## 1 Introduction  
The eddy-current technique has been widely applied to resource exploration [1], environmental investigations [2], and stress measurements [3]. A variant of the eddy-current technique called the pulsed eddy-current (PEC) technique [4, 5] has been widely adopted for the nondestructive evaluation (NDE) of wellbore casings because it accesses the target without making physical contact. Furthermore, it identifies cracks, pits, and corrosion in metallic infrastructures without requiring surface preparation [6, 7, 8]. Casing defects are evaluated from broad-frequency-range data considering the electrical and geometrical parameters of the individually estimated layer [9, 10]. However, unlike surface measurements, NDEs of downhole casings are degraded by high temperatures and metal tool housings against high pressure, and the small operating space downhole can also limit the inspection results [11, 12, 13]. Considering the harsh environment of actual downhole detection, array-based PEC probes and corresponding novel signal processing methods have been proposed to improve the performance of the NDE of downhole casings. References [14, 15, 16, 17, 18] designed and controlled a transmitting array consisting of multiple transmitter coils to achieve magnetic field focusing, thus enhancing the longitudinal resolution of the PEC inspection system. Similarly, the authors of [19; 20] investigated the performance of downhole NDE using a receiving array consisting of multiple receiver coils with different transmitting-to-receiving distances (TRDs). The different TRDs resulted in \"phase shifts\" between the detection curves obtained from different receiving elements, which affected the accuracy of the NDE of downhole casings. To address this problem, the authors of [21] identified the correlations among the receiving elements and weighted the array output using the linear constrained minimum variance (LCMV)-based method, originally developed for radar and sonar detection. The LCMV-based method effectively eliminates the influence of the TRD on the array output. Reference [22] further applied the LCMV-based method to a downhole PEC detection system with a multiple-input-multiple-output (MIMO) structure by increasing the number of channels to reduce the system complexity while maintaining the performance. The above studies show that configuring the transmitting and receiving arrays as uniform linear multicoil arrays (ULMAs) and applying a corresponding signal-processing method can effectively improve the NDE performance and accuracy of downhole PEC systems. Although ULMAs enable the NDE of downhole casings, their degrees of freedom (DoFs) are severely constrained within the limited physical space and borehole conditions in practical applications, thus introducing a pathological problem caused by the DoFs being much smaller than the number of constraints [23]. Using the standard LCMV method for array signal processing can make it difficult to obtain valid weight coefficients due to the irreversibility of the matrix. Diagonal loading (DL) [24; 25; 26] has been shown to help improve the computational accuracy of the weight coefficients, but it is an empirical value-based method with a computationally intensive process and is not practically usable. Recently, sparse optimization has been proven to be effective in reducing computational complexity and beneficial in solving underdetermined problems [27; 28; 29]. To prevent the DoFs falling below the number of constraints, we propose a sparsity-based method for the NDE of downhole casings using the ULMA. First, the ULMA-based borehole PEC signal model was constructed and the Gauss-Legendre quadrature representation of the PEC signal was obtained. Through the structural transformation of the PEC signal, it was observed that the signal power decreased significantly with the increase in the Legendre polynomial stage, and the sparsity of the PEC signal was then analyzed and proven.