# Magnetic Anomaly Detection Based on a Compound Tri-Stable Stochastic Resonance System

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

^{α}(0 < α < 2) [18]. C.B. Wan et al. used a BSR method to detect magnetic anomalies, which can more effectively detect weak magnetic anomalies hidden in the noise background [19]. W. Liu et al. proposed a parallel monostable stochastic resonance system and searched for the optimal system parameters in the SR system to achieve good detection performance under different waveforms. The system improves the monostable stochastic resonance system, whose output is influenced by the peak signal and trough signal [20]. Z.Y. Liu et al. proposed an adaptive stochastic resonance system employing the kurtosis index as the criteria to automatically adjust the system structural parameters, which performs well in the detection of magnetic anomaly signals with background noise [21]. C.B. Wan et al. proposed a novel detection method based on the parallel stochastic resonance (PSR) system, which improves the SNR using the PSR system. The system performs better than the detector of a single stochastic resonance system [22]. H.X. Sun et al. proposed an adaptive cascade weak magnetic anomaly detection method based on the marine predator algorithm-stochastic resonance (MPA-SR) to solve the problem of the effective detection of weak magnetic anomaly signals in complex underwater environments. This detection method includes a cascade detection method with low-pass filtering, stochastic resonance, and threshold detection, which improves the detection probability of magnetic anomaly signals [23].

## 2. Related Work

#### 2.1. Magnetic Anomaly Signal

**M**is located at the origin O of a Cartesian coordinate system, with

**M**lying along the negative direction of the Z-axis. A sensor moves along a straight line with a relative angle θ between the search path and the positive X-axis, l is the closest contact distance between the detection path and the magnetic target, and v represents the sensor moving speed. The total magnetic field consists of the background magnetic field and the anomalous field generated by the ferromagnetic object, namely

**B**

_{t}is the total magnetic field vector,

**B**

_{e}is the background magnetic field vector, and

**B**

_{r}is the magnetic anomaly field vector. In fact, $\left|{B}_{t}\right|\gg \left|{B}_{r}\right|$, so the magnetic distortion field can be expressed as

**B**

_{t}| is the total magnetic field scalar value, |

**B**

_{e}| is the background magnetic field scalar value, and |

**B**

_{r}| is the scalar value of the magnetic anomaly field vector.

**M**= (0, 0, 0.3)A∙m

^{2}, the magnetic sensor moves along the θ = 30°, 200°, and 300° directions, the speed is set to v = 5 m/s, the closest contact distance between the detection path and the magnetic target is l = 5 m, and the sampling frequency is 2 kHz. Combining Equations (1) and (2) and the simulation conditions, the simulation obtains three typical magnetic anomaly signals, as shown in Figure 2.

#### 2.2. Stochastic Resonance Principle

^{2}/(4B).

## 3. Principle of Magnetic Anomaly Detection

#### 3.1. Compound Tri-Stable Stochastic Resonance Model

_{1}(x) is the GP model, a controls the depth of the potential well, b controls the width of the potential well, U

_{2}(x) is the MBSR model, and c controls the barrier height and the steepness of the potential wall. As c gradually increases from 0.1 to 0.5, the height of the potential barrier gradually decreases, but the steepness of the potential wall gradually increases. The CTSR potential function has two potential barriers and three potential wells, as shown in Figure 4.

_{0}

^{−}, and x = x

_{0}

^{+}(where x

_{0}

^{−}and x

_{0}

^{+}are obtained from Equation (5)), and two potential barriers (maxima) separate the three minima zones.

_{0}

^{−}or x = x

_{0}

^{+}, and after the input disappears, the system returns to the initial state at x(t) = 0 under the action of noise energy.

_{t}is the output of the CTSR model. The schematic diagram of the working principle of Equation (6) is shown in Figure 5.

#### 3.2. CTSR Model for Magnetic Anomaly Detection

_{n}is the mixed signal of magnetic anomaly and noise, and x

_{n}is the output of the CTSR model.

_{n}are obtained, and the basic parameters are set in the genetic algorithm. The specific operation is as follows.

