J. Cent. South Univ. (2017) 24: 900-911

DOI: 10.1007/s11771-017-3492-y

Noise cancellation of a multi-reference full-wave magnetic resonance sounding signal based on a modified sigmoid variable step size least mean square algorithm

TIAN Bao-feng(田宝凤)^{1, 2}, ZHOU Yuan-yuan(周媛媛)², ZHU Hui(朱慧)²,

JIANG Chuan-dong(蒋川东)^{1, 2}, YI Xiao-feng(易晓峰)^{1, 2}

1. Key Laboratory of Geo-Exploration Instrumentation, Ministry of Education (Jilin University),

Changchun 130026, China;

2. College of Instrumentation and Electrical Engineering, Jilin University, Changchun 130026, China

Central South University Press and Springer-Verlag Berlin Heidelberg 2017

Abstract: Nano-volt magnetic resonance sounding (MRS) signals are sufficiently weak so that during the actual measurement, they are affected by environmental electromagnetic noise, leading to inaccuracy of the extracted characteristic parameters and hindering effective inverse interpretation. Considering the complexity and non-homogeneous spatial distribution of environmental noise and based on the theory of adaptive noise cancellation, a model system for noise cancellation using multi-reference coils was constructed to receive MRS signals. The feasibility of this system with theoretical calculation and experiments was analyzed and a modified sigmoid variable step size least mean square (SVSLMS) algorithm for noise cancellation was presented. The simulation results show that, the multi-reference coil method performs better than the single one on both signal-to-noise ratio (SNR) improvement and signal waveform optimization after filtering, under the condition of different noise correlations in the reference coils and primary detecting coils and different SNRs. In particular, when the noise correlation is poor and the SNR<0, the SNR can be improved by more than 8 dB after filtering with multi-reference coils. And the average fitting errors for initial amplitude and relaxation time are within 5%. Compared with the normalized least mean square (NLMS) algorithm and multichannel Wiener filter and processing field test data, the effectiveness of the proposed method is verified.

Key words: magnetic resonance sounding signal; multi-reference coils; adaptive noise cancellation; sigmoid variable step size least mean square (SVSLMS)

1 Introduction

Magnetic resonance sounding (MRS) is the most direct and non-invasive geophysical method for the detection of groundwater resources. This method utilizes electromagnetic fields by artificial excitation to make the hydrogen nucleus of groundwater form macro magnetic moment that generates a spiral movement, and detects groundwater by magnetic resonance response produced by hydrogen protons [1-4]. Detection of groundwater by MRS has many advantages compared with traditional methods for geophysical exploration, such as quantitation, high resolution, and information richness. The MRS method can also yield hydro-geological information, such as the depth, thickness, water content of the groundwater zone, and porosity of aquifers, without drilling. Owing to these characteristics mentioned above, the MRS method has been applied in groundwater exploration, hydrogeological survey, evaluation of landslide stability, dam leakage, detection of goaf water in coal mine, early warning of advanced detection of tunnel water inrush and the like [5-10]. However, MRS signals are susceptible to electromagnetic noise, especially strong power-line harmonics, which is the primary limitation to the application of the MRS method.

Many studies concerning the noise cancellation of MRS signals have been conducted by many investigators worldwide. To process free induction decay (FID) signals received by the first generation of MRS instruments [11], data stacking method, laying a “figure-8” shaped surface coil, designing traditional filters, or wavelet transformation are commonly adopted. However, all of these methods can achieve better noise cancellation only under certain conditions or by combining several denoising strategies [12-17]. In 2014, GHANATI et al [18] proposed an MRS signal denoising method based on a statistical optimization and empirical mode decomposition (EMD), but it is useless when SNB is relatively small. Utilising noise cancellation methods and other engineering practices, in 2006, RADIC [19] presented a filtering algorithm in frequency domain using a vertical reference coil as a remote reference. USA and France developed the second generation of multichannel instruments, GeoMRI and NUMIS^poly, in 2008 and 2009, respectively, to detect full-wave MRS signals and proposed the idea of using reference coils to collect noise synchronously and denoising based on a theory of adaptive noise cancellation [20-22]. In 2010, MLLER-PETKE and YARAMANCI [23] proposed a domain method with a single reference coil. In 2012, DALGAARD et al [24] studied a time domain variable step size NLMS algorithm to cancel noise adaptively in multi-channel MRS signals, and for the useful signals contaminated by impulse noise, they adopted the NEO algorithm to detect. MLLER-PETKE et al [25], in 2014, researched the optimal parameter settings and comparisons of the time and frequency domain algorithm for power line noise cancellation based on the method of reference coils, they analysed detailedly how to set parameters can achieve the optimal filtering effect in time and frequency domain methods. In 2014, LARSEN et al [26] from Aarhus University proposed a noise modeling to cancel power-line harmonics, and then adopted multichannel Wiener filter to remove other noise in the MRS signal.

