In view of the disadvantages of the traditional energy supply systems, such as separate planning, separate design, independent operating mode, and the increasingly prominent nonlinear coupling between various sub-systems, the production, transmission, storage and consumption of multiple energy sources are coordinated and optimized by the integrated energy system, which improves energy and infrastructure utilization, promotes renewable energy consumption, and ensures reliability of energy supply. In this paper, the mathematical model of the electricity-gas interconnected integrated energy system and its state estimation method are studied. First, considering the nonlinearity between measurement equations and state variables, a performance simulation model is proposed. Then, the state consistency equations and constraints of the coupling nodes for multiple energy sub-systems are established, and constraints are relaxed into the objective function to decouple the integrated energy system. Finally, a distributed state estimation framework is formed by combining the synchronous alternating direction multiplier method to achieve an efficient estimation of the state of the integrated energy system. A simulation model of an electricity-gas interconnected integrated energy system verifies the efficiency and accuracy of the state estimation method proposed in this paper. The results show that the average relative errors of voltage amplitude and node pressure estimated by the proposed distributed state estimation method are only 0.0132% and 0.0864%, much lower than the estimation error by using the Lagrangian relaxation method. Besides, compared with the centralized estimation method, the proposed distributed method saves 5.42 s of computation time. The proposed method is more accurate and efficient in energy allocation and utilization.