# Modeling of chloride spatial variability in a reinforced concrete wharf from on-site measurements

Romain Clerc, Franck Schoefs, Mestapha Oumouni, Othmen Ines, Stéphanie Bonnet

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Romain Clerc, Franck Schoefs, Mestapha Oumouni, Othmen Ines, Stéphanie Bonnet. Modeling of chloride spatial variability in a reinforced concrete wharf from on-site measurements. ASCE-ASME Journal of Risk and Uncertainty in Engineering Systems, Part A: Civil Engineering, inPress, 10.1061/AJRUA6.RUENG-1214. hal-04494998v2

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1 Modeling of chloride spatial variability in a reinforced concrete wharf from on-site measurements

2 Romain CLERC¹, Franck SCHOEFS¹, Mestapha OUMOUNI¹, Inès OTHMEN², Stéphanie BONNET²

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4 ¹ Nantes Université, École Centrale Nantes, CNRS, GeM, UMR 6183, F-44000 Nantes, France

5 ² Nantes Université, École Centrale Nantes, CNRS, GeM, UMR 6183, F-44600 Saint-Nazaire,

6 France

7 * Corresponding author email: [franck.schoefs@univ-nantes.fr](mailto:franck.schoefs@univ-nantes.fr)

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9

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Chloride ingress by diffusion is the major deterioration process of reinforced concrete (RC) structures exposed to the marine environment. These structures have significant lengths or surfaces exposed to the outside environment. Due to the material variability (different concrete batches, vibrations) and exposure variability, the material experiences a spatial variability of the deterioration process. This paper presents the geostatistical analysis of in-situ chloride profiles, leading to the assessment of the spatial variability (SV) of both the chloride ingress itself and the parameters of the widely used Fick diffusion law (the average surface chloride content, $C_{sa}$, and the average chloride diffusion coefficient, $D_a$). 37 chloride profiles measured on both sides of the same spandrel beam of an RC wharf are studied, as well as the associated estimates of $C_{sa}$ and $D_a$. From an initial selection of random field models, the geostatistical analysis consists in the evaluation of model parameters using a procedure that tests both data and model assumptions on the fly (ergodicity, stationarity, random field modeling). Combined with the calculation of information criteria for each model, this procedure allows to provide relevant geostatistical models for chloride ingress, $C_{sa}$ and $D_a$, which render SV as well as measurement error. It is noteworthy that the estimation error can be neglected when focusing on the SV for the range of chloride content studied in this paper. The SV of the chloride content seems to depend on the depth, with a large variability within the convection zone, and much less and more stable in the diffusion zone with a practical range of about 70 cm. This order of magnitude is consistent with the range of SV calculated for $C_{sa}$ (50 to 73 cm).

32

**Keywords:** Chloride ingress, Reinforced concrete, Marine environment, Spatial variability, Random field, Statistical analysis.

34

1. Introduction

Chloride ingress is among the main causes of degradation of reinforced concrete (RC) structures where the deterioration is more perceptible in marine environment. That leads to major maintenance costs (Bastidas-Arteaga & Schoefs, 2015; Stewart & Val, 2003). These structures offer to the environment important lengths or surfaces. Due to the variabilities of both the material (different concrete batches, vibrations) and exposure, they experience a spatial variability of their deterioration process. Accounting for realistic spatial correlation of the different parameters in structural safety models may lead to more realistic estimates on structural reliability and maintenance planning, knowing that reliability assessment is affected by spatial correlation (Schoefs et al., 2022; Stewart, 2004; Stewart & Mullard, 2007). Moreover, quantifying this property allows optimizing inspection
44 meshing on a structure (Gomez-Cardenas et al., 2015; Oumouni & Schoefs, 2021b; Schoefs et al., 2017a).
45

46 The variability of the degradation parameters and their measurements can be described at
47 different scales:

48 • the point scale, characterizing the repeatability of measurements;

49 • the local scale of the material, characterizing the heterogeneity of the material for a scale
50 lower than the Representative Elementary Volumes (REV) associated with each of the
51 quantities of interest (Hill, 1963);

52 • higher scales (structural elements, structure), characterizing the degree of spatial
53 heterogeneity of the material and/or environmental conditions (exposure, loading, ...) for
54 scales above the REVs.

55 The restitution of point and local variabilities requires a statistical modeling, translating the
56 hypothesis of independence, respectively on (i) the measurements of parameters between distinct
57 measurements and for a same measurement point and (ii) the parameter values between similar
58 structures for the same REV dimension. The spatial variability (taking into account local and larger
59 scales) cannot be solely captured by a statistical modelling because of the hypothesis of the existence
60 of a certain degree of heterogeneity, i.e. spatial correlation, between measurements or between
61 parameter values associated with different measurement points or REV on the same structure. This
62 can only be translated by geostatistical modeling. The major challenge of those models is to determine
63 stochastic properties of the Random Fields (RF), namely their marginal distributions, stationarity and
64 the spatial correlations.

65 Chloride ingress has been investigated deeply since the 70's and it has been shown how
66 complex is the process physically and chemically (Glasser et al., 2008; Kaushik & Islam, 1995; Ragab
67 et al., 2016; Safhi et al., 2019; Wegian, 2010) and the role the material complexity (porosity) is
playing (Qu et al., 2021). For the moment, available models of transfer and knowledge don't allow to represent accurately the spatial variability of pollutant in concrete (Ravahatra et al., 2017). That is the main reason why direct or indirect assessment of chlorides in situ is the only way to measure, analyze and model the spatial variability of chloride ingress. Then, two kinds of quantities of interest can be considered:

* the total chloride content $C$ at several depths;

* their mean diffusion parameters $C_{sa}$ and $D_a$, estimated from profiles of chloride content.

During the last decades sensors have experiences significant progresses (Biondi et al., 2021; Torres-Luque et al., 2014, 2017; Watanabe et al., 2021; Zaki et al., 2015) but there are few concepts that measure the spatial distribution of chloride ingress (Fares et al., 2018; Lecieux et al., 2015, 2019) and their implementation is rare (Priou et al., 2019) even if their interest in terms of the value of information they provide was shown (Schoefs et al., 2022) especially because they allow to filter the signal (Oumouni & Schoefs, 2021a). Moreover, these sensors have to be installed at the building stage and are not suitable for existing structures.

That is the main reason why spatial variability is assessed through Non-Destructive Testing (NDT) tools or Semi-Destructive Techniques (SDT). On-site indirect assessment of chlorides through NDT-techniques is affected by errors of assessment and sensitivity to other parameters (humidity, temperature) which makes it difficult to assess the spatial variability of chlorides (Bonnet et al., 2017; Bourreau et al., 2019; Schoefs et al., 2023). This error affects the optimization of measurements for assessing spatial variability (Oumouni & Schoefs, 2021b). That is why, the rare studies are based on the semi destructive techniques (SDT) i.e. direct weighting of chlorides from cores (O'Connor & Kenshel, 2013; Othmen et al., 2018) reviewed 10 published studies from 2004 to 2016 with only 3 concerning more than 30 measurements and in all cases the distance between measurement was too large in view to assess the spatial correlation. Two main quantities of interest are investigated: chloride content at a given depth and parameters of a model of chloride ingress after post-processing
93 of chloride profiles.

94 Because of the complexity and cost for making a large amount of cores in a beam, only a few
95 amounts of data is available for each and the distance between cores is too large: 5 cores with a
96 distance of about 1 m in the study of (O'Connor & Kenshel, 2013). That leads to a significant
97 numerical error of assessment of the spatial correlation (Schoefs et al., 2017b). Moreover, the
98 underlying assumptions for RF modelling that are stationarity and ergodicity are not checked. Finally,
99 the error of measurement is usually not considered when identifying correlations. That is a drawback
100 because it is well known that SDT measurements for existing structures (Schoefs et al., 2023) and
101 even for concrete in laboratory (Bonnet et al., 2017; Hunkeler et al., 2000) are affected by a significant
102 error. To overcome these limits, this paper relies on two pillars: first, a unique database and second,
103 a geostatistical analysis that allows to provide geostatistical parameters (namely spatial correlation
104 assessments as well as measurement error) by using a recent method of RF identification (SCAP1D,
105 Clerc et al. 2019), that accounts for uncertainty of measurements, checks ergodicity and accounts for
106 the non-stationarity of the RF.

107 Concerning the first pillar, marine structures can be exposed in various environments:
108 atmospheric, spatter or splash, tidal and underwater (BS 6349-1:2000, 2000) which were defined
109 according to the tide range (Bourreau et al., 2020). In this work, 30 cores carried out every 30 cm on
110 the same 28-year beam (Othmen et al., 2018) are analyzed. The beam is located in the worst
111 environment that is splash zone (Angst et al., 2009), and the cores are through its width giving access
112 to profiles from each side i.e., exposure conditions (exposed, sheltered).

113 Concerning the second pillar, two objectives are targeted: the assessment of spatial variability
114 of the total chloride content C, depending on depth and exposure on one hand and the assessment of
115 the spatial variability of the mean diffusion parameters Csa and Da on the second one. First, taking
116 benefit of the availability of C at several depth, the objective is to provide estimates of its geostatistical
117 parameters and to answer the following questions: (i) Is the spatial variability (SV) of C a function of
118 depth? (ii) Is the SV of $C$ a function of the exposure conditions? Second, knowing the age of the
119 structure (28-year), geostatistical parameters of mean diffusion parameters are estimated to answer
120 the following question: Is the SV of $C_{sa}$ and $D_a$ a function of the exposure conditions? The latter
121 question arises because $C_{sa}$ and $D_a$ are directly dependent on the chloride profiles, which vary
122 markedly between the exposed and sheltered faces.

123 Section 2 introduces the geostatistical formalism in a theoretical way before presenting a state
124 of the art of RF modeling, simulation methods and spatial variability estimation methods. An attention
125 is paid to the underlying assumptions of the estimation methods and to the consequences of their
126 neglect on the estimated spatial variability, especially in the frequent case of non-stationary
127 measurements due to spatially variable exposure conditions. Section 3 presents the structure and the
128 available data. Sections 4 and 5 give the results at two levels: total chloride contents, and mean
129 diffusion parameters of total chlorides. Section 6 concludes the paper with some extension to other
130 structures or other exposures.

