# Time-based damage detection of underground ferromagnetic pipelines using complexity pursuit based blind signal separation

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## Abstract

The measured sensor data of underground ferromagnetic pipelines consist of hidden damage information that can be explored by developing output data identification models due to the availability of only output signal responses. The current research findings elaborate output-only modal identification method Complexity Pursuit based on blind signal separation. An attempt is made to apply the complexity pursuit algorithms, for time-based damage detection of underground ferromagnetic pipelines which blindly estimates the modal parameters from the measured magnetic field signals for targeting the pipeline defect locations. Numerical simulations for multi-degree of freedom systems show that the proposed tested method could precisely identify the structural parameters. Experiments are conducted primarily under well-equipped controlled laboratory conditions followed by confirmation in the real field on pipeline magnetic field data, recorded through high precision magnetic field sensors. The measured recorded structural responses are given as input to the blind source separation model where the complexity pursuit algorithms blindly extracted the least complex signals from the observed mixtures guaranteed to be source signals. The output power spectral densities calculated from the estimated modal responses unveiled elegant physical interpretation of the underground ferromagnetic pipeline structures.

## Keywords

Blind source separation Pipeline flaw detection Structure health monitoring Complexity pursuit Output modal identification## Abbreviations

- BSS
Blind Source Separation

- CP
Complexity Pursuit

- CP-BSS
Complexity Pursuit-Blind Source Separation

- CPU
Central Processing Unit

- DC
Direct Current

- DOF
Degree of Freedom

- GPS
Global Positioning System

- GWN
Gaussian White Noise

- MAC
Modal Assurance Criterion

- NDT
Non-Destructive Testing

- PSD
Power Spectral Density

- SDOF
Single Degree of freedom

## 1 Introduction

Pipelines are important channels of oil and gas transportation. They are buried underground for the reason of safety, ease of operation and most important to prevent the influence of traffic and farming activities [1]. They are made of ferromagnetic materials which are often vulnerable to corrosion, fatigue damages and cracks due to surrounding environmental effects such as transmission channels, soil, temperature variations and other mechanical flaws; hence, require early damage detection techniques to confirm safe operation and reliable energy supply [2]. To achieve such goals, the primary aim of this study is to develop a non-contact geomagnetic probe using basic principles of metal magnetic memory testing [3, 4] to detect the geomagnetic field signals. Non-contact pipeline magnetic field testing [5] is a new kind of non-destructive testing (NDT) technique that needs earth magnetic field as the stimulus source to locate underground ferromagnetic pipelines and achieves structural defect information i.e. crack, corrosion and dent etc. without any excavation. However, the magnetic field data recorded for large scale systems like underground pipelines are often contaminated by several factors when in their service environment, as: the interference of parallel communication lines along the pipeline in heavy traffic areas, unusual disturbance often caused by the underground subway passages and overhead high voltage lines. Considerable attention in developing such non parametric methods than can perform quick real-time assessment of the 3-axis magnetic field data is required towards safety and integrity of pipelines.

Several signal processing techniques [6, 7, 8, 9, 10, 11, 12, 13, 14] have been considered in the literature for modal identification and flaw detection. Recently, blind source separation (BSS) techniques have been used as promising signal analysis tools in various fields of science [15, 16, 17, 18, 19]. BSS based algorithms are computational methods used for separating a multivariate signal into additive subcomponents. These methods were successfully applied for damage identification of civil structures [20, 21, 22] and more recently for flaw detection of underground ferromagnetic pipelines [23].

This paper presents a time-domain output data identification model for pipeline magnetic field data using the unsupervised blind source separation technique termed complexity pursuit (CP) [24] that was independently formulated in [25]. CP learning algorithms have been successfully applied for system identification and damage detection in [26, 27, 28, 29, 30]. The main contribution of this paper is to apply the CP algorithms to the pipelines noisy magnetic field data, towards an accurate time-based modal identification. The 3-axis magnetic field data are fed as input into the blind source separation model where the complexity pursuit algorithms are applied for an accurate extraction of mode matrix that is then plotted to obtain the time-domain output modal responses. The power spectral densities calculated from the recovered mode matrix show the abrupt variation in frequency due to the defects occurring in the pipeline. The detailed indoor and outdoor experimental results show the ability of the non-parametric CP-BSS learning algorithms to accurately extract time based modal information of the pipeline structures.

## 2 Blind source separation problem

The process of identifying and extracting the original source signals from a mixture of signals with less amount of information about the original source signals are termed as blind source separation.

