**Uncertainty Quantification Working Group**

December 11, 11:30 AM, CNLS Conf. Room, TA-3, Bldg. 1690

## Uncertainty quantification for Homeland Security applications

Jeffrey Favorite, X-5 (slides)

### Abstract

Given a set of simulated gamma-ray leakage measurements and the unknown object model that best fit these measurements, standard tools of uncertainty quantification in data-fitting were applied to develop an estimate of the uncertainty of the dimensions in the model based on the statistical uncertainty of the measurements. At least two issues were discovered that need further exploration. One is that in standard data analysis, there are usually many more data points than there are model parameters to be fit, so that calculating the inverse of the Hessian (or curvature) matrix to obtain the covariance matrix is straightforward; in Homeland Security applications, however, there are more unknowns than data points, and it is not obvious how the correct covariance matrix should be obtained if the Hessian matrix is singular. Thus, in most problems of interest, it is still unclear how to relate the uncertainty in measurements to the uncertainty in model dimensions. A related issue is how the uncertainty in model dimensions should be used to obtain the uncertainty in other model quantities, such as material masses. The standard procedure is to use the variances and covariances of the dimensions in the simple error propagation formula; an alternative is to randomly sample the model dimensions from Gaussian distributions whose widths are related to the uncertainty in the dimensions, compute the associated masses, and then fit the mass distributions to Gaussians whose widths are related to the uncertainty in the masses. Both of these procedures require knowledge of the covariance matrix for the model dimensions.

Thus, a program to extend the standard methods of uncertainty quantification to develop an estimate of the covariance matrix in the case of a singular Hessian matrix would be of interest for Homeland Security applications.