Investigations of magnetic resonance noise suppression in China began later. In 2001, a team from Jilin University began to study the first generation of MRS instruments for detecting groundwater. They succeeded in developing the MRS instrument JLMRS-I in 2010. Along with the development of MRS instruments, corresponding denoising methods were presented. In 2009, WANG et al [27] extracted FID signals of MRS and cancelled the noise by quadruple Larmor frequency sampling, which can eliminate some random noise and power-line harmonics. In 2009 and 2011, JIANG et al [28, 29] adopted a statistical stacking method to filter the spike and a portion of random noise and employed an adaptive notch filter to eliminate power-line harmonics, respectively. The team from Jilin University has always been devoted to developing second-generation MRS instruments, but research on algorithms has outpaced the progress on MRS instruments. In 2011, TIAN et al [30] proposed the variable step size adaptive filtering algorithm in frequency domain to cancel power-line harmonics containing in MRS signals under conditions of different water content and SNRs. In 2012, the team presented a successful experimental model for multi-channel magnetic resonance sounding instruments, and TIAN et al [31, 32] subsequently presented a method of variable step adaptive noise cancellation for noise suppression. In 2015, TIAN et al [33], for the single channel MRS signal, used a digital orthogonal method to construct input channels to extract the effective MRS signal based on independent component analysis.

The previous comprehensive discussion of the status of noise cancellation research illustrates that the development of the second generation of multichannel MRS instruments and research concerning corresponding denoising algorithms is international in scope, and the corresponding multichannel algorithms in time and frequency domain are also presented incessantly. However, the relevant literature only describes traditional adaptive algorithms or multichannel Wiener filter for denoising. Concerning the non-homogeneous spatial distribution of noise sources, no other studies have addressed the theory of multichannel reference coils for denoising or research on other adaptive algorithms.

From the perspective of the non-homogeneous spatial distribution of environmental noise, this paper describes the construction of a multi-reference coil adaptive noise cancellation system and a theoretical analysis of the feasibility of implementing this system. We present a modified sigmoid variable step size least mean square (SVSLMS) algorithm for denoising and compare it with other traditional algorithms. Furthermore, we discuss the extraction of full-wave MRS signals under different conditions of noise correlation and SNRs.

2 Analysis of data characteristics of MRS signals

The Larmor frequency ω₀ of a full-wave MRS signal is proportional to the local geomagnetic field and ranges from 1.3 kHz to 3.7 kHz worldwide. An MRS signal accompanying with noise can be expressed as follows:

           (1)

where E₀ is defined as the initial amplitude of the MRS signal changing from 10 nV to 4000 nV, denoting the water content of groundwater. Different from other geophysical methods, the value of E₀ cannot be increased by improving the power of the transmitters. is the relaxation time of the MRS signal related to the pore size of the underground medium. φ represents initial phase it is the electrical conductivity of the underground medium [34]. N indicates the noise in the MRS signal. The principal sources of noise are: 1) Power-line harmonics from power lines, power transformers, motors, underground cables and other infrastructure, which theoretically produce fixed frequency of integral multiples of 50 Hz, but the frequency is unstable in reality; 2) Natural noise similar to uniform white noise or Gaussian white noise; 3) Spike noises from solar magnetic storms or sudden discharges from any object, which happen occasionally and have a short duration, a higher amplitude than the MRS signal and a wide spectrum range, which may overlap with the spectrum of the MRS signal. Of these three types of noise, power-line harmonics that affects MRS signals is the most serious and widespread. Figure 1 shows power-line harmonics at 2250, 2300, 2350, 2400 and 2450 Hz. A full-wave MRS signal polluted by power-line harmonics in time-domain is shown in Fig. 1(a) and the corresponding spectra are presented in Fig. 1(b).

