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While purely sparse channel models have been recently investigated for underwater acoustic channels, experimental propagation data suggests that the channel is more complex. Thus, herein we propose a novel channel model based on both diffuse and sparse components. Tailored to this hybrid model, channel estimators are designed for different scenarios which differ in the amount of side information available at the receiver. The proposed channel estimation methods are compared to unstructured and purely sparse estimators. The numerical results show that the new channel estimation schemes considerably improve the estimation accuracy and the bit error rate performance over conventional channel estimators. Further, a mean squared error analysis of the proposed estimators is conducted in two asymptotic regimes (high SNR and low SNR) enabling a simple characterization and thus comparison of the proposed estimators.