Applications of finite element computer modelling and thermal infrared remote sensing to the study of geothermal anomalies

Buried heat sources can be investigated by examining thermal infrared images and comparing these with the results of theoretical models which predict the thermal anomaly a given heat source may generate. Key factors influencing surface temperature include the geometry and temperature of the heat source, the surface meteorological environment, and the thermal conductivity and anisotropy of the rock. In general, a geothermal heat flux of greater than 2% of solar insolation is required to produce a detectable thermal anomaly in a thermal infrared image. A heat source of, for example, 2-300K greater than the average surface temperature must be a t depth shallower than 50m for the detection of the anomaly in a thermal infrared image, for typical terrestrial conditions. Atmospheric factors are of critical importance. While the mean atmospheric temperature has little significance, the convection is a dominant factor, and can act to swamp the thermal signature entirely. Given a steady state heat source that produces a detectable thermal anomaly, it is possible to loosely constrain the physical properties of the heat source and surrounding rock, using the surface thermal anomaly as a basis. The success of this technique is highly dependent on the degree to which the physical properties of the host rock are known. Important parameters include the surface thermal properties and thermal conductivity of the rock. Modelling of transient thermal situations was carried out, to assess the effect of time dependant thermal fluxes. One-dimensional finite element models can be readily and accurately applied to the investigation of diurnal heat flow, as with thermal inertia models. Diurnal thermal models of environments on Earth, the Moon and Mars were carried out using finite elements and found to be consistent with published measurements. The heat flow from an injection of hot lava into a near surface lava tube was considered. While this approach was useful for study, and long term monitoring in inhospitable areas, it was found to have little hazard warning utility, as the time taken for the thermal energy to propagate to the surface in dry rock (several months) in very long. The resolution of the thermal infrared imaging system is an important factor. Presently available satellite based systems such as Landsat (resolution of 120m) are inadequate for detailed study of geothermal anomalies. Airborne systems, such as TIMS (variable resolution of 3-6m) are much more useful for discriminating small buried heat sources. Planned improvements in the resolution of satellite based systems will broaden the potential for application of the techniques developed in this thesis. It is important to note, however, that adequate spatial resolution is a necessary but not sufficient condition for successful application of these techniques.

Uniformly convergent finite element and finite difference methods for singularly perturbed ordinary differential equations

This thesis is concerned with uniformly convergent finite element and finite difference methods for numerically solving singularly perturbed two-point boundary value problems. We examine the following four problems: (i) high order problem of reaction-diffusion type; (ii) high order problem of convection-diffusion type; (iii) second order interior turning point problem; (iv) semilinear reaction-diffusion problem. Firstly, we consider high order problems of reaction-diffusion type and convection-diffusion type. Under suitable hypotheses, the coercivity of the associated bilinear forms is proved and representation results for the solutions of such problems are given. It is shown that, on an equidistant mesh, polynomial schemes cannot achieve a high order of convergence which is uniform in the perturbation parameter. Piecewise polynomial Galerkin finite element methods are then constructed on a Shishkin mesh. High order convergence results, which are uniform in the perturbation parameter, are obtained in various norms. Secondly, we investigate linear second order problems with interior turning points. Piecewise linear Galerkin finite element methods are generated on various piecewise equidistant meshes designed for such problems. These methods are shown to be convergent, uniformly in the singular perturbation parameter, in a weighted energy norm and the usual L2 norm. Finally, we deal with a semilinear reaction-diffusion problem. Asymptotic properties of solutions to this problem are discussed and analysed. Two simple finite difference schemes on Shishkin meshes are applied to the problem. They are proved to be uniformly convergent of second order and fourth order respectively. Existence and uniqueness of a solution to both schemes are investigated. Numerical results for the above methods are presented.

