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In this paper we propose a progressive receiver for orthogonal-frequency-division-multiplexing (OFDM) transmission over time-varying underwater acoustic (UWA) channels. The progressive receiver is in nature an iterative receiver. However, it distinguishes itself from existing iterative receivers in that the system model for channel estimation and data detection is itself continually updated during the iterations. When the decoding in the current iteration is not successful, the receiver increases the span of the inter-carrier interference (ICI) in the model and utilizes the currently available soft information from the decoder to assist the next iteration which deals with a channel with larger Doppler spread. Numerical simulation and experimental data collected from the SPACE08 experiment show that the proposed receiver can self adapt to channel variations, enjoying low complexity in calm channel conditions while maintaining excellent performance in tough channel conditions.