Developments in gamma-ray spectrometry: systems, software, and methods-I. 5. Nuclear Spectral Analysis with Nonlinear Robust Fitting Techniques

A new approach to nuclear spectral analysis based on nonlinear robust fitting techniques has been recently developed into a package suitable for public use. The methodology behind this approach was originally made available to the public as the RobFit command-line code, but it was extremely slow and difficult to use. Recent advances in microprocessor power and the development of a graphical user interface to make its use more intuitive have made this approach, which is quite computationally intensive, feasible for more routine applications. A brief description of some of the fundamental differences in the approach used by RobFit from the more common methods of nuclear spectral analysis involving local peak searches is presented here. Popular nuclear spectral analysis applications generally perform a peak search at their heart. The continuum in the neighborhood of each peak is estimated from local data and is subtracted from the data to yield the area and the energy of the peak. These are matched to a user-selected library of radionuclides containing the energies and areas of the most significant peaks, after accounting for the effects of detector efficiency and attenuation. With these codes, the energy-to-channel calibration, the peak width as a function of energy (or 'resolution calibration'), the detector intrinsic efficiency, and attenuation effects must usually be predetermined and provided as static input for the analysis. Most of these codes focus on regions of interest that represent many small pieces of the sample spectrum. In contrast, the RobFit approach works with an entire continuous spectrum to simultaneously determine the coefficients of all of the user-selected free variables that yield the best fit to the data. Peak searches are generally used only in interim steps to help suggest new radionuclides to include in the search library. Rather than first concentrate on the location of peaks, RobFit first concentrates on the determination of the continuum, treating it as a single, continuous, smooth curve spanning the entire usable spectrum. The continuum is composed of a number of cubic Splines with optimized values for the knots and coefficients. Peaks are avoided during the continuum fit by using robustness, a method originally developed to diminish the effect of spurious outliers. After an initial estimation of the continuum, the residual data above the continuum are analyzed, providing information for a more refined estimate of the continuum, if necessary. Whether fitting the continuum or the residuals above the continuum, the coefficients of each free variable in the analysis are optimized using the method of least squares, in which chi-square is minimized. In the analysis of the residuals, one coefficient is provided for the activity of each of the radionuclides in the library. An expected multiple-peak spectral curve is constructed from all the lines listed in the tables for each suspected radionuclide in the sample spectrum, with modifications to account for the detection system's peak shape, energy calibration, resolution calibration, and intrinsic efficiency, and for any attenuation effects. If the free variables also include the coefficients of the energy calibration, resolution calibration, intrinsic efficiency, and attenuation effects, then these additional coefficients can be optimized simultaneously with the radionuclide vector, provided that there is sufficient structure in the sample spectrum to prevent the optimization from becoming underdetermined. In this way, the sample spectrum data itself is used to dynamically improve and refine the a priori calibrations. As a result, not only does it become possible to achieve much greater sensitivity and more accurate results, but it also becomes possible to perform an analysis without any a priori calibration or characterization measurements except those that can be inferred from the data itself, with only the roughest initial knowledge of the energy calibration and of the sources in the spectrum. (To quantify absolute activities, of course, detector sensitivity, time, and distance must be known). However, with the inclusion of these additional parameters, the values of all the coefficients become strongly dependent upon each other value in a highly nonlinear way, and the solution becomes much more difficult. In the continuum fit in which the knots and coefficients of the Splines are optimized, the optimization problem is highly nonlinear from the start. The greatest challenge in this approach lies in finding the true minimum of chi-square on a multidimensional surface that may contain many local minima. Also, the problems with inversion of large sparse matrices must be overcome. These problems were solved by Coldwell with the development of the RobFit code. Efficient convergence to the vector for the true minimum on the multidimensional chi-square surface is accomplished with Newton-Raphson techniques used to estimate the best Marquardt parameter to add to the diagonal elements of the inversion matrix for the next step in the search. Stability with large, sparse matrix inversion is achieved with Cholesky minimization and numerical techniques to remedy apparent singularities resulting from numerical truncation. Although it requires knowledgeable interactive operation for best results and is computationally intensive, nuclear spectral analysis with nonlinear robust fitting has been shown to be capable of exceptional sensitivity in detecting weak radionuclides in the presence of strong interference and in noisy spectra, sparse spectra, and low-resolution spectra. This increased sensitivity is due to the simultaneous optimization of all the data for all the free variables of the analysis and the iterative construction of a well-determined continuum spanning the entire spectrum. (authors)