The use of machine learning algorithms (SVR) to link ...

The use of machine learning algorithms (SVR) to link volumetric water content and complex permittivity in a wide frequency band (33 to 2000 MHz) of hydraulic concretes Amine IHAMOUTEN(1), Xavier DEROBERT(2), Cedric LE BASTARD(1), Frederic BOSC(1) and Géraldine VILLAIN(2)

(1)

CEREMA, DTO/DLRCA, ERA17-CND, Angers, France

(2)

LUNAM Université, IFSTTAR Centre de Nantes, F- 44344 Bouguenais, France

[email protected], [email protected], [email protected], [email protected], [email protected] *Corresponding author, email address: [email protected], Tel.: +33 2 41 79 13 28

Abstract This paper focuses on the development and validation of an innovative method for estimating volumetric water content in various concrete mixtures. A statistical solution to the inverse problem has been generated using a non-deterministic optimization (Support Vector Regression) of the in-lab calibration curves correlating the controlled water content in various concrete mixtures with the frequency complex permittivity originating from the coaxial electromagnetic transition line. An extrapolation procedure using a frequency power law model has been developed and validated for estimating complex permittivity over a broad frequency bandwidth. Implementation of this extrapolation method allows visualizing various physical phenomena (i.e. polarization vs. water content) that typically affect the dielectric behavior of concrete with respect to frequency.

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The two-step estimation procedure (involving extrapolation and SVR methods) proposed in this paper has been validated on a wide array of moisture-controlled concrete specimens in the laboratory. Its purpose is to build calibration curves that rely on both complex effective permittivity and water content across various concrete mixtures with taking into account frequency dependence.

Keywords Durability, complex permittivity estimation, Support Vector Regression, Jonscher's model, coaxial/cylindrical transition line, water content, concrete.

1. Introduction In excluding the natural aging of concrete, the life cycle of a concrete structure may be limited by the penetration of aggressive agents, such as chlorides [1], or by freezing-thawing cycles that lead to cracks and a scaling of surfaces exposed to de-icing salts. One common factor in all these pathologies is water content. Quantifying this condition parameter as an average value or a surface gradient is hence considered a primary phase in the diagnosis of civil engineering structures. For an in situ assessment of the progression of these degradations, it is essential to adjust the models of water transfer in concrete according to the durability indicators – such as porosity, permeability and water content – as well as to observations recorded during non-destructive (ND) measurements. Numerous research studies have shown the possibility of correlating durability indicators with ND measurement methods on the basis of electromagnetic (EM) wave propagation [2-8]. EM methods actually make it possible to assess the mean water content of an investigated volume (several centimeters deep depending on the device) provided it can be separated from the mix design. For each studied structure, if the concrete mix design changes, the calibration curve correlating, for example, dielectric permittivity and water content also shifts [5]. 2/27

Practically, a part of Ground Penetrating Radar (GPR) users estimate the permittivity values through arrival time picking between successive echoes corresponding to the direct wave in the material with the multi-offset configuration of the antennae. However, GPR wave propagation in very dispersive media (i.e. saturated concrete) causes a significant change in the shape of the radar wavelet according to the propagation distance. This effect introduces a strong bias in the estimation of the dielectric permittivity from the temporal calculation of the "group" velocity. In order to illustrate the effect of this temporal estimation bias, we carried out a parametric study from analytical modeling of radar wave propagation into two dispersive media using a multi-offset configuration for the antennae (i.e. Common Mid Point configuration). To do this, we used the four parameter (4p) variant of Jonscher’s model to build two synthetic wave fields, weakly dispersive (Figure 1a) and highly dispersive (Figure 2a) corresponding respectively to example of EM wave propagation in dry and wet concrete slabs. We have drawn on these two B-scans picked maximum amplitudes of the direct wave in the material (blue curves) and their linear regression (red curves). Furthermore, we have shown in Figure 1b and Figure 2b the frequency variation of the reference velocities (black curves) and the group velocities (red curves).

Figure 1: (a) Synthetic wave field corresponding to GPR direct wave propagation in a low dispersion medium (i.e. dry concrete) - (b) Variation of the reference velocity and the group velocity according to the frequency.