131 **2. Geostatistical methods for spatial variability assessment of concrete structures**

132 **2.1. Modelling of spatial variability**

133 **2.1.1. Random Field modeling**

134 Practical modeling of spatial variability implies modeling spatially variable quantities of interest (QI)
135 as Second-Order Intrinsic Stationary Random Fields (RF) (Chilès & Delfiner, 2012). Following the
136 geostatistical formalism, such an RF Z is defined on a domain $D^n$ of $\mathbb{R}^n$ and on a probability space
137 $(\Omega, A, P)$ such that:

138 • $\forall \mathbf{x} \in D^n, Z(\mathbf{x}, \cdot) = Z_x$ is a multivariate random variable of $\omega \in \Omega$ with a joint probability density
139 function (PDF) $f_Z(\mathbf{m}, \mathbf{\Sigma})$, where $\mathbf{m}$ and $\mathbf{\Sigma}$ are the mean vector and the covariance matrix of $Z_x$,
140 respectively;
141 • $\forall \mathbf{x} \in D^n$ and $\forall \omega \in \Omega$, $Z(\mathbf{x}, \omega) = Z_{\omega}^n$ is a realization of Z georeferenced by the vector $\mathbf{x}$, also called a trajectory (in practice, a set of measurements from a unique structure and at a unique time).

144 • $\forall (h, \mathbf{x}) \in D^2$, $\mathbb{E}[Z_{h+\mathbf{x}} - Z_{\mathbf{x}}] = 0$ and $\mathrm{Var}(Z_{h+\mathbf{x}} - Z_{\mathbf{x}}) = 2\gamma(h)$ where $\gamma(\cdot)$ is the semi-variogram function of Z and $h$ is any distance-lag (distance between two points of D).

146 This definition reflects both the variability of the input parameter between structures and its spatial variability within the same structure, which are embedded in the covariance matrix $\Sigma$ and the semi-variogram.

149 In this paper, data are modeled as (i) unidimensional (ii) noisy (iii) trend-stationary (iv) Gaussian Random Field (GRF). Indeed, (i) the data are unidimensional along the sides of the beam; (ii) if the spacing between measurements is greater than the REV, the RF model must be able to reproduce the point and local variabilities, which have no correlation structure and can be modeled by white noise; (iii) materials and environmental conditions may vary along the sides of the beam. It is then very likely that the mean of the quantities of interest will vary with space; (iv) GRFs are very convenient because they allow modeling most natural variabilities (following normal or lognormal marginal laws) with only a model for the mean and the covariance (Chilès & Delfiner, 2012).

157 Mathematically, $\forall \mathbf{x} \in D^n$ we thus consider Z such that

$$Z_{\mathbf{x}} = \mathbf{m}_{\mathbf{x}} + W_{\mathbf{x}} + Y_{\mathbf{x}} \quad (1)$$

158 with $\mathbf{m}_{\mathbf{x}}$ a deterministic trend, $W_{\mathbf{x}} \sim N(0, t \times \mathbf{1}_n)$ a white noise of variance $t$; $Y_{\mathbf{x}} \sim N(0, \Sigma_{\mathbf{Y}})$ a centered GRF, i.e $Y_{\mathbf{x}}$ a multivariate normal variable such that

$$\forall (h, \mathbf{x}) \in D^2, \Sigma_{Y_{\mathbf{x}+h, \mathbf{x}}} \equiv \mathrm{Cov}(Y_{\mathbf{x}+h, \mathbf{x}} - Y_{\mathbf{x}}) \equiv C_{\mathbf{Y}}(h) \equiv s \times \rho_{\mathbf{Y}}(h) \quad (2)$$

160 where $C_{\mathbf{Y}}(\cdot)$ is the covariance function of $\mathbf{Y}$, $s$ is its variance, $\rho_{\mathbf{Y}}(\cdot)$ is its autocorrelation function, and $h$ is any distance-lag between the data (equivalences due to the Second-Order Stationarity of the GRFs).
This formalism is illustrated in Figure 1 and Figure 2, where trajectories, correlation plots, autocorrelation functions, and semivariograms are plotted for Y and Z respectively, defined such that:

1) $Y_x \sim N(0, \Sigma_Y)$ with $\Sigma_{Y_{x+h,x}} = s \times \exp(-1.73\|h\|^2/r_{95})$, $\rho_Y(h) = \frac{\Sigma_Y_{x+h,x}}{s}$, $\gamma_Y(h) = s - \Sigma_{Y_{x+h,x}}$, $s = 10$ and $r_{95} = 20$;

2) $Z = W + Y$ with $W_x \sim N(0, t \times 1_n)$, $\Sigma_{Z_{x+h,x}} = t \times \delta_{x+h,x} + \Sigma_{Y_{x+h,x}}$, $\rho_Z(h) = \frac{\Sigma_{Z_{x+h,x}}}{s+t}$, $\gamma_Z(h) = s + t - \Sigma_{Z_{x+h,x}}$, and $t = 5$ ($\delta_{ij}$ is the Kronecker delta).

### 2.1.2. Autocorrelation models and parameters
172 **3. Embedded Tables**

173 Table 5 and Figure 1 present and illustrate the exponential and Gaussian autocorrelation functions,
174 widely used in the literature. According to Equation (2), such functions are positive definite, so that
175 they asymptotically decay as a function of the distance-lag *h*. The evolution of this decay is mainly
176 determined by the value of their scale parameter *a* > 0, which parametrizes the extent of the spatial
177 correlation, i.e. the distance threshold below which the QI are strongly correlated.

178 The distance *at* which events are weakly correlated is referred to as the inspection distance threshold
179 with a spatial correlation threshold definition of 30% in (Schoefs et al., 2016). Several notations and
180 parameters coexist in the literature to describe this threshold, sometimes with some confusion. In the
181 case of isotropic fields, the following three notations are commonly used: (i) Cressie (1993, p. 118)
182 proposed the fluctuation range θ, which is defined as $\theta = \int_{-\infty}^{+\infty} \rho(h)dh$ ; (ii) (Cressie, 1993, p. 88)
183 proposed the practical range $r_{95}$, which is defined as $\rho(r_{95}) = 5\%$ (see Figure 1 and Figure 2) ; (iii)
184 many authors proposed the correlation length $l_c$, which is not strictly defined and alternatively taken
185 to be equal to *a*, *θ*, $r_{95}$ or some other particular range. In order to avoid any confusion and to facilitate
186 comparison and interpretation, we give in

187
## 4. Embedded Tables

Table 5 the expression of $\theta$ and $r_{95}$ as a function of $\alpha$ for the exponential and Gaussian models and the corresponding autocorrelation values $\rho$. Note that in Figure 1 Y is obtained from a Gaussian autocorrelation function.

### 4.1. Identification of spatial variability from measurements

#### 4.1.1. Empirical identification

RF models behind a trajectory can be identified in the case of its ergodicity, which can be seen as a mixture of stationarity and asymptotic independence. In this case, the spatial variability can be assessed from empirical representations of the semivariogram, the covariance function, or the autocorrelation function, called the empirical semivariogram, covariogram and autocorrelogram, respectively.

The Semivariogram, denoted $\hat{\gamma}(h)$, is used because its estimation does not require complementary ones (such as mean or variance), and it allows to visually identify (i) the Second-Order Stationarity property, (ii) the ergodicity property, and (iii) the presence and magnitude of local variabilities. Indeed, (i) $\hat{\gamma}(h)$ asymptotically tends to the variance for Second-Order Stationary RF, while otherwise whereas it increases with $h$; (ii) ergodicity is verified when $\hat{\gamma}(h)$ reaches an asymptote in the limit of half of the domain $D$ (Cressie, 1993, p. 57); (iii) white noise is easily identified by a nugget effect, i.e. a non-zero value for $h = 0$ (see Figure 2).

A common estimator of $\hat{\gamma}(h)$ is defined by Equation (3) for a given trajectory $Z_x^\omega$, where $S_h$ is the set of pairs $(x_p, x_q)$ of $D^2$ that are away from $h$ modulo a given tolerance, and $N_h$ is the cardinal of $S_h$.

$$ \hat{\gamma}(h) = \frac{1}{2N_h} \sum_{(x_p, x_q) \in S_h} (Z_{x_p}^\omega - Z_{x_q}^\omega)^2 \quad (3) $$

We point out that when $h$ becomes close to the dimension of $D$, statistical inference is no longer valid because the number of point pairs away from $h$ is very small. Therefore, geostatisticians recommend plotting this graph only for distances $h$ such that $N_h \ge 30$ (Cressie, 1993). In practice,
211 in civil engineering, several authors recommend limiting to values of *h* less than half of the size of
212 the domain (Arnaud & Emery, 2000, Schoefs et al., 2016).

213 **4.1.2. Estimates of model parameters**

214 Once an RF model is selected from empirical identification, its parameters (collected in a vector **p**)
215 must be estimated. Several methods are available:

216 • Maximum Likelihood Estimation (MLE), which allows to estimate **p** from raw data, given an RF
217 marginal distribution model, and as the following properties: (i) asymptotically unbiased, (ii)
218 consistent, (iii) equivariant, (iv) asymptotically efficient, and (v) asymptotically normal
219 (Wasserman, 2004, para. 9.4);

220 • Least-Squares Estimate (LSE), which allows estimating **p** from semivariogram, covariogram or
221 autocorrelogram given models of them;

222 • Weighted LSE (WLSE), similar to LSE, where each of the residuals is weighted by its variance.
223 This allows to increase the weight of the residuals computed from many data (low distance-lag *h* values, for which *N*<sub>*h*</sub> is larger).

225 Usually, LSE and **WLSE** fit the covariograms **well** and the use of **WLSE** implies a better description
226 of **them** for small **distances**, associated with more data<fnref n="1" />. However, covariograms are not necessarily
227 representative of the effective covariance parameters of the field, which can lead to significant relative
228 errors in their estimation. Because it considers **only** raw data, MLE provides more accurate estimates,
229 with **smaller** relative errors. The **trade-off** is **computational** time, as MLE is more time consuming
230 than **WLSE** and LSE, up to 10 times faster. **Therefore,** and **because** it is *a priori* complex to have an
231 initial idea of the scale parameter, a two-phase estimation method of Second-Order Stationary RF
232 parameters is **preferred**:

<footnote n="1">¹ In this case, $\gamma(h) = 1 - C(h)$, so estimates on the covariogram or the semivariogram are equivalent.</footnote>
1. WLSE pre-estimation of the parameters with the following initial values: (i) for s, the empirical variance; (ii) for a, a value such that the fluctuation scale is less than a quarter of the maximum lag-distance (so as to respect the ergodic assumption);

2. MLE estimation of the parameters using the Nelder-Mead simplex minimization algorithm with the WLSE estimates as initial values.

This approach allows to obtain accurate estimates while minimizing the cost of the MLE due to the definition of initial values close to the real values of the parameters (Clerc et al., 2019, para. 4.4).

### 4.1.3. Choice of best RF model

Although the plot of the empirical semi-variogram allows the selection of an RF model, its estimator nature leads to an imperfect representation of the SV. This is all the more true when the number of measurements is small. Therefore, when there is insufficient scientific evidence to support a specific RF model for a QI, which is the case for chloride content and diffusion parameters (Gomez-Cardenas et al., 2015), we can rely on data-based methods for model selection.

