Joint application of a statistical optimization process ...

Journal of Applied Geophysics 111 (2014) 110–120

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Joint application of a statistical optimization process and Empirical Mode Decomposition to Magnetic Resonance Sounding Noise Cancelation Reza Ghanati ⁎, Mahdi Fallahsafari, Mohammad Kazem Hafizi Institute of Geophysics, University of Tehran, North Karegar Avenue, Tehran 1435944411, Iran

a r t i c l e

i n f o

Article history: Received 19 June 2014 Accepted 29 September 2014 Available online 7 October 2014 Keywords: Empirical Mode Decomposition Magnetic Resonance Sounding Non-linear decomposition Statistical analysis

a b s t r a c t The signal quality of Magnetic Resonance Sounding (MRS) measurements is a crucial criterion. The accuracy of the estimation of the signal parameters (i.e. E0 and T2⁎) strongly depends on amplitude and conditions of ambient electromagnetic interferences at the site of investigation. In this paper, in order to enhance the performance in the noisy environments, a two-step noise cancelation approach based on the Empirical Mode Decomposition (EMD) and a statistical method is proposed. In the first stage, the noisy signal is adaptively decomposed into intrinsic oscillatory components called intrinsic mode functions (IMFs) by means of the EMD algorithm. Afterwards based on an automatic procedure the noisy IMFs are detected, and then the partly de-noised signal is reconstructed through the no-noise IMFs. In the second stage, the signal obtained from the initial section enters an optimization process to cancel the remnant noise, and consequently, estimate the signal parameters. The strategy is tested on a synthetic MRS signal contaminated with Gaussian noise, spiky events and harmonic noise, and on real data. By applying successively the proposed steps, we can remove the noise from the signal to a high extent and the performance indexes, particularly signal to noise ratio, will increase significantly. © 2014 Elsevier B.V. All rights reserved.

1. Introduction The method of Magnetic Resonance Sounding (MRS) is a noninvasive hydro-geophysical tool providing information on the distribution of water content in the subsurface and, under favorable conditions, hydraulic conductivity. A major limitation of the MRS technique is high sensitivity to noise. Beside the ubiquitous Gaussian distributed white noise (Costabel and Muller-Petke, 2014), the two important noise sources consist of power-line harmonics and discharges from both natural and man-made sources, that is, thunderstorms, telluric currents and magnetic storms, as well as electrical installations as cars, radio transmitters and electrical fences, etc. (Costabel and Muller-Petke, 2014; Dalgaard et al., 2012; Perttu et al., 2011). Short electrical discharges bring about an impulsive excitation of the band-pass filters with a subsequent near-exponential decay known as spikes (Dalgaard et al., 2012). The MRS signal usually varies between ten to a couple of thousand nV using 100 m square loop and the ambient noise is often higher (Perttu, 2011). Since the MRS signals are originally at nano-volt range, the ambient noise conditions will be very critical for them. In other words, MRS is vulnerably affected by even very low noises. In addition to this, the challenge of characterizing groundwater in some places by MRS is that the returned signals are quite weak due to low

⁎ Corresponding author. Tel.: +98 918 8624200. E-mail address: [email protected] (R. Ghanati).

http://dx.doi.org/10.1016/j.jappgeo.2014.09.023 0926-9851/© 2014 Elsevier B.V. All rights reserved.

water content (Plata and Rubio, 2002; Walsh et al., 2012). Hence, it is vitally significant to implement de-noising on the MRS measurements. So for, several approaches have been developed for removing or at least decreasing the influence of noise during acquisition and data processing (Legchenko, 2007; Legchenko and Valla, 2002). Trushkin et al. (1994) proposed an eight-shaped loop in order to improve the signalto-noise ratio. If the antenna with a figure-of-eight is used signal-tonoise ratio can be increased by a factor of 5 to 10 compared to circle or square loops. Using the same length of wire, the depth of investigation with the eight-shaped antenna is about a half of that the one acquired with the square loop (Bernard, 2007; Plata and Rubio, 2002). However, even with this loop the signal-to-noise ratio may be not sufficiently appropriate for inversion (Legchenko, 2007). Three filtering methods: block subtraction, sinusoid subtraction and notch filtering for removal of power-line harmonics from MRS measurements were studied by Legchenko and Valla (2002). They showed that, the notch filtering was the most effective but it distorts the signal of interest when the frequency offset between the Larmor frequency and one of the power-line harmonics is smaller than 8 Hz. In such conditions, subtraction techniques have preference. Also, a stacking procedure is utilized pffiffiffi during data acquisition so that the signal-to-noise ratio increases n times, where n is number of stacks (Legchenko and Valla, 2002; Perttu, 2011; Plata and Rubio, 2002). But this process is, in fact, timeconsuming. Plata and Rubio (2002) showed that interfering spikes cannot be considerably suppressed by stacking the signal records. Recent developments on hardware design allow overcoming some restrictions

