## 1 Introduction  
With the increasing demand of ocean exploration, pollution monitoring, marine data collection, etc, the role of underwater acoustic (UWA) communication becomes more crucial [1, 2, 3]. However, compared to the terrestrial wireless channel, the UWA channel exhibits serious multipath, Doppler, and time-varying characteristics, which is a time-space-frequency varying channel [4, 5]. The extreme adverse and complex marine environment raises many challenging for the UWA communication technology. The traditional non-sparse channel estimation algorithms, such as least squares (LS) [6, 7], can achieve better channel estimation performance in dense multipath scenarios. The LS algorithm estimates all the tap coefficients directly, which results in the relatively high channel estimation errors, as it is well recognized in the research community that UWA channels are commonly sparse [8]. Under the compressed sensing (CS) framework, it is capable of improving the channel estimation performance by exploiting the inherent channel sparsity. The most popular CS-based algorithms are the \(l_{0}\)-norm constraint algorithms [9, 10, 11] and greedy algorithms [12, 13, 14], such as matching pursuit (MP) [7] and orthogonal matching pursuit (OMP) [13]. Compared with the MP, the main difference of the OMP is that the coefficients are the orthogonal projection of the signal on the selected subspace, and the OMP presents more accuracy on the sparse recovery. However, the time-varying characteristic of the UWA channel would cause the performance degradation for the above algorithms. Dynamic compressed sensing theory (DCS) has attracted considerable attention in recent years, as it provides a new approach to the reconstruction of time-varying sparse signals [15; 16; 17; 18; 19; 20]. Kalman-filtered compressed sensing (KF-CS) [15] was proposed to tackling the time-varying sparsity recovery problem. The key idea of KF-CS is to estimate the initial support set by KF; then, the possible support additions are computed by CS to track the time-varying components. Built on the KF-CS framework, a serial DCS-based works have been proposed to improve the performance of time-varying sparsity reconstruction, such as least squares CS (LS-CS) [17], modified-CS [18], regularized modified-CS [19], and minimum mean square estimation CS (MMSE-CS) [20]. Furthermore, by formulating the hybrid multipath UWA channels as sparse set consisting of static and time-varying supports, a static-dynamic discriminative compressed sensing (SDD-CS) [21] approach was proposed to explore the hybrid sparsity of UWA channels. From the viewpoint of UWA multipath channels, except the channel time variation introduced by the dynamic ocean environment or the platform motion, the paths of the UWA channel often present in a cluster, where it may have several small paths centering around the eigen-paths due to the surface/bottom reflections and the medium refractions [22]. Thus, while the classic DCS-based algorithms only consider the time-varying sparsity recovery problem, the ignorance of block sparsity will limit the estimation performance of the KF-CS algorithm for the block sparse time-varying UWA channel. Moreover, to exploit the block sparsity of channel, [23] proposed the block OMP (BOMP) algorithm, which shows better recovery performance than the OMP for a block sparse channel. A block sparse recursive least squares (BS-RLS) algorithm was proposed to improve the performance of channel estimation under block sparse condition [24], which introduces an approximate mixed \(l_{2,0}\)-norm to take advantage of the underlying block sparse property of the system. However, the time-varying feature of the UWA channel is neglected by the BOMP and the BS-RLS, which may result in a degradation of performance. Moreover, to enhance performance, the adaptive filter based channel estimation algorithm requires a relatively long training sequence and brings extra computational burdens.