## 3 Numerical Simulation Analysis  
To ensure that the simulation closely resembles a real-world environment, the received signal in the simulation process is obtained from WATERMARK [31]. It is noteworthy that WATERMARK reproduces maritime conditions, including hardware effects, and thus, the simulated data are influenced by the small Doppler shifts generated (instantaneous Doppler spreading and time-varying Doppler shifts around the mean) The reception methods discussed in this paper (including the traditional methods used for comparison) primarily focus on addressing the large-scale Doppler shifts caused by the active motion of the vessel. Therefore, the small-scale Doppler effects introduced by WATERMARK mentioned above, do not significantly impact the simulation. In the shallow underwater acoustic channel measured by WATERMARK, we selected the KAU1 and KAU2 channels as the simulation channels, which are the SIMO channels from the Kauai Accomms MURI 2011 (KAM11) experiment. This experiment was conducted from 23 June to 12 July 2011 in the shallow waters off the west side of Kauai, Hawaii. The KAU1 channel was recorded in the frequency range of 4 to 8 kHz, located between the towed source and a vertical suspended array of 16 receivers. The duration of the run was 33 s. We only processed the data received by the first channel receiver, where Data Channel 1 is a seabed hydrophone located 9 m above the seafloor, at a depth of approximately 106 m. KAU2 is another SIMO channel from the KAM11 experiment, which involves the same signal transmission as KAU1, but is recorded on a different array. The main difference from KAU1 is that it has a greater range and a faster rate of attenuation for delayed arrivals. The time-varying channel impulse response (TVCIR) of these channels (see Figure 4). It is evident that the multipath effects of the channel are pronounced, and both channels exhibit time-varying Doppler (TVD) shifts due to the motion of the towing vessel, with relative motion speeds of \(-1.14\) m/s and \(1.28\) m/s, respectively.  
During the simulation process, the parameters of the transmitted signal are shown in Table 1, with the length of the original bit stream being 750 bits. The original bit stream is input into the transmitter structure illustrated in Figure 1, resulting in a total of 150 spread spectrum symbols carrying polar modulation, with the output data organized into frames of 50 symbols each, yielding a total of three frames. Regarding the carrier parameters, the center frequency is set to 6 kHz with a bandwidth of 4 kHz. The spreading code for the  
Figure 4: Time-varying channel impulse response (TVCIR) of KAU1 and KAU2, including both a 3D representation of the channel and its 2D projections  
information sequence is a chaotic spreading code with a length of 511, operating at a rate of 4 kcps. During the simulation process, we incorporated three additional reception methods for comparison: 1. the conventional M-ary spread spectrum reception method without Doppler estimation (WDE-MSS); 2. the conventional M-ary spread spectrum reception method with coarse Doppler estimation (CDE-MSS); 3. the conventional M-ary spread spectrum reception method with fine Doppler estimation (FDE-MSS).