## 2 The Range Localization Method
The average phase speed \(\overline{v}_{p}\) can be approximated by the sound speed in water: \(\overline{v}_{p}\approx c_{w}\), and we can easily obtain the average horizontal wavenumber since we have that \(\overline{k}(f)=2\pi f/\overline{v}_{p}\) for a particular frequency (narrowband case). Furthermore, the average radial velocity within the observation time \(T\) can be obtained as  
\[\overline{v}_{s}=\frac{k_{r}}{\overline{k}}. \tag{7}\]  
In the calculation process, the term \(\sqrt{r_{t}}\) in Equation (3) can be considered to be constant within a short observation period \(T\), and thus can be set to 1 without affecting the subsequent results. Further, Equation (3) can be conveniently calculated using a method such as fast Fourier transform (FFT). After obtaining the radial velocity, we can easily obtain the source speed \(v_{0}\), the range and time of CPA, \(r_{0}\) and \(t_{0}\), according to Equation (1) using the least squares criterion and then calculate the source range as follows  
\[r_{t}=\sqrt{r_{0}^{2}+\left(t-t_{0}\right)^{2}v_{0}^{2}}. \tag{8}\]  
For a broadband case, the frequency processing gain can be obtained in a manner of sub-band division and non-coherent accumulation. Assume that the number of sub-bands divided is \(L\), and the center frequency corresponding to each sub-band is \(f_{l}\), where \(l=1,2,\cdots,L\). Each sub-band is processed according to Equation (3) to obtain the sub-band horizontal wavenumber spectrum \(g(k_{r})_{f_{l}}\), where \(l=1,2,\cdots,L\). Since the average horizontal wavenumbers corresponding to different frequencies are different, the peak positions of the sub-band horizontal wavenumber spectrum are also different. Therefore, the sub-band horizontal wavenumber spectrum cannot be directly added together. Here, we choose any one of the sub-band center frequencies as the reference frequency \(f_{\text{ref}}\), and convert the horizontal wavenumber spectrum of each sub-band into a form relative to the reference frequency by stretching/compression transformation (\(g(k_{r})_{f_{l}}\Rightarrow g(k_{r}(f_{l}/f_{\text{ref}}))_{f_{l}}\)). Finally, we can obtain the following broadband horizontal wavenumber spectrum  
\[G(k_{r,\text{ref}})=\sum_{l=1}^{L}\left|g\left(k_{r,\text{ref}}\frac{f_{l}}{f_{ \text{ref}}}\right)_{f_{l}}\right|^{2}, \tag{9}\]  
where \(k_{r,\text{ref}}\) is the variable corresponding to the reference frequency \(f_{\text{ref}}\). The broadband horizontal wavenumber spectrum is similar to the narrowband horizontal wavenumber spectrum at a frequency \(f_{\text{ref}}\), so that the average radial velocity can be estimated as shown in Equation (10), similarly to the process of obtaining Equation (7)  
\[\overline{v}_{s}=\frac{k_{r,\text{ref}}}{\overline{k}_{\text{ref}}}, \tag{10}\]  
where the average horizontal wavenumber \(\overline{k}_{\text{ref}}=2\pi f_{\text{ref}}/\overline{v}_{p}\). It can be seen from Equation (7) that the approximation of the average horizontal wavenumber by the wavenumber in water causes a bias on the estimation of the average radial velocity, which will in turn affect the accuracy of the range estimation. Assuming that the error of the average horizontal wavenumber is \(\delta k\), then the estimation error of the average radial velocity is given here [14]  
\[ \begin{array}{ll}\Delta\overline{v}_{s}&=\overline{v}_{s}-\overline{v}_{s} \big{|}_{\text{error}}=\frac{k_{r}}{\overline{k}}-\frac{k_{r}}{\overline{k}+ \delta k}\\ &=\overline{v}_{s}\frac{\delta k/\overline{k}}{1+\delta k/\overline{k}}\end{array} , \tag{11}\]  
and the maximum error of the horizontal wavenumber is \(\delta k=k_{\text{max}}-k_{\text{min}}\). According to the previous description, \(\delta k/\overline{k}\) is small since \(\delta k\ll\overline{k}\), so the radial velocity estimation error is also small, ensuring the accuracy of the range localization. So far, we have completed the theoretical derivation of the source range localization. The effectiveness of the method will be verified using simulation and at-sea data in the next two sections.