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The main philosophy underlying this research is that a model should constitute a representation of both what we know and what we do not know about the structure and behavior of a system. In other words it should summarize, as far as possible, both our degree of certainty and degree of uncertainty, so that it facilitates statements about prediction uncertainty arising from model structural uncertainty. Based on this philosophy, the following issues were explored in the dissertation: Identification of a hydrologic system model based on assumption about perceptual and conceptual models structure only, without strong additional assumptions about its mathematical structure Development of a novel data assimilation method for extraction of mathematical relationships between modeled variables using a Bayesian probabilistic framework as an alternative to up-scaling of governing equations Evaluation of the uncertainty in predicted system response arising from three uncertainty types: o uncertainty caused by initial conditions, o uncertainty caused by inputs, o uncertainty caused by mathematical structure Merging of theory and data to identify a system as an alternative to parameter calibration and state-updating approaches Possibility of correcting existing models and including descriptions of uncertainty about their mapping relationships using the proposed method Investigation of a simple hydrological conceptual mass balance model with two-dimensional input, one-dimensional state and two-dimensional output at watershed scale and different temporal scales using the method

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PET (positron emission tomography) scans are still in the experimental phase, as one of the newest breast cancer diagnostic techniques. It is becoming the new standard in neurology, oncology and cardiology. PET, like other nuclear medicine diagnostic and treatment techniques, involves the use of radiation. Because of the negative impact of radioactivity to our bodies the radiation doses in PET should be small. The existing computing algorithms for calculating PET images can be divided into two broad categories: analytical and iterative methods. In the analytical approach the relation between the picture and its projections is expressed by a set of integral equations which are then solved analytically. The Fourier backprojection (FBP) algorithm is a numerical approximation of this analytical solution. Iterative approaches use deterministic (ART = Algebraic Reconstructed Technique) or stochastic (EM = Expectation Maximization) algorithms. My proposed kernel density estimation (KDE) algorithm also falls also into the category of iterative methods. However, in this approach each coincidence event is considered individually. The estimate location of the annihilation event that caused each coincidence event is based on the previously assigned location of events processed earlier. To accomplish this, we construct a probability distribution along each coincidence line. This is generated from previous annihilation points by density estimation. It is shown that this density estimation approach to PET can reconstruct an image of an existing tumor using significantly less data than the standard CT algorithms, such as FBP. Therefore, it might be very promising technique allowing reduced radiation dose for patients, while retaining or improving image quality.

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