Research on the Detection Method of MRS Signal Initial Amplitude Based on Chaotic Detection System

When using magnetic resonance sounding (MRS) method for groundwater, the detection accuracy is often affected by electromagnetic noise. The chaos method can achieve signal extraction at a lower signal-to-noise ratio. However, this method has many problems when used to obtain MRS signals. Aiming at the attenuation characteristics of MRS signals, in this paper we propose a pre-segmentation method of MRS signals to improve the accuracy of chaos detection. In addition, a chaotic detection system model suitable for MRS signal recognition was constructed, and a calculation method for center-covered was used to improve the efficiency of determining the chaotic state. Through a large amount of simulation data, we have verified that this method can reliably extract MRS signals with a signal-to-noise ratio (SNR) of -30 dB, which provides technical support for the MRS method to work with a relatively high electromagnetic noise.


I. INTRODUCTION
Magnetic resonance sounding detection is a geophysical method that can directly detect groundwater [1]. This method can accurately obtain hydrogeological parameters such as the depth and distribution state of groundwater by exciting the hydrogen protons in the groundwater and observing the resulting magnetic resonance signals [2], [3]. It has been widely used in groundwater resource investigation, groundwater environment monitoring, landslide disaster prediction, mine tunnel water inrush disaster warning and other fields [4], [5], [6], [7]. In the measurement of the magnetic resonance method, the signal is received by a loop antenna placed at a certain distance from the body of water, and the signal amplitude is generally in the order of 10 nV-1000 nV [8]. During the signal receiving process, since the loop The associate editor coordinating the review of this manuscript and approving it for publication was Ludovico Minati .
antenna is used as the sensor to receive the signal, the interference generated by a large amount of spatial electromagnetic noise also enters the acquisition device of the detection system through the receiving coil [9], which affects the signal extraction accuracy. These interferences even completely drown the signal in electromagnetic noise, so the groundwater information reflected in the magnetic resonance signal cannot be effectively acquired [10].
There are three main types of spatial electromagnetic noise affecting signals: singular spike noise, power frequency harmonic noise, and random noise [11]. In recent years, many scholars have carried out a lot of research on noise reduction methods [12], and deep learning for denoising has also been applied to geophysical methods [13], [14], [15]. Singular noise and power frequency noise have strong regularity due to their single source and obvious characteristics. At present, reducing these two noise reduction methods can achieve a relatively ideal effect [16], [17], [18], [19], [20]. However, random noise is caused by the spatial electromagnetic interference in the measurement area, the factors of the instrument system itself and so on. Its characteristics show strong irregularity and randomness, and the methods that can be used to reduce it are very limited at present. The method of multiple stacking is usually used to suppress random noise [21], [22], the measurement time will be significantly increased, and the detection efficiency will be reduced [22]. Jiang et al. proposed a noise reduction method based on statistical stacking, which can reduce the measurement time may decrease by nearly 50% [24]. David O. Walsh et al. proposed to use the reference coil of a multi-channel instrument to eliminate random noise to improve the effect of random noise on the signal [25]. However, this method requires a large measurement space to complete the laying of the reference coil. In summary, the random noise removal method often requires more time and larger measurement space, and there is an upper limit on the method to reduce random noise [26]. Lin et al. propose the use of time-frequency peak filtering for random noise attenuation in MRS, and identify the magnetic resonance signal with a SNR of -17 dB [27]. Therefore, it is necessary to develop a new signal extraction method to achieve the extraction of MRS signals at lower SNRs.
In 1992, Birx et al. first proposed a method for weak signal detection using chaotic oscillators [28], and the nonlinear signal detection theory constructed by chaotic systems has received extensive attention. Many studies show that using chaotic systems to estimate parameters of unknown signals is superior to traditional signal identification methods in terms of accuracy and signal identification [29], [30], [31]. Del Marco et al. identify human gait signals using accelerometer data and chaos detection methods in mobile devices [32], Jinfeng Hu et al. used a chaotic detection system to identify weak periodic signals in the diagnosis of mechanical equipment, and the detection effect was better than the stochastic resonance (SR) method [33]. Guozheng Li et al. completed the Chua's circuit design based on the chaos principle, which can directly detect the signal in the strong noise. This design significantly improves the real-time performance in the process of measuring signals [34]. In summary, the detection method of chaotic system to detect weak signals provides a new possibility for extracting signals from strong random noise. It is necessary to apply the chaos detection method to the field of MRS detection signal recognition, and evaluate the recognition ability of this method for magnetic resonance signals.