In the following, the Akaike Information Criterion (AIC) is preferred because it is much more robust than the R² coefficient of determination (Burnham & Anderson, 2010). The AIC allows to compare, with the parsimony criterion, the relevance of any RF model Mz with K parameters p₁,...,K to describe the hidden RF Z at the origin of a set of n data Zₓ^ω (e.g., a trajectory). This criterion assumes that the best model is the one that provides the most accurate description with the fewest possible parameters (Burnham & Anderson, 2010, para. 1.4).

In particular, when n/K < 40 and (which is the case below) we use the corrected information criterion (AICC). When the p estimate $\hat{p}$ is computed with MLE, it writes:

$$ AIC_c = \underbrace{-2 \ln(L(\hat{p}|Z_x^\omega)) + 2K}_{AIC} + \frac{2K(K+1)}{n-K-1} \quad \text{correction} $$

where L(\hat{p}|Z_x^\omega) is the likelihood of \hat{p} with respect to the data. The best model among a set of considered models is then the one that minimizes the AICC, i.e. the one that has both a high goodness of fit (high likelihood) and a limited complexity (limited number of parameters).
In practice, the easiest way to compare the relevance of models based on their AICC is via the Evidence Ratio (ER) (Burnham & Anderson, 2010, para. 2.10). This is defined such that, for any model $M_{Z,i}$ among $N_m$ models,

$$ER_i = \exp\left(\frac{1}{2}\left[AICc_i - \frac{\min_{j=1,...,N_m}{(AICc_j)}}{\text{best a priori model}}\right]\right)$$

The ER is therefore 1 for the best a priori model and grows exponentially as the models become less fit. From an expert perspective, an ER of 2 for a given model does not justify preferring the best a priori model over it, whereas an ER of 18 clearly favors the best a priori model (Burnham & Anderson, 2010, para. 2.10).

### 4.1.4. SCAP1D: a robust estimation procedure

The methods presented assume an adequate RF model to avoid estimation errors: without modeling precautions, they are only valid under the condition that the data can be described by an SSO and ergodic RF. However, since classical regression techniques coupled with generic autocorrelation models can easily approximate autocorrelograms and covariograms, the estimation of RF model parameters is typically performed by LSE on raw data without prior verification of these two hypotheses (O'Connor & Kensel, 2013; Ravahatra et al., 2017; Schoefs et al., 2016). Such a procedure can then lead to significant errors, in especially in the estimation of the scale parameter $\alpha$, and especially in cases of non-stationarity in the mean and/or significant measurement noise. This point is demonstrated in a case study by (Clerc, 2021, p. 143): from a given realization of a standard GRF, adding a noise and/or a mean trend and performing an LSE of $\alpha$ without prior verification of the SSO and ergodic hypotheses led to estimation errors of up to 994%.

To avoid these errors, we use here the SCAP-1D procedure<fnref n="2" /> (Clerc et al., 2019). Based on previously presented tools, it allows to estimate, from a single trajectory with few measurement points, the

<footnote n="2">² Spatial Correlation Assessment Procedure for Unidimensional Data</footnote>
parameters of a unidimensional noisy trend-stationary RF with piecewise polynomial mean, constant variance and constant autocorrelation model.

The concept of SCAP-1D is to ensure the reliability of the parameters' estimates by testing and validating the following hypotheses:

H1: validity of the RF mean model and its parameter estimates, i.e. Second-Order Stationarity of the centered RF;

H2: validity of both the joint PDF and autocorrelation function models of the RF and of the estimation of their parameters; i.e., normality of the standard source GRF, obtained by isoprobabilistic transformation of the RF from estimates considering the a priori model;

H3: ergodicity of the trajectory; i.e., non-correlation of the data at infinity or determination of all geostatistical parameters from this trajectory.

The SCAP-1D flowchart is shown in Figure 3. It is divided into four consecutive steps, which can be repeated if the estimates do not validate the hypotheses on the RF model:

Step I) *Choice of SCAP-1D parameters*, namely piecewise polynomial mean model (regression degree, number of changepoints), RF joint PDF and autocorrelation function. We recommend to plot the trajectory, its histogram and its experimental semivariogram to facilitate this step.

Step II) *Estimation of the mean and geostatistical parameters*: changepoints in the mean and their significance are determined using the PELT (Pruned Exact Linear Time) optimal partitioning algorithm of (Killick et al., 2012). Estimation of the RF parameters is then performed using the approach presented in 4.1.2³.

Step III) *Checking of the underlying mathematical hypotheses*: the plausibility of the estimated model is verified by checking ergodicity and performing stationarity tests (DF-test (Dickey & Fuller, 1981), KPSS-test (Kwiatkowski et al., 1992)) and normality tests (χ2, KS-test (Kolmogorov-Smirnov)) on the standardized trajectory, based on the Step II estimates.

<footnote n="3">³ or by WLSE on the experimental semivariogram in the case of non-explicit density models.</footnote>
304 • *Step IV) Analysis*: based on the Step III tests, the model selected in Step I and the estimates made
305 in Step II are either accepted (algorithm stops) or rejected (modification of Step I parameters is
306 required).

307 In Step III, the standardization of the trajectory is done by the Cholesky transformation, taking into
308 account either:

309 • the empirical covariance matrix: in this case, we only test H1, since the tests results depend only
310 on the data and the estimates of the mean. The tests are then called Σ<sub>exp</sub>-tests;

311 • or the modeled covariance matrix: in this case, we test H1 to H3 together, since the tests results
312 depend on the total estimates. The tests are then called Σ<sub>mod</sub>-tests.

313 4.1.5. *Interpretability criterion*

314 In Step IV of SCAP-1D, the estimates must be analyzed for their physical consistency. In particular,
315 an interpretability criterion can be defined for the estimated practical range. The point is to ensure,
316 with respect to the distance Δx between measurements, that the variability thus described is indeed a
317 spatial variability and not a local or point variability. This is illustrated in Figure 4, where the
318 Gaussian autocorrelogram is plotted on a measurement grid of step Δx = 5 for three practical range
319 values (r<sub>95</sub> < 2Δx, r<sub>95</sub> ≈ Δx, and r<sub>95</sub> > 2Δx):

320 • when r<sub>95</sub> ≥ 2Δx (curves c3 and c2): the variability described is spatial variability, since there is
321 at least one significant intermediate spatial correlation value between the total correlation of a
322 point with itself and the minimum 5% correlation associated with the practical range;

323 • when $r_{95} < 2\Delta x$ (curve c1): the variability described may be local or point variability since there
324 is no significant intermediate spatial correlation value to account for the spatial variability.
Therefore, care will also be taken to ensure that the criterion $r_{95} \ge 2\Delta x$ is met when applying SCAP-1D<fnref n="4" />.

## 5. Presentation of the structure and the measurements

### 5.1. Presentation of the wharf, the beam and its material

The reinforced concrete beam inspected within the framework of the APOS research project is a T hyperstatic beam called "J-beam", whose dimensions and reinforcement are shown in Figure 5. It is located on lane 51 of the gangway 5 of the Montoir-de-Bretagne coal terminal (Loire-Atlantique department, France) with GPS coordinates: 47.32°N, -2.17°E. This terminal, built between 1981 and 1983 on the banks of the Loire and less than 7 km from the Atlantic Ocean, is 234.4 m long and 20.7 m wide and is located at +8.40 m NGF (Figure 6). This proximity to the ocean implies the existence of a tidal phenomenon with an amplitude of 5.80 m and a salinity of the water corresponding to maritime conditions at high tide (about 30 g Cl⁻/L). Thus, at the highest tidal coefficients, the lower face of beam J, located 5.80 m above the level of the Loire, is in contact with the water for about 2 days per month. Finally, one of the peculiarities of the J-beam is its location at the edge of the quay. It has a side exposed to the wetting-drying cycles, marked Ext, and a sheltered side, marked Int (see Figure 6).

Data from the construction archives indicate that the concrete used to build the structure is a Portland cement concrete CEMI 42.5N dosed at 350 kg/m3. The mass fraction of $m_{cl}$ clinker is therefore higher than 90% and that of secondary constituents such as gypsum does not exceed 5%. The aggregates used are sand 0/6, and gravel 5/10 and 10/25. Concrete water porosity $\varepsilon$, measured on the central part of 5 specimens used for chloride titration, was 13.7% on average, with minimum and

<footnote n="4">⁴ This criterion is constructed similarly to that of (Der Kiureghian & Ke, 1987) on the ability of a discretization grid to reproduce the spatial variability properties of an RF.</footnote>
347 maximum values of 11.3% and 15.9%, respectively. The average compressive strength $f_{cm}$, measured
348 on 4 specimens of 5 cm diameter and 10 cm length, was 43.5 MPa, with minimum and maximum
349 values of 38.5 MPa and 48.9 MPa, respectively.

350 **5.2. Total chloride content measurements at 28 years**

351 **5.2.1. Available data set**

352 Chloride content measurements were performed on 30 through cylindrical cores extracted on the J-
353 beam at the same height of 30 cm and with a lateral spacing of 30 cm (see Figure 7) (Othmen et al.,
354 2018). The extraction was performed in 2011-2012, after 28 years of exposure. From initial
355 dimensions of 50 mm in diameter and 400 mm in length, each core was divided into 5 slices
356 (identified in Figure 7). Slices 1, 2, 4 and 5 were used to determine the chloride profiles according to
357 the procedure recommended by the RILEM TC 178-TMC (Vennesland et al., 2013). Slices 1 and 2
358 are related to the outer side (collected in Ext specimens) and slices 4 and 5 are related to the inner
359 side (collected in Int specimens). While deep slices 2 and 4 were 40 mm thick with measurements in
360 10 mm increments, slices 1 and 2 were 45 mm thick with measurements in 5 mm increments to
361 capture the value at the surface. Slice 3 was used for further investigations such as porosity,
362 compressive strength and estimation of the initial chloride content. Out of a total of 30 samples per
363 side, 16 were usable on the Ext Face (53%), and 21 on the Int Face (70%). Furthermore, of the 555
364 total chloride contents of the 30 samples considered, only 460 were usable (83%, or 51% of the total
365 originally planned measurements).
366

366
367 Figure 9 hereafter then represents the remaining set of total chloride measurements at 28 years finally considered (marked *C* in the following). The evolution of the empirical mean and standard deviation
368
369 by depth and by face is shown in Figure 8.
### 5.2.2. Constitution of trajectories

In the following, the evolution of the SV of C is analyzed as a function of depth and exposure conditions. SCAP-1D is then applied to horizontal trajectories of measurements made on the same face and at the same embedding depth z. The estimation of geostatistical parameters for each trajectory then allows plotting their evolutions as functions of z and exposure face. This study principle is illustrated in Figure 9.

