A novel distributed sensor scheduling method for large-scale sensor networks observing space-time continuous physical phenomena is introduced. In a first step, the model of the distributed phenomenon is spatially and temporally decomposed leading to a linear probabilistic finite-dimensional model. Based on this representation, the information gain of sensor measurements is evaluated by means of the so-called covariance reduction function. For this reward function, it is shown that the performance of the greedy sensor scheduling is at least half that of the optimal scheduling considering long-term effects. This finding is the key for distributed sensor scheduling, where a central processing unit or fusion center is unnecessary, and thus, scaling as well as reliability is ensured. Hence, greedy scheduling in combination with a proposed hierarchical communication scheme requires only local sensor information and communication.