DOI: 10.29026/oea.2021.200037

Ultra-high resolution strain sensor networkassisted with an LS-SVM based hysteresis modelTao Liu, Hao Li, Tao He, Cunzheng Fan, Zhijun Yan, Deming Liu andQizhen Sun*

Optical fiber sensor network has attracted considerable research interests for geoscience applications. However, thesensor capacity and ultra-low frequency noise limits the sensing performance for geoscience data acquisition. Toachieve a high-resolution and lager sensing capacity, a strain sensor network is proposed based on phase-sensitive op-tical time domain reflectometer (φ-OTDR) technology and special packaged fiber with scatter enhanced points (SEPs) ar-ray. Specifically, an extra identical fiber with SEPs array which is free of strain is used as the reference fiber, for com-pensating the ultra-low frequency noise in the φ-OTDR system induced by laser source frequency shift and environmenttemperature change. Moreover, a hysteresis operator based least square support vector machine (LS-SVM) model is in-troduced to reduce the compensation residual error generated from the thermal hysteresis nonlinearity between thesensing fiber and reference fiber. In the experiment, the strain sensor network possesses a sensing capacity with 55sensor elements. The phase bias drift with frequency below 0.1 Hz is effectively compensated by LS-SVM based hyster-esis model, and the signal to noise ratio (SNR) of a strain vibration at 0.01 Hz greatly increases by 24 dB compared tothat of the sensing fiber for direct compensation. The proposed strain sensor network proves a high dynamic resolution of10.5 pε·Hz-1/2 above 10 Hz, and ultra-low frequency sensing resolution of 166 pε at 0.001 Hz. It is the first reported alarge sensing capacity strain sensor network with sub-nε sensing resolution in mHz frequency range, to the best of ourknowledge.

Keywords: optical fiber sensing; sensor networking; geoscience research; quasi-static sensing

Liu T, Li H, He T, Fan CZ, Yan ZJ et al. Ultra-high resolution strain sensor network assisted with an LS-SVM based hysteresis model.Opto-Electron Adv 4, 200037 (2021).

 

 

IntroductionDiscovering the physics and dynamics of the solid earthis essential to meet the challenges of natural hazards, en-ergy crisis, and climate changes. Solid earth geoscience isa data-driven filed which demands large datasets of geo-logical events1. For geoscience data acquisition, it is re-quired to sense the earth’s crustal deformation with astrain resolution less than 10-9 ε in a frequency regionfrom static to 100 Hz2. Moreover, in order to refine the

structure image of the errors induced by the Earth crustaldamage, the strain sensing for geoscience research re-quires denser spatial coverage of individual sensors andmore accurate recording instruments3. The broadbandseismometer networks represent the state of arts techno-logy and provide high quality Earth crustal deformationwaveforms with broadband frequency content4.However, the dense seismometer networks deployed inareas where access and power supply are limited willconsume a great effort and resources5. 

School of Optical and Electronic Information and Wuhan National Laboratory for Optoelectronics, Huazhong University of Science and

Technology, Wuhan 430074, China.*Correspondence: QZ Sun, E-mail: qzsun@mail.hust.edu.cnReceived: 17 July 2020; Accepted: 20 August 2020; Published: 20 May 2021

Opto-Electronic Advances

Original Article2021, Vol. 4, No. 5

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License.To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.

© The Author(s) 2021. Published by Institute of Optics and Electronics, Chinese Academy of Sciences.

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Recently, optical fiber strain sensor has attracted con-siderable research interests for geoscience applications,owing to their unique advantages of passive, multiplex-ing, and anti-electromagnetic interference6. For instance,optical fiber Bragg grating (FBG) based sensors havebeen widely exploited to monitor the earth tide, whichpossess a minimum static strain resolution of 270 pε2.However, the multiplexing capacity of the FBG strainsensors is limited by the bandwidth of the reflectionspectrum. Moreover, the signal crosstalk between thesensors in the FBG sensor network will increase the noisefloor of measurement7. Hence, the sensing range andspatial coverage density of the high-quality single pointoptical fiber sensor networks should be further pro-moted.