*n*columns with its \(i{\text{th}}\) column \(a_{i} \in {\text{R}}^{m}\) associated with \(s_{i} (t).\)

## 3 Stone’s theorem for solution of blind source separation problem

Stone [24] proposed that under the effect of some physical laws; the moment of mass in a given time produces possible sources in a system. Likewise, the measured system responses also contains least complex sources, each source is created under the influence of certain physical law. Summarizing, the complexity of a mixture of response signals can be found among the simplest and the most composite constituent sources. It was theoretically proved in [31].

This approach has been used as a solution towards the BSS problem.

The significance of the proposed algorithm is to extract the hidden sources with accurate time-based structure. Since the use of short and long range prediction parameters should be considered carefully; as ‘increasing only the global statistical information \(V\left( {z_{i} } \right)\) will produce a high variance signal.’ While ‘increasing only \(U\left( {z_{i} } \right)\) will produce a smooth DC signal.’ Thus careful selection of parameters is essential to predict a component with reduced local variance (smoothness) as compared with its global (long-range) variance.

### 3.1 System identification by complexity pursuit

The matrices \({\bar{\mathbf{P}}}\) and \({\hat{\mathbf{P}}}\) are calculated only once and the terms \((p_{i} \left( t \right) - \hat{p}_{i} \left( t \right))\) and \((p_{j} \left( t \right) - \bar{p}_{j} \left( t \right))\) are calculated by fast convolution operations. For a given mixture of signals \({\mathbf{x}}(t)\), the complexity pursuit algorithm calculates the de-mixing vector \(w_{i}\) by maximizing the temporal predictability function \({\text{F}}(z_{i} )\);

Using the gradient ascent technique a maximum value of \({\text{F}}\) can be obtained by repeatedly updating \(w_{i} ;\) such that the extracted component \(z_{i} = w_{i} {\mathbf{x}}\), which is “most predictable” is considered as the least complex signal or the simplest source hidden in the mixtures [24].

### 3.2 Modal parameters estimated by complexity pursuit

*n*number of modal responses that can be expressed in Eq. (12) as;

*i*th modal feature column of the mode matrix, and is related with the

*i*th modal response \(q_{i} (t)\) of the modal response vector \({\mathbf{q}}\left( t \right)\). \({\mathbf{q}}\left( t \right)\) is actually the original source signal that can be obtained by multiplying the inverse of the mode matrix with the \((m \; \times \; n)\) matrix of the measured system responses.

The idea of “virtual sources” in [22] states that the recovered modal responses of a system should be considered as independent sources, if the power spectral density is not same or the frequencies are not able to be judged clearly. In such cases the mixing matrix matches with the recovered modal matrix, consequently the hidden sources and unidentified mixing matrix can be obtained by putting the measured system responses from the expanded model in Eq. (12) as known mixtures into the blind source separation framework in Eq. (1); accordingly the desired modal responses and mode matrix can be achieved.

*n*-DOF systems whose motions are given by,

Equation (16) defines the basic idea of the complexity pursuit algorithm by targeting the motion of the decoupled single degree of freedom system on *i*th modal coordinate \(q_{i} \left( t \right)\). The modal parameters of the system i.e. damping ratio is calculated by \(\varsigma_{i} = c_{i}^{*} /2\sqrt {m_{i}^{*} k_{i}^{*} }\) and resonant frequency of the system is calculated in terms of natural frequency \(\omega_{i}\) of the *i*th mode given by \(\omega_{di} = \omega_{i} \sqrt {1 - \varsigma_{i}^{2} } = \sqrt {\left( {1 - \varsigma_{i}^{2} } \right)k_{i}^{*} /m_{i}^{*} }\).

*i*th modal coordinate governed by Eq. (16) can now be written as,

*i*th mode,

The frequency and damping ratio can be readily computed from the recovered time-domain modal response \({\tilde{\mathbf{q}}}\left( t \right)\) using Fourier transform and logarithm-decrement technique, respectively.

## 4 Numerical simulations

The system parameters are adjusted to identify different modal identification problems i.e. proportional damping well separated mode, closely spaced mode and complex mode. Free excitation and random excitation in each case are discussed. Gaussian White Noise (GWN) is used to produce stationary random excitation [25]. Similarly the Gaussian White Noise (GWN) is modulated with a constant exponential decay function to create a non-stationary excitation effect in the system. The time histories of the system responses i.e. the displacement vector, are calculated by the Newmark-Beta solver. The sampling frequency is set to 10 Hz.

