the Creative Commons Attribution 4.0 License.

the Creative Commons Attribution 4.0 License.

# An enhanced correlation identification algorithm and its application on spread spectrum induced polarization data

### Siming He

### Jian Guan

### Xiu Ji

### Hang Xu

### Yi Wang

In spread spectrum induced polarization (SSIP) data processing, attenuation of background noise from the observed data is the essential step that improves the signal-to-noise ratio (SNR) of SSIP data. The time-domain spectral induced polarization based on pseudorandom sequence (TSIP) algorithm has been proposed to improve the SNR of these data. However, signal processing in background noise is still a challenging problem. We propose an enhanced correlation identification (ECI) algorithm to attenuate the background noise. In this algorithm, the cross-correlation matching method is helpful for the extraction of useful components of the raw SSIP data and suppression of background noise. Then the frequency-domain IP (FDIP) method is used for extracting the frequency response of the observation system. Experiments on both synthetic and real SSIP data show that the ECI algorithm will not only suppress the background noise but also better preserve the valid information of the raw SSIP data to display the actual location and shape of adjacent high-resistivity anomalies, which can improve subsequent steps in SSIP data processing and imaging.

Induced polarization (IP) technology operated in both the time domain and the frequency domain is useful in exploration for groundwater mapping, mineral exploration, and other environmental studies (Revil et al., 2012, 2019; Høyer et al., 2018). Since the phenomenon of IP in the time domain was first discovered by Liu et al. (2017b), there has been consistent efforts to explore its utilization in various research efforts. In 1959, the frequency-domain IP (FDIP) approach was proposed by Collett et al. (1959) and Seigel (1959), which became a classic, widely used mapping technique. For example, the first variable-frequency approach was proposed by Wait (1959), then the spectrum approach of the complex resistivity was developed by Zonge and Wynn (1975), and the dual-frequency IP approach was presented and developed by He (1993) and Han et al. (2013). Recently, spread spectrum induced polarization (SSIP) is a popular branch of FDIP which uses pseudorandom current pulses of opposite polarity as an excitation source (Chen et al., 2007; Xi et al., 2013, 2014; He et al., 2015). According to the intrinsic broadband characteristics of the source itself, the spectral response of an observation system can be simultaneously calculated at multiple frequencies in electrical exploration (Liu et al., 2019). Thus, this SSIP technology has been gaining attention and application in electrical prospecting (Xi et al., 2014; Lu et al., 2019; Wang and He, 2020).

In field detection experiments, it is still a major problem that IP data are often contaminated with background noise. The background noise can be mainly categorized into two types: the Gaussian noise and the impulsive interference with different percentage of outliers (Liu et al., 2016; Kimiaefar et al., 2018; Li et al., 2019). If the background noise is not effectively reduced, the remnant noise can affect the calculation of complex resistivity and may mislead subsequent interpretations of the subsurface structure.

The field of FDIP denoising has achieved quite good results through the constant research of experts and scholars. There have been many algorithms that can be used to suppress the FDIP random noise (Mo et al., 2017), such as smooth filter (Guo, 2017), Mean stack (Liu, 2015), digital filter (Meng et al., 2015), and robust stacking (Liu et al., 2016). The smooth can effectively attenuate Gaussian noise, but the impulsive interference with intense energy leaves the effectiveness of this algorithm limited. Therefore, an effective attenuating algorithm for background noise is still a challenging task for traditional noise suppression algorithms (Neelamani et al., 2008; Liu et al., 2017a). SSIP method also faces the same issue (Liu and Chen, 2016; Liu et al., 2017b).

Recently, the new algorithm based on a circular cross-correlation method, time-domain spectral induced polarization based on pseudorandom sequence (TSIP) algorithm, has also been used to suppress the SSIP noise (Li et al. 2013; Zhang et al., 2020). Due to its effective denoising ability, the identification method has gained more attention and development. However, the TSIP algorithm is restricted because the excitation signal is sensitive to the random noise. For this problem, we propose an enhanced correlation identification (ECI) algorithm for reducing the noise in SSIP data. The ECI algorithm obtains cross-correlations between the transmitter output signal, the excitation signal, and the response signal. The performance of the ECI algorithm is demonstrated on both synthetic and field SSIP data. Experimental results show that the ECI algorithm can effectively control the root mean square of noise (NRMS) increase, enhance its denoising performance in background noise and improve the valid signal preservation to display the actual location and shape of high-resistivity anomalies with higher resolution.