### 5.3. Average diffusion parameters at 28 years

The 37 total chloride profiles were post-processed to determine the associated Fick's second law average diffusion parameters at the reference time $t_r$ of 28 years, denoted $C_{sa}$ and $D_a$. The fitting procedure to the peak of the profiles is described in (Othmen et al., 2018) and implemented in (Schoefs et al., 2023). Thus, one trajectory for each diffusion parameter per side is constructed at a height of 30 cm. These include 16 points for the Ext Side and 21 points for the Int Side, with a minimum distance of 30 cm between measurement points.

Table 6 summarizes this information and Figure 10 shows the trajectories. Note that the log function is applied to each parameter to (i) simplify the representation and visual analysis as they are assumed to be lognormally distributed, (ii) use identification procedures (see section 4.2) available for normally distributed random fields.

## 6. Spatial variability estimates of total chloride content in J-beam and physical understanding

### 6.1. Preamble: trajectories conditioning (kriging, debiasing)

Taking into account the non-usable or missing measurements per sample and in order to take into account the maximum number of points per trajectory, these are first enriched by ordinary kriging of the available data (Baillargeon, 2005, Chapter 4; Chilès & Delfiner, 2012, Chapter 3). The preliminary plot of the raw trajectories does not show any obvious non-stationarity (Figure 11), and the expectation of the associated RF is not known.
The kriging is then performed from the Weighted Least-Square Estimate (WLSE) identification on the semivariograms of the initial trajectories. We consider an exponential covariance model and the presence of an additive measurement error, identified by (Bonnet et al., 2017). In addition, (Bonnet et al., 2017) also identified a measurement bias $b = 0.37 \times 10^{-3}$ (kg/kg concrete) associated with the measurement protocol used and with human errors during its implementation. All the measurements are therefore debiased before kriging so that the data considered are now $C + b$. In order to limit the influence of the kriged data on the estimation of the SV of $C$, only the trajectories for which their proportion is less than 25% are considered.

In total, 13 trajectories with 16 points can be considered on the Ext side ($z \in [2.5; 75]$ mm) and 12 trajectories with 21 points on the Int side ($z \in [2.5; 65]$ mm). The number of kriged data per trajectory is detailed in Table 1 in the Appendix, and the selected debiased trajectories are shown in Figure 12. Less than 25% of the data are kriged to 65 mm depth for both sides. We point out that due to the absence of measurements on the abscissae of the non-exploitable specimens, the trajectory grids are irregular, with 30% of measurements missing for the Int side and 53% for the Ext side. The performance of the SCAP-1D normality tests is therefore affected (Clerc et al., 2019), and special care must be taken to ensure the consistency of the estimation results.

## 6.2. Geostatistical pre-study and choice of parameters (Step I):

We recall that the studied trajectories are debiased. We then denote $\tilde{Z}$ the RF of $C$ such that

$$ \tilde{Z} = Z - b \quad | \quad b = -0.37 \times 10^{-3} \text{ kg/kg concrete} \qquad (4) $$

In addition, (Othmen et al., 2018) have previously determined that the marginal law of $C$ is lognormal, and (Bonnet et al., 2017) have identified the "measurement error" introduced by the measurement protocol, but only for a mortar. This is distributed according to a generalized extremum law (GeV, (Johnson et al., 1995, p. 3)) with the parameters $k = 0.16$, $\sigma = 9.3 \times 10^{-5}$, and $\mu = -8.4 \times 10^{-5}$. (Schoefs et al., 2023) recently pointed out that the error model varies with chloride content. Consequently, the error should vary according to each trajectory. For simplicity, it is
considered constant in this paper and its standard deviation is an average of the potential standard deviations (Schoofs et al., 2023).

The best-fitting RF model for $\tilde{Z}$ is then the sum of a log-normal RF Y and a GeV random variable W such that, according to Equation (1), $Z = Y + W$.

However, the joint PDF of such an RF is not explicit, and SCAP-1D cannot simply be applied.

The small number of points per trajectory also precludes an estimation by WLSE on semivariograms (see 4.1.2).

Two modeling compromises are then possible in order to ensure explicitness:

* Model a: consider a normal marginal distribution of the total chloride content and a normal distribution of the error, so that

$$ \tilde{Z} \sim \frac{N(\mu(x), s \cdot R(a)) + t \cdot N(0, I)}{G^1} \quad (5) $$

where s and t are respectively the empirical variance of the signal and the error and R(a) is the autocorrelation matrix.

* Model b: do not consider the measurement error, whose standard deviation (1.55 × 10⁻⁴ kg/kg concrete according to the model of (Bonnet et al., 2017)) would induce a maximum CoV of 8.3% (on trajectory 13; Ext side, z = 75 mm). For this model,

$$ \tilde{Z} \sim \exp\left(\frac{LN}{\left(\frac{N(\mu(x), s \cdot R(a))}{G^2}\right)}\right) \quad (6) $$

$G^1$ and LN are then the RF describing the SV of C, with $G^2$ being the source GRF of LN.

Since the first approximation of the maximum CoV according to the model of (Bonnet et al., 2017) is significant, we then apply SCAP-1D considering both models, so that:

* if the Signal-to-Noise Ratio (SNR) associated with model a is low, the error is not negligible and model a is retained;

* if the SNR associated with model a is high, the error is considered negligible and model b is retained.
Due to the absence of non-stationarities visible in Figure 12, the trajectories are a priori assumed to be stationary, so that $\mu(x) = \mu$. We also recall that the model of the autocorrelation function chosen for $G^1$ and $LN$ is exponential. According to (Chilès & Delfiner, 2012, p. 106), to guarantee such a model for the autocorrelation function $\rho_{LN}(a)$ of $LN$, the autocorrelation function $\rho_{G^2}$ of $G^2$ must be written as follows

$$ \rho_{G^2}(\cdot) = \ln(\rho_{LN}(\cdot) \cdot [e - 1] + 1) \qquad (7) $$

According to (Chilès & Delfiner, 2012, p. 106), the mean $m_{LN}$ and the variance $\sigma_{LN}^2$ of $LN$ are then written as follows

$$ m_{LN} = \exp\left(\mu + \frac{1}{2}s\right) \qquad (8); \qquad \sigma_{LN}^2 = m_{LN}^2(\exp(s) - 1) \qquad (9) $$

### 6.3. Model a: Parameter estimation, hypothesis testing and analysis (Steps II, III and IV)

Preliminary note: Prior to estimation, the trajectory data, of order $10^{-3}$, are multiplied by a factor $k = 10^4$ to facilitate the convergence of the MLE. The estimates of $t$, $s$ and $\mu$ are then transformed accordingly before analysis.

Table 2 in the Appendix details the results of the SCAP-1D implementation considering model a. Only the results of the $\Sigma_{mod}$-tests are considered due to the small number of points per trajectory.

Analyzing these, the estimates for trajectories 4 and 5 are rejected due to the rejection of the stationarity hypothesis by the KPSS-test and because the estimated error and practical range are zero and well below $2\Delta_x$, respectively, so that the spatial correlation of the data cannot be described (see 4.1.5). This is also the case for trajectories 2, 11, 12, and 13 and can be explained by their very small number of points. Similarly, the estimates associated with trajectories 15, 17, and 24 are not considered because the estimated practical range is zero. Finally, no valid and convergent estimate can be obtained for trajectory 14. As a first step, for each selected trajectory, we plot the estimates
462 $\hat{\mu}$ of the mean with their 95% MLE-based confidence intervals in Figure 13 (a)<fnref n="5" />, and the estimates
463 $\sqrt{\hat{s} + \hat{t}}$ of the standard deviation, for which the distribution is unknown, in Figure 13 (b). We then
464 note that these are consistent with the statistical mean and standard deviation profiles previously
465 plotted in Figure 8: the values estimated by kriging (see 4.1) do not produce perturbations of these
466 statistical estimates.

467 As a second step, we plot in Figure 14 (a) the standard deviation estimates $\hat{t} = \sqrt{\hat{s}}$ and $\hat{\sigma} = \sqrt{\hat{s}}$ of
468 the error and signal, respectively, and the standard deviation $\tau_{\varepsilon}$ associated with the error model of
469 (Bonnet et al., 2017). Trajectories for which the estimated error is zero are not considered. We then
470 compare the roots of the experimental SNRs $\hat{t}/\hat{s}$ and the fixed error SNRs $\tau_{\varepsilon}^2/\hat{s}$ in Figure 14 (b).
471 Thus, we find that 75% of the roots of the experimental SNRs are greater than 100 and show no
472 specific trend along. The fixed error SNRs are logically lower than the experimental ones
473 because the error was calculated for mortars, for which the evaluation error is larger (Schoefs et
474 al., 2023).

475 Therefore, in accordance with the previously stated modeling assumptions, we consider the
476 measurement error to be negligible and retain model *b* for the SCAP-1D trajectory study.

### 6.4. Model b: Parameter estimation, hypothesis testing and analysis (Steps II, III, IV)

477 Preliminary note: Since the RF considered in model b is log-normal, the data considered for the
478 SCAP-1D implementation are the logarithms of the debiased data to consider a normal joint PDF.
479
480 The estimators of $m_{LN}$, $\sigma_{LN}^2$ and $r_{95, LN}$ are then calculated from Equations (7) to (9) and from the
481 estimators $\hat{s}$, $\hat{t}$ and $\hat{a}$.

482 Table 3 in the Appendix shows the result of the SCAP-1D implementation considering model *b*. Only
483 the results of the $\Sigma_{\text{mod}}$-tests are considered due to the small number of points per trajectory. The
484 analysis shows that the estimates for trajectories 3, 4, 5, and 16 are rejected due to the rejection of the

<footnote n="5">⁵ some values are missing due to the singularity of the Fisher matrix</footnote>
stationarity hypothesis by the KPSS-test. A model of the mean and covariance function that satisfies
the hypothesis cannot be found, so no geostatistical parameter estimates are available for these
trajectories. 77% of the estimates are then valid on the Ext side (10/13) and 92% on the Int side
(11/12).

The estimates $\hat{m}_{LN}$ and $\hat{\sigma}_{LN}$ of the means and standard deviations are plotted with their confidence
intervals (CIs) for each trajectory in Figure 15. Note that the estimates are consistent with the
statistical mean and standard deviation profiles previously plotted in Figure 8. Meanwhile, the
estimates $\hat{r}_{95,LN}$ of the practical ranges are plotted with their CIs in Figure 16.

In the case of the trajectories associated with the Ext side, only 8 out of 10 estimates retained by
SCAP-1D (i.e., 47% of all estimates) are finally retained after filtering out the outliers. These
estimates all satisfy the interpretability criterion (see 4.1.5) with values between 100 and 300 cm for
a minimum measurement step of 30 cm. However, contrary to the assumptions made at the beginning
of the study, there is no trend in the evolution of $\hat{r}_{95,LN}$ as a function of the depth.