A magnetic resonance signal identification method based on Duffing oscillator chaotic detection system is proposed in this paper. According to the characteristics of magnetic resonance signal decay with time, the detection data is segmented based on relaxation time and detection accuracy, and a detection system model suitable for magnetic resonance signal recognition is constructed. Meanwhile, a center-covered chaotic system state is proposed as the judgment basis to improve the efficiency of detection state judgment.
We have used a large number of simulation data to verify the ability of the chaos detection system to identify the magnetic resonance signal submerged in white noise based on this judgment method in this paper. The research shows that the chaos detection method proposed in this paper can identify the magnetic resonance signal with a SNR of -30 dB, which is superior to the detection ability of the traditional method and meets the needs of practical detection engineering.

A. THE PRINCIPLE OF WEAK SIGNAL DETECTION BY DUFFING
Duffing equation is an important differential equation with typical representative in nonlinear theory, which is used to describe resonance phenomenon, strange attractor and chaos phenomenon. The classic Holmes-Duffing equation form is as follows:ẍ In (1), k is the damping ratio, the set as 0.5, ax + bx 3 is the non-linear restoring force of the system. A cos (ωt) is the driving force of the chaotic system. A is the amplitude of the driving force, and an important parameter in the detection system, the value of which determines the influence of the driving force on the change of the phase diagram of the system. Where ω is the driving frequency of the system itself, the value of which is the same as the target frequency. The Duffing oscillator chaotic system is ''immune'' to random noise, that is, adding random noise to the system will not affect the change of system state. When an external signal is added to the system, if the signal contains a periodic stimulus force with the same components, the system will change from a chaotic state to a large-scale periodic state. We can detect whether the external signal contains the desired target signal we need by judging the change of system state. Based on the theory of magnetic resonance and a lot of experimental research, we make a = −1, b = 5, which is the most suitable detection model for magnetic resonance chaotic system. According to (1), Duffing equation in standard form is: D(t) is the external signal of the system, D (t) = s (t)+n(t), s (t) is the signal to be measured, and n(t) is the noise. In order to detect signals with different frequencies, coordinate transformation is carried out for (2). Let t = ωτ , x (ωτ ) = x τ (τ ), thenẋ(t) andẍ(t) can be expressed as: Take (3) and (4) into (2), and express it as: Equation (5) is solved by the fourth-order Runge-Kutta algorithm, so that the calculation interval, the sampling interval of signal and noise, and the time interval of system operation be the same. Set the initial value of the system [x(0), y(0)]=[0, 0], and obtain the x(n+1) and y(n+1) of the system in turn. Using the output values of these two items as the values of the abscissa and ordinate, the phase diagram of the system motion can be drawn. With the gradual increase of the driving force amplitude A, the system will go through the equilibrium point, homoclinic orbit, bifurcation state, chaotic state and large period state respectively.
When using a chaotic system to detect weak signals, it is necessary to observe the changing state of the phase diagram of the system. In the data D(t) loaded into the system, the signal form is S (t) = S 0 cos ωt, and the noise N(t) form is random white noise. First, it is assumed that the amplitude of the signal is S 0 = 0nV (that is, there is no signal), the average amplitude of the white noise is N 0 = 200nV, and the time domain form of the data is shown in Fig. 1(A).
Loading such data into the system of (5), the resulting phase diagram is shown in Fig. 1(B). Then, keep all the system setting parameters such as the system white noise amplitude unchanged, assuming that the signal amplitude S 0 = 1nV, the data time domain form is shown in Fig. 1(C), and the phase diagram obtained after loading into the system is shown in Fig. 1(D).
