Comparative Study of Electrochemical Noise-Analysis Methods for Corrosion Assessment in Reinforced Concrete

Abstract

In this work, an experimental evaluation was performed using four analytical methods applied to electrochemical noise (EN) signals to estimate the corrosion rate ($C_r$) of reinforced concrete structures. A dataset comprising 10,166 synchronized EN files acquired over approximately 220 days was analyzed. The EN signals were obtained from various natural aqueous media, including seawater and river water, as well as from two laboratory reference media (3.5% NaCl solution and reverse-osmosis water). The Statistical Method (SM), the Fast Fourier Transform (FFT), the Maximum Entropy Method (MEM), and the Stockwell Transform (ST) were used to calculate $C_r$. The resulting corrosion rates were evaluated using a two-way analysis of variance (ANOVA) with full interaction, followed by Tukey HSD post hoc comparisons. Significant effects were found for both the analytical methods and the exposure media ($p<0.001$). Among the methods evaluated, MEM showed the greatest statistical stability and robustness, while ST showed the greatest tolerance to noise and the non-stationary characteristics of the EN signals. Estimated corrosion rates ranged from $0.0366 \mathrm{mm}/\mathrm{year}$ in reverse-osmosis water (MEM) to $0.2022 \mathrm{mm}/\mathrm{year}$ in 3.5% NaCl (MEM). For ST, the corresponding values ranged from $0.0652 \mathrm{mm}/\mathrm{year}$ to $0.3504 \mathrm{mm}/\mathrm{year}$ in the same media. These results demonstrate that both the analytical method and the corrosive medium significantly influence EN-based corrosion rate estimates and highlight the potential of MEM and ST for long-term corrosion monitoring of reinforced concrete.

1. Introduction

Corrosion of reinforcing steel in reinforced concrete (RC) is a leading cause of infrastructure deterioration globally, with significant economic and safety implications. While concrete provides an initial alkaline environment that stabilizes an oxide film (and which passivates the metal surface), chloride penetration, carbonation, and other aggressive agents progressively degrade this protection and trigger active corrosion [1]. The resulting cracking, delamination, and spalling of the concrete cover significantly reduce the structural capacity and service life of RC elements, especially in marine and saline environments. Recent work has applied similar reliability-based approaches to large marine infrastructures, such as the Qingdao Bay Bridge, where a time-varying probabilistic model was used to determine service life under the effects of chloride penetration and corrosion [2].

Common electrochemical methods, such as half-cell potential, linear polarization resistance (LPR), and electrochemical impedance spectroscopy (EIS), are commonly used to assess corrosion rates in RC systems. However, these techniques are often invasive, generating external disturbances to the system and are sensitive to environmental and operational variability. As an alternative, electrochemical noise (EN) analysis has emerged as a passive or non-invasive technique based on spontaneous fluctuations in electrochemical potential noise (EPN) and electrochemical current noise (ECN). Since the first corrosion studies using EN, it has demonstrated good capabilities in terms of determining different types of corrosion, such as pitting, passivation, and passive-layer-breakdown events [3,4,5]. Furthermore, the work of [6,7,8] demonstrated that this analytical method allows the detection of corrosion in reinforced-concrete structures in field work.

Among the advantages that technological development offers today, the fact that EN signals are stochastic and non-stationary presents a significant challenge in determining reliable $C_r$ [9]. For example, the noise resistance ($R_n$) in the time domain, which has a very low computational cost, is sensitive to aspects such as drift in the recordings and instrument noise [10,11]. These limitations are overcome when frequency or time–frequency methods are employed. Studies using Power Spectral Density (PSD) via FFT have allowed the determination of localized corrosion events related to frequency components [12]. Another method for determining PDS, such as MEM, offers better resolution for short or noisy recordings, while ST allows for preserving time–frequency analysis for the detection of transient features, as shown in several studies [13,14,15].

More recent work demonstrates the potential of studying EN signals with more robust analytical methods. Timescale analyses, such as wavelet transforms [16], Hilbert–Huang transforms, and recurrence diagrams, can provide better characterization of chloride damage in steels [15]. Furthermore, studies employing neural networks and unsupervised clustering techniques have classified uniform and localized corrosion [17,18]. These and other analyses demonstrate the potential of EN-based corrosion studies.

In RC systems, EN has been successfully applied under different configurations. Mills et al. [6] correlated noise resistance with actual mass loss of reinforcement in chloride-contaminated mortar, while Nicolás et al. [19] showed that synthetic fibers altered EPN/ECN transients, inducing localized corrosion. García-Contreras et al. [7] reported detection of transients in post-tensioned elements with supplementary cementitious materials, even under partial cathodic protection. Nonetheless, the diversity of methodologies—ranging from time-domain indices to spectral and advanced time–frequency analyses—hinders standardization and limits widespread adoption in structural-health monitoring.

The novelty of this study lies in performing a systematic comparative evaluation of four representative approaches—statistical method (SM), FFT, MEM, and ST—applied to 10,166 synchronized EN files acquired over 220 days in six aqueous environments, including natural river- and seawater as well as two comparative solutions: a 3.5% NaCl solution, used as a benchmark aggressive medium, and reverse-osmosis water, used as a low-aggressiveness control. A two-way analysis of variance (ANOVA) with Tukey post hoc tests was implemented to quantify the influence of both the analysis method and corrosive medium. By integrating statistical, spectral, and time–frequency domains, this work establishes objective criteria for selecting EN methodologies and reinforces the potential of EN as a continuous, sensitive, and non-invasive monitoring tool for reinforced-concrete structures.

2. Methodology

This section covers the mathematical basis of the four analytical methods used to determine the $C_r$ value of the EN recordings obtained from reinforced-concrete samples immersed in aqueous media. These methods include the time-domain SM [9], the frequency-domain FFT and MEM to obtain the PSD [20,21], and the time–frequency ST [13,22].

For accurate reading and recording of EN signals, the analysis was performed according to ASTM G199-09(2020)e1 [23], and to correctly determine $C_r$, ASTM G102-89(2015)e1 was followed [24]. In each of the following subsections, the necessary theoretical foundations are presented, as well as the main equations used in this study. Full mathematical developments can be found in the cited references [9,22,25,26].

2.1. Statistical Method

The statistical method (SM) is the simplest and most widely used approach for EN interpretation due to its low computational demand and applicability to real-time monitoring [9]. It evaluates corrosion activity by analyzing the spontaneous fluctuations of EPN and ECN.

The central parameter is the noise resistance ($R_n$), defined as the ratio of the standard deviation of the potential signal (${\sigma }_v$) to that of the current signal (${\sigma }_i$):

$$R_n={\displaystyle \frac{{\sigma }_v}{{\sigma }_i}},$$

(1)

where ${\sigma }_v$ and ${\sigma }_i$ correspond to the standard deviations of EPN and ECN, respectively.

