## 3 Results and Discussion
### Experimental Results
The RMSEs of the ESPRIT method with fixed thresholds of \(-40\) and \(-30\) dB were 184 and 143 rad/m, respectively. As shown in Figure 9, smaller thresholds tend to make the estimates unstable because the ESPRIT algorithm treated the noise as the desired signal. In contrast, larger thresholds can miss the weaker modes as shown in Figure 10. The threshold should thus be determined manually. The results are sensitive to the threshold value, so it is difficult to determine the threshold. Figure 8: Depiction of the wavenumbers obtained using the proposed method and the spectrum obtained using the conventional method at 1.0 MHz. This is the cross-sectional view at the white dotted line shown in Figure 7. The vertical blue dotted lines indicate the theoretical values. Figure 11 shows the wavenumbers for values of \(N_{\text{sub}}=5\) and 26. The red cross marks and blue circles show the results obtained for \(N_{\text{sub}}=5\) and 26, respectively. The black lines show the theoretical curves. The smaller sub-array had lower resolution and missed many of the modes, while the larger sub-array caused false estimates. We used a frequency window, denoted by \(\omega_{\text{w}}\), of 0.5 MHz. When we used \(\omega_{\text{w}}\) values of 0.1 and 0.75 MHz in the experiments with the bone-mimicking plate, the RMSEs obtained were 144 and 118 rad/m, respectively. Therefore, the dependence on \(\omega_{\text{w}}\) was insignificant and a larger window width provided stable estimation. We chose a DL factor of \(\eta_{2}=-40\) dB. Because this value was added for stabilization when we analyzed the frequency range using a significantly smaller signal intensity, a smaller value was selected. When we used a larger value, more stable estimates were obtained as a result but the weaker modes were missed by the method. Figure 10: Wavenumbers of the 4-mm-thick bone-mimicking plate estimated using the conventional ESPRIT algorithm at a fixed threshold of \(-30\) dB and the proposed method. Red dots show the estimates obtained using the proposed method and blue cross marks show the estimates obtained using the conventional method. Solid gray lines show the theoretical curves. Figure 9: Wavenumbers of the 4-mm-thick bone-mimicking plate estimated using the conventional estimation of signal parameters via rotational invariance technique (ESPRIT) algorithm at a fixed threshold of \(-40\) dB and the proposed method. Red dots show the estimates obtained using the proposed method and blue cross marks show the estimates obtained using the conventional method. Solid gray lines show the theoretical curves. We used the ESPRIT algorithm in this study. The novel advantage of the ESPRIT algorithm when used for wavenumber estimation of a guided wave propagating along the cortical bone is that it can estimate the wavenumber without the need for a peak search process. To extract the wavenumber from the \(f\)-\(k\) spectrum image when using conventional methods such as the Radon and SVD-based methods, multiple one-dimensional peak search processes are required [6; 20]. While the computational cost is not high, parameters such as the intensity threshold and prominence must be set. Thus, the performance of the conventional method such as RMSE depends on the additional parameter setting.