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Figure 2: (a) Synthetic wave field corresponding to GPR direct wave propagation in a high dispersion medium (i.e. wet concrete) - (b) Variation of the reference velocity and the group velocity according to the frequency.

We can note from these results that the spectral bias due to the velocity temporal picking increases strongly with material dispersion (low dispersion for n=0.94 and high dispersion for n=0.60), while the frequency dependency of the velocity (respectively the permittivity) is also important. The increase of this error is due to the instability of the wave field signal phases from a scan to another one. Unfortunately, this bias is directly correlated with the estimation of the dielectric permittivity for dispersive media and consequently correlated with the estimation of water contents in concretes. Hence using frequency methods are needed for complex permittivity measurements. Access to the effective permittivity of concrete depends on the degree of mixture complexity. For simple materials (like siliceous sand) their effective permittivity can be evaluated analytically and physically under certain conditions by applying so-called "mixing laws". However, for media with a complex structure with nanopores and other pores containing an electrolytic solution (like concrete), appear EM dispersion phenomena [2] that establish a frequency dependency of the complex permittivity and consequently no rigorous solutions or systematic methods exist for calculating the complex effective permittivity. A multitude of mixing laws found in the literature attest to these difficulties. The mixing laws commonly used in the field of civil engineering are: the CRIM (Complex Refraction Index Method)

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model [9], the effective medium model based on Maxwell-Garnett theory [10], and the effective medium approximation model based on Bruggeman theory [11]. Given these inverse physical modeling constraints, this study has opted for a statistical resolution of the inverse problem with non-deterministic optimization (machine learning algorithms) of the in-lab calibration curves linking the controlled water content in various concrete mixtures and the complex dielectric permittivity output by the EM characterization device. Machine learning algorithms (i.e. neural networks) have recently been introduced in the GPR community [12-15]. Among this family of algorithms, support vector machines (SVMs) have shown promising results for different applications. Originally proposed for document classification and pattern recognition applications [16], SVMs have been used to identify and classify buried objects from GPR imagery [17] as well as to assess railway ballast conditions [18]. Moreover, SVMs have been extended for regression problems by means of the support vector regression (SVR) method [19]. In the present case, this method has been applied to predict vehicle travel times [20] and estimate the arrival directions of multiple waves via smart antennae [21]. The expected advantage of an SVM is its ability to compensate for model uncertainty by a large experimental training dataset, intended to cover the entire feasible experimental EM characterization with a specified material. In contrast, mixing laws are model-dependent and their performance is related to the simplified EM hypothesis retained by the data model. SVMs provide broad generalization capabilities and typically short calculation times [19]. This paper proposes an application of this method to estimate the volumetric water content of the studied reinforced concrete structure, to avoid the determination of calibration curves on small cores extracted from the studied structure, at controlled water contents in the laboratory as proposed by [22] by using coaxial EM transition lines [23-25]. This EM cell, developed at the IFSTTAR Laboratory [25], assesses relative complex permittivity as a function of frequency (i.e. dispersion curve) between 50 and 1,600 MHz. This frequency range has in fact been narrowed for dispersive materials, such as 5/27

saturated concretes, due to resonance phenomena in the cell relative to material permittivity. The higher the permittivity, the lower the maximum frequency (from about 1.2 GHz for dry concretes to below 900 MHz for saturated concretes, depending to their porosity) [25]. It is thus considered herein that the dispersion curve can be derived with confidence between 50 and 600 MHz. For this reason, it is necessary to extrapolate the EM cell data onto a wider frequency band, encompassing lower and higher frequencies, so as to physically associate inverse method results with the correct frequencies. The 4p variant of Jonscher's model [26] provides an outstanding means for fitting the permittivity dispersion curves on a broad frequency band and extrapolating EM cell results at the required low frequency [1 MHz - 2 GHz]. Use of this extrapolation method serves to take into account various physical phenomena (i.e. polarizations vs. water content) that typically affect the dielectric behavior of concrete as a function of frequency. The two-step estimation procedure (extrapolation and SVR methods) proposed in this paper has been validated across a wide array of concrete specimens controlled for moisture in the laboratory. It is useful for building calibration curves that both rely on complex effective permittivity and water content in various concretes and incorporate frequency dependence. The remainder of this paper will be organized as follows. Section II will introduce the fundamental principle behind the SVR method. Section III will then detail the experimental EM device, and Section IV will present the absorption-dispersion model used to extrapolate coaxial EM transition line data from a limited bandwidth to a wide frequency bandwidth. Next, Section V will carry out the experimental validation of this method on various concrete specimens.(Sentence removed)