As shown in the Fig.1, when the measured data is added to the system, if there is no signal in the data, the system phase diagram is still in a chaotic state. In addition, if there is a signal in the measured data, the system phase diagram rapidly evolves from a chaotic state to a periodic state. Whether there is a signal in the measured data can be effectively observed through the change of the phase diagram, which is the basic principle of the duffing oscillator chaotic system to detect weak signals. In the following, we will demonstrate in detail how the chaotic system can accurately identify the amplitude of the measured signal, the relationship between other values, and the detection ability of the chaotic system.

B. CHAOS CRITERION BASED ON CENTER COVERAG
It can be seen from the above principles that an important step of the chaotic detection system is to determine what state the output of the detection system is in. When the output state is determined to be a chaotic state or a periodic state, it can be further determined whether there is a signal in the measured data and what the characteristics of the signal are. For the chaotic equation criterion of duffing oscillator, there are direct observation methods such as phase diagram direct judgment and time-domain waveform analysis, as well as numerical analysis methods such as Lyapunov exponent method and Poincare section method. The direct observation method is fast, but it needs to rely on subjective judgment, with large error and low efficiency. The numerical analysis method has a large amount of calculation and is not suitable for the verification of massive measured data. Based on the characteristics of the existing methods, we propose a method to judge the state of chaotic system based on the central region of the phase diagram and the number of phase track coverage. This method transforms the state observation of the system phase diagram into numerical analysis, which not only has the fast and efficient characteristics of the direct observation method, but also has the precision of the numerical analysis method, and is more suitable for the rapid implementation of the relevant algorithm written by the computer.
The implementation principle of chaotic system criterion with center-covered is shown in Fig. 2. The central area of the phase diagram output by a set of chaotic detection systems is divided according to a certain size.   range. Therefore, different phase track diagrams have different coverage coefficients under the same central area division and trajectory point statistics. The larger the value is, the more the system tends to chaotic state, and when the value of coverage coefficient is 0, the system is in periodic state. Therefore, this method can be used to judge the state of different chaotic systems after running for a period of time.

III. A CHAOTIC SYSTEM DETECTION MODEL AND METHOD FOR MRS A. THE BASIC PRINCIPLE AND SIGNAL FORM OF MAGNETIC RESONANCE SOUNDING
MRS detection is a geophysical method developed in recent years to directly detect groundwater. It has been widely used in the investigation of groundwater resources distribution and groundwater occurrence monitoring. The principle of groundwater detection by MRS method is shown in Fig. 3 [35]: Hydrogen protons in groundwater will process around an axis at a certain frequency, and the precession frequency is related to the geomagnetic field, called the Larmor frequency. When the energy of this frequency is used to excite hydrogen protons in water, according to the theory of quantum physics, the hydrogen protons will transition to higher energy levels. After a period of time, the excitation energy is removed, and the hydrogen protons transition back to a lower energy level, radiating energy outward at the same time. This phenomenon is called the nuclear magnetic resonance effect. In the process of hydrogen proton transition, the released energy is radiated outward in the form of a magnetic signal. At this time, according to the law of electromagnetic induction, the signal can be captured by the coil laid on the ground, and exists in the form of a weak voltage at both ends of the coil.
As shown in Fig. 4, MRS signal is an alternating current signal attenuated with e index, which is related to the content and distribution state of groundwater, and its expression is: where, E 0 is the initial amplitude of the signal, T * 2 is the spin-spin relaxation time of the signal, q is the excitation pulse moment, t is the signal time, ω 0 is the angular frequency of the signal, ϕ 0 and is the initial phase of the signal. Therefore, it can be seen that the characteristics of magnetic resonance signal are: (1) The signal frequency is fixed, which is only related to the value of geomagnetic field where the measured water body is located. (2) The amplitude of the signal attenuates with e index, and attenuation degree is related to the value of T * 2 in the signal, that is, to the distribution of voids in the groundwater. It can be seen from these characteristics that the chaotic detection system is suitable for signal recognition with fixed frequency, and the MRS signal fully conforms to this characteristic. However, the characteristic of MRS signal attenuation with e index bring some difficulties to the ability of chaos detection system to identify signals. Therefore, it is necessary to establish a specific detection model according to the characteristics of the decay signal in order to effectively identify the MRS signal submerged in random noise.