Fig. 1 Full-wave noise from actual tests and corresponding spectra:

3 Construction of a multi-reference coil adaptive noise cancellation system and corresponding algorithm

3.1 Construction of a multichannel adaptive noise cancellation system

Noise in the reference channels is related to that in the primary channel, so the adaptive noise cancellation method is able to remove noise in useful signal which is the basic theory of ANC [35]. Based on the adaptive principle, the most perfect situation is that only one reference loop (single reference) is adopted to collect the environmental noise without an MRS signal present which can be realized by calculating the minimum relative distance between the reference loop and primary loop, and the reference noise is well-correlated with the primary one. Therefore, the reference loop must be set far from the primary loop and close to noise sources as much as possible. Yet this above mentioned circumstance only can be achieved if there is only one and uniformly distributed noise source. However, this situation does not exist in reality. Instead, the situation of complexity and non-homogeneous spatial distribution of noise sources do exist in application. A diagram of the layout of reference coils with three noise sources is illustrated in Fig. 2.

Fig. 2 Diagram of layout of reference coils with three noise sources in a field

The principle diagram of an adaptive noise cancellation system with multi-reference channels is shown in Fig. 3, and the system has two parts: a primary channel (a measuring channel) and reference channels. The primary channel receives a signal, s, generated by a signal source (groundwater) and disturbed by a noise source, n₀, so the primary channel receives a mixed signal containing s and n₀. Reference channels record signals n₁, n₂, …, n_N from different locations and intensity of noise sources to detect noise interference. The noise signals, n₁, n₂, …, n_N, and n₀, are correlated, whereas n₀, n₁, n₂, …, n_N and the signal of interest, s, are not correlated, which fulfills the premise of adaptive correlative cancelling. Weight coefficients w₁(n), w₂(n), …, w_N(n) are adjusted by adaptive filters whose purpose is to produce an output, y(n), which is the sum of y₁(n), y₂(n), …, y_N(n), which are the corresponding outputs of each adaptive filter. If y(n) is as similar as possible to n₀ by the minimum mean square error, y(n) is the optimal estimation of n₀. By subtraction, n₀ of the primary channel can be cancelled to produce the system output, e(n), which is the optimal estimation of s(n).

Fig. 3 Principle diagram of multi-reference channel adaptive noise cancellation system

3.2 Theoretical calculation and feasibility analysis of multi-reference adaptive noise cancellation.

Considering the characteristic analysis of a full-wave MRS signal, the influence of power-line harmonics near the Larmor frequency is predominant in the case of three typical noise sources. For example, with three reference loops containing strong power-line harmonics, the theoretical calculation of the adaptive noise cancellation by multi-reference coils is as follows:

Noise signals recorded by a primary coil are expressed by

                       (2)

Noise signals recorded by the reference coils are given by

                       (3)

                      (4)

                       (5)

The output of the reference channels is given by

    (6)

where φ₁, φ₂ and φ₃ are determined by

                             (7)

                             (8)

                             (9)

Then, the output, y(n), can be rewritten by turning addition and subtraction into multiplication in the trigonometric functions as follows:

        (10)

If y(n)=n₀(n), the equations can be obtained as follows:

               (11)

                (12)

Both two sides of Eqs. (11) and (12) are squared and added, then we can get the following equation:

         (13)

To prove the previous hypothesis, y(n)=n₀(n), is true, the filter weight coefficients, w₁(n), w₂(n), w₃(n), are adjusted by an adaptive algorithm to make Eq. (13) true, and thus, the ideal output, e(n)=s(n), is obtained.