## 2.1 The TSIP theoretical model

Figure 1 shows a traditional diagram of the electrical resistivity survey.
The transmitter output signal *u*_{T}(*t*) is poured from electrode A to
electrode B, the excitation signal *i*(*t*) flows from electrode A to
electrode B, and the response signal *u*(*t*) between the electrodes M and N
is measured. To simultaneously obtain the spectral response of subsurface at
various frequencies, pseudorandom sequence based the excitation signal
*i*(*t*) is considered. Thus, the spectral response of subsurface be
retrieved by the TSIP algorithm, and its spectral response be expressed as
(Li et al., 2013):

where *P*_{ui}(*ω*) is the cross-power spectral density of
*u*(*t*) and *i*(*t*), *P*_{ii}(*ω*) the auto-power spectral density
of *i*(*t*), and *P*_{S}(*ω*) is the impulse spectral response of the
observing system.

Given this observation mode using low-power signals, the magnetotelluric
system is a time-invariant system and let us suppose that*H*_{S}(*ω*) is 1. Equation (1) can further be expressed as

where fft[.] denotes fast Fourier transform (FFT), *R*_{ui}(*τ*) is the cross-correlation function of *u*(*t*) and *i*(*t*),
*R*_{ii}(*τ*) is the autocorrelation function of *i*(*t*),
*U*(*ω*) and *I*(*ω*) depict the geometric factor defined by the
frequency spectrum of *u*(*t*)and the frequency spectrum of *i*(*t*)
respectively, and *τ*denotes time delay.

In the practical field environment, this observation mode is contaminated by
the background noise, as shown in Fig. 2. The output of the sensors
*A*_{k} ($k=\mathrm{1},\mathrm{2},\mathrm{3}$) can be expressed as

where *n*_{k}(*t*) is the background noise.

Therefore, according to Eq. (2), the formula of the TSIP algorithm is given as

Equation (6) demonstrates that the TSIP algorithm has a weak denoising effect
when *n*_{3}(*t*) is the massive intense noise. In other words,
the TSIP algorithm depends on the energy intensity of *n*_{3}(*t*) present in *i*(*t*).

## 2.2 The ECI theoretical model

That the denoising ability of the TSIP algorithm is limited is caused by
that *i*(*t*) is sensitive to *n*_{3}(*t*). To
solve this problem, the ECI algorithm is proposed in Fig. 3 and its derivation process
is as follows.

Firstly, let us suppose that the telluric system is a time-invariant system
under low-power signals. For three sensor output signals, their
cross-correlation functions are the periodic correlation functions of time
*τ*. When the length of the correlation window NT is specified,
0.0125 s in this experiment. The cross-correlation functions can be expressed
as follows:

where ${R}_{{n}_{\mathrm{1}}{n}_{\mathrm{2}}}\left(\mathit{\tau}\right)$ and
${R}_{{n}_{\mathrm{1}}{n}_{\mathrm{3}}}\left(\mathit{\tau}\right)$ are the cross-correlations
of, *n*_{2}(*t*) and *n*_{3}(*t*) respectively,
and *τ* is time delay that lies in the range of −NT to
NT.

Figure 4 shows the schematic diagram of the ZW-CMDSII instrument (Zhang et al., 2014; He et
al., 2014). As is known from the figure, we can conclude that *u*_{T}(*t*)
is mainly disturbed by the floor noise energy of the instrument, and
*i*(*t*) and *u*(*t*) are mainly contaminated by environmental noise. The
floor noise is relatively very low, while environmental noise possesses a
much higher energy level. Thus we assume that *n*_{1}(*t*)≈0, and can conclude that zero correlation between *n*_{1}(*t*) and *n*_{2}(*t*), *n*_{3}(*t*), ${R}_{{n}_{\mathrm{1}}{n}_{\mathrm{2}}}\left(\mathit{\tau}\right)\approx \mathrm{0}$ and
${R}_{{n}_{\mathrm{1}}{n}_{\mathrm{3}}}\left(\mathit{\tau}\right)\approx \mathrm{0}$.

Based on the above analyses, we can further obtain:

Then the cross-power spectrum of Eqs. (9) and (10) can be written as following

Finally, according to Eqs. (2) and (11), Eq. (12) can be expressed as following

where *φ*_{y1y2}(*ω*) and *φ*_{y1y3}(*ω*) denotes
the difference between *y*_{1}(*t*), *y*_{2}(*t*)
and *y*_{3}(*t*).

So, Eq. (13) is the formula of the ECI algorithm. The derivation process of
this formula clearly describes that the ECI algorithm can effectively
suppress the background noise and be independent on the degree of
*n*_{3}(*t*) present in *i*(*t*).

