ICEEMDAN is an improved algorithm of CEEMDAN, which replaces the Gaussian white noise by the kth-order intrinsic mode function (IMF) component noise resulting from the original EMD during the CEEMDAN process. The process of ICEEMDAN is as follows (Marcelo et al., 2014).

Step 1: Adaptive Gaussian white noise is added to the original signal:

According to the ICEEMDAN process, the C I of IFM components of B x , B y , and B z , shown in Figure 4A, exhibits an initial trend of rapid decline followed by a minor increase, and then a further decline. The local maximum value of B x was located at I M F 7 , and the local maximum values of B y and B z were located at I M F 8 , which is consistent with the characteristics of the IFM components of B x , B y , and B z . Figures 4B–D reveals that the eighth-order components of B y and B z are mainly low-frequency signals with obvious defect signal characteristics, while the first- to seventh-order components are mainly high-frequency noise signals. The seventh-order component of B x has a more obvious characteristic of the micro defect, while the first- to sixth-order components of B x are mainly high-frequency noise signals.

Using I M F 7 as the transition component of B x and I M F 8 as the transition component of B y and B z , the ICEEMDAN denoising results of the enhanced magnetic memory signal of the micro-defect are shown in Figure 5. The results indicate that the signal anomalies of B x , B y , B z , and their gradient values were significantly enhanced at the micro-defect location. B x presents a peak and G B x a peak and valley at the location the micro-defect. Furthermore, B y and B z present peaks and valleys, and G B y and G B z present valleys at the location of the micro-defect. Nevertheless, due to the presence of noise signals in the transition component obtained via ICEEMDAN, a certain amount of noise was still present in the non-defect area, resulting in fluctuations in the signal of this area. When calculating the gradient values for each component of the magnetic field, the noise signals were further amplified, causing a significant increase in the fluctuation amplitudes of G B x , G B y , and G B z in the non-defect area, leading to difficulties in defect identification using G B x , G B y , and G B z . Moreover, the components before the transition component often contained defective feature signals, especially the first component I M F k − 1 before the transition component I M F k . Directly discarding I M F k − 1 often resulted in the loss of useful signals.

VMD is an adaptive denoising method based on EMD. The decomposition process is based on the following formula (Zhang et al., 2022): min u m , ω m ∑ m ∂ t δ t + j / π t ) * u m t e − j ω m t 2 2 s . t . ∑ m = 1 M u m = B t , (7) where m is the number of modal decompositions, u m is the m ^th-order mode corresponding to the decomposition, ω m is the center frequency corresponding to the m ^th-order mode, δ t is the Fermi–Dirac distribution, * denotes the convolution, and B t is the detection signal. To solve the constrained variational problem, an augmented Lagrangian function L u m , ω m , λ is introduced: L u m , ω m , λ = α ∑ m ∂ r δ t + j / π t * u m t e − j ω m t 2 2 + B t − ∑ m ν m t 2 2 + λ t , B t − ∑ m ν m t (8) where λ denotes the Lagrange multiplier, and α denotes the penalty factor. After being updated, the μ m , ω m , λ is calculated as shown in Equation 9: μ ^ m n + 1 ω = f ^ ω − ∑ i ≠ m μ ^ i ω + λ ^ ω 2 1 + 2 α ω − ω m ω ^ m n + 1 ω = ∫ 0 ∞ ω μ ^ m n + 1 ω 2 d ω ∫ 0 ∞ μ ^ m n + 1 ω 2 d ω λ ^ n + 1 ω = λ ^ n ω + γ f ^ ω − ∑ m μ ^ m n + 1 ω . (9)

During the VMD, it is necessary to determine different optimal penalty factors α and the number of decompositions m based on the complexity of the signal. According to experience and the comparison of the decomposition effects for different α and m , the optimal α and m combinations for B x , B y , and B z were respectively determined to be ( α , m ) = (5 × 10⁵, 7), (7 × 10⁵, 7), and (1.2 × 10⁶, 15). The denoising results of the VMD are shown in Figure 6.

The results indicated that the signal anomalies of B x , B y , B z , and their gradient values were more conspicuous at the site of the micro-defect after VMD denoising. B x presents a peak and G B x presented a peak and valley at the location micro-defect. B y and B z present peaks and valleys, respectively, G B y and G B z present valleys at the site of the micro-defect. Compared with the signals after ICEEMDAN denoising, the signal obtained by VMD denoising had smaller signal fluctuations in the non-defect region, suggesting that VMD could eliminate more noise signals. Nevertheless, there were minor continuous fluctuations in the gradient values. The main reason for these fluctuations some extremely small noise signals in B x , B y , and B z after VMD denoising, which would affect the subsequent extraction of micro-defect feature signals.

In response to the deficiencies of the ICEEMDAN and VMD methods, the ICEEMDAN–VMD joint signal denoising method is proposed, the process of which includes main four steps. The steps are shown in Figure 7 and described as follows:

Step 1: ICEEMDAN is performed on the original enhanced magnetic memory signals of the micro-defect to obtain m IMF components.

Compared with ICEEMDAN, ICEEMDAN–VMD can effectively remove noise signals in the transition components, reduce signal noise in non-defect areas, and make the gradient values more stable, as well as minimize the loss of useful signals to the greatest extent. Compared with VMD, ICEEMDAN–VMD can effectively solve the problem of the difficult selection of penalty factors and the number of decomposition layers and avoids the problem of small fluctuations in the signal gradient values caused by the extremely small noise signals after VMD denoising.