The parameters of complexity pursuit based blind source separation method remains the same throughout the process. The long-range parameter \(h_{L}\) = 900,000 and short-range parameter \(h_{S}\) = 1 are taken same as given by [24]. Fast convolution operations are performed to calculate the long-range and short-range covariance matrices. The demixing matrix which is the eigenvector matrix is calculated by conducting eigenvalue decomposition on the obtained covariance matrices. The excitation mode matrix and the time-domain modal responses are calculated by Eqs. (21) and (22) respectively. Fourier transform algorithms and logarithm-decrement technique are used to calculate frequency and damping ratio respectively.

Ranging from 0 to 1, where 0 means no correlation and 1 indicates perfect correlation.

### 4.1 Proportional damping

^{nd}and 3

^{rd}DOFs using stationary and non-stationary Gaussian White Noise. Tables 1 and 2 show the obtained results by CP algorithms and the modal assurance criterion values respectively.

Identified modal parameters (proportional damping)

Mode | Comparison | Frequency (Hz) | Damping ratio (%) | ||||
---|---|---|---|---|---|---|---|

1 | 2 | 3 | 1 | 2 | 3 | ||

α = 0.03 | Theoretical value | 0.0895 | 0.1458 | 0.2522 | 0.8887 | 0.5460 | 0.3155 |

CP identified value | 0.0879 | 0.1465 | 0.2539 | 0.8822 | 0.5691 | 0.3166 | |

α = 0.08 | Theoretical value | 0.0895 | 0.1458 | 0.2522 | 4.4437 | 2.7299 | 1.5775 |

CP identified value | 0.0879 | 0.1465 | 0.2539 | 4.4493 | 2.8233 | 1.5248 |

Identified MAC values (proportional damping)

Mode | Free excitation | Stationary GWN | Non-Stationary GWN | ||||||
---|---|---|---|---|---|---|---|---|---|

1 | 2 | 3 | 1 | 2 | 3 | 1 | 2 | 3 | |

α = 0.03 | 0.9988 | 0.9997 | 0.9992 | 1.0000 | 0.9997 | 0.9998 | 0.9997 | 0.9987 | 1.0000 |

α = 0.08 | 0.9975 | 0.9993 | 0.9990 | 1.0000 | 0.9962 | 0.9997 | 0.9998 | 0.9977 | 0.9981 |

### 4.2 Effect of noise

CP identified values in 10% root-mean-square noise (\(\alpha = 0.08\))

Mode | Frequency (Hz) | Damping ratio (%) | MAC values | ||
---|---|---|---|---|---|

Theoretical value | CP identified | Theoretical value | CP identified | ||

1 | 0.0894 | 0.0885 | 4.4437 | 4.2151 | 0.9989 |

2 | 0.1457 | 0.1459 | 2.7299 | 2.7910 | 0.9617 |

3 | 0.2521 | 0.2529 | 1.5775 | 1.5343 | 0.9979 |

### 4.3 Closely spaced modes

CP Identified results for free excitation (closely spaced modes)

Mode | Comparison | Frequency (Hz) | Damping ratio (%) | ||||
---|---|---|---|---|---|---|---|

1 | 2 | 3 | 1 | 2 | 3 | ||

α = 0.08 | Theoretical | 0.1039 | 0.3425 | 0.3713 | 3.8279 | 1.1618 | 1.0715 |

CP identified | 0.1074 | 0.3418 | 0.3711 | 3.8199 | 1.1434 | 1.0151 | |

α = 0.13 | Theoretical | 0.1039 | 0.3425 | 0.3713 | 9.9526 | 3.0208 | 2.7860 |

CP identified | 0.1074 | 0.3418 | 0.3711 | 9.9770 | 2.9906 | 2.7274 |

Modal assurance criterion results in closely space mode cases

Mode | Free excitation | Stationary Gaussian white noise | Non-stationary Gaussian white noise | ||||||
---|---|---|---|---|---|---|---|---|---|

1 | 2 | 3 | 1 | 2 | 3 | 1 | 2 | 3 | |

α = 0.08 | 1.000 | 0.997 | 0.999 | 0.999 | 1.000 | 0.999 | 1.000 | 0.997 | 0.996 |

α = 0.13 | 1.000 | 0.973 | 0.975 | 0.999 | 0.999 | 1.000 | 0.999 | 0.976 | 0.991 |

### 4.4 Non-proportional damping

Identified results of free excitation in non-proportional high damping

Mode | Frequency (Hz) | Damping Ratio (%) | Modal assurance criterion | ||
---|---|---|---|---|---|