We test the ECI algorithm for attenuating background noise of SSIP data sets
in comparison with the FDIP algorithm and the TSIP algorithm. For the
comparison, the signal-to-noise ratio (SNR), root mean square of noise (NRMS), and relative error
(*ε*) are the objective parameters to judge the performance of
denoising, which are calculated as follows:

where *μ*_{y} and *μ*_{n} denote the mean values of the useful
signal and the noise separately. *y*(*i*) and *n*(*i*) are the useful signal and
the noise separately, *M* is the length, *ρ*_{0} denotes the complex
resistivity calculated without noise, and *ρ*_{1} is the complex
resistivity calculated with the noise added to *ρ*_{0}. *R*_{S} is the
value of the sampling resistor (*R*_{S}=1Ω), and *u*_{i}(*t*) is the
voltage at the sampling resistor.

To validate the effectiveness of the ECI system, we performed a
resistance–capacitance experiment, as shown in Fig. 5. The circuit
parameters are chosen to be *R*_{A}=30.3 Ω/5 W, *R*_{MN}=30.1 Ω/5 W, *R*_{B}=30 Ω/5 W, and *C*_{MN}=470 µF. We recorded the applied
voltage *u*_{T}(*t*), the injected current *i*(*t*), and the measured potential
signal *u*(*t*) as the raw signals. These signals are a three-order spread spectrum
pseudorandom sequence at the clock cycle of 0.0125 s, as shown in Fig. 6a–c and Table 1.

Since our experiment is in a stable environment, we consider the system to be
linear time-invariant, and the noise from the current and voltage measurements
are linearly superpositioned (Pelton and Sill, 1983;
Vinegar and Waxman, 1984; Garrouch and Sharma, 1998).
Therefore, it is actually equivalent whenever the noise is added to the
injected current *i*(*t*), the measured potential signal *u*(*t*), or the applied
voltage *u*_{T}(*t*). Therefore, the injected current *i*(*t*) is only polluted
by the synthetic background noise, including Gaussian and impulsive, as
shown in Fig. 6d and e. Thirdly, the complex resistivity of the main
frequency is considered and discussed because the main energy of the
pseudorandom signal is concentrated on the main frequency (He, 2017).
Finally, for detailed comparisons between the ECI algorithm and the others,
we add the synthetic Gaussian and impulsive noises to the response
signal *i*(*t*), respectively.

We use synthetic Gaussian noise with the deviation and mean values of 0.1
and 1.1 as a standard template. The excitation signal *i*(*t*) is polluted by
synthetic different energy levels of the Gaussian noise. Figure 7 shows that
the denoised results are obtained and compared at the three main frequencies
when the NRMS ranges from 0.12 to 0.25. The figure shows that as the
NRMS increases, the complex resistivity information obtained by each
algorithm decreases. However, the amplitude spectrum after ECI processing
has the slowest-falling speed, and the phase spectrum has the slowest-falling speed at 80 Hz.

Previous literature has shown that if the percentages of outliers in
impulsive noise exceed 50 %, the traditional denoising algorithm will be
limited (Liu and Chen, 2016; Liu et al., 2017a). Thus, synthetic impulsive noise is added
to the excitation signal *i*(*t*) in 10 % steps. Their standard
deviations (SDs) and skewness values (SKs) are shown in Fig. 8. As depicted in
Fig. 9, the three algorithms have a certain degree of denoising
performance versus the different percentages of the synthetic outliers
against the raw data. The figure shows that with the discrete points of
impulse noise growing, the NRMS is different. The amplitude spectrum
and phase spectrum of complex resistivity obtained by each algorithm
fluctuate. The amplitude spectrum after ECI processing remained the slowest-falling speed. Although the noise reduction performance of the phase
spectrum processed by ECI does not stand out, the overall change of the
amplitude spectrum after ECI processing is still slow, especially when the
discrete point is more than 60 %.

To further verify the performance of the ECI algorithm, the Wenner array, which is the traditionally applied system in the field, was selected for performing laboratory tests, as shown in Figs. 10 and 11. SSIP data was acquired with a high-density meter and 20 electrodes at 1 m spacing. A Wenner acquisition sequence was adopted with 55 potential measurements expressed using the green points. The figure shows an example of two high-resistance cavities. The two cavities were presented by the letters A and B, and their calibers were about 1.8 m × 2 m. The two cavities are buried by loess. The loess is measured to have an electronic resistivity of 36 Ωm. The measured excitation signal had a range between 0.04 and 0.19 A approximately. The transmitter output signal is a three-order sequence with 80 Hz frequency, and its voltage is about ±11.8 V. The sampling frequency is 625 kHz. The excitation and response data of 40 periods were recorded at each point.

Figure 12 demonstrates the experimental SSIP data processed by the three algorithms, inverted with Res2DInv (Arifin et al., 2019). It can be observed that the location and shape of two abnormal bodies are distinguished only in the ECI algorithm while recognized as one whole in the other algorithms. We believe the reason that ECI has higher detection precision is due to its higher denoising ability.