Concerning Fig. 3, the system output is

                     (14)

where

                         (15)

              (16)

In order to obtain the optimal output, we should follow the principle of the mean square error of objective function minimum:

                             (17)

The expansion of Eq. (17) is given by

                             (18)

where is unrelated to parameters of adaptive filter, and s(n), n₀(n), y(n) are uncorrelated, so In the actual signal process, because of the influence from other random natural noise, it is difficult to make (the linear combination of reference noise is perfectly correlated with detecting noise). Therefore, when we choose a suitable adaptive algorithm to adjust weight coefficients w_i(n) of filters in order to make is realized, which indicates that e(n) is the optimal estimation of s(n) and noise can be cancelled efficiently.

3.3 Presentation of SVSLMS algorithm

Least mean square (LMS) is a traditional algorithm for the application of adaptive filters and has been widely used in many applications [36, 37]. However, its fixed step size characteristics make a contradiction exist in adaptive convergence speed and steady-state detuning [38]. Thus, corresponding variable step size algorithms are utilised, and the most traditional is the normalised LMS (NLMS) [39, 40], which is expressed as

                      (19)

where the parameter σ is set to avoid the energy of an adaptive input signal that is not sufficiently large to result in a step size that is not sufficiently short to give rise to the algorithm’s divergence, 0≤σ≤1. Thus, the step size μ(n) changes correspondingly and is inversely proportional to the short time energy change of the reference input. Under that condition, the initiation of an adaptive algorithm is slow and inhibits rapid convergence when the environmental noise and input energy are stronger and lead to a smaller initial step size.

Variable step size algorithms have been successively proposed to rectify the situation discussed above, and then have been applied effectively in echo cancellation, speech enhancement, biomedical field and other aspects [41-44]. YASUKAWA et al [45] presented an algorithm in which the step size was proportional to the size of the error signals. This method increased convergence and reduced steady-state detuning, but it was sensitive to the signal input. TAN et al [46] proposed a variable step size LMS algorithm whose step size was a sigmoid function (SVSLMS). The features of this algorithm were that the step size was larger at the beginning, and it was smaller at the steady state, so that the faster convergence speed, faster traced speed and reduced steady-state detuning were achieved. However, the disadvantage of this algorithm was that the step size did not change slowly when the error was smaller. Thus, considering the ideas behind the two algorithms discussed above, we present a modified SVSLMS algorithm in this work whose variable step size expression is defined as

                   (20)

The condition for the convergence of this algorithm in Eq. (20) is

                           (21)

where λ_max is the maximum eigenvalue of autocorrelation matrix of input signals. A conclusion deduced from Eq. (21) is that 0<β<1/λ_max. Figure 4 shows the changing curves of step size μ(n) with different values of α and β. It is concluded that α and β codetermine the size and shape of step size function from Fig. 4. β controls the value range of μ(n), and influences initial step size, and then affects convergence rate. α>0 controls the shape of this function. Figure 4(a) indicates the shape of μ(n) changes a lot with unchanged values of β and different values of α. We expect there is a rapid convergence rate at the adaptive initial phase that means a bigger value of μ(n), and there is a smaller detuning at the steady-state phase that means μ(n) changes slowly when e(n) closes to 0. So, from Fig. 4(a), we can conclude that when α<5, the initial step size changes large, and then affects the convergence rate of algorithm; when , the shape of curves is basically the same; when α>10, the shape of curves changes fast in the process of e(n) closing to 0. Therefore, in this work, we choose α=5, a critical value, which is able to meet the demands of a larger initial step size to ensure a rapid convergence rate and a slow change when e(n) closes to 0 to ensure a smaller steady-state detuning.

Based on the above analysis, Fig. 5 shows the curves of the step size, μ(n) changing with the error function, e(n), adopting methods from the modified SVSLMS algorithm [45, 46]. From the curves of step size function, we find that this proposed algorithm has a larger step size in the initial phase to ensure a rapid convergence rate, and after convergence, the step size varies slowly in a steady-state phase, which is what we expect.

Fig. 4 Changing curves of μ(n) with different values of α (a) and β (b)

Fig. 5 Function curves of μ(n) changing with e(n) in three variable step algorithms

4 Simulation analysis of algorithms

A simulation with MATLAB 7.0 was conducted. An MRS signal in Changchun city with a Larmor frequency of 2326 Hz was chosen for this experiment. Principal noise sources from power-line harmonics of 2350 and 2250 Hz were added to the MRS signal stacked with random noise. According to Eq. (1), E₀ and were set as 200 nV and 150 ms, respectively, and φ could be set as an arbitrary value in [-π, π].