2. Support Vector Regression (SVR) algorithm SVM, developed by V. Vapnik [27], is considered one of the most powerful supervised machine learning algorithms and has been based on the structural risk minimization principle 6/27

[16]. This method is being proposed herein to estimate volumetric water content Wc (considered an output) from frequency complex permittivity measurements (considered an input) in a regression context. This paper makes use of ν-Support Regression [28], where νSVR is a modification of ϵ-SVR [27] and parameter ν is introduced to control the number of support vectors and training errors [28-29]. Interested readers will find further details on this method in [28-30]. z i=( z 1,i … z2 N

For the formalism of -SVR, let's define the feature vector

❑

,i

) of size (2N, 1),

with i the training data index and 2N the number of features. The features included in this paper are the effective complex permittivity vectors (e(f)= e’(f)+je”(f)). Vector v is built as follows: v=(e’(f1), e’(f2),…, e’(fN), e’’(f1), e’’(f2),…, e’’(fN)), where N is the frequency number. To allow for use of -SVR, vector v has been normalized to the [0; 1] range in order to obtain a vector z of size (2N, 1), which in this case represents the feature vector in use. Let TWc = (z0,Wc0),…,(zr-1,Wcr-1) composed of r-1 pairs of training data, with zq being the feature vector associated with water content Wcq, i.e. q=0,…,r-1. The basic idea behind the regression problem is to determine a function as follows: f(z) = a,z + b

(1)

that can accurately approximate future values [29], with the a and b parameters to be determined; .,. denotes the inner product. To estimate a and b from data TWc =  (z0,Wc0),…, (zr-1,Wcr-1) , the authors of [28] proposed the following optimization problem:

(

r −1

1 2 1 ❑ ' min γ ( a , ξ (' ) ,ϵ )= ‖a‖ +C νϵ+ ∑ ( ξ i + ξi ) 2

r

i=0

)

(2)

Subject to the following constraints:

( ⟨ a , z i ⟩ + b ) −Wci ≤ϵ +ξ i

(3)

Wc i − ( ⟨ a , z i ⟩ +b ) ≤ϵ +ξ 'i

(4)

ξ 'i ≥ 0

(5) 7/27

ϵ≥0

(6)

where: C > 0 is a regularization constant,  (0; 1] a constant that allows controlling the ϵ

number of support vectors and training errors,

the tube radius [27], and

ξ❑i

and

ξ 'i

the slack variables. By solving the dual problem, vector a can be written as follows: r−1

a=∑ ( α i − α i ) zi '

❑

(7)

i=0

where

α 'i

and

α ❑i

are Lagrange multipliers. Equation (1) thus assumes the following

form: r −1

' ❑ f ( z )=∑ ( α i −α i ) ⟨ zi , z ⟩❑ +b

(8)

i=0

In the context of nonlinear functions, the function f(z) (from Equation 1) can be written as: f(z) = a,(z) + b, with  being a nonlinear transform from

R

2N

to a higher dimensional

space. Equation (8) then becomes: z Ω (|i ) , Ω ( z ) ¿ z (|i , z )+ b

(9)

r −1

¿ f ( z )=∑ ( α 'i − α ❑ i ) i=0

where

z (|i , z ) Γ

is the kernel function. The sample points that appear with non-zero

coefficients are called supports vectors (SVs) [19]. The dual variable b is computed according to Karush-Kuhn-Tucker conditions [20]. The kernel function may transform the data into a higher dimensional space without an explicit calculation of the nonlinear transformation. Various kernel functions can be used, e.g. a radial basis function (RBF), whether polynomial or other. The RBF kernel and polynomial kernel can be expressed as follows: 2

