For estimating parameters of discrete choice models, observations corresponding to the models are required. In the context of route choice models, we need the information of paths, which are sequences of links and connect between the origin-destination pairs. Passive monitoring with Global Positioning System (GPS) is more and more used to observe trip data, because it contributes to facilitating to observe trip data automatically. However, data from monitoring with GPS is not consistent, in formats, with network and it has the heteroscedasticity of measurement errors dependently on devices and locations. These errors cause the biased observation of route choices, and as the result, the parameter estimation results of route choice models ca be biased. In this study, we propose a sequential link measurement method, which is a bayesian approach and incorporates a Markovian route choice model as the prior. It allows one to infer links based on both measurements and behavioral mechanisms, and at the same time, to estimate the variance of GPS measurement error on each link. Moreover, we propose a structural estimation method for a route choice model to remove biases regarding the initial parameter settings of the prior. The performances of these methods are examined through a numerical example and a case study of applying in a real pedestrian network.