## 4 Simulation Experiments
To analyze the performance of the proposed method, we processed the measured data using the standard LCMV algorithm, the DL-based method, the MIMO-based method, and the proposed S-ICMV algorithm. Figures 5 and 6 show the weighted-sum results of the 8-element ULMA at late and early times, respectively. As can be seen from Figures 5a and 6a, due to the sufficient DoFs for controlling the constraints in S-ICMV, it was possible to obtain an accurate and effective weighting coefficient \(\mathbf{w}_{s}\), which could well compensate for the phase differences between the receivers and formed a smooth EMF curve for detection. Note that the amplitude of the weighted induced EMFs monotonically increased with the thickness of the metal pipe, revealing a relationship between the induced EMFs and pipe thickness. In contrast, for the standard LCMV algorithm (shown in Figures 5b and 6b), it was difficult to obtain the accurate weighting coefficient \(\mathbf{w}\) because Equation (5) included an excessive number of constraints, resulting in poor compensation of the phase difference between the receivers. Figures 5c and 6c demonstrate the processing results of the MIMO-based method. It can be seen that the method improved the weighted-sum effect \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameters** & **Symbol** & **Value** \\ \hline Number of transmitting coil turns & \(N_{\text{T}}\) & 19 \\ Number of receiving coil turns & \(N_{\text{R}}\) & 62 \\ Radius of the multicoil array sensor & \(r_{1}\) & 12 mm \\ Tool housing inner radius & \(r_{2}\) & 18.5 mm \\ Tool housing outer radius & \(r_{3}\) & 21.5 mm \\ Standardized casing inner radius & \(r_{4}\) & 62.13 mm \\ Standardized casing outer radius & \(r_{5}\) & 69.85 mm \\ Cement ring outer radius & \(r_{6}\) & 88.9 mm \\ Excitation current & \(I_{\text{T}}\) & 0.3 A \\ Observation time interval & \(t\) & 0–50 ms \\ Step size & \(\Delta t\) & 1 ms \\ \hline \hline \end{tabular}
\end{table} 
Table 1: _Cont._compared to the standard LCMV, but its consistency with the actual casing variation was not high enough. In addition, due to the structure of the probe with multiple transmitters and multiple receivers, the large TRD made the SNR of the signal poor, and the EMF curve was not smooth enough. Moreover, Figures 5d and 6d demonstrate the experimental results of the DL-based method. It can be seen that the DL-based method could improve the detection effect and obtained a similar performance as the S-LCMV algorithm. However, DL is an empirical-value-based method with a complicated process and tends to result in a huge computational volume, making it not practically usable. In comparison, the S-LCMV algorithm is not only able to avoid the pathological problem by exploiting the sparsity characteristics of the signals, but also possesses strong practical usability, which effectively achieves a good application of array weighting in downhole signal models. Figure 4: The sparsity of \(\mathbf{g}(t,d_{m})\). (**a**) Normalized \(\mathbf{g}(t,d_{m})\); (**b**) percentage contribution of each element. Figure 5: Weighted induced EMFs of an 8-receiver linear multicoil array at a late time. (**a**) S-LCMV; (**b**) LCMV; (**c**) MIMO; (**d**) DL. After obtaining the weighted sum of the induced EMFs, the detection curves needed to be further interpreted to obtain the number and location(s) of the casing changes. For the data processed by the three different methods above-mentioned, the constant false alarm rate (CFAR) [32; 33] detector, which is frequently used for target detection, was employed, and the detection results are demonstrated in Figures 7 and 8. Figures 7a and 8a demonstrate the CFAR detection results based on the proposed S-LCMV method with \(\mathbf{w}_{\text{s}}\), where four target regions (in which the pipe was thicker than the nominal pipe) were clearly detected and their corresponding positions and sizes were highly consistent with the actual situation.