As an alternative approach, optical fiber distributedacoustic sensor (DAS), which offers a long sensing dis-tance without blind area8,9, has demonstrated efficientseismic acquisition ability in seismology application suchas natural hazard prediction and crust exploration10,11. Ingeneral, DAS, which is realized based on phase-sensitiveoptical time domain reflectometry (φ-OTDR) techno-logy, can interrogate the strain and temperature vari-ations along the entire length of an optical fiber12,13. Withcurrent technologies, DAS has achieved a dynamic strainresolution down to nano-strain level14. Moreover, thequasi-DAS assisted with ultra-weak FBG array has beenproposed with pico-strain dynamic resolution above 10Hz frequency range15,16. Due to the high sensitivity, highspatial and temporal resolution, DAS has been de-veloped and used for broadband seismic monitoring inoil/gas exploration17. Nevertheless, the frequency shift ofthe laser source, as well as the cross-sensitivity betweentemperature and strain, severely ruin the signal to noiseratio (SNR) of the DAS in low frequency band18,19. Thenoise in ultra-low frequency band of the DAS systemlimits the availabilities for nature earthquake recordingand earth tide monitoring. In order to improve the sens-ing performance in low-frequency domain, many effortshave been done to suppress the phase drift of the φ-OT-DR. F. Zhu proposed an active compensation method toreduce the laser frequency shift influence by tracking andcomparing the OTDR intensity traces with different fre-quencies19. But the response time of this method will belimited by the number of tuning frequencies. Further-more, Q. Yuan has demonstrated the twice differentialmethod to compensate the influence of laser frequencyshift, and successfully detected the strain variation with

amplitude of 5.9 nε at 0.1 Hz20. However, the strain resol-ution of the DAS in the mHz range is still much worsethan the high-quality single point optical fiber sensor.Actually, the quasi-static measurement from mHz to Hzis essential to the solid earth geoscience research. Tosense the mHz strain information with high sensitivityand dense spatial coverage, it is necessary for the ultra-low frequency noise of the DAS to be greatly suppressed.

In this paper, we demonstrated a high-resolutionstrain sensor network with large sensing capacity basedon φ-OTDR technology and scatter enhanced points(SEPs) array. In the prototype system, the optical fiberwith SEPs array is divided into sensing fiber and refer-ence fiber. The reference fiber packaged with strain isola-tion is used to compensate the ultra-low frequency noiseinduced by laser source frequency shift and environ-mental temperature change. The characteristics of theabove phase noise are systematically analyzed. Then, ahysteresis model based on the hysteresis operator andLS-SVM is introduced to reduce the thermal hysteresisbetween the sensing fiber and reference fiber. Assistedwith the LS-SVM based hysteresis compensation scheme,a strain sensor network with high resolution in the ultra-low frequency and large sensing capacity is successfullyachieved. 

Experimental section 

Experimental setupIn order to achieve a large sensing capacity, the pro-posed high-resolution strain sensing network is interrog-ated by the φ-OTDR technology. The schematic dia-gram of the proposed sensing network for geoscience re-search is illustrated in Fig. 1. The sensing network con-sists of an interrogation office center and a special pack-aged sensing cable. In the interrogation system, an ultra-narrow linewidth laser (NKT x15) with linewidth of 100Hz is used as a light source which is divided into a localoscillator laser and a probe laser by a 99:1 opticalcoupler. The probe laser is modulated into a series ofpulses with frequency shift of 200 MHz by the acousticaloptical modulator (AOM). After amplified by erbiumdoped fiber amplifier (EDFA), the pulses are injected in-to the sensing cable by a circulator. The backscatteredlight from the sensing cable is mixed with the local oscil-lator laser by a 50:50 coupler and then received by a bal-anced photodetector (BPD). Data acquisition card(DAQ) is used to acquire the heterodyne beat signalsfrom the BPD.

