# A Computational Platform for Automatic Signal Processing for Bender Element Sensors +
## 3 Description of the GeoHyTE Platform
### Approximation Bases
#### 3.1.3 Model Updating Technique for Shear Modulus Extraction  
The availability of a numerical model that can correctly recover the experimental output signal endorses the development of a model updating technique for the automatic extraction of the shear modulus. The technique exploits a recent fixed-point iterative procedure [15] that seeks to maximise the correlation between the output signals obtained numerically and experimentally. As these signals are both _output_ signals, the fixed-point technique used in GeoHyTE does not rely on the (arguably faulty) assumption of similarity between input and output signals that underlines the conventional time and frequency domain interpretation methods presented in Section 2. The cross-correlation between experimental and simulated output signals \(E(t_{i})\) and \(S(G_{0},t_{i})\), where \(t_{i},\ i=\{1,2,\ldots N\}\) are the sampling instants and \(G_{0}\) is the shear modulus, is mathematically defined as  
\[\chi(G_{0},\tau)=\sum_{i=1}^{N}\frac{E(t_{i})\times S(G_{0},t_{i}+\tau)}{\|E\| \times\|S\|} \tag{4}\]  
where \(\tau\) is the time lag of the second signal in relation to the first. When \(\tau=0\), the instantaneous correlation between the two signals is obtained. Non-null values of \(\tau\) yield the correlation between the original signal \(E\) and time-lagged copies of signal \(S\). Therefore, cross-correlation (4) is a function of the tentative shear modulus used in the numerical model and the time lag, taking values between \(-1\) and \(1\), with \(1\) corresponding to identical signals, \(-1\) to symmetrical signals, and \(0\) to signals with no correlation whatsoever. The model updating technique is described in full, along with its mathematical apparatus, in reference [15]. Here, the outline of the process is given to support the description of the GeoHyTE platform in the following sections. It consists of the following steps:  
**Step 1:** Run the bender element experiment and analyse the experimental output signal. Select the time interval \(t\in[t_{0},t_{end}]\) to cross-correlate experimental and numerical output signals (correlation window). A typical choice is to correlate the signals from the start of the experiment (\(t_{0}=0\)) to an instant immediately after the first large amplitude in the experimental signal. **Step 2:** Based on the experimental output signal, pick a tentative time for the arrival of the shear wave at the receiver. Based on this arrival time, compute the shear wave velocity and a first shear modulus estimate. The initial assessment of the arrival time is the main uncertainty in the process. However, as shown in Section 5, the fixed-point model updating process is robust to faulty choices of the tentative arrival time, meaning that the consequences of this uncertainty are generally minor. **Step 3:** Based on the shear modulus estimate adopted in Step 2, build a computational model of the experiment and use it to obtain the simulated output signal, that is, the time-history of the displacements in the solid phase near the tip of the receiver. This is the output signal simulated numerically. **Step 4:** Compute cross-correlation (4) between experimental and simulated output signals over the correlation window specified in Step 1. Analyse the cross-correlation plot and identify the time lag \(\tau_{max}\) that corresponds to the maximum correlation. **Step 5:** Shift the arrival time of the shear wave with the time lag \(\tau_{max}\) from the previous estimate and recalculate the tentative shear modulus based on this new arrival time. Check whether the new shear modulus is consistent (within some acceptable tolerance) with the tentative shear modulus computed in Step 2. This step updates the tentative shear modulus according to the update function [15],  
\[\mathcal{G}(G_{0})=\frac{h^{2}\rho}{\left(h\sqrt{\frac{\rho}{c_{0}}}+\tau_{ max}\right)^{2}} \tag{5}\]  
Then, the updated shear modulus is compared with the previous estimate.