Abstract: The research of the ionospheric delay has experienced rapid progress from error elimination, model correction to the terrestrial space environment monitoring, since the last decade. The basic principle and method for the ionospheric delay models universally used are summarized, and almost all the existing models using the first-class-operator-function (FCOF) are mathematically unified. By emphasizing mathematical representation of the ionospheric VTEC and its approaching methods, series of expansion on the methods for establishing the ionospheric delay model, also the potential problems using the newly presented method are mathematically discussed. Finally ,a simple single station model proves the feasibility of the theory.