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Data collection by satellites during and after a natural disaster is of great significance. In this work, a reconfigurable satellite constellation is designed for disaster monitoring, and satellites in the constellation are made to fly directly overhead of the disaster site through orbital transfer. By analyzing the space geometry relations between satellite orbit and an arbitrary disaster site, a mathematical model for orbital transfer and overhead monitoring is established. Due to the unpredictability of disasters, target sites evenly spaced on the Earth are considered as all possible disaster scenarios, and the optimal reconfigurable constellation is designed with the intention to minimize total velocity increment, maximum and mean reconfiguration time, and standard deviation of reconfiguration times for all target sites. To deal with this multiobjective optimization, a physical programming method together with a genetic algorithm is employed. Numerical results are obtained through the optimization, and different observation modes of the reconfigurable constellation are analyzed by a specific case. Superiority of our design is demonstrated by comparing with the existing literature, and excellent observation performance of the reconfigurable constellation is demonstrated.