# A Point Crack Source Location Method without Velocity Information in Anisotropic Plates
## 5 Discussion  
We developed a defect location estimation AE method that uses only the TOA and does not require the physical properties of the object (e.g., modulus of elasticity, wave propagation velocity, and density) for anisotropic materials using the nonlinear least squares method. When we implement the method, it is necessary to choose a parameter \(\lambda\) in Equation (5). In this study, the choice of \(\lambda\) was empirical but not random. We decided that \(\lambda\) was similar to the magnitude of the gradient of \(f\) (i.e., \(|\
abla f|\)). Note that in the minimization process, a large \(\lambda\) provides a small \(\mathbf{s}\), so we needed to decide on an appropriate size for \(\lambda\). The gradient descent method was used to find the minimum value in the SDMM applied to the experiments in this paper. Other optimization techniques (e.g., the Newton method) aside from the gradient descent method can be used to obtain the minimum value of the SDMM. The intermediate process of applying the optimization technique differed depending on which method is used, but there was no significant difference in the application results. Since the result of the optimization technique depends on the initial value, it is important to set the initial value well when applying the SDMM. As a result of applying various initial values to the SDMM, it was confirmed that the estimated position result was close to the source position even if the initial estimated position was not accurate. On the other hand, if the initial velocity ratio \(\alpha\) is not accurate (for example, if the direction of anisotropy is the opposite), then defect location estimation will not work properly. However, when knowing whether the direction of anisotropy is horizontal or vertical, defect location estimation works well. Other studies [19,20] also used minimization of an objective function to locate the source coordinates without any knowledge of the elastic properties of the material. However, those papers required using a group of sensors arranged in an L-shape, while our study did not require any specific sensor alignments. In addition, the novelty of our study lies in devising the concept of time offset for TOA error correction due to signal noise and applying it to the objective function. In [19,20], the number of unknowns was reduced by differentiating the ellipse equation, and the differential elements according to the differentiation were replaced with information from the L-shape. Therefore, the L-shape arrangement in which the three sensors are arranged to form a right isosceles triangle was essentially used to reduce the number of unknowns. In our study, the number of unknowns was reduced by solving the system of two ellipse equations. Therefore, a special sensor arrangement is not required. In addition, the novelty of our study lies in devising the  
Figure 8: Result figures of PLB testing on CFRP with six different PLB positions, which are shown as red marks in (**a**–**f**). The estimated locations from (**a**) to (**f**) are indicated by blue plus signs (+), and the positions of the sensors are indicated by black dots. Relative errors for all six different PLB positions are shown in (**g**). concept of time offset for TOA error correction due to signal noise and applying it to the objective function. Our method did not make any assumptions about sensor placement. In most cases, our method found crack locations for arbitrary sensor deployments, but in some special cases (e.g., sensors arranged in a straight line and a crack not on the line), the proposed method may produce false location results due to symmetry of the sensors. However, except for these special cases, crack location error due to sensor placement does not occur. Please note that the time offset values may have different values for different sensors (i.e., \(s_{i}\
eq s_{j}\) for \(i,j=1,\ldots,N\), where \(s_{i}\) indicates the time offset value at the \(i\)th sensor).