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on applying MRS technique (Dalgaard et al., 2012; Dlugosch et al., 2011; Muller-Petke and Costabel, 2014; Walsh, 2008). Multi-channel devices use a primary loop for excitation and signal receiving, and a number of additional reference loops to simultaneously record the local noise. The references loops provide the possibility of mitigating the noise in the primary loop signal by subtraction through Wiener or adaptive filters (Dalgaard et al., 2012). However, the precondition for the use of adaptive noise cancelation using reference loops depends on the fact that the noise of reference channels has the best correlation with the MRS-signal detection channel. Besides Gaussian noise and power-line harmonics, MRS records may be contaminated by spikes. The conclusion of Dalgaard et al. (2012) observations illustrated that, the presence of spike makes adaptive noise cancelation useless and potentially unreliable. Jiang et al. (2011) suggested a statistical approach called the Romanovsky criterion to discern and discard spiky noises. Their procedure is performed in the stacking process prior to noise canceling. Costabel and Muller-Petke (2014) took advantage of the wavelet-like essence of spiky events to isolate and eliminate spiky signals in the wavelet domain. Moreover, they presented a remote reference based harmonic noise cancelation. They concluded that frequency domain (FD) is preferable to the time domain approaches. However, some drawbacks to the use of the FD method were observed by them. In spite of relatively successful application of the existing de-noising approaches, in our opinion a reliable and efficient technique to cancel both anthropogenic and natural electromagnetic noises from MRS measurements is still missing. The general objective of this study is to use simultaneously advantages and characteristics of Empirical Mode Decomposition (EMD) (Huang et al., 1998) and a statistical optimization process (Shahi et al., 2011) to further enhancement of the signal-to-noise ratio in MRS signals. Despite EMD method offers many promising features for analyzing and processing geophysical data, there are still few applications on geophysics. Magrin-Chagnolleau and Baraniuk (1999) extract seismic time-frequency attributes through EMD method. Chen and Jegen-Kulcsar (2006) applied EMD in Magneto-telluric data processing. Jeng et al. (2007) use EMD for power-line interference and Gaussian white noise elimination from VLF-EM data. Battista et al. (2007) employ EMD to remove cable strum noise in marine seismic data. Han and van der Baan (2013) exploit EMD for seismic time-frequency analysis. The paper is organized as follows. In Section 2, the methods and algorithms used in this paper are described. Next, in Section 3, we discuss the performance of the described methods in synthetic and real examples. A comparison of the results achieved by the proposed algorithm and SMAVOR software for the real data are presented in Section 4. A short conclusion summarizes the main ideas.

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frequency and computed through a sifting process. Any IMFs should satisfy the following two specifications: 1) The number of extrema (maxima and minima) and the number of zero crossings must either equal or differ at most by one. 2) At any given point, the mean value of the envelope defined by the local maxima and the envelope by the local minima should be zero. The EMD algorithm for implementing sifting on MRS signal e(t) is given as follows: 1) Identify all extrema (maxima and minima) of the signal, e(t) 2) Generate the upper and lower envelopes via interpolation among all the maxima and minima points, respectively (in our algorithm, we used Piecewise Cubic Hermit Interpolation). 3) Calculate the mean of the envelopes, m(t) = (envup(t) + evndown(t))/2. 4) Subtract m(t) from the MRS signal to obtain the detail: d1(t) = e(t) − m(t). Step 1 to 4 is one iteration of the sifting process. The signal d1(t) output of the first iteration is tested using the stopping criterion. Two possibilities now exist: I. d1(t) is not an IMF (i.e. it does not satisfy the stopping criterion). In this case, d1(t) is given as input to the next iteration of the sifting process (i.e. step 1 to 4 is repeated) II. d1(t) is found to meet the stopping criterion, so no further iterations are needed. A stopping criterion to the number of sifting iteration is employed to insure that IMF component retain enough physical sense of both amplitude and frequency modulation (Chacko and Ari, 2012; Huang and Wu, 2008). Some stopping criterions are given by Huang et al. (1998). In our algorithm, the stiffing process is continued until a residue error of a standard deviation between consecutive components is met. The standard deviation between component dk − 1 and dk for k number of sifting iterations is given by "

