The diffusion approximation to the equation of transfer (Boltzmann transport equation) is usually applied to media where scattering dominates the interactions. Diffusion approximation helps in significant savings in terms of code complexity and computational time. However, this approximation often has significant error. Error due to the inherent nature of a physics model is called model error. Information about the model error associated with the diffusion approximation is clearly desirable. An indirect measure of model error is a quantity that is related in some way to the error but not equal to the error. In general, indirect measures of error are expected to be less costly than direct measures. Perhaps the most well-known indirect measure of the diffusion model error is the variable-Eddington tensor. This tensor provides a great deal of information about the angular dependence of the angular intensity solution, but it is not always simple to interpret. We define a new indirect measure of the diffusion model error called the diffusion model error source (DME source). When this DME source is added to the diffusion equation, the transport solution for the angular-integrated intensity is obtained. In contrast to the variable-Eddington tensor, our DME source is a scalar that is conceptually easy to interpret. In addition to defining the DME source analytically, we show how to generate this source numerically relative to the Sn radiative transfer equations with linear-discontinuous spatial discretization. This numerical source is computationally tested and shown to reproduce the Sn solution for a number of problems. Our radiative transfer model solves a coupled, time dependent, multi-frequency, 1-D slab equation and material heat transfer equation. We then use diffusion approximation to solve the same problem. The difference due to this approximation can be modelled by a “diffusion source”. The diffusion source is defined as an amount of inhomogeneous source that, when added to a diffusion calculation, gives a solution for the angle-integrated intensity that is equal to the transport solution.

Usage of simulation models is an essential part in many modern engineering disci- plines. Computer models of complex physical systems can be used to expedite the design of control systems and reduce the number of physical tests. Model valida- tion tries to answer the question if the model is a good enough representation of the physical system. This thesis describes techniques to visualize multi-dimensional validation results and the search for an automated validation process. The work is focused on a simulation model of the Primary Environmental Control System of Gripen E, but can be applied on validation results from other simulation models. The result from the thesis can be divided into three major components, static validation, dynamic validation and model coverage. To present the results from the static validation different multi-dimensional visualization techniques are in- vestigated and evaluated. The visualizations are compared to each other and to properly depict the static validation status of the model, a combination of visual- izations are required. Two methods for validation of the dynamic performance of the model are examined. The first method uses the singular values of an error model estimated from the residual. We show that the singular values of the error model relay important information about the model’s quality but interpreting the result is a considerable challenge. The second method aims to automate a visual inspection procedure where interesting quantities are automatically computed. Coverage is a descriptor of how much of the applicable operating conditions that has been validated. Two coverage metrics, volumetric coverage and nearest neigh- bour coverage, are examined and the strengths and weaknesses of these metrics are presented. The nearest neighbour coverage metric is further developed to account for validation performance, resulting in a total static validation quantity.

A probabilistic procedure was applied to the evaluation of predictions from the GLEAMS nonpoint source pollution model. Assessment of both the procedure and model was made by comparing absolute and relative predictions made with both probabilistic and deterministic procedures. Field data used came from a study of pesticide fate and transport in both no-till and conventional tillage plots in a Coastal plain soil. Variables examined were: runoff, sediment yield, surface losses, mass in the root zone, and depth of center of mass for two pesticides and a tracer. Random inputs were characterized with probability distributions. Values for inputs were sampled from these distributions for 5000 model executions to create output distributions in the probabilistic procedure. Central tendency values from the probabilistic input distributions were used as inputs for the deterministic runs. Model predictions generally followed expected trends and were within observed variability. Two exceptions were systematic under-predictions of runoff and pesticide losses and under-predictions of the depth of bromide in the root zone later in the observed period. These exceptions may indicate errors in the runoff and plant uptake components of the model. Neither procedure made relative predictions correctly all the time, however subjective assessment of the model results led to consistent decisions between the two procedures. The probabilistic procedure reduced parameter uncertainty by eliminating arbitrary parameter selection from available data by utilizing the complete range of data, however, it did not eliminate uncertainty in the data itself. / Master of Science

