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

Measuring water-vapour and carbon-dioxide fluxes at field scales with scintillometry

Scintillometry is a measurement technique that has proven itself to be of great value for measuring spatial-averaged fluxes of sensible heat, momentum, and evapotranspiration. Furthermore, for crop fields (field scales), scintillometry has been shown to accurately determine the sensible-heat and momentum flux over time intervals as short as 6 seconds. As a consequence, interests in scintillometry are growing and scintillometers that determine sensible-heat fluxes and momentum fluxes have become commercially available. This thesis deals with two aspects of scintillometry. First, after a general introduction of scintillometry, measurement errors that have been observed in the large-aperture scintillometer from Kipp&Zonen and in the SLS field-scale scintillometer from Scintec are evaluated. For both scintillometer types, we discuss the variability in the measurement errors among different instruments and, where possible, we give solutions to remove these errors. Furthermore, we present the results of a prototype scintillometer that was developed as part of the research project. With our proposed design, we aim to overcome the measurement errors in the Scintec scintillometer and extend the applicability of the field-scale scintillometer to paths that are longer than 200 m. Second, we extend the application of field-scale scintillometry to the flux measurements of latent-heat, carbon-dioxide, and other passive scalars. Until now, scintillometers could not be used for measuring passive-scalar fluxes over crop fields and we show that with our extended methodology these fluxes can be accurately determined over time intervals as short as 1 minute. The methodology is based on a combination of scintillometer measurements and additional high-frequency scalar measurements and works under conditions of homogeneous turbulence, i.e. single crop fields. We introduce four methods, notably the energy-balance method, the Bowen-variance method, the flux-variance method, and the structure-parameter method. Using several validation methods, we show that the energy-balance method is unsuitable for estimating scalar fluxes over 1-min averaging intervals. The Bowen-variance and flux-variance method perform better and the structure-parameter method accurately resolves 1-minute fluxes. Thus, with this methodology fluxes can be resolved with a high temporal resolution, making it possible to study vegetation in a natural environment under non-stationary conditions. This allows us to show that the wheat vegetation affects fluxes upon changes in solar radiation in time periods clearly shorter than 30 minutes and that the canopy resistance can change significantly within several minutes. </p

DUMB ISOTROPIC SENSORS CAN FIND DOAS

Following the SENMA concept, we consider a wireless network of very dumb and cheap sensors, polled by a travelling “rover”. Sensors are randomly placed and isotropic: individually they have no ability to resolve the direction of arrival (DOA) of an acoustic wave. We assume that the communication load must be as limited as possible, so that these times cannot be communicated to the rover. Notwithstanding the lack of transmission of arrival times and the lack of DOA resolution ability of the individual sensors, DOA estimation is possible, and asymptotic efficiency becomes closely approximated after a reasonable number of rover snapshots. Key features are the directionality of the rover antenna, the area it surveys, and the average number of sensors inside that area, as accorded a Poisson distribution. 1