Γ ( x , y )=exp ( − γ ‖x − y‖ )

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(10)

d

Γ ( x , y )=( γ x T y +c )

Γ ( x , y )=tanh ( γ x T y +c )

(11) ❑

(12)

where , c and d are kernel parameters. To use -SVR, several parameters (, C) and kernel parameters ( for the RBF, (,c) for sigmoid and (,c,d) for the polynomial kernel) must first be assigned. In this paper,  is set equal to 0.54 [28], and the other parameters are determined by the well-known k-fold crossvalidation on training data [28]. In a k-fold cross-validation, the data used are partitioned into k equally sized subsets, as shown in Figure 3. Within each iteration (1 to k), a different fold of the data is selected for validation, while the remaining k-1 folds are used for learning purposes. Performance is measured by the mean square error (MSE) on the output Wc from the ith subset taken as a validation set. The procedure is repeated k times and the average MSE is calculated. By searching for different parameters (C,  or other) on a grid, the various parameters providing the best average MSE on this grid will be selected.

Figure 3: Principle of the k-fold cross-validation

For a validation on experimental data, the total database (TD) is composed of M samples, corresponding to entire set of concrete mixtures at M controlled levels of volumetric water 9/27

content (or relative humidity, RH), for which the frequency complex permittivity is obtained from the coaxial EM transition line. For each sample Wcq (%), the feature vector zq (considered as the SVR input) is composed of 2N = 2,000 elements. The frequency bandwidth used contains N = 1,000 frequencies (after extrapolation, see Section IV). The total database is partitioned into two distinct bases: a training database (TGD), and a test database (TTB). The TGD and TTB are composed of 2M/3 and M/3 samples, respectively. In ^ the following discussion, symbols Wc

´ and Wc

and testing procedures, respectively. The value

represent the quantity Wc in the training

~ Wc denotes the estimated Wc value.

To assign the various parameters, the k-fold cross-validation is applied on the TGD with k = 5. This study has opted to use -SVM from the LIBSVM [19]; all results will be presented in Section V.

3. EM device: Coaxial/cylindrical transition line During this research project, the experimental set-up was designed to measure complex dielectric permittivity at radar frequencies; it features a vector network analyzer controlled by a central processing unit and connected to a cylindrical transition line via a coaxial cable [25]. The test sample is placed in a cylindrical waveguide holder connected to the coaxial line (see Figure 4).

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Figure 4: Coaxial/cylindrical transition line included with the network analyzer and several calibration samples

The cell dimensions (75 x 70 mm) correspond to a volume greater than the elementary representative volume of a heterogeneous mixture for a maximum aggregate diameter Dmax = 20 mm. This transition line enables measuring the permittivity of cylindrical samples of heterogeneous materials with large aggregate dimensions (up to 25 mm) over a frequency range extending from 50 MHz to an upper limit that depends on the value of the real part of permittivity. This latter frequency limit has been found to equal 750 MHz for all concrete mix designs and then decreases sharply for saturated concretes displaying high permittivity values. Beyond these values, resonance phenomena disturb the data analysis to a considerable extent. Let's consider therefore that in this study, the dispersion curve is obtained with confidence between 50 and 600 MHz. For this reason, it is necessary to extrapolate data from the coaxial/cylindrical transition line over a large frequency bandwidth, at both lower and higher frequencies in order to accurately describe the polarization phenomena directly tied to the concrete pore solution.

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4. Extrapolation method: Principle and validation using published data Generally speaking, the permittivity is complex with the real part representing the ability to align or induce an electric dipole moment inside the material and the imaginary part being an indicator of polarization and conductivity losses. Moreover, its frequency dependence reflects the presence of different polarization mechanisms that can be correlated with the inherent properties of the studied materials. A model that has proven its validity for the EM characterization of concretes on a large bandwidth is the 4p variant of Jonscher's model [26]: ε e ( ω ) =ε 0 χ r

( ) [ ω ωr

n− 1

1 − jcot

( )]