In this thesis, we produce a rigorous and quantitative analysis of the errors introduced by finite difference schemes into strong constraint 4D-Variational (4D-Var) data assimilation. Strong constraint 4D-Var data assimilation is a method that solves a particular kind of inverse problem; given a set of observations and a numerical model for a physical system together with a priori information on the initial condition, estimate an improved initial condition for the numerical model, known as the analysis vector. This method has many forms of error affecting the accuracy of the analysis vector, and is derived under the assumption that the numerical model is perfect, when in reality this is not true. Therefore it is important to assess whether this assumption is realistic and if not, how the method should be modified to account for model error. Here we analyse how the errors introduced by finite difference schemes used as the numerical model, affect the accuracy of the analysis vector. Initially the 1D linear advection equation is considered as our physical system. All forms of error, other than those introduced by finite difference schemes, are initially removed. The error introduced by `representative schemes' is considered in terms of numerical dissipation and numerical dispersion. A spectral approach is successfully implemented to analyse the impact on the analysis vector, examining the effects on unresolvable wavenumber components and the l2-norm of the error. Subsequently, a similar also successful analysis is conducted when observation errors are re-introduced to the problem. We then explore how the results can be extended to weak constraint 4D-Var. The 2D linear advection equation is then considered as our physical system, demonstrating how the results from the 1D problem extend to 2D. The linearised shallow water equations extend the problem further, highlighting the difficulties associated with analysing a coupled system of PDEs.

Uma grande variedade de modelos é utilizada para estimar a pressão de ruptura de dutos contendo defeitos de corrosão. O presente trabalho tem como objetivo estudar a precisão dos modelos mais comuns e avaliar a pressão de ruptura de dutos submetidos à corrosão. Os modelos avaliados são: ASME B31G, ASME B31G modificado, DNV RP F101 e PCORRC. O estudo é baseado em mais de 400 resultados de ensaios de ruptura em dutos corroídos, todos coletados da literatura. A base de dados contem defeitos de corrosão reais e artificiais. Uma análise estatística foi realizada para a variável erro de modelo. Uma análise de regressão não-linear foi realizada para investigar os efeitos da variável erro de modelo, das variáveis mais relevantes, como profundidade e comprimento do defeito, e tensão de ruptura do aço. Uma análise de confiabilidade foi realizada a partir das estatísticas obtidas da variável erro de modelo, sendo estimado o índice de confiabilidade e a probabilidade de falha do duto com defeitos de corrosão, através do método iterativo de primeira ordem, denominado FORM (First Order Reliability Method). Nesta análise avaliou-se a evolução da probabilidade de falha com o aumento da profundidade do defeito, bem como foram identificadas as variáveis aleatórias mais importantes na falha do duto. O estudo pode ajudar aos operadores a eleger qual modelo utilizar em análises de risco, proporcionando mais segurança às operações dutoviárias. / A variety of models exist to estimate burst pressures of pipelines containing corrosion defects. The objective of this work is to study the accuracy of some of the most popular empirical burst pressure models. The study addresses the models: ASME B31G, ASME B31G Modified, DNV RP-F101 and PCORRC. The investigation is based on over 400 burst test results, all collected from the literature, containing both real and artificial corrosion defects. A statistical analysis is performed for assessing the accuracy of semi-empirical models by using a model error variable. A non-linear regression analysis is performed to identify the influence, on model errors, of the most relevant variables, such as defect depth and length and steels rupture tension. A reliability analysis was carried out, using model error statistics developed herein, in order to evaluate reliability index and probability of failure of pipelines containing corrosion defects, through the iterative first order reliability method, or FORM - First Order Reliability Method. The evolution of failure probabilities, with increasing defect depth, was investigated. The most relevant random variables were identified. The study can help operators choose a proper empirical model to use in their risk analysis, leading to greater safety in pipeline operations.

