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In this paper, we present a silent positioning scheme termed UPS for underwater acoustic sensor networks. UPS relies on the time difference of arrivals locally measured at a sensor to detect range differences from the sensor to four anchor nodes. These range differences are averaged over multiple beacon intervals before they are combined to estimate the 3-D sensor location through trilateration. UPS requires no time synchronization and provides location privacy at underwater vehicles/sensors whose locations need to be determined. To study the performance of UPS, we model the underwater acoustic channel as a modified ultrawideband Saleh-Valenzuela model: The arrival of each path cluster and the paths within each cluster follow double Poisson distributions, and the multipath channel gain follows a Rician distribution. Based on this channel model, we perform both theoretical analysis and simulation study on the position error of UPS under acoustic fading channels. The obtained results indicate that UPS is an effective scheme for underwater vehicle/sensor self-positioning.