## 1 Introduction
The disadvantages of these methods are the large number of calculations and complexity of the implementation scheme. For other interference factors mentioned above, it would also affect the extraction of pipeline defect signal [9]. The non-parametric detection methods have also been studied for MAD, such as high order crossing method [25] and minimum entropy detector [26]. Since both methods do not need actual assumptions about the target signal, this guarantees them a convenient implementation. However, low signal-to-noise (SNR) may limit their detection performance, thus affecting the accuracy of MAD. [PERSON] [27] used an Empirical Mode Decomposition (EMD)-based energy detector to extract magnetic anomaly signal, which is only suitable for the signal under noise. [PERSON] [28] proposed the EMD algorithm for nonlinear and non-stationary signals, which is based on data-driven algorithm and has no reliable mathematical basis. Optimization algorithms such as Ensemble Empirical Mode Decomposition (EEMD) and Complementary Ensemble Empirical Mode Decomposition (CEEMD) are proposed based on EMD, but these methods still cannot completely solve the mode aliasing problem in background noise [29; 30]. The Empirical Wavelet Transform (EWT) [31] algorithm was proposed to extract the different modes from a signal, which combines the advantages of EMD and Wavelet Transform (WT), using WT to compute many wavelet transforms along the time axis. However, when the signal contains mode overlapping or close modes, the ability of separating wide-band signal is still limited for MAD. To analyze the wide-band nonlinear chirp signal, [PERSON] [32] proposed a method called Variational Nonlinear Chirped Mode Decomposition (NVCMD). Therefore, we use this algorithm to analyze the actual signal, and find that there are some problems in the stability and there are certain requirements for the composition of the mixed signal. More recently, the Variational Mode Decomposition (VMD) [33] algorithm was successfully used to filter the pipeline defect signal by the de-trended fluctuation analysis [34]. Since the geomagnetic background signal is a kind of nonlinear and non-stationary signal, it is critical to study the bandwidthestimation based on the modal Instantaneous Frequency (IF) and Instantaneous Amplitude (IA) spectrum. For the extraction of pipeline damage signals, the VMD algorithm still seems too redundant, because it needs to analyze all signal modal components before stopping calculation. Moreover, EMD and VMD can only separate modal components, but cannot directly extract mode characteristics such as IF and IA [32]. Variational Mode Extraction (VME) [35] is a low-frequency modal signal extraction algorithm, which has a similar mathematical basis as VMD. However, the algorithm is based on the assumption that the residual signal after extracting a specific mode should not contain energy at the mode center frequency. We tried to find a more effective technique for MAD to make up for the shortcomings of existing detection methods. Inspired by VMD, NVCMD, and VME algorithms, we propose a novel Variational Specific Mode Extraction (VSME) method to analyze the pipeline defects signals which can easily be submerged in geomagnetic background noise. The proposed method has a reliable mathematical foundation as VMD, which is based on Wiener filtering, variational decomposition, and Hilbert transform [33]. In addition, VSME has another theoretical basis that a wide-band signal can be converted into a narrow-band signal by matching demodulation transform technique [36]. Furthermore, the problem of wide-band signal decomposition is expressed as an optimal demodulation problem, which can be solved by Alternative Direction Method of Multipliers (ADMM) [37]. Therefore, we convert the problem of pipeline defect signals extraction into a demodulation problem. Actually, the proposed algorithm can be regarded as extracting a Time-Frequency filter with a specific center frequency, and its characteristics eliminate the need to determine the number of signal modes.