B. CHAOS DETECTION MODEL FOR MRS SIGNAL RECOGNITION
The detection model of chaotic system is still based on the classical Duffing equation. Different from the usual detection methods of chaotic systems, since the amplitude of the MRS signal exhibits attenuation characteristics with time, the amplitude of the driving force of the chaotic system will be uncertain after being directly sent into the detection system. Therefore, it is necessary to segment the MRS signal to make the segmented signal data stable and improve the detection accuracy of the detection system.
The segmentation strategy of the original data is determined by the relaxation time T * 2 of the MRS signal and the number of iterations required for chaotic system calculation.  When relaxation time T * 2 is large, the signal will not decay rapidly, and the length of data can be appropriately increased to reduce the number of data segmentation and the time for signal identification. When the relaxation time T * 2 is small, the amplitude of the signal changes rapidly, and the length of the data can only meet the needs of iterative calculation, that is, more chaos detection times are required to complete the identification of the signal. The data length of the original data segment is defined as L section , the relationship among it and the required signal detection accuracy ϵ, the initial amplitude E 0 of the magnetic resonance signal and the relaxation time T * 2 is as follows: Based on such segmentation strategy, the principle of sending MRS signal into the chaos detection system to complete the signal identification is shown in Fig. 5.
The data of the magnetic resonance signal are divided into L 1 , L 2 , L 3 ,. . . , Ln according to (7), and then each segment of data is sent into the chaotic system detection equation shown in (5) respectively. Each segment of data is denoted as D L1 , D L2 , D L3 , . . . , D Ln . Before sending the MRS signal into the detection system, firstly, set the driving force amplitude A of the chaotic system equation as the critical value A d of the output state of the chaotic system, which is the critical state at this time. When the driving force amplitude A exceeds the critical value, the phase diagram output by the detection system immediately changes from a chaotic state to a largeperiod state. At the same time, after adding data segment D L with length L to the system, if there is a signal with the same frequency as the driving force in the data segment, let the component of the signal with the same frequency be S A , then the combined value of S A and A must be greater than the critical value of the detection system. Meanwhile, the system state will change from chaotic state to periodic state. Then reduce the driving force value A, the system phase diagram will change from periodic state to chaotic state. The difference between the driving force value A c when the system returns to the chaotic state and the system critical value A d is recorded as S 1 , which can be regarded as the signal component existing in the measured data segment D L . If there is no MRS signal in the data segment D L , that is, there is no amplitude component with the same frequency as the driving force value, then the total amount of driving force value will always be the same as the critical value. The phase diagram of the system does not change abruptly, and the amplitude of the signal in this segment of data is recorded as 0. After each segment of data is segmented and detected, the chaotic system detection model will output the marked signal values S 1 -S n respectively. By combining these values according to the segmentation order, the envelope data of the measured signal can be obtained, and then a MRS signal submerged in random noise data can be extracted. The block diagram of the detection process of the chaotic detection model applicable to the MRS signal is described in Fig. 6. On the basis of Holms-Duffing oscillator model, the improved chaotic system is added MRS signal decomposition and MRS signal waveform reconstruction to complete the MRS signal extraction with amplitude change.

IV. DETECTION PERFORMANCE ANALYSIS AND COMPARISON
Based on the above principles and methods, we carried out the MRS signal extraction experiment based on the principle of chaos detection in the background of electromagnetic noise in this chapter. Random noise is an important noise source in MRS measurements. At present, the detection capability that can be achieved is relatively limited, and the minimum detection SNR is generally −20 dB. In this paper, random white noise is used as the interference source of the detection test to observe the lower limit of the SNR that the detection system can achieve, and analyze the extraction accuracy of the initial amplitude of the MRS signal.

