## 3 Simulation Results and Empirical Results Verifications
### Simulation Results
As a result of nonlinear parameters that were obtained from Figures 5-7, it can be observed that if \(\sigma\) is greater than 0.05, then the amplitudes and phase are complex and varying. When \(\sigma\) = 0.1 and 0.15, different nonlinear parameter curves are observed. Simulated measurement errors (\(\sigma\)= 0.01, 0.05, 0.15) of the nonlinear parameters are shown in Figures 5-7, respectively. The figures show uniformity of the curves at \(\sigma\)= 0.01 and \(\sigma\)= 0.05. However, curves of the \(Bl(\mathbf{x})\) at \(\sigma\)= 0.01 and \(\sigma\)= 0.05 clearly differ from the exact curve at \(\sigma\)= 0.0 (Figures 5a, 6a and 7a). It should be noted that the difference is mainly  
Figure 3: Comparison of exact and inverse values, (**a**) \(Bl(\mathbf{x})\), (**b**) \(K_{m}(\mathbf{x})\), (**c**) \(L_{e}(\mathbf{x})\), and (**d**) \(R_{m}(\mathbf{x})\). Figure 7: _Cont._  
Figure 6: Effect of measurement error \(\sigma\)\(=\) 0.05 on nonlinear parameters, (**a**) \(Bl(\mathbf{x})\), (**b**) \(K_{m}(\mathbf{x})\), (**c**) \(L_{e}(\mathbf{x})\), and (**d**) \(R_{m}(\mathbf{x})\). With reference to the \(K_{m}(\mathbf{x})\) and \(R_{m}(\mathbf{x})\) curves (Figures 5b,d, 6b,d and 7b,d), the \(\sigma\)\(=0.01\) and \(\sigma\)\(=0.05\) simulated measurement error in displacement curves (as shown in Figure 4) were almost identical, however more evident deviation have been observed in the \(\sigma\)\(=0.15\) curve. An error deviation mainly occurred in the central parts of the curves rather than the extremes of the \(K_{m}(\mathbf{x})\) and \(R_{m}(\mathbf{x})\) curves. The \(\sigma\)\(=0.05\) and \(\sigma\)\(=0.15\) error curves overlapped in the extreme parts of the curves and started deviating in the central part of the higher side of \(K_{m}(\mathbf{x})\). A clear difference was observed in the \(L_{e}(\mathbf{x})\) curve at the error \(\sigma\geq 0.05\) (Figures 6c and 7c). The results showed that when the standard deviation was lower than the simulated measurement error of \(\sigma\)\(=0.01\), all of the nonlinear parameters matched the positive curve. The greater changes were observed in the predicted parameters when the simulated measurement error was greater than \(\sigma\)\(=0.05\). The parameters only roughly matched the trend of the positive curve, particularly in the calculation of \(Bl(\mathbf{x})\), \(K_{m}(\mathbf{x})\), and \(R_{m}(\mathbf{x})\). The \(L_{e}(\mathbf{x})\) calculation curve showed that the predicted trend did not match the \(L_{e}(\mathbf{x})\) predicted curve. As indicated by the results that are shown in Figures 5-7, when \(\sigma\geq 0.05\), the estimated nonlinear curves cannot fit the original curves accurately; however, the results remained within the acceptable range when \(\sigma\)\(<0.05\). The comparison of impedance curves in Figure 8a-c show that, unlike BFGS with constraint, the BFGS and BFGS-Brent methods failed to fit the exact curve. However, as the simulated measurement error is increased, the three methods could not accurately match the precise values of the impedance. When the measurement error was small, the changes in the curve were considerable. The results indicated that, in the case of large measurement error, the proposed method could not accurately determine the nonlinear parameters.