By analogy with the polarization resistance ($R_p$), ASTM G102 [24] allows us to estimate the corrosion current density ($i_{\mathrm{corr}}$) as

$$i_{\mathrm{corr}}={\displaystyle \frac{B}{R_n}},$$

(2)

where B is the Stern–Geary constant. In this work, $R_p$ was not directly measured using LPR or EIS; instead, $R_n$ was treated as an analog of $R_p$ following the framework of ASTM G199 and G102. The detailed calculation of the corrosion rate ($C_r$) from $i_{\mathrm{corr}}$ is provided in Section 2.4.

2.2. Power Spectral Density (PSD)

2.2.1. Fast Fourier Transform (FFT)

The FFT is a classical spectral technique that transforms EN signals from the time domain to the frequency domain, allowing evaluation of the energy distribution of EPN and ECN [20,21]. In accordance with ASTM G199, the PSD of a discrete time series $x_n$ of length N, sampled at intervals $\Delta t$, is expressed for component k as follows:

$${\mathrm{\Psi }}_k={\displaystyle \frac{\gamma ·t_m}{N}}{\left|{\displaystyle \sum _{n=1}^N}(x_n-\overline{x})e^{-2\pi ikn/N}\right|}^2,$$

(3)

where $\gamma =2$ for $k=1,\dots ,N/2-1$ and $\gamma =1$ for $k=N/2$. The frequency of each component is given by $f_k=k/(N·\Delta t)$, with limits $f_{min}=1/(N·\Delta t)$ and $f_{\max }=1/(2·\Delta t)$.

From the PSDs of potential (${\mathrm{\Psi }}_v\left(f\right)$) and current (${\mathrm{\Psi }}_i\left(f\right)$), the frequency-dependent noise impedance is calculated as follows:

$$Z_n\left(f\right)=\sqrt{{\displaystyle \frac{{\mathrm{\Psi }}_v\left(f\right)}{{\mathrm{\Psi }}_i\left(f\right)}}}.$$

(4)

The impedance values were averaged within the frequency interval from 3.9 to 9.8 mHz. This range was selected because corrosion-related electrochemical processes are predominantly represented in the lowest-frequency components of electrochemical noise signals, where charge-transfer phenomena contribute most significantly to the noise-impedance response. Frequencies below this interval were not considered because the finite record duration reduces the number of available cycles and increases spectral uncertainty, whereas higher frequencies are progressively influenced by non-corrosion-related fluctuations and instrumental noise. Therefore, the selected interval represents a practical compromise between electrochemical relevance and spectral stability, consistent with previous electrochemical noise studies [5,9,16,17].

The average impedance over the frequency range 3.9–9.8 mHz was then obtained as follows:

$${\overline{Z}}_n={\displaystyle \frac{1}{f_2-f_1}}{\displaystyle \sum _{f=f_1}^{f_2}}{Z}_n\left(f\right)·\Delta f.$$

(5)

The resulting ${\overline{Z}}_n$ was subsequently used to estimate the corrosion current density ($i_{\mathrm{corr}}$), with the corrosion-rate calculation described in Section 2.4.

2.2.2. Maximum Entropy Method (MEM)

The MEM is a high-resolution spectral technique particularly suited for EN signals with short record lengths or high noise levels [9]. Unlike FFT, MEM does not require windowing or specific signal lengths, resulting in smoother and more stable PSD estimates.

In this method, the signal $x\left(n\right)$ is modeled as an autoregressive (AR) process of order M, fitted using the autocorrelation function $R_{xx}\left(\tau \right)$. The PSD is given by:

$$\mathrm{\Psi }\left(f\right)={\displaystyle \frac{2·\Delta t·P_0}{{\left|1+{\sum }_{k=1}^M{P}_k{e}^{-j2\pi fk\Delta t}\right|}^2}},$$

(6)

where $P_0$ is the variance of the AR model, $P_k$ are the autoregressive coefficients, and M is the model order. In this work, $M=200$ was selected as a trade-off between spectral resolution and noise suppression, which is consistent with previous studies [25,26,27].

Applied to EPN and ECN signals, MEM produces the PSDs ${\mathrm{\Psi }}_v\left(f\right)$ and ${\mathrm{\Psi }}_i\left(f\right)$. The corresponding noise impedance is then computed using Equation (4), and its mean value over the frequency range 3.9–9.8 mHz is obtained with Equation (5). As with FFT, the resulting ${\overline{Z}}_n$ was used for the estimation of $i_{\mathrm{corr}}$ and, subsequently, $C_r$.

The Maximum Entropy Method (MEM) has been widely employed for electrochemical noise analysis due to its superior spectral resolution at low frequencies when compared with conventional FFT-based approaches, particularly for short data records and non-periodic signals. However, the MEM power spectral density (PSD) estimation is not unique for a given electrochemical noise record since the resulting spectrum strongly depends on the selected autoregressive-model order (M). Inappropriate selection of (M) may introduce spectral distortion, artificial peak generation, or excessive smoothing, directly affecting the estimation of electrochemical parameters derived from low-frequency impedance behavior. Consequently, the selection of the model order represents a critical aspect in MEM-based corrosion analysis and has been identified as one of the principal limitations of the method in electrochemical noise applications [5,26,28].

2.3. Stockwell Transform (ST)

The ST is a time–frequency analysis tool that provides adaptive resolution for representing non-stationary signals, such as those obtained from EN [13,22]. Unlike the conventional spectral techniques presented earlier, the ST preserves temporal and frequency localization, making it particularly effective for detecting transient and localized corrosion events. While not explicitly included in ASTM G199, the method was implemented following its general recommendations for EN data processing.

Mathematically, the discrete ST of a signal $h\left[kT\right]$ (discrete time series) applies a Gaussian window $G(m,n)=exp(-2{\pi }^2{m}^2/n^2)$ to the Fourier spectrum $H{\displaystyle \frac{n}{NT}}$, yielding

$$S\left[jT,{\displaystyle \frac{n}{NT}}\right]={\displaystyle \sum _{m=0}^{N-1}}H\left[{\displaystyle \frac{m+n}{NT}}\right]e^{-{\displaystyle \frac{2{\pi }^2{m}^2}{n^2}}}{e}^{{\displaystyle \frac{i2\pi mj}{N}}}n\ne 0$$

(7)

$$S\left[jT,0\right]={\displaystyle \frac{1}{N}}{\displaystyle \sum _{m=0}^{N-1}}h\left({\displaystyle \frac{m}{NT}}\right)n=0$$

(8)

were $j,m,\mathrm{and}n=0,1,\dots ,N-1$, and T is the sampling interval.