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The sensing cable is based on the specific packagedoptical fiber with SEPs array21. The SEPs are inscribed bylaser exposure on the single mode fiber (SMF) with a giv-en interval. The scattering efficiency of the SEPs is im-proved by 10 dB compared to the Rayleigh scattering ofthe SMF22. A pair of SEPs along with the fiber betweenthem compose one sensing section. The fiber with SEPsarray is divided in to sensing fiber and reference fiberwith the same SEPs number23. The sensing fiber is for thesensing of strain, and the reference fiber is for the com-pensation of the phase drift induced by the environment-al temperature and laser frequency variation. The sens-ing sections in the sensing fiber and reference fiber arerespectively defined as sensing channels and referencechannels. The SEPs of the sensing channels and the cor-responding reference channels are aligned, and fixedwith the sensing cable by epoxy curing agents. The sens-ing fiber in the cable is applied with a pre-strain for sens-ing of the strain of the cable. While the reference fiber isloosened in the cable to isolate the strain of the cable.Generally, the different packaging method will inducethe length difference between the sensing fiber part andreference fiber part, which will introduce the differentsensitivity for temperature and laser frequency vibrationas well. However, the sensitivity difference between thesensing fiber part and reference fiber part is linear, whichcan be eliminated by linear fitting. When the spatialwidth of the injected probe pulse is shorter than the in-terval between two SEPs of the sensing section, the detec-ted interference signal behaves as a series of beat fre-quency pulse trains, which can be described as: 

I(t) = AN∑

n=0[rect

(t− zn · n0/c

τp

)· cos(2πfdt+φn)] ,

(1)where A is the amplitude of the interference signal, N is

the number of the SEPs in the fiber, n0 is the refractiveindex of the fiber core, c is the light speed in the vacuum,τp is the pulse width of the probe laser, zn is the locationof the n-th SEPs in the fiber, and φn is the phase of thescattering light of the n-th SEP. φn can be demodulatedfrom the interference signal by the digital orthogonalphase discrimination method, which is deduced as: 

φn = arctan(In(t) · cos(2πfdt)In(t) · sin(2πfdt)

) , (2)

where In(t) is the interference signal scattering from then-th SEP. Theoretically, the phase difference betweentwo SEPs is only dependent on the change of optical pathin the sensing section24, which can be obtained by sub-tracting the phase of two SPEs in the sensing section, asexpressed blow: 

φ(k) = φk+1 − φk =4πn0νLεk

c, (3)

where k=1,2, ···, N−1 is the number of the sensing sec-tion in the fiber, ν is the optical frequency of the laser,and εk is the strain applied on the k-th sensing section.When ν and n0 are constant, the phase signal in the k-thsensing section will be only proportional to εk. However,it can be observed clearly that the phase signal of thesensing section φ(k) will be also influenced by variationof the laser frequency ν and refractive index n0. Gener-ally, the laser frequency ν will shift along with the envir-onmental temperature drift, and the refractive index n0 isalso significantly influenced by the temperature vari-ation of the fiber core. Thus, the laser frequency shift andfiber temperature fluctuation are the dominant noisesources in the ultra-low frequency domain.

In the proposed strain sensor network, a referencefiber with strain isolated package is employed to com-pensate the phase noise in low frequency band inducedby environmental temperature and laser frequency shift.

 

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Fig. 1 | Schematic diagram of the proposed high-resolution strain sensor network.

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The compensation efficiency is dependent on the phaserelationship between the sensing channel and corres-ponding reference channel. In order to investigate thephase variation characteristics, the laser frequency vari-ation and temperature change were applied on the sens-ing cable for test. When the laser frequency variationwith amplitude of 600 MHz and frequency of 1 Hz wasintroduced by frequency demodulation, the phasechange of the sensing part and reference part are respect-ively plotted in the Fig. 2(a). The phase waveforms of thesensing channel and reference channel are identical withthe variation of the laser frequency. Moreover, the rela-tionship of the phase change between the sensing chan-nel and reference channel is linear, which is illustrated inFig. 2(b). Therefore, the phase noise induced by laser fre-quency shift can be compensated by direct differenti-ation.

Furthermore, the temperature fluctuation and the cor-responding phase change are plotted in the Fig. 3(a),where the trigonometric temperature fluctuation withthe amplitude of 5 °C was applied by a TEC. From Figs.

