Uncertainty analysis

Biologically based markers (biomarkers) are currently used to provide information on exposure, health effects, and individual susceptibility to chemical and radiological wastes. However, the development and validation of biomarkers are expensive and time consuming. To determine whether biomarker development and use offer potential improvements to risk models based on predictive relationships or assumed values, we explore the use of uncertainty analysis applied to exposure models for dietary methyl mercury intake. We compare exposure estimates based on self-reported fish intake and measured fish mercury concentrations with biomarker-based exposure estimates (i.e., hair or blood mercury concentrations) using a published data set covering 1 month of exposure. Such a comparison of exposure model predictions allowed estimation of bias and random error associated with each exposure model. From these analyses, both bias and random error were found to be important components of uncertainty regarding biomarker-based exposure estimates, while the diary-based exposure estimate was susceptible to bias. Application of the proposed methods to a simple case study demonstrates their utility in estimating the contribution of population variability and measurement error in specific applications of biomarkers to environmental exposure and risk assessment. Such analyses can guide risk analysts and managers in the appropriate validation, use, and interpretation of exposure biomarker information.

Simulation models are routinely used as a powerful tool for reservoir management. The underlying static models are the result of integrated efforts that usually includes the latest geophysical, geological and petrophysical measurements and interpretations. As such, these models carry an inherent degree of uncertainty. Typical uncertainty analysis techniques require many realizations and runs of the reservoir simulation model. In this day and age, as reservoir models are getting larger and more complicated, making hundreds or sometimes thousands of simulation runs can put considerable strain on the resources of an asset team, and most of the times are simply impractical. Analysis of these uncertainties and their effects on well performance using a new and efficient technique is the subject of this paper. The analysis has been performed on a giant oil field in the Middle East using a surrogate reservoir model.

For very long the deterministic view of Nature prevailed in Physics.  Starting from the Illuminism there was the believe that Mechanics was completely described by Newton laws.  If the initial conditions of a mechanical systems were known, then its state could be completely determined in the future as well as in the past.This was precisely the viewpoint of Lagrange and it ruled from the 18th century to around the middle of the 20th century.

However this panglossian view of Nature was disputed, especially with the advent of Thermodynamics that challenged the believe in reversibility and started the study of complex random systems.  Also the discovery of simple systems with chaotic behaviour, that despite being deterministic, showed a very complex behaviour, showed that mechanical problems are far from simple.  The Quantum Theory, on the other hand, showed that the processes in Nature are not deterministic, but stochastic. This change of paradigm is not yet accepted as is well described in the book of Thomas Kuhn, that discusses sociological aspects of sciences.

The main ideas of probability were slow to develop. Perhaps because chance events were interpreted as the wish of the gods and hence were not believed to be random. The first theoretical works on probability were connected with games of chance and since the set of possibilities are finite the studies are combinatorial,  but the big breakthrough is when the continuous case is studied.

Stochastic modelling is, of course, harder than deterministic modelling and the implementation of the model is more costly. Let us see this in a simple example.  In the simplest deterministic continuous model the parameters of the model are constants. The equivalent of this in the stochastic case, is to transform the constants in random variables. For each realisation of the random variable its value is also a constant, but the constant may change with the realisation following a certain distribution of probability. That is, the most simple object of a stochastic model, the random variables, are functions, hence objects of infinite dimension. Also, in a coherent deterministic model  one expects a solution if one fixes suitable conditions. In a stochastic model this is not the case. For each realisation of the stochastic data the parameters are chosen and a deterministic problem is solved.  Then, with the set of solutions obtained, statistics are made and the main result of the problem
is the distribution of probability of the results. The important is to know among the several possible solutions how they  are distributed.  The values obtained from a single realisation are of no importance, the distribution of the values is the important result.  The understanding of this fact was very slow to come for the manufacturing  industry, for example,  and only after long year of application of quality control in the manufacturing came  the realisation that they dealt with a process stochastic in nature.  Since them stochastic modelling is essential in Engineering.  Today Reliability represents a new way of design and it takes into account the inevitable uncertainties of the processes.

The solution of a stochastic model can be decomposed in three steps: generation of the stochastic data following their distribution of probability, solution of a deterministic problem for each sample generated, and finally computation of statistics with the results (for example, to construct a histogram) until they show a certain persistence (do not change accordingly an error criterium). Histograms represent the approximation of the solution of a stochastic problem. When one wants an approximation of the solution of an stochastic problem, histogram is the best approximation. If they are hard to find, or costly to compute, one make do with mean and dispersion, or a certain number of moments. In some situation, as the ones described in Chapter 8, one computes only a certain probability since moments are too hard to compute.

This book intents to show the main ideas of Stochastic Modelling and Uncertainty Quantification and the main tool used is Functional Analysis, more specifically, Hilbert Spaces. With this tool very complex ideas, as Conditional Probabilities, can be developed in a systematic way.

In Chapter 1 are discussed the main ideas of Probability Theory in a hilbertian context. This chapter is designed as a reference, but the main concepts of random variables  and random processes are developed. Chapter 2 discusses how to construct a stochastic model using the Maximum Entropy Principal.  When the data is only independent random variables  the generation is simple. But in the case one has dependent random variables or a random process the generation is harder and one needs more tools. The main ideas of generation is discussed here. Chapter 3 discusses  the problem of approximation of a random variable. The following chapters deals with applications. Chapter 4 discusses linear systems with uncertainties; chapter 5 discusses nonlinear systems with uncertainties; chapter 6 deals with differential equations and the next two chapters, 7 and 8, deals with optimisation with uncertainties.