4.1 Performance analysis of a multi-reference cancellation system relative to a single reference cancellation system

To verify the denoising effect of a multi-reference noise cancellation system was better compared with a single reference system, the simulation experiment was conducted under the conditions of different correlations of noises included in the reference channels and the primary detecting channels and different SNRs (SNR<0 dB). Because SNR is a statistical average of signals and noises, it is inaccurate to use the SNR alone to judge the filtering effect. In the MRS signal, determines the speed of the signal attenuation as well as the lithology of the underground medium hiding in a groundwater signal, so the accuracy of the relaxation time, is the difference between measured relaxation time and ideal relaxation time) combined with the SNR was adopted to assess the denoising performance of systems.

Tables 1 and 2 give the comparison of the SNR and η after adopting one, two, and three reference channels to cancel noise when the correlation coefficients, γ, of the noise of the reference channels and the primary channel are 0.9 and 0.6, respectively. Table 1 shows that when the correlation coefficient is 0.9, the SNRs of the three systems increase to the same level and the maximum improvement in the SNR is more than 19 dB. When the correlation is poor, γ=0.6, the increase in the SNR is low. Additionally, with the same correlation coefficients, a higher SNR is obtained for multi-reference channels (two and three reference channels) rather than for a single reference channel.

Figure 6 shows the denoising results for different reference channels when γ=0.6 and the SNR=-1.0883 dB. A comparison of the traces of the time-frequency domain before and after filtering shows the effect of two and three reference channels is similar, and they yield a better time-domain characteristic and spectrum after filtering compared with a single reference channel.

4.2 Comparative analysis of proposed algorithm and other adaptive methods

To further demonstrate the superiority of the proposed algorithm compared with the traditional fixed step size LMS algorithm and the classical NLMS algorithm that was published in 2012 by DALGAARD et al [24], and with two reference channels, a comparative analysis of the three aforementioned algorithms was conducted when the correlation was poor, γ=0.6, and the starting SNR=-1.0883 dB. Figure 7 shows comparisons

Table 1 SNR outputs of three systems with different correlation coefficients after denoising

Table 2 Accuracy of for three systems with different correlation coefficients after denoising

Fig. 6 Denoising results of different reference channels when SNR=-1.0883 dB and γ=0.6:

Fig. 7 Signal waveform and learning curves after filtering using three algorithms:

Figure 7(a) shows the modified SVSLMS algorithm performs better than the traditional fixed step size LMS algorithm and the NLMS algorithm for filtering results. The signal waveform extracted by the modified SVSLMS algorithm is closer to the ideal MRS signal shape. Figure 7(b) shows that the convergence speed of the algorithm proposed in this paper is faster than the fixed step size LMS algorithm and the NLMS algorithm. After 400 iterations, this algorithm convergence enters a steady state, and its detuning is smaller, which means that the fluctuation changes within a smaller range.

4.3 Comparison between proposed algorithm and multichannel Wiener filter

Further, we carry out comparison between the proposed algorithm and multichannel Wiener filter on the conditions of the following simulation parameters. Adopting the method of double channels, the initial amplitude, E₀, of detecting signal in primary channel is 200 nV, and f₀=2325 Hz. The frequency of power-line harmonics are respectively 2250 Hz and 2350 Hz. Both primary channel and reference channels contain a certain ratio (-2 dB) of uncorrelated white Gaussian noise, and noise correlation coefficient is 0.6 (γ=0.6). Besides, SNB of noisy MRS signal in the primary channel is -7.5809 dB. Figure 8 respectively demonstrates processing results by the proposed algorithm and multichannel Wiener filter.