3(b) and 3(c), the hysteresis phenomenon can be ob-served between the temperature and phase. Moreover,the relationship between the phase change of the sensingchannel and reference channel also exhibits an obvioushysteresis characteristic, as shown in the Fig. 3(d). Thehysteresis stems from different temperature features in-duced by the different package methods, which will bringabout a significant compensation error by direct differ-entiation. Therefore, a precision hysteresis modelbetween the sensing channel and reference channelshould be built to reduce the temperature compensationerror. 

Hysteresis operatorResearch on the hysteresis has a long history about a cen-tury. The Preisach model, which has been widely usedfor the modeling of hysteresis, has been developed basedon incorporates relay operators25,26. Relay operator is thesimplest example for a hysteresis nonlinearity, which ischaracterized by threshold a1<a2 and output values +1and –1, as depicted in Fig. 4(a). For a given input x(t),

 

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the output of the Preisach model y(t) can be expressedas27: 

y(t) =w +∞

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−∞d(r, s)Rs−r,s+r[x](t)dsdr , (4)

where d(r, s) is the density function, Rs−r,s+r[∙] is the re-lay operator, r=(a2−a1)/2 is the half-width value, ands=(a1+a2)/2 is the mean value of the relay operator. A di-viding curve, ψ(r, t), is defined to divide the relay operat-or into two distinct regions. The output value of the re-lay operator is +1 in the region below the curve and -1 inthe region above the curve, respectively. The dividingcurve ψ(r, t) can be considered as the play between twomechanical elements of relay. Therefore, a play operatorPr[x] is determined, and the corresponding input-output

behavior is illustrated in Fig. 4(b). Mathematically, theplay operator can be obtained by linear superposition ofrelay elements as follows27: 

ψ(r, t) = Pr[x](t) =12

w +∞

−∞Rs−r,s+r[x](t)ds . (5)

Therefore, the Preisach operator can be expressed as: 

y(t) =w ∞

0q(r, Pr[x](t))dr , (6)

where 

q(r, s) = 2w s

0w(r, σ)dσ . (7)

According to the general form in Eq. (7), the Preisachmodel can be decomposed into a continuous set of playoperators and a one-to-one mapping function.

 

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channel and sensing channel for temperature fluctuation.

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Generally, the input-output relationship of the Preisachmodel can be defined as: 

y(t) = F(Pr[x](t)) , (8)

where F(∙) is a function that maps the play operators setin the real-valued output y(t). Thus, the Preisach modelcan be implemented by approximating the mappingfunction based on the input and output value of real hys-teresis materials through some identification methods,such as nonnegative least squares. 

Least square support vector machine modelLeast square support vector machine (LS-SVM) has beensuccessfully introduced in modeling application for theglobal optimization property28. Compared to the stand-ard SVM, LS-SVM uses the least squared loss functioninstead of the ε-insensitive loss function. Hence, the ana-lytical solutions of LS-SVM can be obtained by solvinglinear equations instead of quadratic programming prob-lem. For a given training data set {(ui, yi), ui∈Rn, yi∈R},the LS-SVM model is expressed as: 

y(u) = wT · f(u) + b , (9)where f(u) is the nonlinear mapping from the inputspace to higher feature space, w is the weight vector, andb is the bias. The optimization problem in the LS-SVM isobtained through the equation: 

minR(w, ξ) = 12∥w∥2 + C

n∑i

ξ2i

s.t. yi = wT · f(ui) + b+ ξi, i = 1, 2, ..., n ,

(10)

where C is the regularization parameters, and ξi is the er-ror between the actual output and predictive output. Theproblem can be solved by defining the Lagrangian equa-tion as follow: 

L(w, b, ξ, a) = R(w, ξ)−n∑iai(wT · f(ui) + b+ ξi − yi) ,

(11)where ai is the Lagrangian multiplier. The conditions foroptimality are the saddle point of L, which can be foundby solving the partial derivatives. After eliminating the w

and ξi, the solution is given by the following linear equa-tions:  [

0 eTe Ω+ I/2C

] [ba

]=

[ 0y], (12)

where e=[1, ···, 1]T, y=[y1, ···, yn]T, a=[a1, ···, an]T are thesupport vectors, I is the identity matrix, and the Grammatrix Ω is defined as follows: 

