## 1 Introduction  
Vertical underwater acoustic (UWA) communications are critical in deep-sea activities such as scientific exploration with human-occupied vehicles. The communication waveforms are severely distorted with Doppler dilation or compression induced by the platform movement. In practical communication systems [1; 2; 3; 4; 5], the Doppler is usually coarsely estimated using synchronization signals, and then the residual Doppler is tracked by means of the equalizer, with the aid of pilot symbols or decided information symbols. Improving the accuracy of time-varying Doppler synchronization relaxes the burden on the adaptive equalizer and thus prevents the divergence of the adaptive algorithm in response to an impulsive noise. Among the motion sources, the fluctuation of the surface platform is the hardest to estimate because of its randomness. The ocean surface height was modeled as a sinusoid with a period of 8 s in [6], where the time difference between the transmitter and receiver was estimated over 0.5 s with the constant acceleration model. According to the experimental measurements [7] of the channel impulsive response (CIR), the surface motion was a narrow-band random process. A more accurate surface height model was described by the Pierson-Moskowitz (PM) wave height spectrum [8], which was utilized in the analysis of the surface acoustic reflection [9]. However, the PM model has not been involved in the Doppler estimation of UWA communications. Doppler synchronization for UWA communications consists primarily of two steps to accomplish the coarse and fine estimations, respectively [10; 11]. In the first step, the time-invariant motion parameters, such as the timing offsets (displacement), the Doppler scaler (speed), and the Doppler rate (acceleration), are estimated through correlation with the synchronization signal. The correlation can be a single-branch one for Doppler-insensitive signals or a multiple-branch one [12] for Doppler-sensitive signals. The single-branch correlation has low complexity, and the estimation results are biased according to the delay-Doppler ambiguity function. The multiple-branch correlation is unbiased when enough branches are produced; however, this leads to extreme complexity in calculations. In the second step, where the continuous movement is tracked, different methods are carried out according to the modulation structure, such as using an adaptive equalizer combined with a phase-locked loop (PLL) for single-carrier (SC) systems [1] or using intercarrier inference (ICI) cancellation for orthogonal frequency division multiplexing (OFDM) systems. This paper proposes that the motion trajectory can be estimated without the tracking specified above if the relationship among the sequential synchronization results is fully utilized. Deep learning was initially developed in the areas of computer vision and natural language processing, where mathematical description was difficult. Recently, deep learning has been prominent in wireless communications and UWA communications. The deep neural network (DNN)-based OFDM receiver proposed in [13; 14] was shown to be more robust than conventional methods; this model was extended to deal with the time-varying underwater acoustic channel and was verified by simulations [7]. A DNN-based online-training UWA receiver [15] was trained by the adaptive moment estimation (Adam) optimizer for each sub-block for the time-varying UWA channel and was verified by sea trial data. To reduce the training burden, model-driven deep learning [16] used expert knowledge to make the network explainable and predictable. In [17], a DNN-based channel estimation for online UWA communications was proposed that had better simulation performance than the minimum mean squared error (MMSE) algorithm and also had less run time and storage resource consumption. In [18], a convolutional neural network was introduced to fix errors in symbol timing and carrier offset estimation in UWA communications under the condition of flat frequency response.