Abstract: Ionospheric refraction can be expressed by the Total Electron Content (TVEC).In this paper the relationship between ionosphere refraction and dual-frequency GPS observables (code ranges/carrier phases) is derived based on Fermat's theorem.Two ionosphere models are chosen,i.e.Klobuchar model and Georgiadou model. The minimum requirements for solving the model parameters for both models are related to the multiplication of the number of GPS observation sites,the number of observed satellites and the number of epochs for the observation.For Klobuchar model,the multiplication should be smaller than 8,and for Georgiadou model,the multiplication should be smaller than (1+m)(1+n),where m and n are the degree and the order respectivelly. The determination of ionosphere model parameters is derived in detail.Theoretically,it can be proven that:1) for Klobuchar model,GPS data is additionally constrained,i.e.,the data set should be scanned before being put in the estimator,while,for Georgiadou model there is no any constraint to the GPS data;2) to solve the Klobuchar model parameters,the approximate value of parameters should be within certain accuracy.Otherwise,the estimation process is not convergent.On the other hand,there is no such stringent requirement for Georgiadou model parameter estimation;3) GPS data of both multi-stations and single station can be used in both models.On the other hand,many data sets,such as multi-epoch data,single epoch data,single satellite data,multi-satellite data,etc.can fit both models. Test results have shown that the standard deviation of about 1 m can be obtained by using dual-frequency code ranges.