Ωij = K(ui, uj) = f(ui)·f(uj) (i = 1, · · · , n; j = 1, · · · , n) ,(13)

where K(ui, uj) is the kernel function, which can be usedto avoid the computation of the mapping function f(u).Thus, the LS-SVM model for function regression can beformulated as: 

y(u) =n∑

k=1

akK(u, uk) + b , (14)

where ak and b are the solutions of the Eq. (13). The ker-nel function must satisfy the Mercer condition29. The ra-dial base function (RBF) kernel is widely used in nonlin-ear function regression, which is expressed as: 

K(u, uk) = exp(−∥u− uk∥

σ

), (15)

where σ is the width parameter of the KBF kernel. There-fore, for preassigned hyperparameter σ and C, the map-ping function can be determined through Eq. (13).Moreover, the hyperparameters σ and C determine thegeneralization ability and complexity of the LS-SVMmodel, which should be optimized to ensure a high mod-eling accuracy along with high testing accuracy. 

LS-SVM based hysteresis modelTo establish a Preisach model, the density function d(r, s)of the model should be approximated from real hyster-esis loops. In the classical Preisach model, the modelparameters can be tuned by two approaches. One ap-proach is predefined mathematical forms with a few freeparameters30, and another approach is using the numer-ical density function whose values are identified by themethod of nonnegative least squares31. However, both ofthe two methods are complex and time-consuming. Asan alternative approach, the LS-SVM is an adaptive ap-proximation strategy, which can be used as a generalfunction estimator to approximate the Preisach modelmapping function F(∙). The LS-SVM based hysteresismodel consists of two parts: a discrete play operator, anda memoryless mapping function, which are exhibited in

 

y y

xx

1

r

sa1 a2

−1

r r

a b

Fig. 4 | Hysteresis operator. (a) Relay operator. (b) Play operator.

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Fig. 5. The discrete play operator memory is used to con-vert the Preisach model in Eq. (7) into a discrete form,which comprises m play operators with threshold valuesr1, r2, ···, rm. The threshold values are determined by thefollowing equation: 

ri =i

m+ 1· |x|max, i = 1, 2, · · · ,m , (16)

x (t)

x (t)

where |x|max is the maximum absolute value of the inputx(t). The number of the hysteresis play operator m de-termines the modeling accuracy and complexity of thehysteresis model. Additionally, due to that the classicalPreisach model is a rate-independent hysteresis model,the rate of the input signal is included in the modelso that the model can be used to estimate the rate-de-pendent hysteresis behavior. Therefore, the input vectorof the LS-SVM u consists of m play operators, input sig-nal x(t), and the rate of input . Formally, the outputfunction of play operator z(t)=Pr[x](t) on each subinter-val [ti, ti+1], where x(t) is piecewise monotonic on0=t0<t1<t2<…<tN=tE, is inductively defined by27: 

z(0) = pr(x(0), 0),z(t) = pr(x(t), z(ti)),

for ti < t < ti+1, 0 ≤ i ≤ N− 1 , (17)

where 

pr(x, y) = max{x− r,min{x+ r, y}} , (18)

where r is the threshold value of the play operator. Afterconverting the model input x(t) into the discrete paly op-erators, the operator vectors set are used to obtain thememoryless function F(∙) through the memoryless map-ping function, which is established by a LS-SVM. TheLS-SVM is trained though the Eq. (13) with the giventraining data. The hyperparameters σ and C of the LS-SVM are tuned by the k-fold cross-validation to ensurethe high generalization ability of the hysteresis model. 