In the case of the trajectories associated with the Int side, all the estimates retained by SCAP-1D can
be considered after filtering out the outliers (i.e., 92% of all the estimates). In accordance with the
hypotheses formulated at the beginning of the study, we observe a **practical range of zero** near the
surface, then a quasi-constant value around 60 cm-100 cm from 20 mm depth, respecting the
interpretability criterion. This tendency reflects a transition from a purely random quantity of interest
(QI) at the surface to a spatially correlated QI at depth. We propose the following explanation:

* At the surface, many **spatiotemporally random material (microcracks, ...)**, **physicochemical** (local runoff, ...), and environmental phenomena (biofouling, ...) contribute to the supply and convection of chlorides, so that, in coherence with the central limit theorem, the **chloride concentration** at the surface can be considered **completely random**, at least at the scale of the 30 cm spatial discretization studied;

* at depth, the phenomenon of diffusion and fixation of C gradually takes precedence over their distribution within the coating, so that the fixation and diffusion properties of the concrete become
predominant with respect to the external and surface phenomena. Since the concrete has a certain
spatial homogeneity, it is then consistent that C is spatially correlated.

Furthermore, it is noted that the value of $\hat{r}_{95,LN}$ at the rebar depth is of the same order as the as the conventional spacing of the vibrations during concrete pouring, from 30 to 60 cm (Guiraud, 2018; SETRA, 2005, p. 199). This observation then reinforces the idea that the spatial correlation of C is strongly related to the homogeneity properties of the concrete.

## 6.5. Model b: construction of a unique autocorrelation model of chloride content

The results obtained from the study of the Int side trajectories are much more convincing than those obtained from the study of the Ext side trajectories. In order to propose a unique autocorrelation model of C for the diffusion region, we then perform an MLE on the set of reduced centered trajectories of the Int side (without the first 10 mm), considering a unique scaling parameter $\alpha$. Due to the non-stationarity of the mean and variance of C as a function of depth, we consider the trajectories independent of each other (we therefore neglect the SV along the z-axis). Still considering an exponential covariance model, the estimated practical range is $\hat{r}_{95} = 71$ cm, i.e., $\hat{\alpha} = 24$ cm.

We then state that the Gaussian model of identical practical range has the scale parameter $\hat{\alpha} = 41$ cm. However, the calculation of the ER from the AICC does not allow us to distinguish between the two models from the data, either by considering each trajectory or by estimating the MLE over all trajectories. In fact, their values are between 1 and 1.29. They are reported in Table 4 in the Appendix.

Considering only the estimation results from the Int side, which are considered more robust due to the larger amount of data per trajectory, we can partially answer the questions raised in the introduction of this study. The SV of the chloride content seems indeed to depend on the depth, with a large variability within the convection zone, and much less and more stable in the diffusion zone with a practical range of about 70 cm. However, this value is more than two to three times lower than the practical range values of the average diffusion parameters $C_{sa}$ and $D_a$ reported in the literature, between 176 and 264 cm (Engelund & Sørensen, 1998; Karimi, 2002; O'Connor &
Kenshel, 2013). However, **it should be noted** that the latter are obtained with less robust methods than SCAP-1D and with even smaller **data sets**.

However, the question of the influence of the exposure conditions on the SV of the chloride content remains open **due to** the inconsistency of the practical range estimates on the Ext side. However, given the interpretation of the results for the Int side, it can be expected **to have** an influence only near the surface, **where** the chloride contents are already spatially decorrelated. However, this **suggestion** needs to be confirmed by studying an even richer database in order to increase the robustness of the estimates.

## 7. Spatial variability estimation of chloride diffusion parameters of J-beam and physical analysis

### 7.1. Geostatistical pre-study and choice of procedure parameters (Step I):

The trajectories of $log(C_{sa})$ and $log(D_a)$ are shown in Figure 17 to Figure 20, as well as their means and the histograms and experimental semivariograms of centered trajectories. These curves are plotted for each trajectory in three cases: (i) stationary mean (no changepoint); (ii) piecewise stationary mean with one changepoint determined via the PELT algorithm (Killick et al., 2012); (iii) piecewise stationary average at two changepoints determined via the PELT algorithm.

The presence of sills on the experimental semi-variograms allows identifying the *a priori* stationary centered trajectories and thus the most appropriate non-stationarity model for each.

The shapes of the histograms and semi-variograms (nugget effect, tangent to the origin, ...) allows identifying models of joint density and a priori autocorrelation function that are adequate to the geostatistical description of the data.

Since the amount of data per trajectory is limited, especially for the Ext side, Step I could only be executed for the trajectories $log(C_{sa})$ 1 and $log(D_a)$ 1. The observations made as well as the models considered *a priori* are presented in Table 7.
### 7.1.1. Fields models

Although the histogram of the centered trajectory $log(C_{sa})$ 1 is asymmetric, this may be due to the spatial correlation of the data. The histogram of $log(D_a)$ 1 is centered. Moreover, we do not notice any nugget effect. This is due to the fact that the measurement noise of the chloride data is negligible (see 6.3). For simplicity and consistency with the results of the literature, a Gaussian Random Field model (GRF) is used to estimate spatial variability. This model is also retained by default for the $log(C_{sa})$ 2 and $log(D_a)$ 2 trajectories.

### 7.1.2. Averaging models: apparent non-stationarity

The trajectories are a priori globally stationary since the experimental semi-variograms without any changepoint in the mean present a sill. A constant mean model is therefore retained.

### 7.1.3. Covariance function models

Due to the large between measurement points, the spatial resolution is insufficient to visualize the presence of a tangent at the origin of the experimental semi-variograms. We therefore retain both the Gaussian and exponential covariance models without a priori preference.

This choice does not allow accounting for the correlation between $log(C_{sa})$ and $log(D_a)$, well identified by (Othmen et al., 2018) and which should be modeled via a cross-correlated Random Field (XRF) (Chilès & Delfiner, 2012, p. 332). It is however justified as XRF modelling requires additional parameters (cross-correlation coefficient and cross-scale parameter(s)) which leads to **overparameterization** and unsuccessful assessment attempts due to the limited amount of data<fnref n="6" />.

## 7.2. Parameter estimation, hypothesis testing and analysis (Steps II, III and IV)

SCAP-1D steps II and III are applied to the trajectories of $log(D_a)$ considering the models kept in 580 7.1. This allows the most appropriate covariance model to be determined from the calculation of

<footnote n="6">⁶ MLE on raw data and MLE on data enriched via Metropolis-Hastings sampling did not give satisfactory results</footnote>
Evidence Ratios (ER). These are computed from the AICC. Table 8 shows the results of SCAP-1D execution by trajectory. The models are ranked by increasing value of their ER.

First no estimate is rejected by the hypothesis tests. The estimates computed from the exponential and Gaussian models are then consistent for each of the trajectories except for the trajectory $log(C_{sa})$ 2 (relative errors of respectively 21%, 63%, 9% and 21% between the practical ranges of the Gaussian model and those of the exponential model). Furthermore, the exponential and Gaussian covariance models cannot be distinguished from the ERs (maximum difference observed for the $log(C_{sa})$ 1 trajectory with a ratio of 1 for the Gaussian model and a ratio of 2.13 for the exponential model). However, the Gaussian model leads to smaller coefficients of variation of the geostatistical parameters (up to a factor of 2.5 for the scale parameter of trajectory $log(C_{sa})$ 1).

Second, the majority of the practical range estimates do not meet the interpretability criterion (see 4.1.5). All of the estimates of $r_{95}$ made on the $log(D_a)$ trajectories as well as the estimate made on the $log(C_{sa})$ 2 trajectories are indeed of the order of 30 cm, which is the minimum spacing between measurements.

Therefore, only the estimates made on the $log(C_{sa})$ 1 trajectory are retained. The estimated practical range is indeed 50 to 63 cm depending on the chosen covariance model, exponential or Gaussian (i.e., a ratio $r_{95}/L$ of 0.05 to 0.07). The latter can then be preferred *a priori* because of its minimal ER even if exponential model is close.

Thus, due to the too large spacing between measurement points compared to the hidden value of their practical range, the question of whether the SV of the scattering parameters is a function of the exposure conditions cannot be answered here.

Furthermore, the interpretable estimates of practical ranges differ significantly from the 100 to 200 cm values estimated by (O'Connor & Kenshel, 2013) on a bridge exposed to chloride ion penetration.

Such a difference can be explained by the fact that these authors significantly enriched their initial kriging database (number of points multiplied by 7) before producing their estimates, which may have influenced them significantly.
## 8. Conclusion

In this paper, a new database is proposed for analysis of the spatial variability of chloride ingress into concrete. A total of 460 measurements was available, belonging to 2 exposures zones: 12 trajectories of 21 points for each profile on the Internal side (sheltered from wetting-drying cycles) and 13 trajectories of 16 points on the External side. This database was used to analyze the spatial variability of (i) chloride content, (ii) parameters of a model of chloride diffusion. The analysis is performed on the basis of a complete framework (SCAP-1D) which raises, rigorously for the first time, questions of stationarity, marginal distribution and auto-correlation.

The application to the total chloride content data showed that:

* the measurement error on total chloride content is negligible for the estimation of their spatial variability;

* near the surface, spatio-temporally random physico-chemical and environmental phenomena contribute to the diffusion and convection of chlorides so that, in coherence with the central limit theorem, total chloride content at the surface can be considered totally random;

* in depth (from about 20 mm), total chlorides diffusion and fixation gradually take precedence over their distribution within the coating, so that fixation and diffusion properties of concrete become predominant compared to the external and surface phenomena. Total chloride content is then spatially correlated with an estimated general practical range of 71 cm (scale parameter of 24 cm) which is of the same order as the conventional vibration spacing during concrete pouring (30-60 cm (Guiraud, 2018; SETRA, 2005, p. 199)). This supports the fact that total chloride content spatial correlation in depth is strongly related to the homogeneity properties of the concrete resulting from pouring.

The application to the identification of diffusion parameters showed that:

The application to the chloride diffusion parameters data was presented in section 7. It allowed estimation of geostatistical parameters only for the logarithm of the average surface chlorides concentration. This is due to a limited number of points per trajectory (from 16 to 21) and a too large
spacing between measurement points (30 cm). The estimated practical range value is nevertheless
consistent with that of the total chloride content in depth since it is between 50 and 63 cm.

Implementation of SCAP-1D on these two datasets allowed to note that when the number of points
per trajectory is limited but not small (≥ 20), it is more efficient to first consider the results of
statistical tests performed on the standardized trajectory from the empirical covariance matrix, and
then in a second step those performed on the standardized trajectory from the modeled covariance
matrix. This allows to first assess the validity of the mean model, then those of the marginal density
model and the estimates of the geostatistical parameters of the source Random Field. On the contrary,
when the number of points per trajectory is small (< 20), it is preferable to consider only the results
of the statistical tests performed on the standardized trajectory from the modeled covariance matrix,
in order to guarantee its positivity of the latter.