From Fig. 8, we find that obvious denoising results are obtained by two mentioned methods. SNR are 2.0835 dB and 1.3642 dB respectively processed by the proposed algorithm and multichannel Wiener filter, which indicates a better denoising result is able to be obtained by the method in this work. At the same time, in Fig. 8(a), attenuation trend of MRS signal processed by the proposed algorithm is more obvious. In Fig. 8(b), power-line harmonics with correlation in the primary and reference channels are almost cancelled, however random noise without correlation still exists. By the modified SVSLMS algorithm, fitting error of E₀ is 0.2328%, and fitting error of is 3.3021%. By multichannel Wiener filter, amplitude attenuation of the main spectra of MRS signal is larger and there is a certain fluctuation that leads to a bigger fitting error of E₀ and (fitting error of E₀ is -13.3104% and of is 18.5339%) which does not meet the demands of practical application.

Fig. 8 Contrastive analysis of proposed algorithm and multichannel Wiener filter:

5 Real-world cases

To verify the performance of the multi-reference noise cancellation system and the modified SVSLMS algorithm, test data were collected at Wenhua square of Changchun city. In this experiment, an analogue MRS signal was transmitted by 25 m of single-turn coil added with the signal sources, and three 25 m double-turn coils were laid out; one was the primary channel used to record analogue MRS signals containing noise, and the other two were the reference channels used to collect environmental noise. The distance of each coil calculated theoretically was more than 50 m, ensuring that the reference channels did not record MRS signals. Figures 9 and 10 show the time and frequency domains of three datasets before and after filtering using a single reference and double reference noise cancellation methods respectively. The increase in the SNR and characteristic parameters extracted via the single and double reference methods are shown in Table 3.

Figures 9 and 10 show the two reference channel method performs better when considering both the time domain plots and spectra after filtering.

Fig. 9 Comparison plots of time and frequency domains of test data processed by a single reference channel noise cancellation method:

Fig. 10 Comparison plots of time and frequency domains of test data processed by double reference channel noise cancellation method:

Table 3 Results processed by a denoising method using a single reference channel and two reference channels

Table 3 illustrates that a method using double reference channels yields a better result than that using a single reference channel for both the increase in the SNR and the deviation of extracted characteristic parameters relative to the original values. Calculation yields average fitting errors for the initial amplitude E₀ and relaxation time within 2.53% and 4.01%, respectively.

6 Conclusions

1) Under conditions of different noise correlations and SNRs (SNR<0 dB), the multi-reference noise cancellation method yields a higher increase in the SNR and smaller average fitting errors in the initial amplitude E₀ and relaxation time than seen with a single reference noise cancellation system. When the noise correlation of the reference channels and a primary channel is poor, the multi-reference noise cancellation method performs better with respect to the signal shape and spectra after denoising than does the single reference noise cancellation system. In addition, the denoising effect processed by two and three reference channels is similar, which lays a good foundation during subsequent applications to avoid wasting manpower and material resources to install more reference channels.

2) The modified SVSLMS algorithm has more advantages in convergence rate during the initial phase and steady-state detuning than a traditional variable step NLMS algorithm. Through comparative analysis between the proposed algorithm and multichannel Wiener filter, it is concluded that the modified SVSLMS algorithm performs better on signal shape after filtering and data fitting error.

3) The test data demonstrate that a good denoising effect can be obtained with a two reference channel noise cancellation method combined with the modified SVSLMS algorithm for processing test data that has different SNRs, and the average fitting errors of initial amplitude and relaxation time are smaller, which meet the requirements of practical application.

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(Edited by YANG Bing)

Cite this article as: TIAN Bao-feng, ZHOU Yuan-yuan, ZHU Hui, JIANG Chuan-dong, YI Xiao-feng. Noise cancellation of a multi-reference full-wave magnetic resonance sounding signal based on a modified sigmoid variable step size least mean square algorithm [J]. Journal of Central South University, 2017, 24(4): 900-911. DOI: 10.1007/s11771-017-3492-y.

Foundation item: Projects(41204079, 41504086) supported by the National Natural Science Foundation of China; Project(20160101281JC) supported by the Natural Science Foundation of Jilin Province, China; Projects(2016M590258, 2015T80301) supported by the Postdoctoral Science Foundation of China

Received date: 2015-09-28; Accepted date: 2016-08-01

Corresponding author: YI Xiao-feng, PhD; Tel: +86-13843143707; E-mail: yixiaofeng@jlu.edu.cn