http://hdl.handle.net/10379/1944 2017-10-29T23:53:57Z http://hdl.handle.net/10379/1961 Three-dimensional Magnetotelluric Inversion Avdeeva, Anna The main objective of my PhD research is developing a better understanding of the mathematics, physics and numerical aspects of 3D MT inversion. This should lead to a working software that allows inversion of MT datasets in three dimensions. It is well known that reliable inversion includes three important and well-developed constituents: forward modelling, optimization and regularization methods. Fortunately, I am lucky to have full access to an up-to-date forward modelling code, x3d. In addition there is a large number of optimization and regularization techniques. Having all those components in place greatly facilitate my task. It should be understood however that developing such complicated software as 3D inversion is not by any means a simple mix of the three constituents mentioned above. Even if all three mentioned constituents are ready to be used, the development of reliable inversion software still requires many years of hard work, as is shown by experience of many other inversion software developers. So why is it not just a simple mix? First of all, all three main constituents should be adjusted to our specific case - 3D MT data. As an example, it is unclear a priori which optimization method better suits 3D MT inversion. The theory and algorithm of our chosen optimization method - the limited memory quasi-Newton (QN) method - is presented in Section 2.2. To investigate what specific parameters of QN optimization are optimal for 3D MT inversion we, as a start, studied thoroughly the 1D MT case (Chapter 3). Another important point is that optimization methods require the calculation of derivatives (such as gradients, Jacobians, Hessians) of some penalty function. In my work this important question is considered in Sections 3.1.1 and 4.1.1. Even more simple than this, it is unclear a priori what specific form of the penalty function is effective for 3D MT inversion. We introduce the chosen penalty function for the 3D case in Section 4.1. Finally, of course, the whole approach has to be verified on synthetic examples. We performed this verification for several representative MT models and present the results in Section 4.2 and Chapter 5. In summary we have developed a working 3D MT inversion approach and verified it on synthetic test examples. Of course, as usual some unresolved problem remains. This problem is related to the physics of the 3D MT inversion itself, rather than to bugs in the numerical software. This problem manifests itself in the synthetic experiment presented in Section 4.2.4. There we encountered a problem that the recovered anomaly image shows erratic behaviour in conductivity in the upper part of the model. For the moment the reason for this behaviour is clear - sizes of inversion cells are much smaller than an average distance between MT sites and at the same time the gradient is much higher for cells located just below the sites. This problem is not new and some remedies has been proposed. For example, usage of model covariance matrix to address this problem is reported in Siripunvaraporn et al. (2005) or usage of preconditioning of Newton system by an approximation to the inverted Hessian matrix (Newman, personal communication). This is discussed in Chapter 5 and is subject of my ongoing research. Regardless of this unresolved issue, our solution can be used to recover the true resistivity of the lower parts of realistic Earth models. Proper recovery of the true resistivity for the upper part of the models depends on many factors, such as the geometry and resistivity of the structures inside the Earth, coverage of the region by MT sites and many others. 2008-01-01T00:00:00Z http://hdl.handle.net/10379/1862 Features and purposes of mathematical proofs in the view of novice students: observations from proof validation and evaluation performances Pfeiffer, Kirsten This thesis describes a comprehensive exploratory study of the approaches taken by novice students to the validation and evaluation of mathematical proofs. A theoretical framework based on sociocultural learning theories was considered suitable as a basis to develop a new terminology and schema for observations and interpretations of proof validations and evaluations. In this theoretical framework learning is seen as accessing and participating in the practice of an expert community. The theoretical considerations of this thesis build on Hemmi's conception of proofs as artifacts in the community of practice. Philosophical theories about the evaluation of artifacts are specialized to the case of mathematical proof. A result of these considerations is a schema that extends Hemmi's model of proofs as artifacts, and provides both a theoretical basis and an analytic tool for consideration of the practice of proof evaluation and for interpretation of specific instances of proof evaluation. The study is based on a series of tests and interviews with first year honours mathematics students at the National University of Ireland, Galway. The students were asked to evaluate and criticize numerous proposed (correct and incorrect) proofs of mathematical statements. The participants' written comments and the interview discussions on different and partly incorrect proofs give insights into their criteria for valuing a mathematical proof, their habits when performing proof validations and evaluations and their knowledge about features and purposes of mathematical proofs. This thesis describes the theoretical framework as well as the design, observations and findings of the written and oral exercises. It also includes a discussion about advantages and shortcomings of the schema that has been developed and used for the interpretation of evaluations of mathematical proofs and a discussion about possibilities for further research. 2010-12-22T00:00:00Z