## 3 Applications
### Geoacoustic Parameter Inversion
#### 3.4.2 Inversion of Geoacoustic Parameters by Genetic Algorithm  
In order to compare the linear inversion results of the grid search method, the six parameters in Table 3 were inverted simultaneously by a genetic algorithm (GA) [28]. The values of the genetic operator were as follows: the selection probability was 0.5, the mutation probability was 0.08, the crossover probability was 0.8, and the population number was 100. To ensure that Equation (16) converges to the minimum, we set the genetic algebra to 5000 and analyzed the collected samples by posterior probability distribution (PPD). Table 3 shows the optimization range and result, the truevalue, and the error of each inversion parameter. The reliability of the multi-dimensional inversion results were analyzed by calculating its PDD. Firstly, we sorted the sampling values generated by GA according to the cost func  
Figure 11: Grid search inversion result; (**a**) the minimum cost function value under different fringe orders; (**b**) the normalized two-dimensional inversion results of the sound speed and thickness of the sediment layer with the fringe order [7,9,11] (the reciprocal of Equation (7)). \begin{table}
\begin{tabular}{c c c c c} \hline
**Inversion Parameters** & **Optimization Range** & **Optimization Result** & **True Values** & **Error** \\ \hline \(d\) (m) & 0 to 50 & 20.23 & 20.0 & 1.15\% \\ \(c_{\rm s}\) (m/s) & 1450 to 1700 & 1601.52 & 1600.00 & 0.095\% \\ \(\rho_{\rm s}\) (g/cm\({}^{3}\)) & 1.1 to 2.0 & 1.47 & 1.50 & 4.50\% \\ \(c_{\rm b}\) (m/s) & 1700 to 2200 & 2018.18 & 1800.00 & 12.12\% \\ \(\rho_{\rm b}\) (g/cm\({}^{3}\)) & 1.5 to 3.0 & 2.48 & 2.00 & 24.00\% \\ \(n\) & [1,3,5] to [15,17,19] & [7,9,11] & [7,9,11] & 0 \\ \hline \end{tabular}
\end{table} 
Table 3: Optimization range, optimization result, truevalue, and inversion error. tion value, and then utilized the Boltzmann function to weigh the probability distribution. The probability distribution of the parameters of the \(k\)th group is [29]:  
\[\sigma(m_{k})=\frac{\exp(-[E(m_{k})]/T)}{\sum\limits_{l=1}^{N_{\text{obs}}}\exp(- [E(m_{l})]/T)} \tag{22}\]  
where \(N_{\text{obs}}\) is the number of samples, and \(T\) is the average value of the best cost function in the optimization process. The edge probability distribution of the i-th parameter value \(\kappa\) in the sample vector is:  
\[\sigma(m^{i}=\kappa)=\frac{\sum\limits_{l=1}^{N_{\text{obs}}}\exp(-[E(m_{l})] /T)\delta(m_{l}^{i}=\kappa)}{\sum\limits_{l=1}^{N_{\text{obs}}}\exp(-[E(m_{l})] /T)} \tag{23}\]  
where \(\delta\) is the Dirac function. The one-dimensional (1D) PPD of each geoacoustic parameter is shown in Figure 12. The red arrow shows the optimal value of the cost function. Figure 12 shows that Equation (16) was extremely sensitive to \(c_{\text{s}}\), \(d\), and \(n\). The 1D-PPD distribution of these three parameters was narrow, and the optimal value of the cost function coincided with the 1D-PPD maximum value. Meanwhile, \(c_{\text{s}}\), \(d\), and \(n\) could reach the optimal value with a higher probability, which shows that the inversion result was reliable. For the three parameters of \(\rho_{\text{s}}\), \(c_{\text{b}}\), and \(\rho_{\text{b}}\), their 1D-PPD distributions were comparatively dispersed and all of them converged to false parameter values. This result is consistent with the sensitivity analysis. That is, the insensitivity of Equation (16) to these three parameters caused greater uncertainty and unreliable in the inversion results of the three parameters. In summary, it was shown that Equation (16) has a good inversion effect on \(c_{\text{s}}\) and \(d\) through the comparison of the inversion results using the grid search method and genetic algorithm. When the inversion target was \(c_{\text{s}}\) and \(d\), we could quickly obtain relatively accurate (compared to the multi-dimensional optimization process) inversion results of \(c_{\text{s}}\) and \(d\) based on the grid search method and other parameter assumptions combined with empirical formulas and hypothetical empirical values.