645
<table><caption>647 Table 1- Definition of trajectories and proportion of kriged data per trajectory – Ext side and Int side</caption><thead><tr><th colspan="3">kriged data - Ex side</th><th colspan="3">kriged data - Int side</th></tr><tr><th>trajectory</th><th>depth (mm)</th><th>number percentage</th><th>trajectory</th><th>depth (mm)</th><th>number percentage</th></tr></thead><tbody><tr><td>1</td><td>2.5</td><td>0</td><td>16</td><td>2.5</td><td>0</td></tr><tr><td>2</td><td>7.5</td><td>0</td><td>17</td><td>7.5</td><td>1</td></tr><tr><td>3</td><td>12.5</td><td>0</td><td>18</td><td>12.5</td><td>1</td></tr><tr><td>4</td><td>17.5</td><td>0</td><td>19</td><td>17.5</td><td>1</td></tr><tr><td>5</td><td>22.5</td><td>0</td><td>20</td><td>22.5</td><td>2</td></tr><tr><td>6</td><td>27.5</td><td>0</td><td>21</td><td>27.5</td><td>1</td></tr><tr><td>7</td><td>32.5</td><td>0</td><td>22</td><td>32.5</td><td>2</td></tr><tr><td>8</td><td>37.5</td><td>2</td><td>23</td><td>37.5</td><td>2</td></tr><tr><td>9</td><td>42.5</td><td>3</td><td>24</td><td>42.5</td><td>4</td></tr><tr><td>10</td><td>47.5</td><td>4</td><td>25</td><td>47.5</td><td>3</td></tr><tr><td>11</td><td>55</td><td>1</td><td>26</td><td>55</td><td>4</td></tr><tr><td>12</td><td>65</td><td>1</td><td>27</td><td>65</td><td>4</td></tr><tr><td>13</td><td>75</td><td>3</td><td>28</td><td>75</td><td>9</td></tr><tr><td>14</td><td>85</td><td>9</td><td>29</td><td>85</td><td>15</td></tr><tr><td>15</td><td>95</td><td>10</td><td>30</td><td>95</td><td>17</td></tr></tbody><tfoot><tr><td>Total</td><td></td><td>33</td><td>Total</td><td></td><td>66</td></tr><tr><td></td><td></td><td>14%</td><td></td><td></td><td>21%</td></tr></tfoot></table>

<table><caption>648 Table 2 - Estimation of geostatistical parameters of total chloride concentration trajectories by SCAP-1D - model a
649</caption><thead><tr><th colspan="2">trajectory</th><th colspan="2">model</th><th colspan="3">Σexp-tests</th><th colspan="3">Σmod-tests</th><th colspan="4">estimates (data multiplied by k=10e4)</th><th colspan="4">Coefficient of Variation (CoV)</th></tr><tr><th>#</th><th>Side z (mm)</th><th>Npts</th><th>model</th><th>cov</th><th>reg</th><th>chg</th><th>chi2</th><th>KS</th><th>KPSS</th><th>DF</th><th>chi2</th><th>KS</th><th>KPSS</th><th>DF</th><th>μ</th><th>s</th><th>t</th><th>a</th><th>r95</th><th>CoV μ</th><th>CoV s</th><th>CoV t</th><th>CoV a</th></tr></thead><tbody><tr><td>1</td><td>Ext</td><td>2,5</td><td>16</td><td>GW</td><td>expon</td><td>0</td><td>0</td><td>0</td><td>0</td><td>1</td><td>0</td><td>0</td><td>0</td><td>1</td><td>18,66</td><td>19,63</td><td>0,00</td><td>0,01</td><td>0,03</td><td>NaN</td><td>NaN</td><td>NaN</td><td>NaN</td></tr><tr><td>2</td><td>Ext</td><td>7,5</td><td>16</td><td>GW</td><td>expon</td><td>0</td><td>0</td><td>0</td><td>0</td><td>1</td><td>0</td><td>0</td><td>0</td><td>1</td><td>25,86</td><td>64,36</td><td>0,00</td><td>0,00</td><td>0,00</td><td>2,0E-01</td><td>1,2E+00</td><td>5,7E-01</td><td>1,7E+00</td></tr><tr><td>3</td><td>Ext</td><td>12,5</td><td>16</td><td>GW</td><td>expon</td><td>0</td><td>0</td><td>1</td><td>0</td><td>1</td><td>0</td><td>0</td><td>0</td><td>1</td><td>40,04</td><td>49,27</td><td>36,47</td><td>2,57</td><td>6,28</td><td>1,2E-01</td><td>8,9E-01</td><td>5,3E-01</td><td>1,3E+00</td></tr><tr><td>4</td><td>Ext</td><td>17,5</td><td>16</td><td>GW</td><td>expon</td><td>0</td><td>0</td><td>0</td><td>0</td><td>1</td><td>0</td><td>0</td><td>0</td><td>1</td><td>41,59</td><td>91,14</td><td>0,00</td><td>0,01</td><td>0,01</td><td>5,7E-02</td><td>7,6E+02</td><td>1,8E+02</td><td>3,8E+05</td></tr><tr><td>5</td><td>Ext</td><td>22,5</td><td>16</td><td>GW</td><td>expon</td><td>0</td><td>0</td><td>0</td><td>0</td><td>1</td><td>0</td><td>0</td><td>0</td><td>1</td><td>39,44</td><td>96,31</td><td>0,00</td><td>0,01</td><td>0,01</td><td>NaN</td><td>NaN</td><td>NaN</td><td>NaN</td></tr><tr><td>6</td><td>Ext</td><td>27,5</td><td>16</td><td>GW</td><td>expon</td><td>0</td><td>0</td><td>1</td><td>0</td><td>1</td><td>0</td><td>0</td><td>0</td><td>1</td><td>36,65</td><td>53,47</td><td>0,00</td><td>0,52</td><td>1,55</td><td>NaN</td><td>NaN</td><td>NaN</td><td>NaN</td></tr><tr><td>7</td><td>Ext</td><td>32,5</td><td>16</td><td>GW</td><td>expon</td><td>0</td><td>0</td><td>1</td><td>0</td><td>1</td><td>0</td><td>0</td><td>0</td><td>1</td><td>34,95</td><td>59,57</td><td>0,00</td><td>0,32</td><td>0,96</td><td>NaN</td><td>NaN</td><td>NaN</td><td>NaN</td></tr><tr><td>8</td><td>Ext</td><td>37,5</td><td>16</td><td>GW</td><td>expon</td><td>0</td><td>0</td><td>0</td><td>0</td><td>1</td><td>0</td><td>0</td><td>0</td><td>1</td><td>30,95</td><td>62,31</td><td>0,00</td><td>0,37</td><td>1,12</td><td>NaN</td><td>NaN</td><td>NaN</td><td>NaN</td></tr><tr><td>9</td><td>Ext</td><td>42,5</td><td>16</td><td>GW</td><td>expon</td><td>0</td><td>0</td><td>0</td><td>0</td><td>1</td><td>0</td><td>0</td><td>0</td><td>1</td><td>29,20</td><td>18,89</td><td>7,15</td><td>2,17</td><td>5,80</td><td>9,1E-02</td><td>7,5E-01</td><td>7,5E-01</td><td>1,1E+00</td></tr><tr><td>10</td><td>Ext</td><td>47,5</td><td>16</td><td>GW</td><td>expon</td><td>0</td><td>0</td><td>0</td><td>0</td><td>1</td><td>0</td><td>0</td><td>0</td><td>1</td><td>27,69</td><td>19,49</td><td>0,00</td><td>0,71</td><td>2,12</td><td>NaN</td><td>NaN</td><td>NaN</td><td>NaN</td></tr><tr><td>11</td><td>Ext</td><td>55</td><td>16</td><td>GW</td><td>expon</td><td>0</td><td>0</td><td>0</td><td>0</td><td>1</td><td>0</td><td>0</td><td>0</td><td>1</td><td>20,98</td><td>16,46</td><td>0,00</td><td>0,01</td><td>0,02</td><td>NaN</td><td>NaN</td><td>NaN</td><td>NaN</td></tr><tr><td>12</td><td>Ext</td><td>65</td><td>16</td><td>GW</td><td>expon</td><td>0</td><td>0</td><td>1</td><td>0</td><td>1</td><td>0</td><td>0</td><td>0</td><td>1</td><td>18,89</td><td>11,95</td><td>0,00</td><td>0,01</td><td>0,03</td><td>4,6E-02</td><td>1,5E+02</td><td>9,6E+09</td><td>5,6E+05</td></tr><tr><td>13</td><td>Ext</td><td>75</td><td>16</td><td>GW</td><td>expon</td><td>0</td><td>0</td><td>0</td><td>0</td><td>1</td><td>0</td><td>0</td><td>0</td><td>1</td><td>18,61</td><td>8,47</td><td>0,00</td><td>0,01</td><td>0,03</td><td>3,9E-02</td><td>1,9E+02</td><td>9,0E+09</td><td>1,5E+05</td></tr><tr><td>14</td><td>Int</td><td>2,5</td><td>21</td><td>GW</td><td>expon</td><td>0</td><td>0</td><td>NaN</td><td>NaN</td><td>NaN</td><td>NaN</td><td>NaN</td><td>NaN</td><td>NaN</td><td>NaN</td><td>NaN</td><td>NaN</td><td>NaN</td><td>NaN</td><td>NaN</td><td>NaN</td><td>NaN</td></tr><tr><td>15</td><td>Int</td><td>7,5</td><td>21</td><td>GW</td><td>expon</td><td>0</td><td>0</td><td>0</td><td>0</td><td>1</td><td>0</td><td>0</td><td>0</td><td>1</td><td>22,94</td><td>135,81</td><td>0,00</td><td>0,00</td><td>0,00</td><td>NaN</td><td>NaN</td><td>NaN</td><td>NaN</td></tr><tr><td>16</td><td>Int</td><td>12,5</td><td>21</td><td>GW</td><td>expon</td><td>0</td><td>0</td><td>0</td><td>0</td><td>1</td><td>0</td><td>0</td><td>0</td><td>1</td><td>40,27</td><td>38,35</td><td>111,41</td><td>3,19</td><td>5,21</td><td>1,2E-01</td><td>1,3E+00</td><td>3,6E-01</td><td>1,5E+00</td></tr><tr><td>17</td><td>Int</td><td>17,5</td><td>21</td><td>GW</td><td>expon</td><td>0</td><td>0</td><td>0</td><td>0</td><td>1</td><td>0</td><td>0</td><td>0</td><td>1</td><td>53,50</td><td>180,15</td><td>0,00</td><td>0,00</td><td>0,00</td><td>NaN</td><td>NaN</td><td>NaN</td><td>NaN</td></tr><tr><td>18</td><td>Int</td><td>22,5</td><td>21</td><td>GW</td><td>expon</td><td>0</td><td>0</td><td>0</td><td>0</td><td>1</td><td>0</td><td>0</td><td>0</td><td>1</td><td>55,01</td><td>150,55</td><td>0,00</td><td>0,34</td><td>1,01</td><td>6,5E-02</td><td>1,1E+00</td><td>1,8E+07</td><td>1,0E+00</td></tr><tr><td>19</td><td>Int</td><td>27,5</td><td>21</td><td>GW</td><td>expon</td><td>0</td><td>0</td><td>0</td><td>0</td><td>1</td><td>0</td><td>0</td><td>0</td><td>1</td><td>50,09</td><td>154,45</td><td>0,00</td><td>0,24</td><td>0,72</td><td>6,5E-02</td><td>1,8E+00</td><td>3,2E+05</td><td>1,1E+00</td></tr><tr><td>20</td><td>Int</td><td>32,5</td><td>21</td><td>GW</td><td>expon</td><td>0</td><td>0</td><td>0</td><td>0</td><td>1</td><td>0</td><td>0</td><td>0</td><td>1</td><td>47,18</td><td>168,62</td><td>0,00</td><td>0,31</td><td>0,93</td><td>7,8E-02</td><td>9,9E-01</td><td>8,2E+07</td><td>8,8E-01</td></tr><tr><td>21</td><td>Int</td><td>37,5</td><td>21</td><td>GW</td><td>expon</td><td>0</td><td>0</td><td>0</td><td>0</td><td>1</td><td>0</td><td>0</td><td>0</td><td>1</td><td>39,46</td><td>126,85</td><td>4,31</td><td>0,09</td><td>0,26</td><td>6,4E-02</td><td>2,6E+01</td><td>7,8E+02</td><td>7,8E+00</td></tr><tr><td>22</td><td>Int</td><td>42,5</td><td>21</td><td>GW</td><td>expon</td><td>0</td><td>0</td><td>0</td><td>0</td><td>1</td><td>0</td><td>0</td><td>0</td><td>1</td><td>39,57</td><td>103,23</td><td>2,06</td><td>0,29</td><td>0,86</td><td>7,2E-02</td><td>1,2E+00</td><td>5,9E+01</td><td>1,1E+00</td></tr><tr><td>23</td><td>Int</td><td>47,5</td><td>21</td><td>GW</td><td>expon</td><td>0</td><td>0</td><td>1</td><td>0</td><td>1</td><td>0</td><td>0</td><td>0</td><td>1</td><td>34,59</td><td>72,53</td><td>0,00</td><td>0,35</td><td>1,06</td><td>7,3E-02</td><td>9,4E-01</td><td>2,7E+07</td><td>8,5E-01</td></tr><tr><td>24</td><td>Int</td><td>55</td><td>21</td><td>GW</td><td>expon</td><td>0</td><td>0</td><td>0</td><td>0</td><td>1</td><td>0</td><td>0</td><td>0</td><td>1</td><td>26,46</td><td>28,31</td><td>0,00</td><td>0,00</td><td>0,00</td><td>NaN</td><td>NaN</td><td>NaN</td><td>NaN</td></tr><tr><td>25</td><td>Int</td><td>65</td><td>21</td><td>GW</td><td>expon</td><td>0</td><td>0</td><td>0</td><td>0</td><td>1</td><td>0</td><td>0</td><td>0</td><td>1</td><td>24,73</td><td>25,35</td><td>0,00</td><td>0,25</td><td>0,74</td><td>5,4E-02</td><td>1,9E+00</td><td>6,9E+05</td><td>1,2E+00</td></tr></tbody></table>