The ICEEMDAN–VMD denoising method was used to process the detected signals of a micro-defect. The selection of penalty factor α and the number of decomposition layers m for the VMD method was made according to the predictions outlined in Xiaoya et al. (2022), with the stipulated values set to α = 3,600 and m = 4, with the results shown in Figure 8. The results indicate that the ICEEMDAN–VMD denoising method could effectively reduce the influence of noise signals on the defect characteristic signals B x , B y , and B z as well as their gradient values, and it could significantly reduce abnormal signal fluctuations in non-defect areas, making the signal anomalies at defect locations clearer.

Comparing the signals processed by ICEEMDAN, VMD, and ICEEMDAN-VMD, the signal components and their gradient curves processed by ICEEMDAN-VMD are smoothest, especially in non-defect areas where the signal fluctuation amplitude is smallest (Figure 9).

Furthermore, to quantitatively compare the improved effect of the ICEEMDAN–VMD method, the SNR and FE were selected to evaluate its effect. The SNR is the ratio of useful signal energy to noise energy; the larger the SNR, the better the denoising effect. Since the original pure signal could not be obtained, the maximum amplitude of the useful signal was selected for comparison with the maximum amplitude of the noise signal for calculation. The SNR formula is shown as follows (Ren et al., 2021): S N R = 20 ⁡ lg B t N t , (10) where B t represents the maximum magnitude of the useful signal, and N t represents the maximum magnitude of the noise signal.

FE serves as a measurement of the probability that a time series will generate new patterns when the dimensionality varies. The higher the probability that the time series generates new patterns and the higher the complexity of the time series, the higher the FE of the time series. Therefore, the smaller the FE of the denoised signal, the better the denoising effect (Zhu et al., 2024) The FE calculation formula is: F E m , n , r = lim N → ∞ ln ⁡ φ m n , r − ln ⁡ φ m + 1 n , r φ m n , r = 1 N − m ∑ i = 1 N − m 1 N − m − 1 ∑ j = 1 , j ≠ i N − m D i j m , n , r D i j m , n , r = e − d X i m , X j m n r , (11) Where N denotes the test sample data, and X i m is the vector of the signal in m dimensions. X i m = x i , x i + 1 , … , x i + m − 1   i = 1 , 2 , … , N − m + 1 , d X i m , X j m n r is the maximum absolute value of the difference between the element values, r denotes the similarity tolerance, and n denotes the similarity tolerance boundary gradient.

According to the results in Table 1, for B x and B y , the SNRs after VMD denoising were the highest, followed by those after ICEEMDAN–VMD denoising, and the SNRs after ICEEMDAN denoising were the lowest. In terms of B z , the SNR after ICEEMDAN–VMD denoising was the highest, followed by that after VMD denoising, and the SNR after ICEEMDAN denoising was the lowest. The SNR of B z after ICEEMDAN–VMD denoising was improved by 51.25% and 10.94%, respectively, compared with those after ICEEMDAN and VMD denoising. For the gradient values G B x , G B y , and G B z , the SNRs after ICEEMDAN–VMD denoising were higher than those after ICEEMDAN denoising, increasing by 17.03%, 67.73%, and 445.85%, respectively, and the ICEEMDAN–VMD yielded an order-of-magnitude improvement in the SNR compared with VMD.

According to the results in Table 2, the FEs of B x , B y , B z , and their gradient values after ICEEMDAN–VMD denoising were the smallest. After ICEEMDAN–VMD denoising processing, the FE of B x decreased by 31.142% and 48.412%, and the FE of G B x decreased by 28.59% and 71.89% compared with those after ICEEMDAN and VMD denoising, respectively. Similarly, the FE of B y respectively decreased by 20.47% and 48.73%, and the FE of G B y respectively decreased by 24.20% and 81.01% compared with those after ICEEMDAN and VMD denoising. Finally, the FE of B z respectively decreased by 35.61% and 49.94%, and the FE of G B z respectively decreased by 29.20% and 69.16% compared with those after ICEEMDAN and VMD denoising. Therefore, the above results indicated that ICEEMDAN–VMD yielded the fewest new modes after processing and the denoising effect was improved significantly. Based on the comparison of the SNR and FE, the denoising effect of ICEEMDAN–VMD was best for the enhanced magnetic signals of micro-defects.

The VMD demonstrates the highest SNR at B x and B y , which can be attributed to the fact that the Gaussian white noise added by the ICEEMDAN method is not fully removed, resulting in some of the noise being retained. However, an examination of G B x and G B y reveals that the ICEEMDAN-VMD method yields a substantial enhancement in SNR. This can be attributed to the fact that the signal obtained by the VMD is characterized by a lack of smoothness and is amplified following the solution of the gradient values. This phenomenon is further substantiated by the observed change in FE. The efficacy of the method for ICEEMDAN with VMD enhancement has been demonstrated.

To verify the applicability of ICEEMDAN-VMD to various paths, it was used to process detection signals from different paths within a 20 mm × 20 mm area near the defect, and the processed signals were used to create a pseudo-color map (Figure 10). Figures 10A–F display the unprocessed detection signals B x , B y , B z and their gradient signals G B x , G B y , and G B z , and Figures 10G–L show the processed detection signals and their gradient signals obtained after applying ICEEMDAN-VMD. It becomes evident that slight signal anomalies are observed in the defect area in the absence of denoising methods, rendering it a challenge to identify and locate defects through these anomalous signal regions. However, after denoising with ICEEMDAN-VMD, the noise signals are significantly eliminated, revealing distinct abnormal signal areas near the defect. This enables effective identification and localization of defects through these abnormal signal regions. In summary, the proposed ICEEMDAN-VMD method demonstrates effectiveness in processing magnetic field signals from various paths.