## 4 Simulation  
A simulation in a typical shallow-water waveguide was conducted to demonstrate our proposed method based on 2D deconvolution. The KRAKEN toolbox was used. The environmental parameters of the employed Pekeris [2,17] waveguide are summarized in Figure 2a. The water depth was 100 m and the speed of sound was assumed to be 1500 m and the same at different depths. The seabed was typically sand bottom.  
A horizontal array with 16 elements was employed. The array length was 2.8 m. The bandwidth of the noise radiated by the source of opportunity was 2-4 kHz. The sampling frequency was 20 kHz. The existing array-invariant method [11] was used as the reference method. The source and the horizontal array were placed at 3 m and 97 m under water, respectively. The azimuth angle of the sound source was 65\({}^{\circ}\). Additive white noise was used. The signal-to-noise ratio at each array element was set at 10 dB. The length of simulation data is 2 s. The Green's function of the 16 th element at the range of 1000 m was extracted with blind deconvolution and is shown in Figure 2b.  
The beam-time migrations obtained using the reference method and proposed methods are shown in Figure 2c,d, respectively, which have the same scale. The peak position of beam-time migrations is marked by the red square. In Figure 2c, it can be seen that the relative arrival time of the two paths was 13.95 ms and 40.1 ms, respectively, and the sine of the beam angle of the two paths was 0.879 ms and 0.839 ms, respectively. In Figure 2d, it can be seen that the relative arrival time of the two paths was 13.2 ms and 40.1 ms, respectively, and the sine of the beam angle of the two paths was 0.883 ms and 0.849 ms, respectively.  
It is clear that our proposed method based on 2D deconvolution, shown in Figure 2d, had a much higher resolution in both the bearing and time domains, compared to the reference method. Based on the high-resolution beam-time migration, it is clear that the source-range estimation error can be decreased.  
For passive shallow-water localization with a horizontal array, the relative error of range estimation determines the localization accuracy. The source range was estimated according to Equation (14). In the case of the true source range of 1000 m, the range estimated with the reference method, shown in Figure 2c, was 850.2 m. The range estimated  
Figure 2: (**a**) The Pekeris waveguide with sand bottom, where \(c_{w}\), \(\rho_{w}\), and \(\kappa_{w}\) are the sound speed, density of the water, and attenuation of the water, and \(c_{b}\), \(\rho_{b}\) and \(\alpha_{b}\) correspond to those in the sand. (**b**) The Green’s function was obtained with blind deconvolution at a distance of 1000 m between the array and source. (**c**) Beam-time migration obtained using the reference method. (**d**) Beam-time migration obtained using the proposed method.  
with the proposed method, shown in Figure 2d, was 1038.7 m. The relative estimation errors of the proposed and reference methods were 3.9% and 15.0%, respectively. The estimation performances of the two methods at different ranges are summarized in Table 1. On average, the estimation error of our proposed method was only one third of that of the reference method. The results also prove that the wideband beam-time migration obtained using conventional plane-wave beamforming in the reference array-invariant method [11] is one of the main factors influencing the accuracy of the source range estimation.