# Analysis of Elastic Nonlinearity Using Continuous Waves: Validation and Applications
## 4 Efficiency and Robustness of the Approach
### Dependence on Calibration Errors  
One of the major sources of possible errors in the proposed approach is related to the numerical calibration procedure. In Figure 8, the effects of errors on the calibration parameters \(U_{0}\) and \(\phi_{0}\) (indicated with \(w_{U_{0}}\) and \(w_{\phi_{0}}\), respectively) in the estimation of quadratic nonlinear parameters are shown. Here, the relative errors in velocity and attenuation coefficient are plotted, obtained from the analysis of the simulation data with respect to those theoretically expected, in the case of quadratic nonlinearities imposed both on modulus and damping.  
The error in the quantitative determination of the nonlinear parameters increases roughly proportionally to the errors in the calibration procedure. It is interesting to note that errors in the calibration amplitude \(U_{0}\) have negligible influence on the attenuation coefficient (Figure 8d). On the contrary the estimation of the attenuation nonlinear parameter \(\delta_{\alpha}\) is extremely sensitive to errors in the calibration phase \(\phi_{0}\) (Figure 8b). Instead, the influence of errors on \(U_{0}\) and \(\phi_{0}\) in the estimation of the velocity nonlinear quadratic parameter \(\delta_{c}\) is of the same order of magnitude.  
It is worth pointing out here that in experiments calibration of the phase is always very accurate, thus errors induced by calibration could be confined to a range of few per cent. Concluding this section,  
\begin{table}
\begin{tabular}{c c c c c c} \hline
**NL** & & **Case 1** & **Case 2** & **Case 3** & **Case 4** \\ \hline \multirow{2}{*}{\(\delta_{c}\)} & \(\delta_{c}^{th}\) & \(-5.00\times 10^{6}\) & 0 & \(-5.05\times 10^{6}\) & \(-5.06\times 10^{6}\) \\  & \(\delta_{c}^{fit}\) & \(-5.01\times 10^{6}\) & \(\approx 0\) & \(-5.07\times 10^{6}\) & \(-4.23\times 10^{6}\) \\ \hline \multirow{2}{*}{\(\gamma_{c}\)} & \(\gamma_{c}^{th}\) & 0 & 0 & 0 & \(-5.07\times 10^{15}\) \\  & \(\gamma_{c}^{fit}\) & \(\approx 0\) & \(\approx 0\) & \(\approx 0\) & \(-6.31\times 10^{15}\) \\ \hline \multirow{2}{*}{\(\delta_{\alpha}\)} & \(\delta_{h}^{th}\) & 0 & \(3.39\times 10^{8}\) & \(3.51\times 10^{8}\) & \(3.51\times 10^{8}\) \\  & \(\delta_{a}^{fit}\) & \(\approx 0\) & \(3.38\times 10^{8}\) & \(3.56\times 10^{8}\) & \(3.54\times 10^{8}\) \\ \hline \end{tabular}
\end{table}
Table 1: Theoretical nonlinear parameters (input of the simulations) and derived nonlinear parameters obtained as described in the main text: Equation (22). We analysed the data shown in Figure 3.  
it is also notable to highlight that errors in the calibration do not affect at all the determination of the functional dependence of velocity and attenuation coefficient on strain.  
Figure 8: Influence of the calibration errors in the quantification of quadratic nonlinearity parameters. Plots of variations of the nonlinear parameters \(\delta_{c}\) and \(\delta_{a}\), respect to their theoretical values: (**a,b**) varying the calibration error \(w_{U_{0}}\) on \(U_{0}\); (**c,d**) varying the calibration error \(w_{\phi_{0}}\) on \(\phi_{0}\).  
Figure 7: Results of the analysis in the case of quadratic nonlinearities \(S\) and \(\lambda\), varying frequencies of the source: (**a,b**) amplitudes and phases of the signals vary in different ways depending on the source frequency; and (**c,d**) \(\Delta c/c_{0}\) and \(\Delta\alpha/\alpha_{0}\) calculated with the MoDaNE approach, are frequency independent as theoretically expected for the simulations performed.