The subject of this dissertation is use of computer models as data analysis tools in several different geoscience settings, including integrated surface water/groundwater modeling, tephra fallout modeling, geophysical inversion, and hydrothermal groundwater modeling. The dissertation is organized into three chapters, which correspond to three individual publication manuscripts. In the first chapter, a linear framework is developed to identify and estimate the potential predictive consequences of using a simple computer model as a data analysis tool. The framework is applied to a complex integrated surface-water/groundwater numerical model with thousands of parameters. Several types of predictions are evaluated, including particle travel time and surface-water/groundwater exchange volume. The analysis suggests that model simplifications have the potential to corrupt many types of predictions. The implementation of the inversion, including how the objective function is formulated, what minimum of the objective function value is acceptable, and how expert knowledge is enforced on parameters, can greatly influence the manifestation of model simplification. Depending on the prediction, failure to specifically address each of these important issues during inversion is shown to degrade the reliability of some predictions. In some instances, inversion is shown to increase, rather than decrease, the uncertainty of a prediction, which defeats the purpose of using a model as a data analysis tool. In the second chapter, an efficient inversion and uncertainty quantification approach is applied to a computer model of volcanic tephra transport and deposition. The computer model simulates many physical processes related to tephra transport and fallout. The utility of the approach is demonstrated for two eruption events. In both cases, the importance of uncertainty quantification is highlighted by exposing the variability in the conditioning provided by the observations used for inversion. The worth of different types of tephra data to reduce parameter uncertainty is evaluated, as is the importance of different observation error models. The analyses reveal the importance using tephra granulometry data for inversion, which results in reduced uncertainty for most eruption parameters. In the third chapter, geophysical inversion is combined with hydrothermal modeling to evaluate the enthalpy of an undeveloped geothermal resource in a pull-apart basin located in southeastern Armenia. A high-dimensional gravity inversion is used to define the depth to the contact between the lower-density valley fill sediments and the higher-density surrounding host rock. The inverted basin depth distribution was used to define the hydrostratigraphy for the coupled groundwater-flow and heat-transport model that simulates the circulation of hydrothermal fluids in the system. Evaluation of several different geothermal system configurations indicates that the most likely system configuration is a low-enthalpy, liquid-dominated geothermal system.

The purpose of this study is to determine if element decimation of a 3-D anatomical model affects the results of Finite Element Analysis (FEA). FEA has been increasingly applied to the biological and medical sciences. In order for an anatomical model to successfully run in FEA, the 3-D model’s complex geometry must be simplified, resulting in a loss of anatomical detail. The process of decimation reduces the number of elements within the structure and creates a simpler approximation of the model. Using the National Library of Medicine’s Visible Human Male dataset, a virtual 3-D representation of several structures of the hip were produced. The initial highest resolution model was processed through several levels of decimation. Each of these representative anatomical models were run in COMSOL 3.5a to measure the degree of displacement. These results were compared against the original model to determine what level of error was introduced due to model simplification.