When applied to EPN and ECN signals, ST generates time–frequency spectra ${\zeta }_v(t,f)$ and ${\zeta }_i(t,f)$. The instantaneous noise impedance is then obtained as follows:

$${\psi }_n(t,f)={\displaystyle \frac{{\zeta }_v(t,f)}{{\zeta }_i(t,f)}}.$$

(9)

The mean impedance over the frequency range $3.9$ mHz–$9.8$ mHz was calculated as follows:

$$\overline{{\psi }_n}={\displaystyle \frac{1}{f_2-f_1}}{\displaystyle \sum _{f=f_1}^{f_2}}\left({\displaystyle \frac{1}{N}}{\displaystyle \sum _{t=t_1}^{t_N}}{\psi }_n(t,f)\right)\Delta f.$$

(10)

The resulting $\overline{{\psi }_n}$ was used to estimate the corrosion current density ($i_{\mathrm{corr}}$), with the subsequent corrosion rate ($C_r$) calculation detailed in Section 2.4.

2.4. Corrosion Rate ($C_r$) Calculation

The corrosion current density ($i_{\mathrm{corr}}$) was estimated through the Stern–Geary relationship using the equivalent electrochemical-resistance parameter obtained from each EN-analysis method. The Stern–Geary constant (B) depends on the electrochemical condition of the steel–concrete system, including the active or passive state of the reinforcement, electrolyte composition, oxygen availability, and environmental-exposure conditions. Therefore, although constant B values are frequently adopted for practical corrosion-rate estimation, the parameter should be interpreted as an electrochemical approximation associated with the polarization behavior of the system rather than as a universal constant [24,29,30,31]. In this study, B was determined from the anodic and cathodic Tafel slopes.

The Stern–Geary constant (B) was calculated as follows:

$$B={\displaystyle \frac{{\beta }_a·{\beta }_c}{2.303({\beta }_a+{\beta }_c)}},$$

(11)

where ${\beta }_a$ and ${\beta }_c$ are the anodic and cathodic Tafel slopes.

A representative value of $B=26 \mathrm{m}\mathrm{V}$ was adopted because direct determination of anodic and cathodic Tafel slopes was not available [29].

It should be noted that the constant B depends on the electrochemical conditions of the steel–concrete system and can vary according to factors such as the passivation state, chloride concentration, oxygen availability, and moisture content. For this reason, the $C_r$ values reported in this work should be interpreted as comparative estimates based on the EN standard rather than independently validated absolute corrosion rates.

The corrosion current density was then determined as follows:

$$i_{\mathrm{corr}}={\displaystyle \frac{B}{R_p}}\approx {\displaystyle \frac{B}{R_n}},{\displaystyle \frac{B}{\overline{Z_n}}},{\displaystyle \frac{B}{\overline{{\psi }_n}}},$$

(12)

depending on the analysis method applied.

Finally, the corrosion rate was obtained using a Faraday-based expression:

$$C_r=k·i_{\mathrm{corr}}·\left({\displaystyle \frac{W_{\mathrm{eq}}}{\rho }}\right),$$

(13)

where $C_r$ is expressed in $\mathrm{mm}/\mathrm{year}$, $i_{\mathrm{corr}}$ in $\mathrm{\mu }\mathrm{A} \mathrm{c}{\mathrm{m}}^{-2}$, $W_{\mathrm{eq}}=27.925$ g/mol is the equivalent weight, $\rho =7.85 \mathrm{g}/{\mathrm{cm}}^3$ the density, and $k=3.27\times {10}^{-3}$ the conversion factor.

These equations, based on ASTM 102, were applied to all the analytical methods described above in order to be able to perform a comparison of $C_r$ in the following statistical analysis.

2.5. Statistical Methodology

After determining the $C_r$ value according to ASTM G102 [24], and in order to determine the influence of both the analytical methods and the aqueous media used, a two-way ANOVA with full interaction was performed. This analysis allows us to determine both the main effects and their interaction.

The general model is expressed as follows:

$$C_{r,ijk}=\mu +{\alpha }_i+{\beta }_j+{\left(\alpha \beta \right)}_{ij}+{\epsilon }_{ijk},$$

(14)

where $C_{r,ijk}$ is the corrosion rate in the k-th trial under method i and medium j; $\mu$ is the overall mean; ${\alpha }_i$ and ${\beta }_j$ represent the effects of method ($i=1,\dots ,I$) and medium ($j=1,\dots ,J$); ${\left(\alpha \beta \right)}_{ij}$ is the interaction term; and ${\epsilon }_{ijk}\sim \mathcal{N}(0,{\sigma }^2)$ is the random error.

The null hypotheses tested were

$$\begin{array}{cc}\hfill H_0^{\mathrm{method}}& :{\alpha }_1=\cdots ={\alpha }_I=0,\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill H_0^{\mathrm{medium}}& :{\beta }_1=\cdots ={\beta }_J=0,\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill H_0^{\mathrm{interaction}}& :{\left(\alpha \beta \right)}_{ij}=0,\forall i,j.\hfill \end{array}$$

(17)

The ANOVA table was obtained by partitioning the total sum of squares into method, medium, interaction, and residual components. The corresponding F statistics were calculated as the ratio between the mean square of each source of variation and the residual mean square.

To perform the two-way ANOVA, we used the MATLAB R2025a (The MathWorks, Inc., Natick, MA, USA) command anovan in MATLAB with an $\alpha =0.05$. Subsequently, Tukey HSD post hoc tests were performed to obtain the main characteristics.

$$q={\displaystyle \frac{|{\overline{x}}_i-{\overline{x}}_j|}{\sqrt{M{S}_E/n}}},$$

(18)

where ${\overline{x}}_i$, and ${\overline{x}}_j$ correspond to the group means, $M{S}_E$ is the mean square error, and n is the sample size per group.

Once these results were obtained, and in order to verify them, box plots of the $C_r$, graphs showing the interaction between methods and means, as well as graphs for multiple comparisons and 95% confidence levels were used. All this statistical analysis allowed us to obtain the necessary criteria to determine the differences between methods and means.

Prior to interpretation of the ANOVA results, diagnostic analyses were performed to evaluate the validity of the model assumptions. Residual normality was assessed using histograms and quantile–quantile (Q–Q) plots, variance homogeneity was examined through Brown–Forsythe-type dispersion analyses, and residual independence was investigated using Durbin–Watson statistics and autocorrelation functions (ACF). Because electrochemical noise measurements were collected over extended exposure periods and under markedly different corrosion conditions, departures from strict normality, homoscedasticity, and independence were expected. Consequently, the ANOVA was employed as an exploratory comparative framework to quantify the effects of analytical method and exposure medium, and their interaction, while the interpretation of statistical significance was complemented by graphical diagnostics and effect-size considerations [32].

2.6. Computation of Normalized Performance Indicators

To more objectively present the qualities of the different analytical methods used in this work, we will present five standardized indicators. These metrics, presented as a complement, are not part of the ASTM standards, but were established to achieve method reproducibility [9,32,33,34,35].

These indicators are overall statistical stability (the behavior across different media), intra-medium robustness (consistent behavior with changes in media), normalized range (which allows us to measure the dispersion of the results), computational efficiency (the speed or time the algorithm takes to obtain the $C_r$), and noise tolerance (residual signal-to-noise ratio), as shown in Table 1.