SD ¼

TX −1 t¼0

2

jdk−1 ðt Þ−dk ðt Þj d2k−1 ðt Þ

# ;

ð1Þ

2. Methods 2.1. EMD algorithm The recovery of a signal from observed noisy data remains a challenging problem in both signal processing and statistics. A number of filtering methods have been proposed, particularly for the case of nonstationary signals. Due to non-linear and non-stationary nature of the geophysical data (Mohebian et al., 2013) (e.g. MRS signals), use of an adaptive method for analysis of such data is absolutely necessary. EMD, as an adaptive method means that the basis is defined based on and derived from the data (Huang and Wu, 2008; Jeng et al., 2007), decomposes a signal into a set of mono-component functions called intrinsic mode functions (IMFs) (Huang et al., 1998). A mono-component function indicates an oscillating function close to the most common and basic elementary harmonic functions. Therefore, IMFs contain frequencies ranging from the highest to the lowest ones of the signal presented as amplitude and frequency modulated (AM–FM) signal, where AM carries the envelope and FM is the constant amplitude variation in

Fig. 1. Ideal (red) and noisy MRS signal (black) with SNR = 5.3 dB. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.).

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First step Noisy MRS signal Decomposition via EMD

Noise-only Signal Decomposition via EMD Detect Noisy IMFs based on amounts of the IMF energies

Eliminate the Noisy IMFs for Signal Reconstruction Second step De-noise and Recover the Signal from the First Step via the Variance Criterion Fig. 2. Flowchart of MRS signal noise cancelation by the proposed approach. Noise-only signal decomposition via EMD. Eliminate the noisy IMFs for signal reconstruction.

where T is the time duration. Typically the value of SD is set between 0.2 and 0.3 (Huang et al., 1998). Once dk(t) is accepted as first IMF, h1(t), hence h1(t) = dk(t) 5) The residue is calculated as r 1 ðt Þ ¼ eðt Þ−dk ðt Þ;

ð2Þ

r1(t) is given as the input to the next round of sifting process. For extracting other IMFs, the performance from step 1 to step 5 is repeated on the residue. Extracting procedure is stopped if residue is constant or a function with no extrema. After extracting all IMFs, MRS signal e(t) can be expressed as eðt Þ ¼

N X k¼1

hk ðt Þ þ r N ðt Þ;

ð3Þ

where rN is the final residue, hk(t) is kth IMF and N is the number of extracted IMFs. The IMFs h1,h2,…,hN represent the finally obtained amplitude and frequency modulated output set (Saurabh and Mitra, 2012). It should be noted that, numerically, the sum of all IMFs may slightly differs from the original signal as a result of rounding error caused by the computation. Decomposition of a signal using EMD method leads to n-empirical modes and a residual rN, so that the higher frequencies are ordinarily found in the initial IMFs and lower frequencies in subsequent IMFs (Boudraa and Cexus, 2007; Huang et al., 1998; Jeng et al., 2007; Karagiannis and Constantinou, 2011). The components with the higher frequencies carry clutter energy and contain noise. Thus, some IMF components may be eliminated selectively (Boudraa et al., 2005; Jeng et al., 2007). The logic of selecting the components depends on the basic assumption made by the interpreter and the fundamental concept of the MRS signal and noise. The initial attempt at using EMD as a de-noising tool emerged from the demand to realize whether a specific IMF contains

Fig. 3. Illustration of resulting IMFs after Empirical Mode Decomposition of the noisy MRS signal shown in Fig. 1.

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Fig. 4. Noise added to the MRS signal and resulting IMFs after Empirical Mode Decomposition of the noise-only signal.

useful information or primarily noise. Flandrin et al. (2004), and Wu and Huang (2004) proposed procedures based on the statistical analysis of modes to evaluate the noisiness of IMFs. If the energy of the IMFs resulting from the decomposition of a noise-only signal with certain features is known, then in actual cases of signals compromising both information and noise following the specific characteristics, a significant discrepancy between the energy of a noise-only IMF and the corresponding noisy signal IMF indicates the presence of useful information (Flandrin et al., 2004; Kopsinis and McLaughlin, 2009; Wu and Huang, 2004). Therefore, in a de-nosing process, the IMFs that contain useful information are kept for reconstructing the signal and those that carry primarily noise, i.e. the IMFs that share similar amounts of energy with the noise-only case, are discarded. Whereas, the NUMIS equipment directly records the field noise prior to the signal record thus the noise-only signal is available in order to apply EMD and estimate the IMFs energies. As

a result, comparing the IMF energies derived from the noisy signal and only-noise signal based on the similarity of amounts of the IMF energy, the noisy IMFs are detected (Fallahsafari et al., 2014). The total energy of a discrete signal e(t) over [n1,n2] is defined as

Fig. 5. Noise-only signal (red) and noisy MRS signal (blue) IMF energies with respect to IMF number. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.).