Description
Uncertainty Quantification (UQ) is a relatively new research area which describes the methods and approaches used to supply quantitative descriptions of the effects of uncertainty, variability and errors in simulation problems and models. It is rapidly becoming a field of increasing importance, with many real-world applications within statistics, mathematics, probability and engineering, but also within the natural sciences. Literature on the topic has up until now been largely based on polynomial chaos, which raises difficulties when considering different types of approximation and does not lead to a unified presentation of the methods. Moreover, this description does not consider either deterministic problems or infinite dimensional ones.
This book gives a unified, practical and comprehensive presentation of the main techniques used for the characterization of the effect of uncertainty on numerical models and on their exploitation in numerical problems. In particular, applications to linear and nonlinear systems of equations, differential equations, optimization and reliability are presented. Applications of stochastic methods to deal with deterministic numerical problems are also discussed.
Matlab® illustrates the implementation of these methods and makes the book suitable as a textbook and for self-study.
Contents
1. Elements of Probability Theory and Stochastic Processes. 2. Maximum Entropy and Information.
3. Representation of Random Variables.
4. Linear Algebraic Equations Under Uncertainty.
5. Nonlinear Algebraic Equations Involving Random Parameters. 6. Differential Equations Under Uncertainty.
7. Optimization Under Uncertainty.
8. Reliability-Based Optimization

ISBN: 978178540058

The paper describes a study concerning point-by-point sampling of complex surfaces using tactile CMMs. A four factor, two level completely randomized factorial experiment was carried out, involving measurements on a complex surface configuration item comprising a sphere, a cylinder and a cone, combined in a single assembly. An investigation into the source and effects of different uncertainty contributors during this complex surface measurement was carried out. The factors involved were machine, probe, operator, and procedure dependent. The results obtained from the experiments were analysed in terms of deviations from CAD model, reproducibility, and measurement uncertainty. Statistical analysis and uncertainty budget are presented.

A systematic assessment, based on an extensive literature review, of the impact of gaps and uncertainties on the results of quantitative risk assessments (QRAs) for CO 2 pipelines is presented. Sources of uncertainties that have been assessed are: failure rates, pipeline pressure, temperature, section length, diameter, orifice size, type and direction of release, meteorological conditions, jet diameter, vapour mass fraction in the release and the dose-effect relationship for CO 2 . A sensitivity analysis with these parameters is performed using release, dispersion and impact models. The results show that the knowledge gaps and uncertainties have a large effect on the accuracy of the assessed risks of CO 2 pipelines. In this study it is found that the individual risk contour can vary between 0 and 204 m from the pipeline depending on assumptions made. In existing studies this range is found to be between <1 m and 7.2 km. Mitigating the relevant risks is part of current practice, making them controllable. It is concluded that QRA for CO 2 pipelines can be improved by validation of release and dispersion models for high-pressure CO 2 releases, definition and adoption of a universal dose-effect relationship and development of a good practice guide for QRAs for CO 2 pipelines.

A review of existing analysis codes for nuclear fuel cycle systems was performed to determine if any existing codes meet technical and functional requirements defined for a U.S. national program supporting the global and domestic assessment, development, and deployment of nuclear energy systems. The program would be implemented using an interconnected architecture of different codes ranging from the fuel cycle analysis code, which is the subject of the review, to fundamental physical and mechanistic codes. Four main functions are defined for the code. Function 1 is the ability to characterize and deploy individual fuel cycle facilities and reactors in a simulation while discretely tracking material movements. Function 2 is the capability to perform an uncertainty analysis for each element of the fuel cycle and an aggregate uncertainty analysis. Function 3 is the inclusion of an optimization engine able to optimize simultaneously across multiple objective functions. Function 4 is open and accessible code software and documentation to aid in collaboration between multiple entities and to facilitate software updates.

When conducting sensitivity and uncertainty analysis, most of the global sensitivity techniques assume parameter independence. However, it is common that the parameters are correlated with each other. For models with correlated inputs, we propose that the contribution of uncertainty to model output by an individual parameter be divided into two parts: the correlated contribution (by the correlated variations, i.e. variations of a parameter which are correlated with other parameters) and the uncorrelated contribution (by the uncorrelated variations, i.e. the unique variations of a parameter which cannot be explained by any other parameters). So far, only a few studies have been conducted to obtain the sensitivity index for a model with correlated input. But these studies do not distinguish between the correlated and uncorrelated contribution of a parameter. In this study, we propose a regression-based method to quantitatively decompose the total uncertainty in model output into partial variances contributed by the correlated variations and partial variances contributed by the uncorrelated variations. The proposed regression-based method is then applied in three test cases. Results show that the regression-based method can successfully measure the uncertainty contribution in the case where the relationship between response and parameters is approximately linear. r

Life Cycle Assessment is a tool to assess the environmental impacts and resources used throughout a product's life cycle, i.e., from raw material acquisition, via production and use phases, to waste management. The methodological development in LCA has been strong, and LCA is broadly applied in practice. The aim of this paper is to provide a review of recent developments of LCA methods. The focus is on some areas where there has been an intense methodological development during the last years. We also highlight some of the emerging issues. In relation to the Goal and Scope definition we especially discuss the distinction between attributional and consequential LCA. For the Inventory Analysis, this distinction is relevant when discussing system boundaries, data collection, and allocation. Also highlighted are developments concerning databases and Input-Output and hybrid LCA. In the sections on Life Cycle Impact Assessment we discuss the characteristics of the modelling as well as some recent developments for specific impact categories and weighting. In relation to the Interpretation the focus is on uncertainty analysis. Finally, we discuss recent developments in relation to some of the strengths and weaknesses of LCA.