The Application of Euclidean Space for the Assessment of Randomly Distributed Radioactive Material Utilizing Multiple Detector Measurements

If a detector response is proportional to a fixed distribution of radioactive material, the measured radioactivity can be assessed from a single detector response. Unknown or random spatial distributions of activity, however, results in an uncertainty associated with the assessment. This spatial assessment uncertainty can be reduced by utilizing multiple detector measurements. Conventional techniques such as averaging or summing of multiple detector responses combined possibly with rotation and segmentation of the source container avoid the complexity of directly interpreting a multiple detector measurement and result in a loss of information by ignoring the additional dimensions of the problem. The vector representation of a multiple detector response allows for the maximum utilization of a limited amount of information. By considering the mathematical relationships between the detectors, formulations have been developed in this dissertation which allow for the assessment of both the amount and associated spatial uncertainty of the radioactive material measured from N detector responses. The analysis of the proportional relationships between the detector responses results in the following: the calculation of the possible activities from a detector response of unknown activity and distribution, the calculation of the largest possible uncertainty from N measurements for a constrained distribution of activity, the property that local optimization over an unconstrained distribution is a global optimization. The analysis of the first and second derivatives of a response sets results in a technique for reducing both the number of optimization steps and the number of independent variables necessary for the calculation of the largest uncertainty. Increasing the number of symmetric detectors surrounding a container results in the largest uncertainty approaching some asymptotic limit. A relationship between an infinite number of symmetric measurements and a definite integral is developed allowing the solution to the asymptotic limit to be obtained by analytical means. The optimization of a detector system by minimizing the largest uncertainty with respect to detector position is dependent on the amount of activity actually present. Utilizing the above mathematical relationships, a detector system which optimizes the (N + 1)th detector based on the results of the interpretation of the previous N detectors is developed.