Table 1. Normalized performance indicators and their mathematical formulations.

Indicator	Equation
overall statistical stability	${\displaystyle 1-\frac{{\sigma }_{C_r}}{\max \left({\sigma }_{C_r}\right)}}$
Intra-medium robustness	${\displaystyle 1-\frac{\overline{{\sigma }_{C_r,medium}}}{\max \left(\overline{{\sigma }_{C_r,medium}}\right)}}$
Normalized range	${\displaystyle 1-\frac{R_{C_r}}{\max \left(R_{C_r}\right)}}$
Computational efficiency	${\displaystyle 1-\frac{T_{proc}}{\max \left(T_{proc}\right)}}$
Noise tolerance	${\displaystyle 1-\frac{{\mathrm{SNR}}_{res}}{\max \left({\mathrm{SNR}}_{res}\right)}}$

This framework does not replace standardized electrochemical parameters but provides a complementary, holistic comparison of stability, robustness, amplitude response, computational efficiency, and resilience to non-stationary signals, supporting the selection of the most appropriate EN-analysis method for reinforced-concrete applications.

3. Experimental Design

A total of ten reinforced-concrete specimens were fabricated to evaluate the electrochemical behavior of embedded steel under different natural aqueous media. The specimens incorporated ribbed carbon steel rebars conforming to ASTM A615 Grade 60 (yield strength ≈ 420 MPa) as working electrodes. According to ASTM A615 [36], the typical chemical composition of this steel includes C ≤ 0.30%, Mn 0.60–1.00%, Si 0.15–0.30%, p ≤ 0.06%, and S ≤ 0.05–0.06%, with Fe as the balance. The geometric design of the specimens is shown in Figure 1.

Figure 1. Design and preparation of the reinforced-concrete specimens.

The concrete mix design was performed according to standards NMX-C-159-ONNCCE-2016 and NMX-C-160-ONNCCE-2016, using a total of $816\mathrm{kg}$ of aggregates: 40% gravel (maximum size $1.905 \mathrm{c}\mathrm{m}$) and 60% sand with a fineness modulus of 3.93. The water-to-cement ratio was 0.64.

Each cylindrical specimen had a diameter and height of $15 \mathrm{c}\mathrm{m}$. The steel bars, which were $20 \mathrm{c}\mathrm{m}$ in length, were partially coated with acrylic paint, leaving an active section of $5 \mathrm{c}\mathrm{m}$ length, equivalent to an effective area of approximately $14.96 \mathrm{c}{\mathrm{m}}^2$. The bars were positioned radially at 120° intervals and at a distance of $2.54 \mathrm{c}\mathrm{m}$ from the geometric center of the specimen.

3.1. Data Acquisition

A cell composed of three nominally identical working electrodes (${\mathrm{WE}}_1$, ${\mathrm{WE}}_2$, and ${\mathrm{WE}}_3$) was embedded in reinforced-concrete specimens with an approximate concrete-cover depth of 3.8 cm and a water-to-cement ratio of 0.64. No external reference electrode was employed; therefore, the electrochemical potential was defined with respect to the third working electrode. The electrical configuration used during the electrochemical noise measurements is presented in Figure 2.

Figure 2. ECN/EPN connection scheme with three nominally identical working electrodes.

ElectrochemicalCurrent Noise (ECN). ${\mathrm{WE}}_1$ and ${\mathrm{WE}}_2$ were connected through a Keysight 34461A digital multimeter (Keysight Technologies, Santa Rosa, CA, USA) configured as a zero-resistance ammeter (ZRA), operating within a range of ±100 μA, with $NPLC=1$ and an aperture time of $t_{\mathrm{ap}}=0.0167$ s. During the experiments, the potential difference between ${\mathrm{WE}}_1$ and ${\mathrm{WE}}_2$ was verified to remain negligible relative to the electrochemical noise fluctuations.

Electrochemical Potential Noise (EPN). Simultaneously, the electrochemical potential’s noise signal was acquired according to the following:

$$EPN\left(t\right)=E_{\left({\mathrm{WE}}_1\right|\left|{\mathrm{WE}}_2\right)}-E_{{\mathrm{WE}}_3}$$

using a Keysight 34460A digital multimeter (Keysight Technologies, Santa Rosa, CA, USA) configured for 1 V DC measurements, with $NPLC=1$ and input impedance $\ge 10$ M$\mathrm{\Omega }$.

Synchronization and data-acquisition structure. Both digital multimeters were synchronized and controlled through LabVIEW 2022 (National Instruments Corporation, Austin, TX, USA) using USB/LAN communication. The ECN and EPN signals were simultaneously sampled using a sampling interval of

$$\Delta t=0.125\mathrm{s}$$

Each synchronized acquisition generated one independent electrochemical noise file (EN file) composed of 8192 ECN samples and 8192 EPN samples, corresponding to an acquisition duration of approximately

$$8192\times 0.125\approx 1024\mathrm{s}$$

per file.

In this study, the term “sample” refers to each individual digital point contained within the ECN or EPN time series, whereas the term “EN file” refers to one complete synchronized electrochemical noise acquisition. The hierarchical structure employed for electrochemical noise processing and statistical analysis is summarized in Table 2.

Table 2. Hierarchical structure used for electrochemical noise acquisition, processing, and statistical analysis.

Term	Definition in this Study	Use in Analysis
Exposure medium	Aqueous environment used for reinforced-concrete exposure.	Experimental factor in ANOVA
Specimen	Reinforced-concrete cylinder containing embedded working electrodes.	Experimental unit
EN file	One complete synchronized acquisition containing 8192 ECN samples and 8192 EPN samples.	Processing unit
Sample	One individual digital point acquired every 0.125 s within ECN or EPN signals.	Raw data point
Derived corrosion-rate observation	One corrosion-rate value obtained from one EN file using one analytical method (SM, FFT, MEM, or ST).	Statistical observation

The experimental dataset was additionally organized according to exposure medium, reinforced-concrete specimen, monitoring duration, number of acquired EN files, and derived corrosion-rate observations. Table 3 summarizes the complete dataset structure used throughout the study, including specimen identifiers, monitoring periods, acquired files, and the total number of corrosion-rate observations generated from each exposure condition.

Table 3. Experimental dataset organization by exposure medium, including specimen identifiers, monitoring duration, acquired EN files, and corrosion-rate observations used in the statistical analyses.

