## 4 Simulation Experiments  
The ULMA-based borehole PEC system for downhole casing inspections was validated through field experiments conducted in standardized \(5^{1/2}\)-inch metal casings (thickness 7.72 mm; outer diameter 139.7 mm). To illustrate the effectiveness of the proposed method, the thickness of the metal casing was changed to varying degrees from the nominal thickness, with wall thickness variations from top to bottom of 3 mm, 1 mm, 2 mm, and 4 mm, corresponding to the lengths of the varied segments of 15 cm, 10 cm, 10 cm, and 15 cm, respectively (as shown in Figure 3). Standard PEC measurements of the above metal casing were performed using an 8-element linear multicoil array in accordance with the standardized experimental procedure. The relevant models and experimental results were processed using the MATLAB platform on a system using a 16-core processor with 32 GB of RAM and a 1 TB hard disk. The parameters of the experiment and the multicoil array sensor are listed in Table 1. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameters** & **Symbol** & **Value** \\ \hline Number of receiving coils & \(M\) & 8 \\ Inter-element spacing & \(\Delta z\) & 20 mm \\ Transmitting-receiving distances & \(z_{1}-z_{M}\) & 20–160 mm \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of the simulation experiment. Figure 3: Configuration of the experimental metal casing. By analyzing the PEC signals in this experiment, the distribution of each element in \(\mathbf{g}(t,\,d_{m})\) (normalized by the maximum value) as a function of \(p\) is obtained, as shown in Figure 4a. It can be seen that the term \(\mathbf{g}(t,\,d_{m})\) decays in an exponential-like manner as the stage \(p\) increases (i.e., the signal characteristics are dominated by the first several elements), verifying that \(\mathbf{g}(t,\,d_{m})\) is sparse. To make it easier to understand, we provide the percentage contribution of each element with different \(p\)-values to the overall signal power (as shown in Figure 4(b)). Obviously, for the vector \(\mathbf{g}(t,\,d_{m})\), the contribution of the first eight elements could account for more than 90%, while the contribution of the other elements was basically negligible. Therefore, by exploiting this property, it is possible to reduce the number of constraints from \(P\) to \(K\), thus providing enough DoFs to obtain accurate weight coefficients. Initially, we determined \(K<8\) to avoid the pathological problems that would lead to the failure of the weight coefficients. Combining the balanced relationship between model approximation accuracy and computational cost, the original full-dimensional LCMV with \(P=20\) could be reduced to an S-ICMV with \(K=6\) in this experiment. In other words, the number of constraints was reduced by 70% from \(P\) for LCMV to \(K\) for S-ICMV. In addition, according to the relevant definitions in Equations (24) and (25) in Section 3, the proposed method was capable of reducing the computational cost by about 87% from the original. It should be noted that the selection of the value of \(K\) is mainly related to the calculation accuracy of the weight coefficients and the model approximation accuracy. Specifically, the larger the value of \(K\), the more the signal model approximates the actual situation, but it introduces a larger amount of computation, and even when \(K>M\), it is difficult to obtain effective weight coefficients. Conversely, the smaller the value of \(K\), the amount of computation is greatly reduced, which is conducive to the accurate calculation of the weight coefficients, however, an excessively small \(K\) leads to a large deviation between the signal model and the actual signal, thus creating a large inspection error. Therefore, the selection of \(K\) is the process of realizing the balance between the calculation accuracy of the weight coefficients, the calculation speed, and the model approximation accuracy.