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Physical phenomena such as temperature, humidity, and wind velocity often exhibit both spatial and temporal correlation. We consider the problem of tracking the extremum value of a spatio-temporally correlated field using a wireless sensor network. Determining the extremum at the fusion center after making all sensor nodes transmitting their measurements is not energy-efficient because the spatio-temporal correlation of the field is not exploited. We present an optimal centralized algorithm that utilizes the aforementioned correlation to not only minimize the number of transmitting sensors but also ensure low tracking error with respect to the actual extremum. We use recent order statistics bounds in the formulation of the cost function. Since the centralized algorithm has high time complexity, we propose a suboptimal distributed algorithm based on a modified cost function. Our simulations indicate that a small fraction of sensors is often sufficient to track the extremum, and that the centralized algorithm can achieve about 70% energy savings with almost perfect tracking. Furthermore, the performance of the distributed algorithm is comparable to that of the centralized algorithm with up to 25% more energy expenditure.