In this paper, we propose a model-mediated tele-operation (MMT) system using an octree-based model (OBM) to spatially map the environment impedance for the emerging use cases in Tactile Internet. Different from the existing just-noticeable-difference (JND) based MMT, our method avoids continuous transmission of environment impedance. Moreover, it allows the local model to generate accurate force feedback and reduces the number of model updates. Furthermore, the OBM can be deployed with or without previous knowledge of the environment. An online estimation of the OBM is proposed using a JND and a rate-of-change threshold. An offline estimation method is also proposed when the geometry and impedance parameters of the remote environment are known. In addition, a point cloud-based force rendering algorithm is tailored to use the OBM, thereby allowing the generating of force feedback for complex environments. An experiment without human-in-the-loop was conducted, showing that for an online estimated OBM, the accuracy of the force feedback was improved by up to 44 percent while using less than half the number of model updates when compared to JND-based MMT. Another experiment with a human operator interacting with a virtual environment showed that using an offline estimated OBM improves the accuracy of the force feedback and is reliable against packet loss and short temporal breakdown of the communication link.