## 2 Depth-Bistatic Bottom Reverberation Model  
The active towed array system of this study consists of a depth variable source and a triplet line array, operated with cable in a research vessel. The operation depths of the source and receiver arrays can be controlled by the ship speed and the employed cable length. The geometry of this array is nominally considered bistatic. However, the range between the source and receiver arrays in our system is usually within 1 km. This small range suggests the applicability of the depth-bistatic model, which takes into account that the source and receiver are collocated in range but separated in depth, in order to predict the reverberation of an active towed array. Reverberation in the ocean is expressed as the sum of the time signals scattered from the randomly distributed scatterers in volume and boundary. It means that the reverberation signal is essentially a random process. To model a random pressure signal directly is numerically expensive and meaningless. Thus, a heuristic approach for modeling the ensemble-averaged intensity signal has been widely used in the underwater acoustics community [22]. We consider an \(N\times 2D\) ocean environment with a depth-bistatic geometry of the source and receiver, as shown in Figure 1. Assuming the positions of the source and receiver are fixed, the impulse response of the incoherent bottom reverberation for the \(i\)-th bottom cell is formulated with the approach of [14] as the following equation:  
\[h_{nm}^{i}(t)=\Delta\phi\Delta\tau_{nm}\big{|}p_{n}\big{|}^{2}\big{|}p_{m} \big{|}^{2}\sigma(\theta_{n},\theta_{m})\prod(\frac{t-\tau_{n}-\tau_{m}}{ \Delta\tau_{nm}}), \tag{1}\]  
where \(\Delta\phi\) is the azimuthal interval, the subscripts n and m indicate the eigenray index from the source to the scatterer and from the scatterer to the receiver, respectively, \(p_{n,m}\) and \(\tau_{m,n}\) are the propagation losses and the time delays for a one-way path, respectively, and \(\sigma\) is the bottom scattering cross-section as a function of the incident angle \(\theta_{n}\) and the scattered angle \(\theta_{m}\). In addition, \(\Delta\tau_{nm}\) is defined as \(\Delta\tau_{nm}=(\Delta R/c_{0})(\cos\theta_{n}+\cos\theta_{m})\) with a range interval \(\Delta R\) and a sound speed \(c_{0}\) near the bottom. \(\prod(t)\) is a rectangular function defined as 1 inside \(-\)1/2 to 1/2 and 0 outside it. Note that Equation (1) is called the ensemble-averaged impulse response. To account for Doppler effects resulting from the motions of the source and receiver, we apply a Doppler shift obtained from the straight-line propagation between the source, receiver, and the \(i\)-th bottom cell into Equation (2). The Doppler angular frequency is written as:  
\[\omega_{d,i}=\omega\frac{c_{r}+\overrightarrow{V_{r,i}}\cdot\overrightarrow {n_{r,i}}}{c_{t}-\overrightarrow{V_{t,i}}\cdot\overrightarrow{n_{t,i}}}, \tag{2}\]  
where \(\omega\) is the original angular frequency of the transmitted waveform, \(c_{r}\) and \(c_{t}\) are the sound speeds in the receiver and the source positions, respectively, and \(\overrightarrow{V_{r,i}}\) and \(\overrightarrow{V_{t,i}}\) are the velocities of the receiver and the source relative to the \(i\)-th scatterer, respectively. Here, \(\overrightarrow{n_{r,i}}\) and \(\overrightarrow{n_{t,i}}\) are the unit vectors in the direction of the receiver to the \(i\)-th scatterer and in the direction of the source to the scatterer, respectively.