To mitigate the influence of strong unknown disturbances and instrument faults on observations in practical applications, and to alleviate the degradation caused by random and unmodeled interferences on the system, so as to improve the state estimation accuracy of the system in non-Gaussian noise environment and the robustness of the filter, a Gaussian-heavy-tailed switching distribution based robust Kalman filter (GHTSRKF) is proposed. Firstly, the noises are modeled as a GHTS（Gaussian-heavy-tailed switching）distribution by adaptively learning the switching probability between the Gaussian distribution and the newly designed heavy-tailed distribution. The designed GHTS distribution can model non-stationary heavy tail noise by adjusting the switching probability between the Gaussian distribution and the new heavy-tailed distribution online. The Gaussian distribution with a virtual covariance is used to deal with Gaussian noise with inaccurate covariance matrix. Secondly, two auxiliary parameters following the category distribution and the Bernoulli distribution are introduced to express the GHTS distribution as a hierarchical Gaussian form. Furthermore, the GHTSRKF is derived by utilizing the variational Bayesian method. Finally, a simulation scenario is used to compare and verify several different robust Kalman filters (RKFs). The results show that the accuracy of the proposed GHTSRKF algorithm is insensitive to the selection of initial state and exhibits higher estimation accuracy compared to other RKFs. Its root mean square errors（RMSEs）are closest to those of KF with true noise covariances（KFTNC）with accurate noise information. Compared with existing filters, GHTSRKF has better estimation performance when the system and measurement noise are unknown time-varying Gaussian noise, thus verifying the effectiveness of GHTSRKF.