Medium	Specimen IDs	Specimens	Monitoring Period	EN Files	${\mathit{C}}_{\mathit{r}}$ Observations per Method	Total ${\mathit{C}}_{\mathit{r}}$ Observations
CSW	CSW01, CSW03	2	∼220 days	2100	2100	8400
ADSW	ADSW05, ADSW07	2	∼220 days	2016	2016	8064
CRW	CRW02, CRW04	2	∼220 days	2016	2016	8064
TRW	TRW09, TRW10	2	∼160 days	2016	2016	8064
NaCl	NaCl08	1	∼217 days	1008	1008	4032
RO	RO06	1	∼220 days	1010	1010	4040
Total	–	10	–	10,166	10,166	40,664

Each EN file was independently processed using four analytical methods:

• Statistical Method (SM)
• Fast Fourier Transform (FFT)
• Maximum Entropy Method (MEM)
• Stockwell Transform (ST)

As summarized in Table 3, a total of 10,166 synchronized EN files were acquired during the monitoring campaign. Since each EN file was independently processed using four analytical methods, the final statistical dataset comprised 40,664 derived corrosion-rate observations.

$$10,166\mathrm{EN}\mathrm{files}\times 4\mathrm{methods}=40,664\mathrm{derived}\mathrm{corrosion}-\mathrm{rate}\mathrm{observations}$$

In contrast, the complete raw database comprised more than $1.6\times {10}^7$ digital data points considering both ECN and EPN signals.

Electromagnetic interference mitigation. Shielded twisted-pair wiring was employed throughout the acquisition system. The shielding was grounded at a single point (chassis ground) to avoid ground loops, while cable lengths were minimized. Additionally, the acquisition computer operated under galvanic-isolation conditions.

Quality-control procedures. The following quality-control procedures were implemented during acquisition:

• Short-circuit input tests to quantify instrumental noise;
• Verification of $\Delta E_{{\mathrm{WE}}_1-{\mathrm{WE}}_2}\approx 0$ during acquisition;
• Confirmation of absence of saturation within the selected measurement ranges.

These procedures allowed the exclusion of significant electromagnetic-interference artifacts, instrumental saturation, and aliasing effects under the experimental conditions employed.

3.2. Exposure Media

The reinforced-concrete specimens were exposed to six aqueous environments: seawater from Coatzacoalcos (Universal Transverse Mercator (UTM): $X=347,836$ and $Y=2,007,824$), seawater from Agua Dulce ($X=374,577$ and $Y=2,012,797$), river water from Coatzacoalcos ($X=349,485$ and $Y=2,004,077$), river water from Las Choapas (Tancochapa River, $X=385,160$ and $Y=1,976,172$), a laboratory-prepared 3.5% NaCl solution used as an aggressive reference medium, and reverse-osmosis water used as a low-aggressiveness control medium.

The physicochemical properties of the exposure media were experimentally determined and are summarized in Table 4. The evaluated parameters included pH, hardness, and chloride concentration. During the entire monitoring period, the working temperature was maintained at 25 ± 2 °C.

Table 4. Physicochemical properties of the aqueous media used in this study.

Medium	pH	Hardness	Chlorides
		(mg/L as CaCO3)	(g/L as NaCl)
NaCl	7.74	38.00	4.61
ADSW	7.85	6916.67	6.28
CSW	7.79	5750.00	7.31
CRW	7.74	78.00	0.36
TRW	7.49	58.00	0.36
RO	6.14	32.00	0.36

It should be noted that dissolved oxygen concentration, electrical resistivity, and sulfate concentration were not experimentally determined. These parameters may significantly influence corrosion kinetics and oxygen transport within reinforced-concrete systems; therefore, their absence represents a limitation of the present study and should be considered when interpreting the corrosion behavior observed in the different aqueous media.

The following abbreviations are used throughout this work to identify the exposure environments: Coatzacoalcos Seawater (CSW), Agua Dulce Seawater (ADSW), Coatzacoalcos River Water (CRW), Tancochapa River Water (TRW), 3.5% NaCl solution (NaCl), and Reverse-Osmosis Water (RO).

4. Results and Discussion

4.1. Initial Characterization of Electrochemical Noise Signals

As a first step, we will perform a basic analysis of the time-domain signals from the first two recordings (the first from 0 to 1023 s and the second from 1024 to 2048 s), of the electrochemical noise signals EPN and ECN. In this initial analysis, dynamic corrosion patterns can be identified. The complete results of the statistical analyses discussed below are presented in Table 5 and Table 6.

Table 5. Complete statistical results for Rec1 (0–1023 s) and Rec2 (1024–2048 s) EPN signals.

Medium	$\overline{\mathit{E}\mathit{P}\mathit{N}}$		${\mathit{\sigma }}_{\mathbf{EPN}}$		${\mathbf{CV}}_{\mathbf{EPN}}$		${\mathbf{Skew}}_{\mathbf{EPN}}$
	Rec1	Rec2	Rec1	Rec2	Rec1	Rec2	Rec1	Rec2
CSW	−1.9 × ${10}^{-3}$	−3.7 × ${10}^{-3}$	3.4 × ${10}^{-3}$	1.83 × ${10}^{-4}$	1.7682	0.0492	2.6108	−0.2506
ADSW	9.93 × ${10}^{-4}$	1.1 × ${10}^{-3}$	1.6 × ${10}^{-3}$	2.89 × ${10}^{-4}$	1.6466	0.2637	1.8598	0.2619
NaCl 3.5%	2.0 × ${10}^{-3}$	2.9 × ${10}^{-3}$	1.6 × ${10}^{-3}$	1.27 × ${10}^{-4}$	0.7950	0.0431	−1.4813	−0.8550
CRW	−1.48 × ${10}^{-2}$	−1.73 × ${10}^{-2}$	3.5 × ${10}^{-3}$	9.02 × ${10}^{-5}$	0.2363	0.0052	1.5380	−0.3383
TRW	−4.2 × ${10}^{-3}$	−2.5 × ${10}^{-3}$	2.9 × ${10}^{-3}$	1.96 × ${10}^{-4}$	0.6997	0.0787	−0.7720	0.7209
RO	−1.2 × ${10}^{-3}$	5.03 × ${10}^{-4}$	2.7 × ${10}^{-3}$	2.94 × ${10}^{-4}$	2.3452	0.5858	−1.4418	−0.0241

Table 6. Statistical results of ECN signals and noise resistance.

Medium	$\overline{\mathit{E}\mathit{C}\mathit{N}}$		${\mathit{\sigma }}_{\mathbf{ECN}}$		${\mathbf{CV}}_{\mathbf{ECN}}$		${\mathbf{Skew}}_{\mathbf{ECN}}$		${\mathbf{R}}_{\mathbf{n}}$
	Rec1	Rec2	Rec1	Rec2	Rec1	Rec2	Rec1	Rec2	Rec1	Rec2
CSW	1.96 × ${10}^{-8}$	4.57 × ${10}^{-9}$	8.31 × ${10}^{-7}$	7.02 × ${10}^{-7}$	42.4827	153.6963	0.3909	−0.0195	4118	261
ADSW	−1.51 × ${10}^{-8}$	−6.91 × ${10}^{-9}$	9.00 × ${10}^{-7}$	8.82 × ${10}^{-7}$	59.3850	127.7521	0.0148	0.0117	1817	328
NaCl 3.5%	4.33 × ${10}^{-9}$	5.86 × ${10}^{-9}$	8.52 × ${10}^{-7}$	8.04 × ${10}^{-7}$	197.0259	137.1781	−0.0134	−0.0394	1849	157
CRW	−2.14 × ${10}^{-8}$	4.89 × ${10}^{-10}$	8.65 × ${10}^{-7}$	8.03 × ${10}^{-7}$	40.4568	1640.0000	0.0162	−0.0119	4047	112
TRW	1.40 × ${10}^{-8}$	7.56 × ${10}^{-9}$	1.04 × ${10}^{-6}$	9.51 × ${10}^{-7}$	74.2450	125.7847	−0.0486	−0.0221	2791	206
RO	6.58 × ${10}^{-9}$	−3.00 × ${10}^{-10}$	8.48 × ${10}^{-7}$	7.60 × ${10}^{-7}$	128.9461	2530.0000	0.1359	−0.0024	3214	387

4.1.1. Initial Transient Behavior (0–1023 s)

The time interval from 0 to 1023 s can be considered the initial transient stage, and in this stage differences between the different corrosive media can already be found.

