## 4 Comparison with Previous Studies
The new regression model for the mid-frequency band from the present study is shown as a blue line in Figure 4. Given that the study \(\sigma_{eW}\) and \(\sigma_{eL}\) are unknown, the regression model was calculated using the measured data in the present study. Quadratic regression analysis was performed for these well-converged data only for the mid-frequency band. Thus, the regression curve of \(\alpha\) with the mid-frequency band was proposed using fish schools (Japanese horse mackerel) in the present paper (see Equations (6) and (7); Figure 4, blue lines). \[\sigma_{eW(mid-frequency)}\,\left(m^{2}/kg^{2/3}\right)=1.94\times 10^{-3}+4.20 \times 10^{-4}f+1.76\times 10^{-5}f^{2}\left(r^{2}=0.84\right), \tag{6}\]  
\[\sigma_{eL(mid-frequency)}=1.09\times 10^{-2}+1.66\times 10^{-3}f+7.39\times 10^{-5}f ^{2}\left(r^{2}=0.87\right), \tag{7}\]  
where \(f\) represents the frequency in units of kilohertz (kHz). The coefficients of determination (\(r^{2}\)) are 0.84 and 0.87 from the regression curves of \(\sigma_{eW}\) and \(\sigma_{eL}\), respectively [28]. Furthermore, when including the present study to the previous ones, the new regression models were calculated from all the data, ranging from the mid-to-high frequencies based on the previous and present data (see Equations (8) and (9); Figure 4, red lines). Figure 4: Results of normalized extinction cross-section of (**a**) the wet weight (\(\sigma_{eW}\)) and (**b**) the total length (\(\sigma_{eL}\) ) of fish with frequency in previous and present studies. The black line is the previous regression model [6], the blue lines are the new quadratic regression models for mid-frequency from the present study, and the red lines are the total quadratic regression models for all-frequency. \[\sigma_{e\textit{W}(all-frequency)}\,\left(m^{2}/kg^{2/3}\right)=6.92\times 10^{-3}+3.6 0\times 10^{-5}f+4.48\times 10^{-11}f^{2}\,\left(r^{2}=0.39\right), \tag{8}\]  
\[\sigma_{eL(all-frequency)}=3.99\times 10^{-2}+2.00\times 10^{-4}f+4.60\times 10^{-13} f^{2}\,\left(r^{2}=0.45\right). \tag{9}\]  
In Figure 4, the curve of the Japanese horse mackerel (blue lines) differed considerably from the curve of other fishes (black lines). In the results of previous studies, an increasing tendency was observed for each frequency, but \(\sigma_{e\textit{W}}\) and \(\sigma_{eL}\) varied depending on the species even at the same frequency. In these results, the resonance effect was not considered because the resonance of horse mackerel fish should be in the lower frequency band. The results of the attenuation experiments in fish schools depending on fish species, frequency, \(W\), \(L\), \(W\)/\(L^{3}\), number density (\(n\)), \(\sigma_{e}\), \(\sigma_{e\textit{W}}\), \(\sigma_{eL}\), and references from previous and present studies are shown in Table 3. The attenuation of \(\sigma_{e\textit{W}}\) and \(\sigma_{eL}\) slightly increased with the frequencies of 3-420 kHz. Because \(r^{2}\) was low because of a broadband frequency ranges, \(\sigma_{e\textit{W}}\) and \(\sigma_{eL}\) should be used by each frequency band. Comprehensively, \(\sigma_{e}\) and \(\alpha\) can be calculated through Equations (6)-(9), the fish school attenuation can be numerically calculated according to densities and frequencies.