## 2 The Range Localization Method  
We consider a range-independent shallow ocean environment and a moving source with a depth \(z_{s}\) and constant speed \(v_{0}\) during the observation time \(T\). The radial velocity is given by  
\[ \begin{array}{ll}v_{s}(t)&=\frac{\delta r}{\delta t}(t)\\ &=\left|\frac{\sqrt{t_{0}^{2}+(t-t_{0})^{2}v_{0}^{2}}-\sqrt{t_{0}^{2}+(t-t_{0 }-\delta t)^{2}v_{0}^{2}}}{\delta t}\right|\end{array} , \tag{1}\]  
where \(r_{0}\) and \(t_{0}\) are the range at the closest point of approach (CPA) and the time of CPA, respectively, and \(\delta t\) is the variation of time associated to \(\delta r\), the radial range variation for each acquisition time \(t\). The normal mode theory expression of the pressure field received at a receiver at a depth \(z_{r}\) and time \(t\) is  
\[p(t)=\sum_{m=1}^{M}\sqrt{\frac{2\pi}{k_{m}r_{t}}}\phi_{m}(z_{s})\phi_{m}(z_{r })e^{-ik_{m}r_{t}-\alpha_{m}r_{t}-i\pi/4}, \tag{2}\]  
where \(r_{t}\) is the source range at time \(t\), \(\phi_{m}(z)\) is the \(m\)th mode depth function, \(k_{m}\) is the \(m\)th horizontal wavenumber, and \(\alpha_{m}\) is the \(m\)th mode attenuation coefficient. \(M\) is the number of modes that can be propagated in the current environment. The pressure field at different times is beamformed as follows [17; 18]:  
\[g(k_{r})=\frac{e^{i\pi/4}}{\sqrt{2\pi k_{r}}}\int_{t_{1}}^{t_{2}}p(t)e^{ik_{r }t}\sqrt{r_{t}}dt, \tag{3}\]  
where \(t_{1}\) and \(t_{2}\) represent the start and end moments of the observation time \(T\), respectively, and \(e^{ik_{r}t}\) is the steering vector and \(k_{r}\) is a variable. In [17] the observation time \(T\) should set to be long enough to separate different horizontal wavenumbers of modes. Here, we set the observation time in the opposite way. When the observation time \(T\) is short, the radial velocity can be considered to be constant, which is represented by \(\overline{v}_{s}\). Within the observation time \(T\), the source range \(r_{t}\) can be expressed as  
\[r_{t}=r_{t_{1}}+\overline{v}_{s}t, \tag{4}\]  
where \(r_{t_{1}}\) represents the range at time \(t_{1}\). After substituting the normal mode expression into Equation (3), one obtains  
\[ \begin{array}{ll}g(k_{r})&\simeq\sum\limits_{m=1}^{M}\frac{\phi_{m}(z_{s}) \phi_{m}(z_{s})}{\sqrt{k_{r}k_{m}}}\int_{t_{1}}^{t_{2}}e^{ik_{r}t-ik_{m}r_{t}- \alpha=n}dt\\ &=\sum\limits_{m=1}^{M}a_{m}\frac{\phi_{m}(z_{s})\phi_{m}(z_{s})}{k_{r}-k_{m} \overline{v}_{s}+i\alpha_{m}\overline{v}_{s}}\\ &=\sum\limits_{m=1}^{M}a_{m}\frac{\phi_{m}(z_{s})\phi_{m}(z_{s})}{k_{r}- \left(\overline{k}+\delta k_{m}\right)\overline{v}_{s}+i\alpha_{m}\overline{ v}_{s}}\end{array} , \tag{5}\]  
where  
\[a_{m}=\frac{e^{ik_{r}t_{2}-ik_{m}v_{s}t_{2}-\alpha_{m}v_{s}t_{2}}-e^{ik_{r}t_{ 1}-ik_{m}v_{s}t_{1}-\alpha_{m}v_{s}t_{1}}}{i\sqrt{k_{r}k_{m}}}e^{-ik_{m}r_{t_{ 1}}-\alpha_{m}r_{t_{1}}},\]  
and the wavenumbers can be expressed as \(k_{m}=\overline{k}+\delta k_{m}\), where \(\overline{k}\) is the average horizontal wavenumber of the modes since \(\delta k_{m}\prec\overline{k}\) (typically, \(\delta k_{m}/\overline{k}\sim\mathcal{O}(10^{-2})\) for discrete mode wavenumbers, which are between \(k_{\min}=\omega/c_{b}\) and \(k_{\max}=\omega/\min(c_{w})\) where \(c_{b}\) and \(c_{w}\) are the sound speeds at the bottom surface and in the water, respectively [14]. For some lower frequencies' cases, \(\delta k_{m}/\overline{k}\sim 10^{-1}\)). Then \(g(k_{r})\) can be written as  
\[g(k_{r})\simeq\sum\limits_{m=1}^{M}\frac{a_{m}\phi_{m}(z_{s})\phi_{m}(z_{r})} {k_{r}-\overline{k}\overline{v}_{s}+i\alpha_{m}\overline{v}_{s}}. \tag{6}\]  
It can be seen from Equation (6) that the horizontal wavenumber spectrum \(g(k_{r})\) will have maxima for \(k_{r}=\overline{k}\overline{v}_{s}\), noting that the mode attenuation coefficient is generally orders of magnitude smaller than the horizontal wavenumber.