A. ANALYSIS OF DETECTION ABILITY OF SNR
Firstly, establish a group of white noise data N(t), and the noise intensity is represented by the mean square error σ . Suppose that the amplitude of the noise is 200 nV, the mean square error of random white noise is σ = 0.005, the data length is 1000 ms, and the data interval is 0.01 ms, that is, the sampling frequency is 100 kHz. The assumed random noise data is shown in the red curve in Fig. 7(a). A group of MRS signal data is established. The initial signal amplitude E 0 is 5 nV, the relaxation time T * 2 is 500 ms, the signal frequency is 2000 Hz, and the sampling frequency is 100 kHz, which is the same as the white noise. The signal waveform is shown in the blue curve in Fig. 7(a). Using the chaotic system detection model proposed in section III-B, the noise data and signal data shown in Fig. 7(a) are sent into the chaotic detection system together. The interval of data segments is 10 ms, the number of single-point chaotic iterations is 20,000, the central-covered of the chaotic system criterion is [±0.4, ±0.4], and the critical amplitude of the driving force of the chaotic system is 0.3695. After the detection of the chaotic system, the output result is shown in Fig. 7(b), and the red curve in the diagram is the output result of the chaotic system. It can be seen from the result diagram that the envelope curve of the MRS signal submerged in white noise can be depicted in the envelope curve output by the detection system. The formula for evaluating the SNR using the mean square error is as follows: σ signal is the standard deviation of the MRS signal, and σ noise is the standard deviation of the random noise. According to the above results, the detection SNR of this group of data is −29 dB. It should be pointed out that the detection output result cannot be attenuated strictly according to the envelope curve of the MRS signal, and there is a certain detection accuracy deviation, which are mainly caused by the output accuracy of the chaotic detection system and the calculation deviation caused by the motion of the nonlinear dynamic system. Three sets of combined data of random noise and MRS signals under different SNRs are constructed for detection, and the detection effect of the system on MRS signals under different SNRs are observed, as shown in the Fig. 8.  In order to further verify the detection effect of chaos detection system, we have verified the suppression effect of stacking method and chaos detection system on MRS full wave signal random noise through numerical simulation of algorithm, and quantifies the initial amplitude fitting error. Simulation signal model parameters are as follows: the initial amplitude of MRS full-wave signal E 0 = 15 nV, relaxation time T * 2 = 500 ms, signal frequency is 2000 Hz, sampling frequency and white noise are both 100 kHz. The amplitude of noise is 200 nV, the data segment length is 1000 ms, the data interval is 0.01 ms, that is, the sampling frequency is 100 kHz. The time-domain simulation of noise signal with 100,000 equal precision sampling points is shown in Fig. 9.
Stacking method and chaos detection system are respectively used to de-noise the simulated random noise signals in Fig. 9. The noise elimination results after 32 times of stacking are shown in Fig. 10. (a), where the red line represents the signal containing random noise and the blue line represents the signal after superposition. The denoising results and fitting results of the stacking method of pure signal waveform and data are shown in Fig. 10. (b) (c) respectively.
It can be clearly seen that the signal after 32 stacked still contains some random noise, the initial amplitude E 0 = 53.95 nV, the initial amplitude fitting error IAPE = 259%. Noisy signals are input into the duffing chaos detection system, and the initial amplitude E 0 = 14nV is obtained, and the fitting error of the initial amplitude IAPE = 6.6%. Comparing the two denoising results shown in Fig. 11. The pink line represents the envelope curve of the signal, the red line represents the denoising curve processed by the duffing chaotic system,  . Pure MRS oscillating signal fitting curve, fitting curve after random noise attenuation by stacking method and fitting curve after random noise attenuation by chaos detection method. and the blue line represents the curve after noise elimination by 32 stacked. It can be seen that the detection model of duffing chaotic system can effectively suppress random noise, and the method is better than the stacking method.

B. STATISTICAL ANALYSIS OF DETECTION ACCURACY
We have constructed 100 sets of random noise data, and the power Np is 0.07. MRS signals with E 0 amplitudes of 3 nV, 9 nV, and 15 nV were added to the noise, and the signal decay  time T * 2 was 500 ms to form MRS signal detection data with different SNRs. The above data are respectively sent to the chaos detection system, and the average detection error of the initial amplitude E 0 under different SNRs is evaluated. The average detection error is calculated using the arithmetic mean, and the calculation formula is shown in (9). In (9), n is the number of calculations, and x i is the output value of the chaotic system after each calculation.
