Improving the Numerical Stability of an In-Situ Gamma Ray Spectroscopy Method using Multiple Measurements for the Determination of Activity Concentration as a Function of Depth
S.C. Dewey and K.J. Kearfott (University of Michigan)
In-situ gamma ray spectroscopy is used to assess radioactive contamination. The instrumentation response is dependent upon the source material distribution. Current methods for determining detector response make assumptions about the source distribution, which can lead to erroneous results. In the proposed method, which involves determining the source distribution as a function of depth, multiple measurements are made with varying geometries sampling the media with different efficiencies at different depths in the object. These efficiencies are pre-calculated through a combination of analytical and Monte Carlo calculations. The media is subdivided into a number of discrete layers, each assumed to have uniform activity. The measurement series produces a system of linear equations, which can be solved using standard mathematical methods to determine the activity in each discrete layer. This results in an estimate of the vertical activity distribution. The primary challenge is that due to the exponential nature of the response of the detector with increasing depth the equations are poorly conditioned. Small errors in input variables can lead to large errors in calculated solutions. Increasing the number of discrete layers used increases the condition number of the system. Current efforts are focused on improving the conditioning of the system to a point such that the media can be sub-divided into four or more discrete layers and limiting solution errors to levels practical for field measurements. The method has been explored using analytical solutions, Monte Carlo modeling and backward error analysis. Methods to improve the conditioning of the system have included using non-uniform thickness layers when the system is discretized, optimizing sampling geometries, and forcing conditions onto the solutions such as requiring the solution to be continuous at the layer boundaries. Efforts to date have improved the conditioning of the system by a factor of ten or more.