Fig. 6. Noisy MRS signal (black) and de-noised signal from EMD (blue) with SNR = 21 dB. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.).

Es ¼

n2 X

2

jeðt Þj :

ð4Þ

n¼n1

Thus, according to Eq. (4), the energy of the IMFs resulting from the decomposition of the MRS signal and noise-only signal is calculated based on the following expression. EIM F k ¼

m    X 2  hk t j 

ð5Þ

j¼1

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Table 1 SNR and RRMSR performance of EMD method applied on a synthetic MRS signal (E0 = 150 nV and T⁎2 = 200 ms). Signal 2 (SNR = 5.3 dB)

Signal 3 (SNR = 8 dB)

Evaluation parameters

Signal 1 (SNR = 3.1 dB) SNR (dB)

RRMSRa [%]

SNR (dB)

RRMSR [%]

SNR (dB)

RRMSR [%]

EMD method

19

11

21

7.3

23.7

6.4

a

RRMSR: Relative Root-Mean-Square Error.

Where m indicates the number of signal samples and hk(t) is kth IMF. Then, semilog diagram of the IMFs energies log2EIMF with respect to IMF number for the MRS signal along with noise-only signal is calculated.

EDetection ðt Þ ¼ E0 exp

2.2. Variance criterion algorithm Shahi et al. (2011) proposed a time-based noise removal method by defining an optimization problem based on the variance criterion. We initially express a brief review on MRS basics. MRS energizes the protons in groundwater by transmitting a resonance electromagnetic pulse with the Larmor frequency (the resonance frequency of the water molecules in the geomagnetic field). The energized protons then generate a secondary magnetic resonance signal which is an exponentially decaying function of time and according to Legchenko and Valla (2002) is given by    Eðt; qÞ ¼ E0 ðqÞ exp −t=T 2 ðqÞ cos ð2π f L t þ ΘðqÞÞ;

ð6Þ

where E0 is the initial maximum voltage induced in the antenna and is directly linked to the water content, q denotes the pulse moment, the transverse relaxation time T2⁎ is correlated with the mean size of the water-saturated pores (Hertrich, 2008; Legchenko and Valla, 2002; Yaramanci and Muller-Petke, 2009) and hydraulic conductivity (Kenyon, 1997; Legchenko et al., 2002; Perttu et al., 2011), fL is the Larmor frequency and Θ is the phase shift between the returned signal and the excitation current. The Eq. (6) explains the presence of only one relaxation regime in the pore space. However, MRS relaxation data are of a multiexponential nature as a consequence of distribution of different pore sizes within a rock layer in MRS (Mohnke and Yaramanci, 2005, 2008). The detection of the signal is realized by the synchronous detection technique. Thereby the original signal is multiplied by a mono-harmonic reference wave, generated with the same frequency fR as the excitation pulse, and then low-pass filtered (Legchenko and Valla, 2002; Strehl, 2006). Mathematically, this procedure can be expressed as a multiplication with the complex term 2e j2π f R t (Levitt, 1997), which gives EMultiplied ðt Þ ¼ E0 exp

And after applying the low-pass filter in which the subtracted signal is however desired because its frequency being (f0 = fR-fL) is approached to zero as much as possible, and therefore

 t −  ½ exp ð jð2π f R þ 2π f L Þt þ jΘ0 Þ T2

ð7Þ

þexp ð jð2π f R −2π f L Þt−jΘ0 Þ:

 t −  exp ð jð2π f 0 t−Θ0 ÞÞ: T2

ð8Þ

Note that the frequency offset f0 does not exceed the range of −5 Hz to 5 Hz (Strehl, 2006). The Eq. (8) indicates the ideal form of the MRS signal. But generally, this is not true and MRS signal is, in fact, contaminated with noise. Thus, the real noisy MRS signal can be expressed as follows.  t Noisy EDetection ðt Þ ¼ E0 exp −  ½ exp ð jð2π f 0 t−Θ0 ÞÞ T2 X þ P K exp ð j2π f K t þ jΘK Þ þ eComplex ðt Þ:

ð9Þ

K

The second term of the above equation is related to the harmonic noise part and eComplex(t) denotes the stochastic noise part containing the background noise, i.e. Gaussian distributed white noise (Costabel and Muller-Petke, 2014), and spiky events. Spiky signals appear randomly, so that these noise features are considered as parts ofeComplex(t) (Strehl, 2006). The main objective is to remove the noise from Eq. (9) and access to Eq. (8). Noise cancelation from the MRS signal requires perception about the noise parameters and the ideal signal features. The mean or the area under the curve is an exclusive characteristic of these two components. The mean of two signals, i.e. the ideal signal and noise signal, is equal to the summation of the mean of each one. On the other hand, the mean of the real signal is equal to the summation of the mean values of the ideal signal and noise (Shahi et al., 2011). Mathematically, this can be defined as A ¼ AS þ AN

ð10Þ

Where A denotes the area under the curve of the real signal, AS is the area of the ideal signal of Eq. (9) and AN is the noise area. As the mean value of the noise (or its area under the curve) is approximately zero (Carlson, 1987; Shahi et al., 2011; Strehl, 2006), therefore the signal mean will not be changed after being noisy (Shahi et al., 2011). Integrating Eq. (8) between 0 and t:  Z t Z t       exp −t=T 2 dt:  EDetection ðt Þdt  ¼ E0 0   0

ð11Þ

Table 2 Estimated parameters and SNR and RRMSR performance of variance criterion and joint application EMD and variance criterion implemented on a synthetic MRS signal (E0 = 150 nV and T⁎2 = 200 ms). Signal 1 (SNR = 3.1 dB)

Signal 2 (SNR = 5.3 dB)

Signal 3 (SNR = 8 dB)

Evaluation parameters

Esta. E0 (nV)

Est. T⁎2 (ms)

SNR (dB)

RRMSRa [%]

Est. E0 (nV)

Est. T⁎2 (ms)

SNR (dB)

RRMSR [%]

Est. E0 (nV)

Est. T⁎ 2 (ms)

SNR (dB)

RRMSR [%]

Variance criterion Joint application of EMD & variance criterion

140 141

245 240

22 24

8.1 6.04

141 144

217 207

27 29

4 3.2

158 151

192 199

33 40

2.01 0.9

RRMSR: Relative Root-Mean-Square Error. a Est: estimated.

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different additive noises which contain stochastic and harmonic noises. The vector noise is modeled via the expression e = eh + es, where the stochastic noise consists of the background noise and spiky events (outliers) (Costabel and Muller-Petke, 2014; Strehl, 2006). In order to add outliers in random time samples to the simulated MRS signal, we use the following formulation applied by Gholami and Sacchi (2012). h

espike

(

i N

¼

h i ℜ  eg if N∈M 0

N

if N∉M

ð14Þ

Where eg is the Gaussian noise, ℜ ≥ 0, N = 1, …, n, and ℳ ⊂ {1, …, n}. Here, we suppose that the set ℳ is comprised of 0.03n elements. The performance of the proposed method is evaluated based on the signalto-noise ratio (SNR) and Relative Root-Mean-Square Error (RRMSR). The SNR can be presented as the following 2

SNR ¼ 10 log10 Fig. 7. Ideal signal, Noisy MRS signal (SNR = 5.3 dB), de-noised signal from EMD (SNR = 21 dB) and estimated signal using EMD and variance criterion (SNR = 29 dB).

And considering the mentioned assumption, meaning that the area of the ideal signal after being noisy remains constant (which is a precondition for the estimation of the ideal signal), we get   t  A ¼ −E0 T 2 exp −  −1 : T2 Then we have      E0 ¼ A= 1−exp −t=T 2 T 2

