## 1 Introduction  
Underwater acoustic (UWA) sensor networks can be applied to a variety of underwater activities, including environment monitoring, resource exploration and equipment navigation. Reliable high-rate communication is significant for massive data transmission between sensor nodes [1]. However, the UWA channel is widely known as one of the most difficult communication media, and high-rate communication with high reliability is quite challenging. Since the sound attenuation rapidly increases with frequency, the bandwidth available for communication is extremely limited. Due to the reflection and refraction of sound in the ocean, the multipath spread in a medium-range shallow water channel can extend over tens or even hundreds of milliseconds [2], resulting in severe frequency-selective signal distortion. Due to the dynamic environment and moving transceivers in ocean, the UWA channel is time-varying. In particular, the normalized frequency offset factor induced by Doppler can be on the order of \(10^{-3}\) compared with \(10^{-7}\) for high-speed mobile radio channels [2]. Due to high spectral efficiency and good anti-multipath performance, orthogonal frequency division multiplexing (OFDM) and some OFDM-based modulation techniques are widely adopted for UWA communications [3; 4; 5; 6]. In UWA-OFDM systems, accurate channel state information (CSI) is vital for channel equalization and adaptive transmission [7; 8], which can significantly affect the reliability and throughput. However, it is difficult to estimate the time-varying channel impulse responses (CIRs) with excessive delay spread. To overcome this problem, some structural information such as sparsity can be exploited to improve channel estimation performance. The paths of the UWA channel tend to be clustered-sparse, which carry comprehensive structural information [9; 10; 11; 12; 13]. As shown in Figure 1, there are many weak paths centering around the eigen-paths associated with the medium refractions and surface/bottom bounces, which form the clustered-sparse structure. As different clusters are usually derived from different transmission routes, the channel coherence time usually varies across clusters. Thus, the path variation for different clusters can reflect the channel variation characteristics [14], and can be used for predicting the time-varying UWA channel [15]. However, the fading process of the UWA channel may randomly vary [2], which makes it difficult to accurately obtain the path variation for different clusters between adjacent OFDM blocks. To obtain the clustered-sparse structure of the UWA channel, we proposed a clustered-sparse Bayesian channel estimation method in [16], which is the preliminary version of this work (This work extends the conference paper [16]) by (i) adding a Bayesian learning-based channel estimation architecture; (ii) adding a noise-corrected clustered-sparse channel estimation algorithm to lessen the effect of strong colored noise; (iii) giving the detailed derivation and performance analysis of the proposed algorithms; and (iv) presenting more complete performance results related to modulation order, model hyperparameters and noise distribution). In addition to the time-varying CIRs with excessive delay spread, the complex distribution of noise in the UWA channel also makes the channel estimation difficult. Generally, there are two kinds of channel noise, including the ambient noise and site-specific noise. The former type is presented in the background of the quiet deep sea, and exhibits a colored characteristic when approximated by Gaussian distribution, while for the site-specific noise, significant non-Gaussian components should be carefully considered [2]. Some experiments have confirmed that the noise is non-uniform across different locations, times and frequencies [17]. As a popular noise model in most UWA-OFDM systems, additive white Gaussian noise (AWGN) is straightforward for modeling and solving the noise distribution of the UWA channel, but it deviates from the actual underwater noise.