fredholm integral equation of the first-kind (IFK). The inversion of coupled IFKs is necessarily ill-posed, however, meaning that solutions are either unstable or non-unique. Measurements are thus insufficient in themselves to generate a realistic image of the gas and additional information must be incorporated into the reconstruction procedure. Statistical inversion is an approach to inverse problems in which the measurements, experimental parameters, and quantities of interest are treated as random variables, characterized by a probability distribution. These distributions reflect uncertainty about the target due to fluctuations in the flow field, noise in the data, errors in the forward model, and the ill-posed nature of reconstruction. The Bayesian framework for tomography features a likelihood probability density function (pdf), which describes the chance of observing a measurement for a given distribution of gas, and prior pdf, which assigns a relative plausibility to candidate distributions based on assumptions about the flow physics. Bayes’ equation updates information about the target in response to measurement data, combining the likelihood and prior functions to form a posterior pdf. The posterior is usually summarized by the maximum a posteriori (MAP) estimate, which is the most likely distribution of gas for a set of data, subject to the effects of noise, model errors, and prior information. The framework can be used to estimate credibility intervals for a reconstruction and the form of Bayes’ equation suggests procedures for improving gas tomography. The accuracy of reconstructions depends on the information content of the data, which is a function of the experimental design, as well as the specificity and validity of the prior. This thesis employs theoretical arguments and experimental measurements of scalar fluctuations to justify joint-normal likelihood and prior pdfs for gas-phase tomography. Three methods are introduced to improve each stage of the inverse problem: to develop priors, design optimal experiments, and select a discretization scheme. First, a self-similarity analysis of turbulent jets—common targets in gas tomography—is used to construct an advanced prior, informed by an estimate of the jet’s spatial covariance. Next, a Bayesian objective function is proposed to optimize beam positions in limited-data arrays, which are necessary in scenarios where optical access to the flow area is restricted. Finally, a Bayesian expression for model selection is derived from the joint-normal pdfs and employed to select a mathematical basis to reconstruct a flow. Extensive numerical evidence is presented to validate these methods. The dissertation continues with two novel experiments, conducted in a Bayesian way. Broadband absorption tomography is a new technique intended for quantitative emissions detection from spectrally-convolved absorption signals. Theoretical foundations for the diagnostic are developed and the results of a proof-of-concept emissions detection experiment are reported. Lastly, background-oriented schlieren (BOS) tomography is applied to combustion for the first time. BOS tomography employs measurements of beam steering to reconstruct a fluid’s optical density field, which can be used to infer temperature and density. The application of BOS tomography to flame imaging sets the stage for instantaneous 3D combustion thermometry. Numerical and experimental results reported in this thesis support a Bayesian approach to gas-phase tomography. Bayesian tomography makes the role of prior information explicit, which can be leveraged to optimize reconstructions and design better imaging systems in support of research on fluid flow and combustion dynamics

Recent Advances in Signal Processing

The signal processing task is a very critical issue in the majority of new technological inventions and challenges in a variety of applications in both science and engineering fields. Classical signal processing techniques have largely worked with mathematical models that are linear, local, stationary, and Gaussian. They have always favored closed-form tractability over real-world accuracy. These constraints were imposed by the lack of powerful computing tools. During the last few decades, signal processing theories, developments, and applications have matured rapidly and now include tools from many areas of mathematics, computer science, physics, and engineering. This book is targeted primarily toward both students and researchers who want to be exposed to a wide variety of signal processing techniques and algorithms. It includes 27 chapters that can be categorized into five different areas depending on the application at hand. These five categories are ordered to address image processing, speech processing, communication systems, time-series analysis, and educational packages respectively. The book has the advantage of providing a collection of applications that are completely independent and self-contained; thus, the interested reader can choose any chapter and skip to another without losing continuity

Acoustic tomography imaging for atmospheric temperature and wind velocity field reconstruction

Owing to its non-invasive nature, fast imaging speed, low equipment cost, scalability for a variety of measurement ranges, and ability to simultaneously monitor both temperature and wind velocity fields, acoustic tomography has attracted considerable interest in the field of atmospheric imaging. This thesis aims to improve the reconstruction quality of the acoustic tomography system for temperature and wind velocity field imaging. Focusing on this goal, the contribution of the thesis can be summarised from the perspectives of data collection system development, robust and accurate TOF estimation method, and high-quality scalar and vector tomographic image reconstruction methods for temperature and wind velocity fields respectively. Details are given below. Firstly, in order to facilitate the experimental study of acoustic tomography imaging, the design and evaluation of the data collection system and TOF estimation method was presented. The evaluation results indicate that the presented data acquisition system and TOF estimation method has good quantitative accuracy in the lab-scale experiments. The temporal resolution is of great significance for the real-time monitoring of the fast-changing temperature field. To improve the temporal resolution, a novel online time-resolved reconstruction (OTRR) method is presented, which can reconstruct high quality time-resolved images by using fewer TOFs per frame. Compared to state-of-the-art dynamic reconstruction algorithms such as the Kalman filter reconstruction, the proposed algorithm demonstrated superior spatial resolution and preferable quantitative accuracy in the reconstructed images. These features are necessary for the real-time monitoring of the fast-changing temperature field. The forward modelling of most acoustic tomography problems is based on a straight ray model, which may result in large modelling errors due to the refraction effect under a large gradient temperature field. In order to reduce the inaccuracy of using the straight ray model, a bent ray model and nonlinear reconstruction algorithm is applied, which allows the sound propagation ray paths and temperature distribution to be reconstructed iteratively from the TOFs. Using acoustic tomography to reconstruct large-scale temperature and wind velocity fields, a fully parallel TOF measurement scheme is necessary. To achieve this goal, a set of orthogonal acoustic waveforms based on the filtered and modulated Kasami sequence is designed and a cross-correlation based TOF estimation method is used for data collection. Besides, to overcome the invisible field problem and improve the image quality of the wind velocity reconstruction, a divergence-free regularised vector tomographic reconstruction algorithm is studied. The proposed method is able to provide accurate tomographic reconstruction of the 2D horizontal wind velocity field from the TOF measurements. In summary, this thesis focuses on the improvement of acoustic tomography techniques for temperature and wind velocity fields, including the phase corrected Akaike information criterion (AIC) TOF estimation for accurate and robust TOF estimation, the online time-resolved reconstruction method for real-time monitoring of the fast changing temperature field, the nonlinear reconstruction based on the bent ray model to reconstruct the temperature field with a large gradient, and the divergence-free regularised reconstruction method to visualise the 2D horizontal wind velocity field