The goal of this paper is to develop an efficient and accurate computation scheme based on mixed-order curved moment method for computing scattering from complex targets. To this end, the numerical performances of different kinds of higher-order basis functions are presented and compared numerically. The formulae in the improved Duffy method for higher-order curved moment method are derived and given explicitly. An efficient and accurate computation scheme is presented and implemented on the basis of the optimized triangular hierarchical higher-order basis function and the improved Duffy method. Numerical experiments are performed to show accuracy and efficiency of the presented computation scheme for complex targets. Copyright © 2012 John Wiley & Sons, Ltd.