## 3 Applications
### Comparison of Inversion Results of Different Cost Functions  
Equation (18) is the cost function of matching physical quantities based on the ocean bottom reflection coefficient interference period (Equation (17)). We could compare Equation (16) and Equation (18) to invert the normalized two-dimensional cost function value of \(c_{\rm s}\) and \(d\). The simulation parameters are given in Table 2.  
\[T=\frac{c_{\rm s}}{2d\sin\varphi_{\rm s}} \tag{17}\]  
\[E_{T}(c_{\rm s},d)=\frac{1}{\sqrt{\frac{1}{N}\sum\limits_{i=1}^{N}\bigl{(}T_{ \rm exp}(\varphi_{\rm wi})-T_{\rm cal}(\varphi_{\rm wi},c_{\rm s},d)\bigr{)}^{2 }}} \tag{18}\]  
where \(T_{\rm exp}\) is the interference period extracted from the measured data, \(T_{\rm cal}\) is the interference period calculated by Equation (17), \(\varphi_{\rm w}\) is the ocean bottom grazing angle, and \(N\) is the number of \(\varphi_{\rm w}\).  
It can be seen from Figure 7 that the two-dimensional results of \(c_{\rm s}\) and \(d\) in Equation (16) were more concentrated compared to the results of Equation (18). Although Equation (18) could get a more concentrated result of \(d\), the estimation result of \(c_{\rm s}\) was more dispersed.  
Figure 6: The normalized value of the two-dimensional cost function of each inversion parameter (the reciprocal of Equation (7)). Red and white “+” is the true value.  
There are two reasons for the inversion results of Equation (18). One is that even if the interference period at multiple grazing angles was used for matching, the coupling between \(c_{\mathrm{s}}\) and \(d\) in Equation (17) still exists. Equation (18) was comparatively insensitive to \(c_{\mathrm{s}}\) near the true value of \(d\). Equation (17) can be written as:  
\[T=\frac{1}{2d}\sqrt{\frac{c_{\mathrm{s}}{}^{4}}{c_{\mathrm{s}}{}^{2}-c_{ \mathrm{w}}{}^{2}\cos\varphi_{\mathrm{w}}{}^{2}}} \tag{19}\]  
The other reason is shown by Equation (19). When \(c_{\mathrm{w}}\) and \(\varphi_{\mathrm{w}}\) are known, the interference period changes approximately linearly with \(c_{\mathrm{s}}\), but changes inversely with \(d\). As shown in Figure 8, within the range of the parameter, \(d\) had a much greater influence on the interference period than \(c_{\mathrm{s}}\), and it resulted in a more dispersed estimation of \(c_{\mathrm{s}}\).  
For Equation (16), within the range of the parameter, the spatial position of the interference fringe of the same order changed drastically with the parameter. When the receiving range exceeded the range corresponding to the critical angle of the seawater-sedimentary layer, the spatial position of the fringe was also affected by the additional phase \(\frac{Q\varphi_{\mathrm{s}}}{T}-1\). Accordingly, the influence of \(c_{\mathrm{s}}\) on the spatial position of the fringe was enhanced, and Equation (16) became more sensitive to \(c_{\mathrm{s}}\) than Equation (18). Figure 5 shows that the sensitivity of Equation (16) to both \(c_{\mathrm{s}}\) and \(d\) was similar. Compared with Equation (18), the influence of \(d\) on \(c_{\mathrm{s}}\) in the inversion process of Equation (16) was reduced.  
Figure 8: Variation of T with the sound speed and thickness of the sediment within the parameter range.  
Figure 7: The normalized value of the two-dimensional cost function of the sound speed and thickness of the sediment under two matching physical quantities (the reciprocal of Equation (7)). (**a**) The interference fringes of the shadow zone in range-frequency domain; (**b**) the ocean bottom reflection coefficient interference period.