The output results calculated 100 times by the chaotic system detection model are shown in Fig. 12. As can be seen from the Fig. 12, when the initial amplitude E 0 is 15 nV, the maximum error amplitude is 17 nV, and the average detection error after calculation is η =5.8%. When the initial amplitude E 0 is 9 nV, the maximum error value is 12 nV, and the average detection error after calculation is η =7.9%. When the initial amplitude E 0 is 3 nV, the maximum error value is 5.4 nV, and the average detection error after calculation is η =12.3%. It can be concluded that the smaller the initial amplitude is, the lower the SNR of the input signal of the system is, and the detection error is gradually increased when the noise of the detection system is unchanged. The MRS signals under different SNRs are constructed respectively, and the calculation is repeated 100 times for each group of data and the average detection error is obtained by statistics of the detection error. The calculation results are shown in Table 1. When the SNR is −30 dB, the detection error of the initial amplitude is 4.7%, and when the SNR is as low as −35 dB, the detection error of the initial amplitude is 5.7%. Taking the detection error of the initial amplitude required by the existing detection engineering as the standard of 5%, the detection lower limit of signal to noise ratio that can be reached by the detection method proposed in this paper is −30 dB. VOLUME 11, 2023

V. CONCLUSION
In this paper, we propose a method to extract MRS signals submerged in electromagnetic noise by using chaos detection system. According to the attenuation characteristics of MRS signal with time, a signal segmentation method based on signal characteristic parameters and detection accuracy is proposed to complete the preprocessing of MRS signal. Moreover, the detection model of chaotic system suitable for MRS signal recognition is established, and the state of the chaotic system is determined by the calculation method of center-covered to improve the detection efficiency. Compared with the stacking method, the advantages of this method are verified. The verification of a large number of simulation data shows that the method proposed in this paper can effectively identify the MRS signal with a SNR of -30 dB, which expands the application range of the MRS method in the high electromagnetic noise environment.
The simulation verification is carried out in an ideal noise environment in this paper, which proves the feasibility of the detection method. In the field environment, the electromagnetic noise data is more complex and changeable. In the follow-up work, we will test and analyze the actual measured noise data to further verify the signal extraction ability of the chaos detection method for the actual MRS data. XIAOFENG YI received the Ph.D. degree from Jilin University, Changchun, China, in 2014. He is currently an Associate Professor and the Doctoral Supervisor with Jilin University. In recent years, he has presided over and participated in the completion of a number of national and provincial scientific research projects, including the National Natural Science Foundation of China Youth Fund, the Jilin Provincial Outstanding Young Talent Fund, and other scientific research projects. He also participated in the national major instrument and equipment development project as the main finisher. In 2021, he was selected into the High-Level Scientific and Technological Innovation Talent Project of the Ministry of Natural Resources and won the title of Young Talent. His current research interests include geophysical detection methods and equipment, urban and engineering safety perception equipment, and early warning technology.
PENGYU WANG was born in Liaoyuan, Jilin, China. He is currently pursuing the Ph.D. degree with the College of Instrumentation and Electrical Engineering, Jilin University. His research interest includes geophysical exploration methods.
YUE WANG was born in Harbin, Heilongjiang, China. He is currently pursuing the master's degree with the College of Instrumentation and Electrical Engineering, Jilin University. His research interest includes geophysical exploration methods.
KAIHUA LU was born in Zhoukou, Henan. He is currently pursuing the master's degree in electrical theory and new technology with Jilin University. His research direction is microseismic monitoring in underground coal mine. During his master's degree, he participated in the research project of microseismic monitoring equipment development and early warning method for water inrush in mine wells.
YONG SUN received the B.S. degree in survey and control technology and instrument from Jinlin University, Jilin, China, in 2012 and 2016, where he is currently pursuing the master's degree with the College of Instrumentation and Electrical Engineering. His current research interest includes application of nuclear magnetic resonance detection technology in underground engineering.
BAIZHOU AN is currently a Geophysics Engineer. He is currently pursuing the Ph.D. degree with the Jilin University. He is also engaged in geophysical research, with the research direction of geothermal, ground penetrating radar, and other geophysical methods for engineering and environmental applications. VOLUME 11, 2023