ð13Þ

where the upper limit t of the integral is the signal recording time. Whereas, NUMIS equipment records the noise automatically prior to the MRS measurement thus the parameters of the environmental noise can be measured. Variance is one of such parameters which fully identified before the original signal record (IRIS Instruments, 2004; Strehl, 2006). According to Eq. (13), much of T2⁎ and E0 can satisfy the constant area for the exponential signal. Therefore, if the area under the curve is correctly estimated, in case there are wrong T2⁎ or E0, the other component will be incorrect too. In other words, if E0 or T2⁎ be wrong, the mean of the estimated noise can be possibly zero, while its variance is incorrect (the variance of the estimated noise) (Shahi et al., 2011). Thus, comparison of the variance of the estimated and recorded noises makes possible to discern the precision of the values of E0 as well as T2⁎ . In order to achieve correct E0 and T2⁎ values, an optimization problem based on the variance of the estimated and recorded noises is defined. The calculated variance is compared with the variance of the recorded noise in order to realize the E0 variations, then the estimation and alteration of E0 are continued until the difference between the variance of the estimated noise and the variance of the recorded noise will be approached to approximately zero. When E0 is flawlessly calculated, we will be ensured that the value of T2⁎ is accurate as well as the estimated ideal signal is considerably noiseless. In the next section the efficiency of our procedure is investigated on synthetic and real MRS signals.

ð15Þ

Where E(t) denotes the ideal signal and Y(t) is the estimate of the signal of interest. In this study, RRMSR is used to measure the quality of the information which is preserved in the de-noised MRS signal. RRMSR is defined as follows: RRMSR ¼

ð12Þ

kEðt Þk2 kY ðt Þ−Eðt Þk22

RMSðEðt Þ−Y ðt ÞÞ 100½% RMSðEðt ÞÞ

ð16Þ

Fig. 1 shows the simulated MRS signal (SNR = 5.3 dB) contaminated with harmonics at multiples of 50 Hz (Here, we consider harmonics with the frequency of 2000 Hz), spiky events ℜ = 300 nV and Gaussian noise with standard deviation σ = 30 nV and mean value m = 0. The removal of the noise is carried out in two stages here: the EMD method is just implemented at the first stage, at the second stage the variance criterion is implemented as well. The different steps of the proposed method are illustrated in Fig. 2. Fig. 3 shows the IMFs derived from the decomposition of the synthetic MRS signal. Also, the IMFs of the noise-only signal added to the corresponding MRS signal are illustrated in Fig. 4. Based on the IMFs energies of the MRS signal and noise-only signal, the noisy IMFs are identified. Hence, semilog diagram of the IMFs energies log2 EIMF with respect to IMF number for the MRS signal as well as noise-only signal is calculated, as shown in Fig. 5. We observe that, after the sixth IMF, the energies significantly diverge

3. Numerical results 3.1. Application to synthetic example In the following, we deal with the performance of our de-noising approach by using the modeling of a synthetic MRS signal. Firstly, an exponentially decaying signal with initial amplitude E0= 150 nV and a decay time T2⁎ = 200 ms is generated and then corrupted by three

Fig. 8. The first real MRS signal with q = 947.82 used to evaluate the performance of the proposed procedure.

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Fig. 9. Illustration of resulting IMFs after Empirical Mode Decomposition of the real MRS signal shown in Fig. 8.

from the noise-only signal, indicating the presence of significant amounts of no-noise signal. The partial signal reconstruction including only IMF numbers 7 and 8 results in the de-noised signal shown in Fig. 6. It can be seen that the harmonic and Gaussian noises as well as outliers have been considerably removed through the EMD algorithm. In addition, the signal-to-noise ratio increases from 5.3 dB (related to the noisy signal) to 21 dB. Afterwards, the signal obtained from the previous stage enters the variance criterion to cancel the remaining noise in the signal and estimate the concerning parameters. So, by implementing the second stage, the signal-to-noise ratio increases to

29 dB and the MRS signal parameters E0 and T2⁎ are estimated 144 nV and 207 ms with RRMSR 3.2. Figure 7 displays the outcome of applying the EMD filtering scheme (blue) and the proposed combined method (black) to the simulated MRS signal (5.3 dB). The results of implementing EMD, variance criterion and joint application of EMD and variance criterion on the simulated MRS signal with three different SNRs (i.e. the first signal = 3.1 dB, the second signal = 5.3 dB and the third signal = 8 dB) are reported in Tables 1 and 2. One can see that, the EMD-based de-noising algorithm slightly improves SNR in the three artificial signals. Furthermore, comparing to merely use of

Fig. 10. Noise-only signal recorded prior to the MRS signal measurement and resulting IMFs after Empirical Mode Decomposition of the noise-only signal.