Uma grande variedade de modelos é utilizada para estimar a pressão de ruptura de dutos contendo defeitos de corrosão. O presente trabalho tem como objetivo estudar a precisão dos modelos mais comuns e avaliar a pressão de ruptura de dutos submetidos à corrosão. Os modelos avaliados são: ASME B31G, ASME B31G modificado, DNV RP F101 e PCORRC. O estudo é baseado em mais de 400 resultados de ensaios de ruptura em dutos corroídos, todos coletados da literatura. A base de dados contem defeitos de corrosão reais e artificiais. Uma análise estatística foi realizada para a variável erro de modelo. Uma análise de regressão não-linear foi realizada para investigar os efeitos da variável erro de modelo, das variáveis mais relevantes, como profundidade e comprimento do defeito, e tensão de ruptura do aço. Uma análise de confiabilidade foi realizada a partir das estatísticas obtidas da variável erro de modelo, sendo estimado o índice de confiabilidade e a probabilidade de falha do duto com defeitos de corrosão, através do método iterativo de primeira ordem, denominado FORM (First Order Reliability Method). Nesta análise avaliou-se a evolução da probabilidade de falha com o aumento da profundidade do defeito, bem como foram identificadas as variáveis aleatórias mais importantes na falha do duto. O estudo pode ajudar aos operadores a eleger qual modelo utilizar em análises de risco, proporcionando mais segurança às operações dutoviárias. / A variety of models exist to estimate burst pressures of pipelines containing corrosion defects. The objective of this work is to study the accuracy of some of the most popular empirical burst pressure models. The study addresses the models: ASME B31G, ASME B31G Modified, DNV RP-F101 and PCORRC. The investigation is based on over 400 burst test results, all collected from the literature, containing both real and artificial corrosion defects. A statistical analysis is performed for assessing the accuracy of semi-empirical models by using a model error variable. A non-linear regression analysis is performed to identify the influence, on model errors, of the most relevant variables, such as defect depth and length and steels rupture tension. A reliability analysis was carried out, using model error statistics developed herein, in order to evaluate reliability index and probability of failure of pipelines containing corrosion defects, through the iterative first order reliability method, or FORM - First Order Reliability Method. The evolution of failure probabilities, with increasing defect depth, was investigated. The most relevant random variables were identified. The study can help operators choose a proper empirical model to use in their risk analysis, leading to greater safety in pipeline operations.

The aim of this diploma thesis is to analyze the development of the total volume of debt in the Czech Republic and the analysis separately for the household sector and non-financial corporations. From economic theoretical assumptions it can be concluded that there is a correlation between the amount of loans and GDP development or between credit and economic cycle. The thesis is divided into three parts. The first part made up of chapters 1 to 4, describes the theory used further in the text. The second part, Chapter 5, describes the specific time series used in the thesis, i.e. The time series of the volume of debt for the Czech Republic, GDP and interest rates. Interest rates and the volume of debt are further broken down by maturity and also by two selected sectors. The last part, Chapter 6, focusing on co-integration analysis, ADL and error correction models, attempts to capture short-term and long-term relationships between the time series.

Methods for Direction of Arrival, DOA estimation of multiple objects based on phased arrayantenna technology have many advantages in for example electronic warfare and radarapplications. However, perfect calibration of an antenna array can seldom be achieved. Thepurpose of this report is to study different methods for DOA estimation and how calibration-/modelerrors affect the results. Possible methods for quantifying these kinds of errors using measurement data are suggested. This thesis consists of essentially five parts. The different studies have been carried out using MATLAB simulations as well as theoretical considerations, i.e., calculations. In the first study, examples of the possible performance of four DOA algorithms, MUSIC, TLS-ESPRIT, WSF, and DML are provided. Results are given both with and without applying spatial smoothing. The latter scheme is used for handling correlated, or even coherent, sources. The results show that, for the considered scenarios, MUSIC performs the most consistently well, while the performance of DML is inferior. ESPRIT is well-performing when spatial smoothing is applied and performs the best when the angles of two signals are very close. It has been observed that WSF with weighting matrices for optimal asymptotic performance as well as spatial smoothing applied doesn’t perform well. When applying model errors to the systemin the second study, the corresponding conclusions about the algorithms can be drawn. That separation distance between the angles and that higher SNR results in better estimates are also confirmed. Quantification of certain array errors is also considered using methods inspired by a scheme proposed in the context of nonlinear system identification. The results show that the DOA algorithms are very good at dealing with noise and that the attempted method works well when the model error is like the true signals, but different enough that it is not confused with a problem with more signals. The model error that results in the worst results is when it only affects some ofthe channels in the antenna array. The fourth study explores DOA estimation using extended Kalman filtering and concludes that it is a very good tracker of the angle over time for the considered scenarios. All of this is then applied to measured data, but due to either extensive model error, errors with processing the data, or both, the results are worse than expected. Simulations that try to replicate the measured data results in good angle estimation for the DOA algorithms. The Kalman filter also performs well in simulations.