σ nπ + ε ∞ − j dc 2 ω

(13)

where  is the instantaneous dielectric response, dc the direct current conductivity, r the relative dielectric susceptibility, and n the dispersion parameter. The reference angular frequency r is a constant arbitrarily chosen. Since the EM cell is limited by a minimum frequency of 50 MHz and a maximum frequency of 600 MHz, a new method has been developed to overlap the validity ranges of these two techniques. This method is based on extrapolating EM cell data using the 4p variant of Jonscher's model. Ihamouten et al. [26] demonstrated that the 4p variant reveals no limitation over the entire EM cell frequency range, in accurately tracking the permittivity variations. This finding led to inverting the complex permittivity dispersion curve of the EM cell data using the 4p variant of Jonscher's model in order to extract the 4 parameters for a limited bandwidth. These parameters are determined simultaneously by numerically minimizing the cost function (14) using Matlab functions (lsqnonlin or fminsearch): 2

ε experimental − ε model | | e e χ ,n , ε , σ min

r

∞

(14)

dc

Using parameters [r, n, , dc] extracted from (14), the frequency variation of e on a large bandwidth can now be estimated using Eq. (13). 12/27

The experimental validation is performed using the concrete characterization results published in [24] within the frequency range [1 MHz - 2 GHz]. The first step of this parametric study consists of extracting the four parameters by inverting a part of these experimental data on a narrower bandwidth [50 MHz - 600 MHz]. The second step entails estimating the frequency variation of e in the frequency interval [1 MHz - 2 GHz] in using Eq. (13). The result of this e estimation is then compared with the raw

experimental

εe

data from

[24] in the range [1 MHz - 2 GHz] according to the normalized residual, as given by: N

residu=

2 1 ε model ω i ) −ε experimental ωi )| ∑ | ( ( e e N i=1 i

i

(15)

where N is the number of frequencies. The results of this parametric study are presented in Figure 5.

Figure 5: Extrapolation of EM cell data (concrete C1, water content Wc = 9%) from frequency band (a) [50 MHz - 600 MHz] to band (b) [1 MHz - 2 GHz] using the 4p variant of Jonscher's model

The tested material corresponds to concrete with a volumetric water content equal to 9%. This specification makes it possible to consider the validation medium as being highly dispersive. Let's note in Figure 5b that the extrapolation model provides the best fit over the entire frequency range for either the real or imaginary part of the permittivity. This result is confirmed by a very low normalized residual (mean of the residual values) of about 0.12. From this parametric study, it can be deduced that the complex permittivity estimation at both very low and very high frequencies using an extrapolation of raw data (limited in their lower band to 50 MHz and in their upper band to 600 MHz) is indeed consistent with the

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physical characteristics of the tested material. The following section will consider the data stemming from this extrapolation method as raw inputs for the SVR processing step.

5. Experimental validation Concrete mixtures and moisture conditioning Let's recall that the primary objective of this study is to evaluate, through EM measurements, the volumetric water content in concrete, a parameter that is directly correlated with porosity. One of the key parameters controlling porosity is the water-to-cement ratio (w/c). Moreover, non-destructive EM techniques may be sensitive to aggregate size, which mainly influences measurement uncertainties. Consequently, the following set of requirements has been adopted in order to design six concrete mixtures, whose samples will be cast for further testing; this design process includes [31]: •

2 water-to-cement ratios (w/c) of approx. 0.35 and 0.65, covering the range of

medium-to-high strength concretes classically encountered in bridge construction; •

3 maximum aggregate diameters Dmax: 4 mm, 10 mm and 20 mm. In all, four

concretes, labeled B1 to B4, and two mortars, B5 and B6, have been mixed as part of this design process (see Table 1); •

To eliminate the influence of paste volume (i.e. volume of water + cement +

sand particles less than 80 µm) on measurements and highlight the influence of both porosity and maximum skeleton size D max, the ideal situation would have been to maintain the paste volume nearly constant across the various mixes. Such a situation however is infeasible since holding the past volume constant for both the mortar and concrete would have led to concrete segregation or unworkable mortars. In conclusion, specimens have been designed using a past volume between 350 and 374 l/m 3 for concretes (B1 to B4) and between 503 and 531 l/m 3 for mortars (B5 and B6). 14/27