For CSW and ADSW marine media, the EPN signals show $\overline{EPN}$ values of $-1.9\mathrm{mV}$ for CSW and $1.0\mathrm{mV}$ for ADSW, together with ${\mathrm{CV}}_{\mathrm{EPN}}$ values of $1.77$ and $1.65$, respectively, indicating substantial signal variability during the initial exposure stage. Furthermore, ${\mathrm{Skew}}_{\mathrm{EPN}}$ values of $2.61$ and $1.86$ suggest asymmetric fluctuations that may be associated with localized electrochemical activity.

For fluvial environments, CRW exhibits a more corrosive response with a $\overline{EPN}$ of $-14.8\mathrm{mV}$ ($\mathrm{CV}=0.24$). TRW, on the other hand, has a $\overline{EPN}$ of $-4.2\mathrm{mV}$ ($\mathrm{CV}=0.7$), which could be considered a mixed response. In all cases, $R_n$ values between 1800 and $4100 \mathrm{\Omega }$, suggest that this system is predominantly controlled by resistance.

For the ECN signal analysis, $\overline{ECN}$ values are close to zero in all cases $\pm 35\mathrm{nA}$. However, the ${\mathrm{CV}}_{\mathrm{ECN}}$ values are extremely high (40 to 200), which is consistent with the close average values to zero. For ${\mathrm{Skew}}_{\mathrm{ECN}}$ values ranging from $-0.05$ to $0.39$, we could consider it a balanced distribution.

4.1.2. Stabilization Phase (1024–2048 s)

In this stage, where the system stabilized, the following characteristics are present:

• The values of ${\sigma }_{\mathrm{EPN}}$ decreased in the order of 10 to 100×, as well as there being a decrease in ${\mathrm{CV}}_{\mathrm{EPN}}$ from $0.005$ to $0.26$.
• The drastic decrease in resistance values $R_n$, which fell to 50–$500 \mathrm{\Omega }$, indicates a substantial change in the electrochemical response compared with the initial transient stage.
• The decrease in the ranges of the ${\mathrm{Skew}}_{\mathrm{EPN}}$ values. The difference between $-0.86$ and $0.72$ suggests normalization of distributions.

Some behaviors show that in the case of CRW, where $\overline{EPN}$ values range from $-14.8\mathrm{mV}$ to $-17.3\mathrm{mV}$, activation continues to increase, which is consistent with a more active electrochemical response during this early stage of exposure. On the other hand, for the NaCl sample, the behavior can be considered more stable, with values of $C{V}_{EPN}=0.04$ and $Ske{w}_{EPN}=-0.85$.

4.1.3. Aggressiveness and Stability Hierarchy

Based exclusively on the first two EN records, qualitative differences between media can already be observed. However, these observations should be considered preliminary because they only represent the initial stages of exposure. Consequently, at this time a definitive classification of corrosivity is not established, and it is necessary to analyze the entire dataset using different spectral and time-frequency methods before drawing general conclusions about the aggressiveness of the medium. The most stable media in the records acquired from 1024 to 2048 s are NaCl and ADSW ($C{V}_{\mathrm{EPN}}<0.1$), while the most unstable, according to the results, is CRW with divergent behavior, which may also suggest greater unpredictability. The CV gives us evidence that the analyzed systems are changing from high variability ($CV>1.0$) towards stability ($CV<0.1$). And all this is based on the first two records analyzed. These preliminary time-domain observations indicate that additional spectral and time–frequency analyses are required to properly characterize the electrochemical processes occurring in each medium. Therefore, FFT, MEM and ST analyses were subsequently applied to the complete dataset in order to evaluate corrosion-rate evolution under long-term exposure conditions [13,18,22].

4.2. Estimation and Temporal Analysis of Corrosion Rate Using Analytical Methods

For a complete analysis of all the records obtained for each medium, the $C_r$ value was determined using the four analytical methods employed: SM, FFT, MEM, and ST. For each method, the noise resistance was determined for the SM method, or the noise impedance for FFT, MEM, and ST. Finally, the $C_r$ value was determined following the ASTM G102 standard.

The analysis of the graph in Figure 3 shows more compact distributions and less variability for the ST method, which may suggest greater robustness to noise and non-stationary signals. On the other hand, the MEM method also yielded stable values with slightly wider tails. The SM and FFT methods showed greater dispersion and larger interquartile ranges, suggesting reduced robustness when non-stationary components dominate the signal.

Figure 3. Boxplot distribution of $C_r$ by analysis method across all aqueous media.

Furthermore, the time-domain graphs in Figure 4 highlight the influence of chloride content on the temporal evolution of $C_r$. In the environments with higher chloride concentrations (CSW, ADSW, and 3.5% NaCl), the MEM and ST methods exhibit smoother transitions and controlled variability throughout the monitoring period. In contrast, the SM and FFT methods show greater dispersion and a higher occurrence of isolated peaks, indicating greater sensitivity to transient fluctuations and non-stationary components of the EN signals. For the CRW and TRW media, moderate variability is observed, with SM and FFT again exhibiting greater fluctuations than MEM and ST.

Figure 4. Temporal evolution of $C_r$ by method and aqueous medium.

Finally, in the RO, we have the $C_r$ values with the least dispersion when analyzed with MEM and ST, which demonstrates their stability and robustness under low-corrosivity conditions. Overall, the results support the suitability of MEM and ST under the evaluated conditions for evaluating EN signals, showing that these methods are appropriate for environments with high chloride content as well as those without. The SM and FFT methods are more sensitive to noise and instability.

Table 7 does not constitute a direct validation because the materials, exposure conditions, specimen geometry, monitoring duration, and measurement techniques differ among studies. Nevertheless, the corrosion-rate values estimated from electrochemical noise measurements fall within the same order of magnitude reported for reinforced concrete subjected to chloride-induced corrosion. This agreement is consistent with the physical interpretation of the results obtained and suggests that the proposed methodology provides realistic estimates of corrosion activity.

Table 7. Contextual comparison between corrosion rates reported in the literature and those estimated from electrochemical noise measurements in the present study.

