## 1 Introduction  
Achieving reliable and efficient communication has always been the core goal in underwater acoustic communications (UACs), which is the best method for medium-to-long-range communication underwater [1]. It is well known to be the most complicated wireless channel, and the characteristics of UWA channels include strong multipath, time variance, large time delays, and Doppler shifts. Despite some high-rate transmission schemes, including single-carrier frequency domain equalization [2; 3; 4] and orthogonal frequency division multiplexing (OFDM), having been proposed [5; 6; 7; 8] recently, they still are unsuitable for underwater acoustic channels with characteristic of time-frequency double selective fading. Orthogonal time frequency space (OTFS) modulation, which can convert the time-varying multipath channels into the delay-Doppler (DD) domain, in which the channels tend to be time-invariant, seems to be an optional technique to tackle the above problem [9; 10]. Although OTFS shows merits in anti-time-varying channels [11; 12; 13] compared with that of OFDM, it may suffer severe inter-Doppler interference (IDI) and intersymbol interference (ISI) [11] when OTFS works in doubly dispersive channels. Thus, in this case, equalization definitely needs to be employed in the OTFS system [11; 14]. Since prior channel information achieved by channel estimation is needed in most equalizers [15; 16], channel estimation highly affects performance and plays a key role in the OTFS system. There has been some research on channel estimation of OTFS systems. In [12], [PERSON] et al. estimated the OTFS channel by using a PN pilot sequence, achieving lower channel estimation errors with longer pilot sequences. In [17; 18], the impulse pilot was utilized for channel estimation in the DD domain. Its main idea is to insert a big value in the middle of the DD domain, and there are enough zero pilots as the guard pilots. Then, the channel parameters are filtered out at the receiver by setting an appropriate threshold. In [19], the pilot design is improved to reduce pilot overhead during in estimating channel of the MIMO system with the OTFS technique. In [20], a joint channel estimation and equalization method related to the OTFS system is proposed by using expectation propagation (JCEE- EP). Utilizing the channel sparsity of the DD domain seems to be feasible to lighten the computation burden induced by long pilot [21]. For example, [PERSON] proposed a 3D orthogonal matching pursuit algorithm (OMP) to efficiently estimate the downlink channel of the MIMO OTFS system. Since OMP has shown bad performance in bit error rate (BER) [22], and modified subspace pursuit (MSP) suffers from excessive computational burden [23], the two-choice hard thresholding pursuit (TCHTP) algorithm was proposed for the MIMO and MultiUser (MU) OTFS systems [23; 24]. In [25], by designing a Dolph-Chebyshev window to enhance the sparsity of the channel, the estimation accuracy of the OTFS system channel was improved effectively. Some methods related to exploiting channel sparsity of the DD domain are discussed too. For example, the authors applied sparse Bayesian learning in channel estimation [26], while a novel scheme also proposed to estimate the channel in the UWA OTFS system for the purpose of avoiding the disadvantage caused by fractional Doppler [27]. Different from the above work focused on exploiting the channel sparsity of the DD domain, some previous work tends to estimate channel OTFS systems in the time domain, which was first proposed in [28]. [PERSON] et al. presented a time-domain equivalent channel matrix estimation method based on energy threshold and spline interpolation [29]. Meanwhile, exploiting the time-frequency (TF) domain sparsity of the channel seems to be feasible too [30]. In [31], [PERSON] et al. proposed a channel estimation scheme that can effectively restrain the leakage on the OTFS system by exploiting smoothness regularization in TF space. The compressed sensing (CS) method proposed by [PERSON].