Table 1: Concrete characteristics under dry conditions [31] B B1 Water-to-cement ratio

B B2

B B3

B B4

B B5

B B6

w/c

0.36

0.68

0.36

0.69

0.35

Maximum aggregate diameter

Dmax (mm)

20

20

10

10

4

4

Mean saturated density by gammadensimetry Mean saturated density by water saturation Bulk porosity by water saturation Paste volume

sat (kg/m3) sat (kg/m3)  (%) l/m3

2406  9 2437  5 12.5  0.2 374.6

2312  13 2346  1 16.6  0.3 351.4

2370  10 2396  14 13.5  0.3 369.2

2275  11 2302  9 17.5  0.6 350.0

2257  4 2290  4 19.5  0.3 531.2

2152  10 2176  12 26.4  0.5 503.1

Compressive strength at 360 days

Rcsat (MPa)

82.4 2.2

44.2 1.9

82.5 2.8

41.6 1.6

82.1 2.2

39.4 1.3

Static Young's modulus at 360 days

Esat (GPa)

42.2 1.9

33.8 1.8

41.6 2.2

30.7 1.6

32.9 2.1

24.8 1.5

Dynamic Young's modulus at 2 years

Edyn (GPa)

53.4  2.2

41.7  1.6

51.8  2.5

40.1  1.4

43.0 1.8

30.6  1.2

4.7816

16.973

4.7194

24.066

7.1628

22.915

Liquid water permeability

Kl (10

-20

2

m)

0.67

For each concrete mixture, five cylindrical samples 75 mm in diameter and 70 mm long were cored. The choice of sample dimensions is justified by the results of the research work in [25], which demonstrated through numerical simulations relative to the reflection of EM waves in a coaxial cylindrical cell that stabilized measurements of the frequency variation in complex permittivity are only obtained for maximum aggregate diameters of less than 30 mm in the chosen frequency bandwidth [50 MHz - 600 MHz]. Cylindrical cores (∅75 × 70 mm) tested by this EM cell thus constitute a representative elementary volume for concrete mixtures with a maximum aggregate diameter equal to 20 mm. Furthermore, gammadensimetry was applied on cylindrical samples (∅100 × 120 mm) to control water content homogeneity. The principle of this technique is based on the absorption of gamma rays by matter, whereby absorption is dependent on the medium thickness, its specific mass absorption coefficient and the material density. This method is considered seminon-destructive and details of its operating principle are reported in [32]. After a 2-year curing period in water, all cylindrical samples corresponding to the 6 mix designs under study were subjected to drying-wetting cycles in order to obtain homogeneous water conditions at various water content levels. (Paragraph removed)

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Figure 6 displays all the relative humidity (RH) levels imposed in a climatic chamber during after isothermal cycles at 67°C for the studied materials. This protocol shows various conditioning periods depending on the humidity level; their final durations are obtained either from the results of water homogeneity control using gammadensimetry (samples ∅100 × 120 mm in size) or from weighing. (Paragraphs removed)

Figure 6: Experimental isothermal conditioning (in relative humidity) of the studied concretes in a climate controlled chamber maintained at 67°

It should be noted that the conditioning of the studied concretes was disturbed on occasion by several climate chamber failures, which explains the few outliers not anticipated in the original protocol (Figure 6). (Sentence removed) Figures 7 and 8 provide an example of gammadensimetry results for samples No. 1 of concrete B1 and mortar B6 for all RH conditioning steps. An analysis of these gammadensimetry results (including those not discussed in this paper concerning mixes B2 through B5) yields the conclusion that hygrometric equilibrium is more easily approximated in the high w/c (B2, B4 and B6 because more porous) concretes than in the low w/c samples.

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Figure 7: Evolution in the bulk density obtained by the gammadensimetry device when following the conditioning steps in terms of the RH targeted for concrete B1 sample no. 1 ( w/c = 0.35)

Figure 8: Evolution in the bulk density obtained by the gammadensimetry device when following the conditioning steps in terms of the RH targeted for mortar B6 sample no. 1 (w/c = 0.65)

Since the calculation of saturation rate and actual volumetric water content for concrete samples depends on their dry weights, we have estimated the dry weights of cylinders ∅120 × 100 mm from: water porosity, saturated mass, and core volume. For the ∅75 × 70 mm samples, their dry weight was measured for each studied concrete in following the protocol described in [31, 33].

