One of the authors (JAQ) has recently introduced a method of modelling random materials using excursion sets of Gaussian random fields. This method uses convex quadratic programming to find the optimal admissible field autocorrelation function, providing both theoretical and computational advantages over other techniques such as simulated annealing. In this paper, we discuss the application of this algorithm to model various aerogel systems given small-angle neutron scattering data. We also present new results concerning the robustness of this method.