Reference	Material/Environment	Technique	${\mathit{C}}_{\mathit{r}}$ (mm/Year)	Observation
[37]	Steel in chloride-contaminated concrete	LPR + Gravimetric	0.015–0.14	Corrosion rate increased with chloride content; LPR agreed closely with gravimetric measurements.
[38]	Steel rebar in concrete exposed to 3.89% NaCl	Gravimetric	0.003–0.153	Values representative of active corrosion under severe chloride exposure.
[38]	Steel rebar in concrete exposed to 3.89% NaCl	Electrochemical methods	0.007–0.022	Electrochemical estimates were lower than gravimetric results.
Present study	Reinforced concrete exposed to chloride-containing aqueous environments	EN-FFT, EN-MEM, EN-ST	0.03–0.35	Values fall within the same order of magnitude reported for chloride-induced corrosion.

4.3. Statistical Analysis of Variance (ANOVA)

To evaluate the influence of different aqueous exposure media (CSW, ADSW, NaCl, CRW, TRW, and RO) and different analytical methods (SM, FFT, MEM, and ST) on the corrosion rate ($C_r$), a two-way ANOVA with interaction was performed. Table 8 presents the results, including the sum of squares, degrees of freedom (df), mean squares, F statistic, and associated p-values.

Table 8. Two-way ANOVA results: Effect of method and medium, and their interaction, on corrosion rate ($C_r$).

Source of Variation	Sum of Squares	df	Mean Square	F	p-Value
Method	4539.5	3	1513.2	4986.3	<0.001
Medium	1447.9	5	289.6	954.2	<0.001
Method × Medium	1185.8	15	79.1	260.5	<0.001
Error	12,333.2	40,640	0.3035
Total	19,710.0	40,663

The results of the ANOVA analysis indicate statistically significant effects associated with both the analytical method and the exposure media ($p<0.001$). Likewise, the interaction between these factors was also statistically significant, indicating that the relative performance of the analytical methods depends on the corrosive medium in which they are applied.

The high F values determined for the analytical methods ($F=4986.3$) suggest that the approach used for the different signal processing analytical methods substantially influences the estimated corrosion rates. Similarly, the corrosive media ($F=954.2$) reflects the marked differences between the corrosive media evaluated. The significant interaction of method × medium ($F=260.5$) shows that the relative performance of the methods used is not constant across all media, justifying the application of post hoc comparisons to identify statistically distinguishable groups.

In addition to the above, diagnostic analyses were performed to determine the assumptions associated with the ANOVA model. Residual histograms and Q–Q plots showed deviations from strict normality, while Durbin–Watson statistics and autocorrelation analyses revealed a positive temporal correlation between the observations. Brown–Forsythe analyses also indicated heterogeneity of variances between groups. These characteristics are consistent with the long-term electrochemical noise datasets acquired under different environmental conditions. Therefore, the ANOVA results are interpreted within a comparative and exploratory framework, while the overall statistical trends remain clear and highly significant due to the large number of samples considered (N = 40,664) [32,39].

4.4. Post Hoc Multiple Comparisons (Tukey HSD)

Given the significant effects identified by the two-way ANOVA, a post hoc Tukey HSD test was performed to determine which of the specific pairs of analytical methods and corrosive media contributed to the observed variability in the corrosion rate ($C_r$).

4.4.1. Comparisons Between Methods

Tukey’s developed analysis confirmed that all pairwise comparisons between analytical methods were highly significant ($p<0.001$). As summarized in Table 9, the Maximum Entropy Method consistently determined the lowest $C_r$ estimates, followed by the Stockwell Transform and Fast Fourier Transform, while, on the other hand, the Statistical Method systematically determined the highest values. Thus, this classification (MEM < ST < FFT < SM) can be considered to underline the superior robustness of spectral and time-frequency methods against non-stationary fluctuations, compared to the sensitivity of SM to baseline drift and transient amplification. These results highlight the importance of selecting an advanced analytical method when accurate and reproducible results are required for corrosion monitoring and control.

Table 9. Mean and standard deviation of corrosion rate $C_r$ (mm/year) by method and medium, with supporting references for each method.

	SM [9]		FFT [20]		MEM [26,27]		ST [13]
Medium	Mean	SD	Mean	SD	Mean	SD	Mean	SD
CSW	1.339	1.314	0.330	0.412	0.133	0.149	0.229	0.282
ADSW	1.138	1.044	0.233	0.240	0.110	0.100	0.172	0.179
NaCl	1.880	1.918	0.465	0.586	0.202	0.241	0.350	0.442
CRW	0.574	0.608	0.095	0.118	0.047	0.048	0.075	0.091
TRW	0.440	0.474	0.088	0.108	0.038	0.038	0.066	0.074
RO	0.423	0.374	0.088	0.080	0.037	0.030	0.065	0.057

4.4.2. Comparisons Between Media

Regarding corrosivity levels, the Tukey test allowed us to observe three statistically different groups based on their corrosivity.

• Group 1 (high corrosion): Corresponds to marine environments represented by CSW, ADSW, and 3.5% NaCl. These corrosive media exhibited the highest $C_r$ values and were statistically grouped using the Tukey test, indicating that there were no significant differences between them ($p>0.05$). On the contrary, they were significantly different from the corrosive media included in Groups 2 and 3 ($p<0.001$).
• Group 2 (moderate corrosion): The corrosive media CRW, and TRW, which were statistically similar to each other ($p>0.05$) but significantly different from the high and low corrosion groups ($p<0.001$).
• Group 3 (low corrosion): Reverse-osmosis water (RO), which consistently had the lowest $C_r$ estimates and was significantly different from all other corrosive media ($p<0.001$).

All the relations obtained can be summarized as follows:

$$\mathrm{CSW}\approx \mathrm{ADSW}\approx \mathrm{NaCl} 3.5\% > \mathrm{CRW}\approx \mathrm{TRW} > \mathrm{RO}$$

In general, these findings demonstrate that chloride-rich environments constitute the most aggressive exposure conditions evaluated in this research; on the other hand, river waters produced intermediate corrosion levels and RO served as a reference medium of low aggressiveness. The Tukey analysis further demonstrates that corrosive media with similar chloride availability tend to exhibit comparable corrosion behavior, while marked statistical differences emerge between environments exhibiting high, moderate and low levels of aggression. In methodological terms, these results reinforce the robustness of the analytical methods MEM and ST under a wide range of corrosive conditions, highlighting that both the aqueous medium and the selected analytical technique strongly influence the interpretation of the EN data.

4.5. Corrosion-Rate Distribution by Method and Medium (Violin Plots)

As an additional element to the statistical methods (ANOVA and Tukey), the violin plot, is shown in Figure 5, which presents the distribution of the $C_r$ values. This plot shows both the probability-density estimate and the dispersion of the individual data points, allowing for a good comparison of all the analytical methods in each of the corrosive media.

