## 4 Numerical Simulation  
In this section, numerical simulations are presented to evaluate the estimation performance. In this paper, we assume that the non-zero coefficients' time-varying set of the UWA channel changes slowly over time, namely, the UWA channel is a slow time-varying channel. Moreover, the addictive noise \(\mathbf{w}\) is assumed to be Gaussian. We compare the estimation performance of the proposed algorithm to the following benchmark algorithms. 1. LSQR: The classic non-sparse channel estimation algorithm. 2. OMP: The classic CS channel estimation algorithm. 3. BOMP: The classic block sparse-based CS channel estimation algorithm. 4. BS-RLS: The novel block sparse-based channel estimation algorithm, which was proposed to exploit the block sparsity of UWA channels. 5. KF-CS: The classic DCS channel estimation algorithm. 6. SDD-CS: The novel DCS-based channel estimation algorithm, which was proposed to explore the hybrid sparsity of UWA channels. (7). BS-CS: The proposed channel estimation algorithm, which is designed for the block sparse time-varying UWA channels. To verify and compare the estimation performance of the BS-CS algorithm with the benchmarks, a known signal is transmitted through a simulated block sparse time-varying channel, and the channel estimation is performed on the received signal. Our numerical simulation performance matrix is the channel-to-reconstruction error ratio (CRER). The CRER is defined by:  
\[\text{CRER}=10\text{log}_{10}\frac{||\mathbf{h}||_{2}^{2}}{||\mathbf{h}- \mathbf{\tilde{h}}||_{2}^{2}}, \tag{20}\]  
where \(\mathbf{h}\) is the true CIR and \(\mathbf{\tilde{h}}\) is the estimated CIR. The simulations were performed using MATLAB R2019b running on a computer with an Intel i7-1165G7 processor and 8 GB of memory. We firstly use the PE propagation model to generate a block sparse time-varying channel with a water depth of 30 m and a distance of 2000 m. Our propagation model adopts a zero or negative gradient sound speed profile with an uneven seabed, while the sound speed on the surface is given by a uniform distribution between 1500 m/s and 1540 m/s. The transmitter and the receiver are deployed at the depth of 5 m and 10 m under the sea surface, respectively. Figure 3 shows the sound speed profile and the corresponding simulated UWA channel, which has time-varying and block sparse characteristics. Note that the delay and the magnitude of two surface paths are imposed with zero-mean random variations and sinusoidal variations, respectively, to artificially simulate the time-varying multipath induced by the sea surface, while the remaining paths remain static. In our numerical simulation, the lengths of the CIR \(N\) and the observation window \(M\) are set to 200 and 150, respectively, with a sampling rate of 4000 Hz. The received signal-to-noise ratio (SNR) is set to 15 dB. The length of the training sequence for BS-RLS is set to 1000. The continuous measurement number for SDD-CS is set to 2, and the addition threshold for KF-CS, SDD-CS, and BS-CS is set to 0.03. Specifically, larger block size \(d\) would exploit better the intra-block correlation of the channel, which is more appropriate for the UWA channel with less sparsity. When the block size \(d=1\), the BS-CS becomes a classic DCS-based algorithm. Thus, in this paper due to the sparse simulated channel, a small block size \(d\) is adopted for our simulation. The sparsity factor and block sparsity factor \(\kappa\) are set to 16 and 8, respectively, while the block size is set to \(d=2\) for BOMP, BS-RLS, and BS-CS. Figure 4 shows the CRER results obtained by different estimators, while the iteration number is set to 1000 for each 1 ms time slot. We observe that the LSQR estimator has significantly lower CRER output than all the six sparse estimators, namely OMP, SOMP, BS-RLS, KF-CS, SDD-CS, and BS-CS, due to the considerable estimation noise.