Results and discussionIn the experiment, an optical fiber with 112 SEPs array is

enclosed in the sensing cable with the packaging methodas shown in Fig. 1. The SEPs array possess -45 dB re-flectivity with an interval of  5 m. When a probe laserpulse is injected into the sensing cable, the coherent beatsignals backscattering from the sensing cable are recor-ded in Fig. 6(a). It is obvious the backscattering signalhas 112 stable beat frequency pulses with high SNR, cor-responding to 112 SEPs. The former 56 beats frequencypulses scattering from the SEPs belong to the sensingfiber and the latter part come from the reference fiber. Asthe sensing principle describes above, the 56 beat fre-quency pulses from the sensing fiber or reference fibercan compose into 55 sensing channels Si (1≤i≤55) and 55reference channels Ri (1≤i≤55), respectively. The sensingchannel Si and the corresponding reference channel Rj

(j=56-i) are combined into a sensor element. Therefore,the proposed strain sensor network can interrogate asensing cable with 55 sensing elements. To calibrate thestrain sensitivity of the sensor elements, the static strainfrom 2 με to 20 με with a step of 2 με is applied to thesensor element 28 at the end of the sensing cable by aprecision displacement stage. The relationship betweenthe phase change and the applied strain amplitude is il-lustrated in Fig. 7, which presents a high sensitivity of -47.04 rad/με and high precision with R-square over0.999.

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nal of sensor element 55 and 54. 

To train the hysteresis model between the sensingchannel and reference channel, a triangular-wave

 

Input

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operator

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Pr1[x](t)

Prm[x](t)

dx:tx(t)

x(t)

ψ(t, rm)

ψ(t, r1)

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…

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Fig. 5 | Block diagram of the LS-SVM based hysteresis model.

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temperature with the same period and variable rate wasapplied to the sensor element 28, which is shown as thered line in Fig. 8(a). The triangular-wave temperaturefluctuation used to train the LS-SVM model can providea temperature change with different input rate, whichcan construct a complete set of training data. The corres-ponding phase of the sensing channel and referencechannel are also plotted in the Fig. 8(a), respectively. Therelationship between the sensing channel and referencechannel is illustrated as the color scatter distribution inFig. 8(b). The hysteresis loops curve reveals that thethermal hysteresis characters of the strain sensor is de-pendent on the change rate of the applied temperature.The larger change rate of the temperature will introducethe wider hysteresis loop. Therefore, the thermal hyster-esis of the proposed strain sensor is a rate-dependenthysteresis model. The phase signal of the reference chan-nel is set as the input of the proposed LS-SVM based hys-teresis model, while the phase signal of the sensing chan-nel is the output. The regression result of the LS-SVMbased hysteresis model is plotted as the black line in Fig.8(b). The results prove the highly regression accurate forthe thermal hysteresis modeling. The modeling error ofthe LS-SVM based hysteresis model is also exhibited inthe Fig. 8(c). Compared to the large compensation errorof the direct differential method, the root mean squarederror (RMSE) between the LS-SVM model output andphase signal of the sensing channel is less than 0.095 rad.

After training the hysteresis model by the hysteresisoperator LS-SVM, the phase signal of the sensing chan-nel is compensated by the trained hysteresis model. Toestimate the compensation accuracy of the hysteresismodel, the sensor element 28 is applied with a temperat-

ure fluctuation with amplitude of 5 °C, and strain signalwith frequency of 0.01 Hz and amplitude of 20 nε. Theoriginal signals of the sensing channel and referencechannel are plotted in the Fig. 9(a). The compensationresult of the directly differential method and hysteresismodel are illustrated in Fig. 9(b), respectively. It can beseen that the measured strain signal by the direct differ-entiation method is almost buried in the phase noise in-duced by the temperature fluctuation. However, thestrain signal can be observed clearly in the phase of thesensing channel compensated by the hysteresis model.Moreover, the power spectrum density (PSD) of the ori-ginal signal, compensated signals by direct differentialmethod and hysteresis model is exhibited in the Fig. 9(c).In the PSD of the original signal and signal compensatedby direct differentiation, the signal peak of strain vibra-tion at 0.01 Hz is almost buried in the noise peaks in-duced by the temperature fluctuation. However, aftercompensating by LS-SVM based hysteresis model, thesignal of the strain vibration can be obtained moreclearly, and the signal to noise ratio (SNR) of a strain vi-bration at 0.01 Hz greatly increases by 24 dB comparedto that of the sensing fiber for direct compensation.Moreover, the bias drift induced by the laser frequencyshift is also suppressed compared with the uncom-pensated original signal.