GW : Gaussian + additive white noise; reg : regression degree of mean model; chg : maximum number of changepoints of mean model
658 **11. Data Availability Statement**

659 Some or all data, models, or code generated or used during the study are available from the

660 corresponding author by request:

661 • chloride content dataset;

662 • whole MATLAB code generated during the study.

663 **12. Acknowledgment**

664 The authors would like to thank all the partners of iMAREO2 project: Keops Automation (D. Follut,

665 D. Olivier), Université de Nantes (M. Roche), Nantes – Saint-Nazaire Port (P. Lijour, M. Labegorre).

666 The authors would like to thank the Pays de la Loire region for its financial support.

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792 spatial variability assessment of stationary random fields. *Engineering Structures*, **152**, 35–
793 44. https://doi.org/10.1016/j.engstruct.2017.08.070

794 Schoefs, F., Oumouni, M., Clerc, R., Othmen, I., & Bonnet, S. (2017). Statistical analysis and
795 probabilistic modeling of chloride ingress spatial variability in concrete coastal
796 infrastructures. *Edition 4, Split, Croatia*, 229–234. https://doi.org/10.5150/cmcm.2017.042

797 Schoefs, F., Oumouni, M., Follut, D., Lecieux, Y., Gaillard, V., Lupi, C., & Leduc, D. (2022).
798 Added value of monitoring for the maintenance of a reinforced concrete wharf with spatial
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801 SETRA. (2005). *Mémoar—Mémento pour la mise en oeuvre sur ouvrages d'art*. Cerema (ex-Setra).
802 https://www.cerema.fr/fr/centre-ressources/boutique/memoar-memento-mise-oeuvre-
803 ouvrages-art

804 Stewart, M. G. (2004). Spatial variability of pitting corrosion and its influence on structural fragility
805 and reliability of RC beams in flexure. *Structural Safety*, **26**(4), 453–470.
806 https://doi.org/10.1016/j.strusafe.2004.03.002

807 Stewart, M. G., & Mullard, J. A. (2007). Spatial time-dependent reliability analysis of corrosion
808 damage and the timing of first repair for RC structures. *Engineering Structures*, **29**(7),
809 1457–1464. https://doi.org/10.1016/j.engstruct.2006.09.004

810 Stewart, M. G., & Val, D. V. (2003). Multiple Limit States and Expected Failure Costs for
811 Deteriorating Reinforced Concrete Bridges. *Journal of Bridge Engineering*, **8**(6), 405–415.
812 https://doi.org/10.1061/(ASCE)1084-0702(2003)8:6(405)

813 Torres-Luque, M., Bastidas-Arteaga, E., Schoefs, F., Sánchez-Silva, M., & Osma, J. F. (2014).
814 Non-destructive methods for measuring chloride ingress into concrete: State-of-the-art and
815 future challenges. *Construction and Building Materials*, **68**, 68–81.

816 Torres-Luque, M., Osma, J. F., Sánchez-Silva, M., Bastidas-Arteaga, E., & Schoefs, F. (2017).
817 Chlordetect: Commercial Calcium Aluminate Based Conductimetric Sensor for Chloride
818 Presence Detection. *Sensors*, **17**(9), 2099. https://doi.org/10.3390/s17092099

819 Vennesland, P. Ø., Climent, M. A., & Andrade, C. (2013). Recommendations of RILEM TC 178-
820 TMC: Testing and modelling chloride penetration in concrete. *Materials and Structures*, **8**.
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823 21736-9

824 Watanabe, A., Tokuda, S., Mizuta, Y., Miyamoto, S., Nakanishi, T., Furukawa, H., & Minagawa, H. (2021). Toward automated non-destructive diagnosis of chloride attack on concrete structures by near infrared spectroscopy. *Construction and Building Materials*, 305, 124796.

827 https://doi.org/10.1016/j.conbuildmat.2021.124796

828 Wegian, F. M. (2010). Effect of seawater for mixing and curing on structural concrete. *The IES Journal Part A: Civil & Structural Engineering*, 3(4), 235-243.

830 https://doi.org/10.1080/19373260.2010.521048

831 Zaki, A., Chai, H. K., Aggelis, D. G., & Alver, N. (2015). Non-Destructive Evaluation for Corrosion Monitoring in Concrete: A Review and Capability of Acoustic Emission

833 Technique. *Sensors*, 15(8), Article 8. https://doi.org/10.3390/s150819069

834

835
836 **14. Embedded Tables**

<table><caption>Table 5 - Exponential and Gaussian autocorrelation functions, and relations between their scale parameter, fluctuation scale and practical range</caption><thead><tr><th>Name</th><th>Model</th><th>θ</th><th>ρ(h = θ)</th><th>r<sub>95</sub></th><th>ρ(h = r<sub>95</sub>)</th></tr></thead><tbody><tr><td>exponential</td><td>ρ(h) = exp (−||h||/a)</td><td>2a</td><td>0.135</td><td>3a</td><td>0.05</td></tr><tr><td>Gaussian</td><td>ρ(h) = exp (−||h||<sup>2</sup>/a)</td><td>√πa</td><td>0.043</td><td>1.73a</td><td>0.05</td></tr></tbody></table>

a: scale parameter; θ: fluctuation parameter; r<sub>95</sub>: practical range; ρ(·): autocorrelation function

839

840

<table><caption>Table 6- Numbering of the trajectories of average scattering parameters (J-beam)</caption><thead><tr><th>trajectory</th><th>Side</th><th>number of measurement points</th></tr></thead><tbody><tr><td>log (C<sub>sa</sub>) 1</td><td>Int</td><td>21</td></tr><tr><td>log (D<sub>a</sub>) 1</td><td>Int</td><td>21</td></tr><tr><td>log (C<sub>sa</sub>) 1</td><td>Ext</td><td>16</td></tr><tr><td>log (D<sub>a</sub>) 2</td><td>Ext</td><td>16</td></tr></tbody></table>

841

<table><caption>Table 7- Geostatistical pre-study of the trajectories of the logarithms of the average diffusion parameters – J-beam</caption><thead><tr><th rowspan="2">trajectory</th><th colspan="5">analysis of experimental semi-variograms and histograms</th><th colspan="5">models considered</th></tr><tr><th>chgpt(*)</th><th>nugget</th><th>oscillations</th><th>original tangent</th><th>histogram shape</th><th>average</th><th>field</th><th>covariance</th><th>r<sub>95</sub> (cm)</th><th>r<sub>95</sub>/L</th></tr></thead><tbody><tr><td>log (C<sub>sa</sub>) 1</td><td>0</td><td>no</td><td>no</td><td>insufficient resolution</td><td>asymmetrical</td><td>cste</td><td>GRF</td><td>Gauss or expon</td><td>50</td><td>0,05</td></tr><tr><td>log (D<sub>a</sub>) 1</td><td>0</td><td>no</td><td>no</td><td>insufficient resolution</td><td>symmetrical</td><td>cste</td><td>GRF</td><td>Gauss or expon</td><td>63</td><td>0,07</td></tr><tr><td>log (C<sub>sa</sub>) 2</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>cste</td><td>GRF</td><td>Gauss or expon</td><td>57</td><td>0,06</td></tr><tr><td>log (D<sub>a</sub>) 2</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>cste</td><td>GRF</td><td>Gauss or expon</td><td>35</td><td>0,04</td></tr></tbody><tfoot><tr><td colspan="11">(*) : changepoints necessary for the experimental semi-variogram to have a sill</td></tr></tfoot></table>