CP identified | Theoretical | CP identified | Theoretical | ||

1 | 0.137 | 0.135 | 10.95 | 10.88 | 0.985 |

2 | 0.239 | 0.244 | 6.731 | 6.875 | 0.952 |

3 | 0.497 | 0.508 | 4.674 | 4.872 | 0.981 |

Identified MAC values for non-proportional high damping

Mode | Free excitation | Stationary Gaussian white noise | Non-Stationary Gaussian white noise | ||||||
---|---|---|---|---|---|---|---|---|---|

1 | 2 | 3 | 1 | 2 | 3 | 1 | 2 | 3 | |

α = 0.13 | 1.000 | 0.973 | 0.975 | 0.999 | 0.999 | 1.000 | 0.999 | 0.976 | 0.991 |

### 4.5 Modal identification of a 12-DOF system

MAC values for 12-DOF system proportional damping

Modes | Free excitation | Stationary GWN | Non-stationary GWN | ||||||
---|---|---|---|---|---|---|---|---|---|

α = 1 | α = 2 | α = 3 | α = 1 | α = 2 | α = 3 | α = 1 | α = 2 | α = 3 | |

1 | 0.9950 | 0.9972 | 0.9904 | 0.9989 | 0.9946 | 0.9837 | 0.9973 | 0.9975 | 0.9938 |

2 | 0.9974 | 0.9977 | 0.9939 | 0.9989 | 0.9951 | 0.9857 | 0.9970 | 0.9979 | 1.0000 |

3 | 0.9994 | 0.9952 | 0.9922 | 0.9982 | 0.9981 | 0.9937 | 0.9989 | 0.9948 | 0.9921 |

4 | 0.9997 | 0.9977 | 0.9959 | 0.9995 | 0.9982 | 0.9927 | 0.9996 | 0.9974 | 0.9958 |

5 | 0.9999 | 0.9992 | 0.9985 | 0.9999 | 0.9995 | 0.9991 | 0.9998 | 0.9991 | 0.9985 |

6 | 0.9998 | 0.9997 | 0.9995 | 0.9999 | 0.9996 | 0.9996 | 0.9998 | 0.9997 | 0.9995 |

7 | 0.9999 | 0.9998 | 0.9997 | 1.0000 | 0.9998 | 0.9997 | 0.9999 | 0.9998 | 0.9997 |

8 | 1.0000 | 0.9999 | 0.9999 | 1.0000 | 1.0000 | 0.9999 | 1.0000 | 0.9999 | 0.9999 |

9 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 |

10 | 1.0000 | 1.0000 | 0.9999 | 1.0000 | 1.0000 | 0.9999 | 1.0000 | 1.0000 | 0.9999 |

11 | 1.0000 | 1.0000 | 0.9999 | 1.0000 | 1.0000 | 0.9999 | 1.0000 | 1.0000 | 0.9999 |

12 | 1.0000 | 1.0000 | 0.9999 | 1.0000 | 1.0000 | 0.9999 | 1.0000 | 1.0000 | 0.9999 |

## 5 Experimental analysis

### 5.1 Measurement probe to detect magnetic field of a pipeline

### 5.2 Indoor experimental procedure

### 5.3 Outdoor experimental setup

## 6 Concluding remarks

Time based damage detection of underground ferromagnetic pipelines is presented in this paper. Complexity Pursuit based blind signal separation algorithms are implemented to identify structural damage from the measured magnetic field sensor data. Using Fast Fourier transform the power spectral densities are calculated from the approximated modal responses. Numerical simulations for multi-DOF systems are carried out to elaborate the CP based BSS method under different damping conditions. The complexity pursuit based BSS model is implemented on the indoor and outdoor experimental data comprised of 3-axis magnetic field signals; it offers excellent results about the pipeline structural information. The proposed method requires minimum user interaction because the parameters of the model remain same throughout the process of targeting the input data. Similarly length of the recorded sensor data does not influence the accuracy of the CP model. The performance of the unsupervised CP-BSS model to identify structural information makes it more suitable for real time as well as for off-line inspection of underground ferromagnetic pipeline structures.

## Notes

### Funding

The work is supported by The National Key Research and Development Program of China (2017YFC0805005), Joint Program of Beijing Municipal Natural Science Foundation Commission and Beijing Municipal Education Commission (18JH0005), China Postdoctoral Science Foundation (2018T110018), Collaborative Innovation Project of Beijing Chaoyang District (CYXC1709) and “Rixin Scientist” of the Beijing University of Technology.

### Compliance with ethical standards

### Conflict of interest

The authors declare that they have no conflict of interest.

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