R. Ghanati et al. / Journal of Applied Geophysics 111 (2014) 110–120

Fig. 11. Noise-only signal (red) and the first real MRS signal (blue) IMF energies with respect to IMF number. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.).

variance criterion, the proposed method (joint application of the EMD algorithm and variance criterion) can provide more satisfactory results so that the SNR values corresponding to all the simulated signals exhibit better enhancement. It can be also seen that the values of RRMSR obtained by using the proposed algorithm are lower than those of variance criterion. It should be noted that in the second stage of the proposed method (i.e. variance criterion), the calculation of the E0 value is dependent on the first sample of the MRS signal which may be destroyed by the noise. Therefore, based on the above assumption as well as our investigation, if the first sample of the corresponding signal would be larger than the mean value of the signal and the variance criterion is merely applied to the MRS signal (i.e. the second stage of the proposed method is only used for the noise removal), the concerning parameters E0 and T2⁎

117

Fig. 13. The second real MRS signal with q = 510.85 used to evaluate the performance of the proposed procedure.

the ideal values. In other words, the implementation of the EMD method prior to the variance criterion easily compensates the mentioned weakening of the variance criterion. 3.2. Application to field data example

cannot be correctly estimated (Fallahsafari et al., 2014; Ghanati and Fallahsafari, 2015); it leads to overestimate E0 and underestimate T2⁎. In case the variance criterion is applied following the implementation of the EMD technique, it is observed that the performance of the signal de-noising improves by increasing the signal-to-noise ratio to a great extent and the estimated values of E0 and T2⁎ will greatly approach to

Two real MRS signals recorded by the NUMIS equipment are used to show the functionality of the proposed method. The aquifer where the MRS measurements were performed is in Quaternary gravels with clayey matrix that lies over Miocene impermeable marls. Data acquisition was taken with an 8-square 50 m side length together with a signal stacking of 64 stacks. Moreover, the geomagnetic field intensity was about 45220.66 nT and the inclination of the field was 24°. Figs. 8 and 13 show the recorded MRS data with two different pulse moments q = 947.82 and 510.85 A.ms, respectively. The proposed method initially implemented on the first MRS curve. As seen the data is intensely noisy, consequently the decaying exponential form is partially observable. As explained the de-noising procedure of the signal includes two stages based on the proposed method. Initially, by applying the EMD algorithm

Fig. 12. The first real MRS signal (black), de-noised signal from EMD (red) and estimated signal using EMD and variance criterion (blue). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.).

Fig. 14. The second real MRS signal with q = 510.85 A.ms (black), de-noised signal from EMD (red) and estimated signal using EMD and variance criterion (blue). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.).

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it will be proceeded the de-noising and in the second stage by applying the variance criterion, then the remaining noise is canceled as well as the signal parameters E0 and T2⁎ are estimated. Thus the realistic MRS signal enters the EMD algorithm to be decomposed into IMFs (Fig. 9). Fig. 10 illustrates the noise-only signal recorded prior to the MRS signal measurement and its decomposition to 7 IMFs. Then, Semilog diagram of the IMFs energies with respect to IMF number associated with the MRS signal and noise-only signal is computed (Fig. 11). It is observed that, after the fifth IMF, the energies considerably deviate from the noiseonly signal, implying the presence of significant amounts of no-noise signal. Hence, the partially de-noised signal is reconstructed through only IMF numbers 6 and 7. In the second stage, the remaining noise from the first stage is discarded using the variance criteria as well as the signal parameters E0 and T2⁎ are estimated 55.65 nV and 193.1 ms, respectively. Fig. 12 shows the measured MRS signal (black), de-noised signal from EMD (red) and estimated signal using joint application of EMD and variance criterion (blue). The graphical results of applying the proposed method to the second MRS curve with q = 510.85 A.ms are presented in Fig. 14. The E0 and T2⁎ values associated with the second MRS signal are estimated 49.22 nV and 201 ms, respectively. As it is seen (Figs. 12 and 14) after application of the proposed approach the present noise over the signals has been eliminated as much as possible and the decaying exponential form is easily observable.

4. Discussion The presence of noise, may have a destructive influence in the interpretation of the final values of a MRS, in the acquisition of MRS data is inevitable. In the previous section, the performance of the proposed algorithm was demonstrated by presenting the results of implementing EMD, variance criterion and a hybrid approach of EMD and variance criterion on the two MRS curves with different pulse moments. In the following, a comparison between the signal parameters T2⁎ and E0 estimated using the proposed method and the ones calculated by the commercial software SAMOVAR (IRIS instruments, 2001) will be shown. The inversion algorithm used in the SAMOVAR software is based on the minimization of the Tikhonov functional (Legchenko and Shushakov, 1998). During the inversion process using the NUMIS inversion software, the regularization parameter, which controls the tradeoff between the data fidelity and regularization term in the cost function of optimization problem, can be chosen automatically or by user (Legchenko, 2007). In our study, the parameter of regularization was equal to 366.2. Fig. 15 illustrates the graphical results of the MRS sounding inversion using SAMOVAR, where a top aquifer between about 7 m and 15 m and a second aquifer below 40 m are detected. Table 3 indicates the T2⁎ and E0 values corresponding to the two MRS curves computed using the NUMIS inversion software and the proposed method,