844

<table><caption>Table 8- Geostatistical parameter estimates of the trajectories of the logarithms of the mean scattering parameters by SCAP-1D</caption><thead><tr><th rowspan="2">traj</th><th rowspan="2">#</th><th colspan="4">model</th><th colspan="4">Σ<sub>exp</sub>-tests</th><th colspan="4">Σ<sub>mod</sub>-tests</th><th colspan="3">estimates</th><th colspan="2">CoV</th><th colspan="2">model selection</th><th colspan="2">range</th></tr><tr><th>model</th><th>cov</th><th>reg</th><th>χ<sup>2</sup></th><th>KS</th><th>KPSS</th><th>DF</th><th>χ<sup>2</sup></th><th>KS</th><th>KPSS</th><th>DF</th><th>α1</th><th>s</th><th>a1 (cm)</th><th>α1</th><th>s</th><th>a1</th><th>AICc</th><th>ER</th><th>r<sub>95</sub> (cm)</th><th>r<sub>95</sub>/L</th></tr></thead><tbody><tr><td rowspan="2">log(C<sub>sa</sub>) 1</td><td>1</td><td>GRF</td><td>expon</td><td>0</td><td>0</td><td>0</td><td>0</td><td>1</td><td>0</td><td>0</td><td>0</td><td>1</td><td>-4,82</td><td>0,13</td><td>17</td><td>0,02</td><td>0,31</td><td>0,81</td><td>-14,08</td><td>2,13</td><td>50</td><td>0,05</td></tr><tr><td>2</td><td>GRF</td><td>gauss</td><td>0</td><td>0</td><td>0</td><td>0</td><td>1</td><td>0</td><td>0</td><td>0</td><td>1</td><td>-4,81</td><td>0,15</td><td>37</td><td>0,02</td><td>0,39</td><td>0,33</td><td>-15,59</td><td>1,00</td><td>63</td><td>0,07</td></tr><tr><td rowspan="2">log(C<sub>sa</sub>) 2</td><td>1</td><td>GRF</td><td>expon</td><td>0</td><td>0</td><td>0</td><td>0</td><td>1</td><td>0</td><td>0</td><td>0</td><td>1</td><td>-5,20</td><td>0,07</td><td>19</td><td>0,01</td><td>0,36</td><td>1,20</td><td>-19,22</td><td>1,00</td><td>57</td><td>0,06</td></tr><tr><td>2</td><td>GRF</td><td>gauss</td><td>0</td><td>0</td><td>0</td><td>0</td><td>1</td><td>0</td><td>0</td><td>0</td><td>1</td><td>-5,20</td><td>0,07</td><td>20</td><td>0,01</td><td>0,36</td><td>0,80</td><td>-19,11</td><td>1,06</td><td>35</td><td>0,04</td></tr><tr><td rowspan="2">log(D<sub>a</sub>) 1</td><td>1</td><td>GRF</td><td>expon</td><td>0</td><td>0</td><td>0</td><td>0</td><td>1</td><td>0</td><td>0</td><td>0</td><td>1</td><td>-27,48</td><td>0,31</td><td>10</td><td>0,00</td><td>0,31</td><td>1,65</td><td>3,78</td><td>1,01</td><td>29</td><td>0,03</td></tr><tr><td>2</td><td>GRF</td><td>gauss</td><td>0</td><td>0</td><td>0</td><td>0</td><td>1</td><td>0</td><td>0</td><td>0</td><td>1</td><td>-27,48</td><td>0,31</td><td>18</td><td>0,00</td><td>0,31</td><td>0,77</td><td>3,76</td><td>1,00</td><td>32</td><td>0,03</td></tr><tr><td rowspan="2">log(D<sub>a</sub>) 2</td><td>1</td><td>GRF</td><td>expon</td><td>0</td><td>0</td><td>0</td><td>0</td><td>1</td><td>0</td><td>0</td><td>0</td><td>1</td><td>-27,56</td><td>0,29</td><td>13</td><td>0,01</td><td>0,35</td><td>1,40</td><td>4,19</td><td>1,00</td><td>40</td><td>0,04</td></tr><tr><td>2</td><td>GRF</td><td>gauss</td><td>0</td><td>0</td><td>0</td><td>0</td><td>1</td><td>0</td><td>0</td><td>0</td><td>1</td><td>-27,56</td><td>0,29</td><td>19</td><td>0,01</td><td>0,35</td><td>0,72</td><td>4,21</td><td>1,01</td><td>33</td><td>0,04</td></tr></tbody></table>

847

848
849 **15. List of Figure Captions**

850 Figure 1 - Illustration of the geostatistical formalism: GRF Y

851 Figure 2 - Illustration of the geostatistical formalism: noisy GRF Z

852 Figure 3 - General flowchart of SCAP-1D

853 Figure 4 - Interpretability of practical range (Gaussian autocorrelograms $\rho(h)$ with three practical range values, measurement step $\Delta x = 5$)

855 Figure 5 - Size and position of rebars of the “J-beam” (distance unit: cm; diameter unit: mm)

856 Figure 6 - Different views: a) Aerial view, b) Sketch of bridge 3, 4 and 5, c) Beam J, d) Exposed (Ext) and sheltered (Int) side of beam J.

858 Figure 7 - Samples taken for chloride titration (J-beam): a) Locations, b) Cross-section, c) Slices

859 Figure 8 - Profiles of the evolution of the statistical mean and standard deviation of total chloride concentrations according to the depth of embedding

861 Figure 9 - Total chloride concentration database and illustration of the principle of implementing SCAP-1D on total chloride concentration data

863 Figure 10 - Trajectories and statistical averages of the logarithms of the mean diffusion parameters at 28 years (J-beam)

865 Figure 11 - Raw trajectories and associated statistical means - trajectory number / depth correspondences presented in Table 5

867 Figure 12 - Selected trajectories and associated statistical means

868 Figure 13 - Profiles of the evolution of the mean and standard deviation of total chloride content according to the depth - model a

870 Figure 14 - Study of the Signal to Noise Ratio - model a

871 Figure 15 - Profiles of the evolution of the mean and standard deviation of total chloride concentrations according to the depth of embedding - model b

873 Figure 16 - Practical range profiles of total chloride concentration trajectories as a function of embedment depth - model b
875 Figure 17- Data and averages of trajectory 1 of log mean surface concentration; histograms of trajectories so centered; experimental semivariograms of trajectories so centered

877 Figure 18- Data and averages of trajectory 1 of logarithm of mean diffusivity; histograms of trajectories so centered; experimental semivariograms of trajectories so centered

879 Figure 19- Data and averages of logarithm of mean surface concentration trajectory 2; histograms of trajectories so centered; experimental semivariograms of trajectories so centered

881 Figure 20- Data and averages of trajectory 2 of logarithm of mean diffusivity; histograms of trajectories so centered; experimental semivariograms of trajectories so centered

883

884
<figure><img src="image_2.png" /><figcaption>Figure 1 - Illustration of the geostatistical formalism: GRF Y</figcaption></figure>

885

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887
<figure><img src="image_3.png" /><figcaption>Figure 2 - Illustration of the geostatistical formalism: noisy GRF Z</figcaption></figure>

888

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890
<figure><img src="image_4.png" /><figcaption>Figure 3 - General flowchart of SCAP-1D</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>Figure 4 - Interpretability of practical range (Gaussian autocorrelograms $\rho(h)$ with three practical range values, measurement step $\Delta x = 5$)</figcaption></figure>

891

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893

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895
<figure><img src="image_6.png" /><figcaption>Figure 5 - Size and position of rebars of the “J-beam” (distance unit: cm; diameter unit: mm)</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Figure 6 - Different views: a) Aerial view, b) Sketch of bridge 3, 4 and 5, c) Beam J, d) Exposed (Ext) and sheltered (Int) side of beam J.</figcaption></figure>
<figure><img src="image_8.png" /></figure>

<figure><img src="image_9.png" /><figcaption>Figure 7 - Samples taken for chloride titration (J-beam): a) Locations, b) Cross-section, c) Slices</figcaption></figure>

<figure><img src="image_10.png" /></figure>

<figure><img src="image_11.png" /><figcaption>Figure 8 - Profiles of the evolution of the statistical mean and standard deviation of total chloride concentrations according to the depth of embedding</figcaption></figure>
<figure><img src="image_12.png" /><figcaption>Figure 9 - Total chloride concentration database and illustration of the principle of implementing SCAP-1D on total chloride concentration data</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>(a) Trajectories and statistical means of logarithms of the 28-year average surface concentrations of total chlorides - beam J</figcaption></figure>

<caption>(b) trajectories and statistical means of the logarithms of the mean diffusivities of total chlorides at 28 years - beam J</caption>

<caption>Figure 10 - Trajectories and statistical averages of the logarithms of the mean diffusion parameters at 28 years (J-beam)</caption>

909

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911

912

913

914
<figure><img src="image_14.png" /><figcaption>Figure 11 - Raw trajectories and associated statistical means - trajectory number / depth correspondences presented in Table 1</figcaption></figure>

<figure><img src="image_15.png" /><figcaption>Figure 12 - Selected trajectories and associated statistical means</figcaption></figure>
<figure><img src="image_16.png" /><figcaption>Figure 13 - Profiles of the evolution of the mean and standard deviation of total chloride content according to the depth - model a</figcaption></figure>

923

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925

<figure><img src="image_17.png" /><figcaption>Figure 14 - Study of the Signal to Noise Ratio - model a</figcaption></figure>

926

927

928
<figure><img src="image_18.png" /><figcaption>Figure 15 - Profiles of the evolution of the mean and standard deviation of total chloride concentrations according to the depth of embedding - model b</figcaption></figure>

<figure><img src="image_19.png" /><figcaption>Figure 16 - Practical range profiles of total chloride concentration trajectories as a function of embedment depth - model b</figcaption></figure>
<figure><img src="image_20.png" /><figcaption>Figure 17- Data and averages of trajectory 1 of log mean surface concentration; histograms of trajectories so centered; experimental semivariograms of trajectories so centered</figcaption></figure>

<figure><img src="image_21.png" /><figcaption>Figure 18- Data and averages of trajectory 1 of logarithm of mean diffusivity; histograms of trajectories so centered; experimental semivariograms of trajectories so centered</figcaption></figure>
<figure><img src="image_22.png" /></figure>

Figure 19- Data and averages of logarithm of mean surface concentration trajectory 2; histograms of trajectories so centered; experimental semivariograms of trajectories so centered

<figure><img src="image_23.png" /></figure>

Figure 20- Data and averages of trajectory 2 of logarithm of mean diffusivity; histograms of trajectories so centered; experimental semivariograms of trajectories so centered

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945

946

947

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949