The six concrete mixtures were thus studied at five relative humidity (RH) values, i.e. 30%, 50%, 70%, 80%, 94% and 100%, which yielded 6 x 36 volumetric water content (Wc)

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recordings (with six Wc values equal to zero, corresponding to dry state), ranging from 1.9% to 26.4% (Table 2). Table 2: Volumetric water content in the concrete samples (average values), corresponding to the relative humidity imposed in the climate chamber

Wc [%] Concretes

RH = 30 %

RH = 50 %

RH = 70 %

RH = 80 %

RH = 94 % RH = 100%

B1

3.07

3.55

4.26

5.48

7.89

12.5

B2

1.96

3.37

4.26

5.64

7.67

16.6

B3

3.70

4.33

5.11

6.18

8.79

13.5

B4

3.68

5.08

5.95

7.29

9.38

17.5

B5

4.84

5.79

6.97

8.77

12.10

19.5

B6

4.96

6.75

7.96

9.96

12.62

26.4

With this extent of production and moisture conditioning of concrete samples, an experimental design could be built so that the volumetric water content is controlled for a wide range of dispersive materials, for the purpose of deriving a dielectric characterization in a coaxial EM transition line. These water contents constitute the first part of databases feeding the SVR methods and are essential to optimizing laboratory calibration curves. The second part of databases corresponds to the estimated dispersion curves in complex permittivity correlated with the water content of each conditioned concrete sample. These dispersion curves are obtained by measurements in the coaxial EM transition line, followed by broadband extrapolation using the 4p variant of Jonscher's model.

Results and discussion EM data analysis For a better illustration of the effect of EM dispersion on water content in concrete, Figure 9 depicts the evolution of the real and imaginary parts of permittivity vs. water content at various extrapolated frequencies (33, 334, 948 MHz and 1.84 GHz) after testing both concrete B1 (w/c = 0.35) and mortar B6 (w/c=0.65) in the EM cell.

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The initial observation from the curves in Figure 9 is the practically constant complex permittivity for the first two, or even three, water content values, irrespective of the studied frequency. This effect has been verified for all studied materials. The phenomenon involved could be is explained by the fact that at very low degrees of saturation, the water present in the porous network is physically bound to the hydrates by van der Waals forces; moreover, the weak mobility of these particles in no way modifies the complex relative permittivity [5, 31, 34-36].

Figure 9: Evolution of real and imaginary parts of the relative permittivity obtained with the EM cell at 33, 334, 948 and 1,840 MHz vs. water content - (a) for concrete B1 - (b) for mortar B6

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The second observation is the nonlinear increase in permittivity vs. water content, which trend is stronger at lower rather than higher frequencies, which confirms the analysis in [4, 35, 36]. A difference can also be noted in the slopes of these linear regressions beyond the point of dry conditions, which for a Wc of around 0.5% means, on the one hand, between the real and imaginary parts of permittivity (ε'r and ε''r) of the same material and, on the other, between the same observables for two different materials. By analyzing the full set of slopes for all six studied concretes (not included in this paper), it is recognized that the slopes of ε'r and ε''r for the concretes with w/c = 0.35 always display lower values than those samples with w/c = 0.65. The corresponding interpretation can be summarized in two points:  Under nearly dry conditions, the permittivities are almost the same for the concretes with a high or low w/c since the aggregates are the same. The higher the material porosity, the higher the permittivity variation. The behavior of B5 (  19.5%) is more similar to high w/c materials because it is a mortar. These results are in accordance with those of the literature [5, 36].  Moreover, the studied concretes contain a high quantity of cement paste with pores filled by a very ion-rich interstitial solution. Under saturated conditions therefore, MaxwellWagner's physical phenomenon plays a major role for the relative permittivity at low frequencies: at 33 MHz, it increases considerably compared with just a slight decrease at 948 and 1,840 MHz [31, 36].