Fig. 15. a) E0(q) sounding curve versus pulse moment (MRS curves corresponding to the fifth and seventh pulse moments were used as real data examples). b) Decay time versus pulse moment and c) interpretation results for water content and T⁎1 , versus depth. A top aquifer between about 7 m and 15 m and a second aquifer below 40 m are detected.

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Table 3 Comparison between the estimated signal parameters for the two MRS curves with q = 947.82 and 510.85 A.ms through the SAMOVAR inversion software and proposed procedure. Signal parameters

Estimated parameters for MRS curve with q = 947.82 via SAMOVAR inversion software

Estimated parameters for MRS curve with q = 947.82via joint application of EMD and variance criterion

Estimated parameters for MRS curve with q = 510.85 via SAMOVAR inversion software

Estimated parameters for MRS curve with q = 510.85 via joint application of EMD and variance criterion

E0 (nV) T⁎2 (ms)

48.16 151.27

55.65 193.1

48.16 133.35

49.22 201

where a close agreement is noticed between the E0 values obtained by SAMOVAR inversion program and those obtained by the proposed procedure. On the contrary, a notable difference exists between the T2⁎ values. This difference can be expected because the MRS signals generally exhibits a multi-exponential behavior due to a signal superposition of layers or volume units having different decay times and due to a possibly multi-model decay time distribution with layers or volume units. Moreover, the conventional inversion approaches (e.g. SAMOVAR inversion algorithm) assume mono-exponential relaxation. As a result, such unsophisticated data analysis can yield erroneous decay time distribution with depth (Yaramanci and Hertrich, 2007). 5. Conclusion remarks This paper was focused on the joint application of EMD and a statistical optimization technique to further enhancement of signal-to-noise ratio in the MRS signal. In the first stage, the EMD method was applied to the MRS signal to decompose it into n-empirical modes. Whereas, the NUMIS equipment directly measures the field noise prior to the signal record and based on the decomposition of the noise-only signal thus we used this feature to identify the noisy IMFs sharing similar amounts of energy with the noise-only case. Afterwards, the obtained signal from the previous stage enters the variance criterion to mitigate the remaining noise. Experimental results from applying the proposed de-noising approach to the synthetic and real MRS signals confirmed relatively high capability of the proposed method in noise removal that allows more accurate retrieval of the signal parameters. Furthermore, we found that in the case where the first sample of the MRS signal (i.e. the E0 value) is larger than the mean value of the signal, if the variance criterion is merely applied to the MRS signal, it yields overestimation of the initial amplitude E0 and underestimation of the decay timeT2⁎. The implementation of the EMD algorithm prior to the variance criterion corrects the mentioned shortcoming of the variance criterion. We also compared the estimated E0 values associated to the two MRS curves with those calculated by SAMOVAR, where an acceptable agreement was found between the values of E0 while there was a relatively high difference between the values of T2⁎. It seems that there are still many open possibilities for future studies based on hybrid application of the nonlinear decomposition methods and statistical analysis to monoand multi-exponential MRS signal. Acknowledgments The authors would like to express sincere thanks to Prof. J. L. Plata for providing the MRS data and his guidance. We are also grateful to the referee and the editor Prof. Klaus Holliger for their invaluable comments and suggestions, which helped to significantly improve the manuscript. References Battista, B.M., Knapp, C., McGee, T., Goebel, V., 2007. Application of the empirical mode decomposition and Hilbert–Huang transform to seismic reflection data. Geophysics 72 (2), 29–37. Bernard, J., 2007. Instruments and field work to measure a magnetic resonance sounding. Bol. Geol. Min. 118 (3), 459–472. Boudraa, A.O., Cexus, J.C., 2007. EMD-based signal filtering. IEEE Trans. Instrum. Meas. 56 (6), 2196–2202. Boudraa, A.O., Cexus, J.C., Saidi, Z., 2005. EMD-based signal noise reduction. Int. J. Signal Process. 1 (1), 33–37.

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