Uniformly convergent finite element methods for singularly perturbed parabolic partial differential equations

This thesis is concerned with uniformly convergent finite element methods for numerically solving singularly perturbed parabolic partial differential equations in one space variable. First, we use Petrov-Galerkin finite element methods to generate three schemes for such problems, each of these schemes uses exponentially fitted elements in space. Two of them are lumped and the other is non-lumped. On meshes which are either arbitrary or slightly restricted, we derive global energy norm and L2 norm error bounds, uniformly in the diffusion parameter. Under some reasonable global assumptions together with realistic local assumptions on the solution and its derivatives, we prove that these exponentially fitted schemes are locally uniformly convergent, with order one, in a discrete L∞norm both outside and inside the boundary layer. We next analyse a streamline diffusion scheme on a Shishkin mesh for a model singularly perturbed parabolic partial differential equation. The method with piecewise linear space-time elements is shown, under reasonable assumptions on the solution, to be convergent, independently of the diffusion parameter, with a pointwise accuracy of almost order 5/4 outside layers and almost order 3/4 inside the boundary layer. Numerical results for the above schemes are presented. Finally, we examine a cell vertex finite volume method which is applied to a model time-dependent convection-diffusion problem. Local errors away from all layers are obtained in the l2 seminorm by using techniques from finite element analysis.

Assessment of dietary background exposure of the Irish adult population to dioxins and PCBs particularly taking into account additional exposure due to the 2008 Irish dioxin food contamination incident

Irish monitoring data on PCDD/Fs, DL-PCBs and Marker PCBs were collated and combined with Irish Adult Food Consumption Data, to estimate dietary background exposure of Irish adults to dioxins and PCBs. Furthermore, all available information on the 2008 Irish pork dioxin food contamination incident was collated and analysed with a view to evaluate any potential impact the incident may have had on general dioxin and PCB background exposure levels estimated for the adult population in Ireland. The average upperbound daily intake of Irish adults to dioxins Total WHO TEQ (2005) (PCDD/Fs & DLPCBs) from environmental background contamination, was estimated at 0.3 pg/kg bw/d and at the 95th percentile at 1 pg/kg bw/d. The average upperbound daily intake of Irish adults to the sum of 6 Marker PCBs from environmental background contamination ubiquitous in the environment was estimated at 1.6 ng/kg bw/d and at the 95th percentile at 6.8 ng/kg bw/d. Dietary background exposure estimates for both dioxins and PCBs indicate that the Irish adult population has exposures below the European average, a finding which is also supported by the levels detected in breast milk of Irish mothers. Exposure levels are below health based guidance values and/or Body Burdens associated with the TWI (for dioxins) or associated with a NOAEL (for PCBs). Given the current toxicological knowledge, based on biomarker data and estimated dietary exposure, general background exposure of the Irish adult population to dioxins and PCBs is of no human health concern. In 2008, a porcine fat sample taken as part of the national residues monitoring programme led to the detection of a major feed contamination incidence in the Republic of Ireland. The source of the contamination was traced back to the use of contaminated oil in a direct-drying feed operation system. Congener profiles in animal fat and feed samples showed a high level of consistency and pinpointed the likely source of fuel contamination to be a highly chlorinated commercial PCB mixture. To estimate additional exposure to dioxins and PCBs due to the contamination of pig and cattle herds, collection and a systematic review of all data associated with the contamination incident was conducted. A model was devised that took into account the proportion of contaminated product reaching the final consumer during the 90 day contamination incident window. For a 90 day period, the total additional exposure to Total TEQ (PCDD/F &DL-PCB) WHO (2005) amounted to 407 pg/kg bw/90d at the 95th percentile and 1911 pg/kg bw/90d at the 99th percentile. Exposure estimates derived for both dioxins and PCBs showed that the Body Burden of the general population remained largely unaffected by the contamination incident and approximately 10 % of the adult population in Ireland was exposed to elevated levels of dioxins and PCBs. Whilst people in this 10 % cohort experienced quite a significant additional load to the existing body burden, the estimated exposure values do not indicate approximation of body burdens associated with adverse health effects, based on current knowledge. The exposure period was also limited in time to approximately 3 months, following the FSAI recall of contaminated meat immediately on detection of the contamination. A follow up breast milk study on Irish first time mothers conducted in 2009/2010 did not show any increase in concentrations compared to the study conducted in 2002. The latter supports the conclusion that the majority of the Irish adult population was not affected by the contamination incident.