SVR method results With respect to SVR processing, TD in this study is composed of r-1 = 192 samples, whose water content lies between 1.9% and 26.4%. For each sample Wcq (%), the feature vector zq is composed of 2N = 2,000 elements. The post-extrapolation frequency bandwidth used is [0.1 - 2] GHz, with N = 1,000 frequencies. The total database has been partitioned into two: a training database (TGD), and a test

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database (TTB). Both the TGD and TTB are composed of 128 and 64 samples, respectively. As discussed below, the k-fold cross-validation has been used on the TGD with k = 5. Various kernels have been tested using the k-fold cross-validation: the linear, RBF and sigmoid kernels. Table 3 lists the results obtained from TGD with the k-fold cross-validation. Only the results with RBF, linear and sigmoid (where c = 0) kernels are shown. From these tests, as indicated in Table 3, the best results were obtained by the RBF kernel with C = 78.79 and  = 0.088. Table 3: Optimal parameters obtained using the k-fold cross-validation Kernel Parameters

 C

Average of MSE

Linear

RBF

724

0.088 78.79

2.24

1.33

Sigmoid (with c = 0) 0.0063 0.707

2.33

Following the training phase, the test phase is carried out, as shown in Figure 10. For each sample, the method estimates water content. The "square" marker represents the actual water content, whereas the "cross" marker denotes the estimated water content. Figure 10 presents the

~ Wc

values from the TTB. These results reveal that

~ Wc

tends to be well estimated. The SVR method estimates water content quite accurately. Figure 11 provides the absolute value of the absolute error between the estimated ´ the true Wc value in TTB as follows: absolute Error =

~ Wc

value and

´ −~ |Wc Wc| . Figure 11 also shows

that the absolute error is relatively small, i.e. less than 3.5% (and less than 1% in average), which is acceptable for this application knowing that only seven estimates have an absolute error beyond 1.5% on 64 samples. These results have been generated at a high computation speed. To estimate the 64 water contents, only about 0.03 s are needed using a computer equipped with a 2.3-GHz processor and 8 Go of RAM.

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Figure 10: SVR results on the test database

Figure 11: SVR method bias on the test database

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From all these results, a good correspondence can be deduced between experimental data and the inverse SVR model data. They also highlight the capability of Jonscher's 4p model to extend the frequency bandwidth at low frequencies, where dispersion phenomena appear (Maxwell-Wagner effects), allowing for their incorporation in SVR processing. Consequently, the EM cell associated with Jonscher's extrapolation and SVR algorithm can be considered as constituting a valuable means to obtain the calibration curve relating dielectric constant with water content on the cores.

6. Conclusion This paper has studied an innovative method for estimating the volumetric water concrete in various concrete mixtures. The inverse problem was solved statistically by applying a nondeterministic optimization (machine learning) algorithm to the in-lab calibration curves that tie the controlled water content in various concrete mixtures with the complex dielectric permittivity output by the EM characterization device. An extrapolation procedure using the 4p variant of Jonscher's model was developed and validated to estimate the complex permittivity at very low and very high frequencies. Introduction of this extrapolation method made it possible taking into account complex permittivity dispersion that typically affect the dielectric behavior of concrete as a function of frequency. The two-step estimation procedure (extrapolation and SVR methods) proposed in this paper has been validated on a broad array of moisture-controlled concrete specimens in the laboratory, thus serving to build calibration curves based on complex effective permittivity and water content in various concrete mixes and in accounting for frequency dependence. Although the prediction of volumetric water content obtained via the proposed method is acceptable relative to the needs expressed by infrastructure owners, it is still essential to optimize the inverse approach with new settings for kernels (i.e. a polynomial kernel), featuring a new parallel learning process on a limited frequency bandwidth ([1 MHz - 100

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MHz], [101 MHz - 200 MHz] ... [1,900 MHz - 2,001 MHz]) and other attributes. This optimization strategy will allow encompassing various EM wave dispersion phenomena associated with the polarization of free ions present in the concrete pore solution. In looking towards the future, the SVR calibration curves, combined with a step-frequency radar for in situ measurements, could lead to an accurate characterization of complex cover concrete permittivities as well as volumetric water contents with a limited number of cores still needed for the calibration on site.

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