# A Computational Platform for Automatic Signal Processing for Bender Element Sensors +
## 5 Robustness of GeoHyTE Predictions
### Robustness to Erroneous Initial Shear Modulus Estimate  
We consider, again, the testing setup used in Section 4.2 to illustrate the conventional and model updating interpretation techniques. To evaluate the convergence properties of the fixed-point method implemented in GeoHyTE, the iterative process is performed starting from 15 points within the correlation window, as shown in Figure 24.  
The objective of the study is not only to show the ability of GeoHyTE to converge to consistent solutions, but also to sketch a plot of the update function (5) in order to study the mathematical grounds of this ability. Therefore, not all initial points in Figure 24 are practically reasonable. For instance, points \(A\) to \(G\) precede all significant oscillations in the output signal and therefore their choice as plausible arrival times is probably not reasonable. Likewise, point \(O\) is posterior to large amplitudes in the output signal, so the arrival of the shear wave probably occurs earlier. Nevertheless, the fixed-point iterative process is convergent for all starting points, within no more than four iterations. The values of the shear moduli obtained during the iterative processes are shown in Table 5. It is noted that notation \(G_{0}^{i}\) designates the shear modulus used in iteration \(i\). The convergence process is graphically depicted in Figure 25. The vertical axis, where the values of the shear moduli are plotted, is logarithmic to endorse a clearer observation of the convergence patterns. The convergence process is successfully completed for all starting points and the final shear modulus predictions vary between 51.70 MPa and 53.20 MPa. The average shear modulus value is 52.20 MPa and the standard deviation of the predictions is 0.541 MPa. To better understand the mathematical justification of the speed of the convergence process, the update function defined by Expression (5) is plotted with a solid line for 25 values of \(G_{0}\) in Figure 26. The tentative shear modulus at the start of an iteration is represented on the abscissa. The updated shear modulus after the iteration is completed is plotted on the vertical axis. Bots axes are logarithmic. The fixed point axis defined by \(f(G_{0})=G_{0}\) is also plotted with a dashed line. Figure 24: Starting points for the fixed-point procedure. For reader's convenience, the iterative process starting from point \(A\) (see Table 5) is illustrated with blue arrows in Figure 26. It starts from the tentative shear modulus \(G_{0}^{1}=2321.10\) MPa which is assumed to lay on the fixed point axis. After the first iteration, GeoHyTE delivers a better approximation of the shear modulus, \(G_{0}^{2}=1019.40\) MPa, which is, again, assumed to be a fixed point. The process is repeated, yielding, successively, \(G_{0}^{3}=66.10\) MPa and \(G_{0}^{4}=52.90\) MPa, with the last point being confirmed as a fixed point within the stipulated tolerance. The fixed point is marked with a red dot in Figure 26.  
The fixed point iterative process is convergent if the (absolute value of the) derivative of the update function in the fixed point is less than one. The lower the value, the faster the convergence. For our case, the update function has a single fixed point, as the solid line intersects the fixed point axis in a single point. This is an important advantage of the fixed-point technique implemented in GeoHyTE over conventional correlation maximisation methods. Indeed, the correlation function is known to have multiple local extrema [15], but a single fixed point is identified in this study. This explains why (roughly) the same fixed point is found regardless of the starting point of the iterative process, while the presence of local correlation extrema means that a conventional maximisation algorithm cannot be guaranteed to reach the global maximum.