Additionally, to evaluate the strain resolution of theproposed strain sensor network, the sensing cable isplaced in a quiet environment and isolated with vibra-tion and strain. The original phase signals of the sensingchannel and reference channel from sensor element 28are plotted in Fig. 10(a), where a severe phase drift in-duced by the environment temperature change and laserfrequency variation is observed. The compensated phasein time domain and frequency domain are illustrated inFigs. 10(b) and 10(c) respectively. The sine wave phasenoise in the sensing channel and reference channel withfrequency nearly 0.001 Hz is introduced by the air condi-tioner system in the lab, which will control the roomtemperature with fixed period. The bias drift of the sens-ing channel in time domain is eliminated by compensat-ing with the LS-SVM based hysteresis model. In the fre-quency domain, the noise PSD below 0.1 Hz is sup-pressed clearly by compensation. Compared to the com-pensation method by direct differentiation, the noise ofthe compensated sensor by LS-SVM hysteresis model is

 

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Fig. 7 | The relationship between phase change and strain.

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further suppressed by 20 dB below 0.01 Hz. For com-pensation by hysteresis model, the phase power spectraldensity of the compensated sensor element is better thanSp(f)=5.01×10-4 rad·Hz-1/2 in the frequency range above10 Hz, corresponding to a minimum dynamic strain res-olution of 10.5 pε·Hz-1/2. Moreover, the phase noisepower spectral density below 0.01 Hz even at 0.001 Hz issuperior to 0.251 rad·Hz-1/2, which suggests the ultra-lowfrequency strain resolution of 166 pε @ 0.001 Hz. Re-markably, the quasi-static sensing resolution in mHzrange of the strain sensor network can compete with the

high-quality single point fiber sensor. Moreover, thesensor network proves a 55 sensor elements access capa-city in the experiment and possesses a potential sensingcapacity up to thousands of sensing channels which canmatch with the DAS system. It is the first reported strainsensor network to have sub-nε resolution in mHz fre-quency range along with ultra-large sensing capacity tothe best of our knowledge. The large sensing capacityalong with ultra-high quasi-static sensing resolutionmakes the proposed strain sensor network has potentialapplications in geoscience research. 

 

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the regression result. (c) Compensation error.

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ConclusionsWe have demonstrated an ultra-high resolution strainsensor network based on φ-OTDR technology and SEPsarray fiber. To reduce the ultra-low frequency noise in-duced by the laser frequency shift and temperature fieldchange, the phase of the sensing fiber is compensated bythe reference fiber with strain isolated package. The hys-teresis operator based LS-SVM hysteresis model is intro-duced to reduce the thermal hysteresis nonlinearitybetween the sensing fiber and reference fiber. After com-pensated by the hysteresis model, the SNR of a sinusoid-al strain vibration with a frequency of 0.01 Hz increaseby 24 dB compared with direct compensation, whichproves the compensation accuracy of the LS-SVM hys-teresis model. Moreover, the phase noise level in the quiltenvironment presents a high dynamic resolution of10.5 pε·Hz-1/2 in the frequency range above 10 Hz, and ul-tra-low frequency sensing resolution of 166 pε at 0.001Hz. The high resolution in the ultra-low frequency and

the large sensing capacity make the strain sensor net-work play an irreplaceable role in the geoscience re-search.

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AcknowledgementsWe are grateful for financial supports from the National Natural ScienceFoundation of China (NSFC) (No. 61922033 & 61775072); Major Techno-logy Innovation of Hubei Province (2019AAA053); Foundation for Innovat-ive Research Groups of Hubei Province of China (2018CFA004); and In-novation Fund of WNLO.

Author contributionsT. Liu, H. Li and Q. Z. Sun proposed the original idea of the project. Q. Z.Sun supervised the project. Z. J. Yan and D. M. Liu participated the discus-sion of the research. T. Liu, T. He and C. Z. Fan carried out the experimentsand collected the data. T. Liu analysed all the data. T. Liu and Q. Z. Sunwrote the paper. All authors discussed the results and commented on themanuscript.

Competing interestsThe authors declare no competing financial interests.

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