To verify the reason for the improved detecting precision, the SDs of data points are calculated from 18 to 50 (Fig. 11), as shown in Fig. 13. This figure shows that the 33 SD in ECI processing the SSIP data is the lowest at all points. The average SD values in ECI processing of the SSIP data are 7 % and 3.8 % lower than the FDIP and TSIP methods, respectively. Also, the maximum value of SDs with the ECI method is 5 % and 1.4 % lower than the others, and the minimum value is 8 % and 10 % lower, respectively.

Meanwhile, amplitude–frequency |*ρ*(*f*)| and
phase–frequency *φ*(*f*) characteristics of complex resistivity are
calculated by the three algorithms (one period) in survey point no. 38 in
Fig. 11.

For example, Fig. 14a1 and a2 show that the amplitude and phase of
the complex resistivity spectrum for this point at 80 Hz processed by FDIP are
39.7 Ωm and −0.0881 rad, the amplitude and phase are 40.9 Ωm and 6.12 rad when at 160 Hz, and the amplitude and phase are
38.7 Ωm and −0.253 rad when at 320 Hz. As depicted in Fig. 14, the complex resistivity processed by the ECI shows a linear trend with
the three main frequencies. Also, the SD of the amplitude–frequency |*ρ*(*f*)| characteristic is 0.10 and 0.49 lower than the others,
and the SD of the phase–frequency *φ*(*f*) is 3.56 and 0.03 lower.
Therefore, we believe that the ECI algorithm has an advantage in suppressing
background noise, which benefits the subsequent steps in SSIP data
processing and imaging.

The simulation results indicate that the ECI algorithm has very good performance in noise reduction and robustness. Along with the increase of the Gaussian noise level, we found that the ECI algorithm can, to some extend, overcome the shortcomings of the TSIP algorithm has, i.e., being susceptible to the noise of the current. This result coincided with Eqs. (6) and (13), which provides a novel approach for correlated identification noise reduction. In the impulsive noise experiment, we found that the ECI algorithm still has good noise reduction when the discrete point is more than 60 %, which compensates for the disadvantage of the traditional denoising algorithm. Moreover, these simulation results also reveal that the ECI algorithm should have high robustness.

The standard deviation analysis of the real data indicates that the ECI algorithm improves the accuracy and robustness of the collected data, which are compatible with the simulation analyses. This consistency shows that the ECI algorithm can obtain the location and shape of two abnormal bodies by improving the SNR of SSIP data, which can increase the resolution of inversion results.

We propose the ECI algorithm that effectively attenuates the background noise in SSIP data and improves the complex resistivity spectrum. This algorithm uses the correlation function to neutralize the influence of the background noise in the SSIP data, and the spectrum complex resistivity can be calculated at multiple frequencies by the formula of the complex resistivity. Simulation results show that the ECI algorithm can effectively attenuate the background noise and improve the SNR. Subsequently, the practicability of the ECI algorithm is further verified by a field test. The results demonstrate that the SD of the SSIP data is improved, which benefits the accuracy and stability of the collected data. There is a good agreement between the complex resistivity and the geological target body with high resistance, which indicates that the ECI algorithm can help to improve the quality of interpretation and inversion in the survey area. For the amplitude spectrum, the ECI algorithm can more effectively suppress the background noise, including the Gaussian random and impulsive noises. Still, its effect is very limited for the phase spectrum. Therefore, a denoising algorithm based on pseudorandom sequence correlation identification is still left open for more investigation.

The code is a collection of routines in MATLAB (MathWorks) and is available upon request to the author (e-mail: hsmfly1982@163.com).

All the SSIP data are collected by the ZW-CMDSII and are available upon request to the author (e-mail: hsmfly1982@163.com).

SH and YW designed the study, performed the research, analyzed data, and wrote the paper. JG contributed to language polishing and response. XJ and HX contributed to refining the ideas, carrying out additional analyses, and finalizing this paper.

The authors declare that they have no conflict of interest.

We are grateful for the help of Jun Wang, Shi Zhu, Hui Wang, and Jinshi Cui. We thank the editors and the reviewers for the constructive comments that helped to improve this article.

This research has been supported by the Key Technology Projects of Science and Technology Department of Jilin Province Scientific (grant no. 20190303015SF), a research project of Jilin Provincial Department of Education (grant nos. JJKH20210692KJ and JJKH20211053KJ), and the Fundamental Research and Theme Funds for Changchun Institute of Technology, China.

This paper was edited by Richard Gloaguen and reviewed by three anonymous referees.

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