- (1)
- The range of parameters to be optimized is set to $a,b,c\in \left[0,3\right]$. Set basic parameters in the quantum genetic algorithm are set to P = 40, B = 20, and G = 50. This initialization is set to ensure a high convergence rate and an acceptable calculation time as a rule of thumb.
- (2)
- The maximum output SNR by the QGA and crystal-optimized parameters a, b, and c are found.
- (3)
- The optimized parameters are substituted into the CTSR system to obtain the best output waveform.

#### 3.3. Statistical Filtering-Based Judgment System

_{n}output of the CTSR model response is used as the basis for the final verdict on the presence or absence of the magnetic anomaly signal, which is prone to false verdicts or false alarms. Therefore, a filter and a judgment system need to be added to the output of the CTSR model to improve the detection probability. We designed a statistical filtering-based judgment system, which performs statistical analysis on data within a certain domain, filters out noisy data that do not meet the conditions, and sets thresholds d

^{−}

_{threshold}and d

^{+}

_{threshold}as judgment thresholds.

_{n}with the statistical filter in Figure 6 is shown in Equations (10) and (11).

_{n}within the sliding window, σ is the standard deviation of the response signal x

_{n}within the sliding window, d

^{−}

_{threshold}dthreshold-and d

^{+}

_{threshold}are the standard deviation thresholds, z(n) is the output of the statistic filter, and k is the adaptive standard deviation threshold factor, which is set according to the rule of thumb to ensure that the judgment is more accurate.

## 4. Experiment

## 5. Discussions

^{α}(0 < α < 2) and is similar to the real geomagnetic noise power spectral density distribution.

^{α}(0 < α < 2). Low frequency 1/f

^{α}noise is ubiquitous in various electronic devices. Due to the difference in sensitivity and background noise of the magnetic sensor, the energy barrier of the stochastic resonance system of the CTSR system is different. Finally, the system parameters (a, b, and c) are changed.

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 7.**Experimental system: (

**a**) data acquisition system; (

**b**) internal structure diagram of the detection and storage module.

**Figure 9.**(

**a**) Background magnetic field noise; (

**b**) background magnetic field power spectral density.

**Figure 10.**(

**a**) Simulated background magnetic field noise; (

**b**) simulated background magnetic field power spectral density.

**Figure 13.**Responses of the CTSR system to the field test. (

**a**) The response of the CTSR system to the peak signal; (

**b**) The response of the CTSR system to the trough signal; (

**c**) The response of the CTSR system to the peak and trough signal.

Saturated Field | Sensitivity | Background Noise |
---|---|---|

±8Oe | 100 mV/V/Oe | <150pT/rt(Hz)@1 Hz |

Detector | Detection Probability | |
---|---|---|

SNR = −8 dB | SNR = −6 dB | |

PSR system | <0.01 | 0.191 |

New PSR system | 0.546 | 0.775 |

TTSR system | 0.726 | 0.912 |

CTSR system | 0.794 | 0.996 |

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**MDPI and ACS Style**

Huang, J.; Zheng, Z.; Zhou, Y.; Tan, Y.; Wang, C.; Xu, G.; Zha, B.
Magnetic Anomaly Detection Based on a Compound Tri-Stable Stochastic Resonance System. *Sensors* **2023**, *23*, 9293.
https://doi.org/10.3390/s23229293

**AMA Style**

Huang J, Zheng Z, Zhou Y, Tan Y, Wang C, Xu G, Zha B.
Magnetic Anomaly Detection Based on a Compound Tri-Stable Stochastic Resonance System. *Sensors*. 2023; 23(22):9293.
https://doi.org/10.3390/s23229293

**Chicago/Turabian Style**

Huang, Jinbo, Zhen Zheng, Yu Zhou, Yuran Tan, Chengjun Wang, Guangbo Xu, and Bingting Zha.
2023. "Magnetic Anomaly Detection Based on a Compound Tri-Stable Stochastic Resonance System" *Sensors* 23, no. 22: 9293.
https://doi.org/10.3390/s23229293