A polynomial chaos based minimum variance method is introduced to solve inverse problems. Two different set of quadrature points, Conjugate Unscented Transform and Gauss Legendre quadrature points, are being used to perform the estimation process. The main concentration of this paper is to show the efficiency of Conjugate Unscented Transform points versus Gauss Legendre quadrature scheme. For validation purposes, an advection-diffusion simulation code, SCIPUFF (Second-order Closure Integrated PUFF) is being used as a modeling testbed to study the effect of both these quadrature schemes on solution of an inverse problem. Simulation results show efficiency of Conjugate Unscented Transform versus Gauss-Legendre quadratures.