Figure 5. Corrosion rate ($C_r$) distribution by medium and analytical method, represented as violin plots.

From a methodological perspective, the violin plots in Figure 5 show the same trends observed in the statistical analyses. In the SM plots, the distributions are wider, with pronounced tails, which is especially marked in media with high chloride levels, such as marine environments and 3.5% NaCl, demonstrating sensitivity to noise amplification and transient events. The FFT plot shows an intermediate $C_r$ dispersion, an aspect that can be considered limited because this transform assumes stationary signals, which potentially reduces its capacity, as evidenced in the plot. The MEM method shows thin and symmetrical profiles in the graph, mainly in low-aggressiveness media such as RO. Finally, the ST method shows compact distributions as well as reduced variance in each of the corrosive media, highlighting and demonstrating the ability of this integral transform to analyze non-stationary and noisy signals.

Once we have these graphs, we should analyze them from an electrochemical perspective. The most aggressive media, such as marine environments and 3.5% NaCl, exhibit broader distributions and higher $C_r$ values; this is observed in each of the analytical methods. River water and RO water show more compact distributions, especially when evaluated with the MEM and ST analytical methods. All this graphical evidence reinforces the results obtained using Tukey’s HSD test: that marine environments are the most corrosive, river water exhibits intermediate corrosion, and RO water corresponds to the least aggressive medium evaluated.

In general, the violin-shaped graphs show that MEM and ST reduce variability and preserve diagnostic sensitivity in different corrosive environments. The results obtained with these analytical methods are promising for long-term structural monitoring as they demonstrate their ability to differentiate between various corrosive media, which translates into a reliable corrosion assessment.

4.6. Comparative Statistical Performance of Analytical Methods

To quantitatively assess both the robustness and stability of the analytical methods used in this work, Table 9 shows both the mean and standard deviation of $C_r$ for the different method–mean combinations. This numerical analysis complements the analysis of variance and the violin plots, providing numerical confirmation of the previously obtained results.

This analysis, presented in Table 9, demonstrates that the SM method yields the highest $C_r$ values as well as the largest standard deviations. For the aggressive media (CSW, ADSW, and 3.5% NaCl), the deviations are of the same order as the obtained means (e.g., $1.880\pm 1.918$ mm/year in NaCl), which corresponds to extreme variability. This also shows that the SM method has low robustness to noise and baseline drift [9,40]. Even under low-aggressiveness conditions RO, with SM, the $C_r$ values tend to be overestimated, which could be inadequate and misleading if only this analytical method is used.

On the other hand, the MEM and ST analytical methods exhibit statistical stability. These two analytical methods consistently exhibit lower averages and less dispersion, especially in water with low chloride concentrations, such as the MEM: $0.037\pm 0.030$ mm/year in RO, or river samples. The fact that these methods can minimize variance demonstrates their robustness to noise and their adaptability to the nature of non-stationary signals, such as electrochemical noise signals [13,20]. These results support the potential of MEM and ST for long-term monitoring under the evaluated conditions.

Finally, the FFT can be placed in an intermediate position. While this transform shows attributes that improve upon the results obtained with the SM, the fact that it treats the evaluated signals as stationary makes it particularly vulnerable to transients, which are characteristic of electrochemical noise signals. These characteristics are highlighted and clearly displayed in marine environments (CSW and ADSW), where we observe relatively high standard deviations even though the means decrease when compared to the SM method.

From all of the above, it follows that in highly conductive media, the MEM and ST methods provide compact and controlled $C_r$ distributions, thus supporting their suitability under severe conditions. These findings are consistent with failure analyses of structural steels exposed to marine and tropical atmospheres, which showed differences in oxide film stability as well as in the kinetics of corrosion phenomena. Conversely, the SM method consistently presents high corrosion values even in benign environments, while the FFT provides reasonable values, but with limited robustness. Table 9 indicates that ST and MEM provided the most stable and robust estimates under the evaluated conditions, followed by FFT, with the limitations of being a transform for stationary and non-noisy signals and, finally, by SM, with the least reliable $C_r$ values.

A sensitivity analysis was performed to evaluate the influence of the autoregressive order (M) on the corrosion-rate estimates obtained using the MEM approach. Figure 6 shows the mean absolute relative difference of $C_r$ using $M=200$ as the reference condition for all exposure media. The largest deviations were observed for $M=100$, with relative differences close to 15%, whereas the interval $150\le M\le 300$ exhibited substantially lower variations, generally below 8%.

Figure 6. Mean absolute relative difference of MEM-derived corrosion rate ($C_r$) as a function of autoregressive order (M) and exposure medium, using $M=200$ as the reference configuration.

The results obtained indicate that the selected value ($M=200$) falls within a stable operating region that preserves the relative corrosion trends among the different corrosive media, while avoiding excessive spectral sensitivity.

On the other hand, additional calculations were performed using Stern–Geary constants of $B=13\mathrm{mV}$, $B=26\mathrm{mV}$, and $B=52\mathrm{mV}$ to evaluate the influence of this parameter on the estimated corrosion rates. As predicted by the Stern–Geary formulation, the $C_r$ values obtained showed a linear variation proportional to B. This meant that there were no changes in the relative ranking of the exposure media or in the comparative performance of the analytical methods. These results show that, while B influences the absolute magnitude of the $C_r$ estimates, it does not alter the main comparative trends analyzed in this work. For this reason, $B=26\mathrm{mV}$ was maintained as the reference value for determining the corrosion rate, since this value is widely used in corrosion studies of reinforcing steel in reinforced concrete [41] and is consistent with the exposure conditions and the experimental evidence observed during the $C_r$ period. Furthermore, the sensitivity analysis performed confirms that the main conclusions of the study do not depend on the specific selection of the Stern–Geary constant.

4.7. Computational Efficiency Analysis by Method

One of the aspects we must consider today is computational efficiency, as this is a crucial factor when developing embedded systems or real-time applications. Table 10 presents a summary of the $C_r$ calculation times obtained for each EN record ($N=8192$ data points). Three computing platforms with different capabilities were used for this analysis.

Table 10. Average execution time per method and relative comparison across platforms.

Method	PC1	PC2	PC3	Mean	PC2/PC1	PC3/PC1	PC3/PC2
SM	$31.6\mathsf{\mu }$s	$88.5\mathsf{\mu }$s	$36.0\mathsf{\mu }$s	$52.0\mathsf{\mu }$s	2.80	1.14	0.41
FFT	$0.616\mathrm{ms}$	$0.636\mathrm{ms}$	$0.623\mathrm{ms}$	$0.625\mathrm{ms}$	1.03	1.01	0.98
MEM	$19.6\mathrm{ms}$	$24.6\mathrm{ms}$	$21.4\mathrm{ms}$	$21.9\mathrm{ms}$	1.26	1.09	0.87
ST	$2.72\mathrm{s}$	$3.83\mathrm{s}$	$3.65\mathrm{s}$	$3.40\mathrm{s}$	1.41	1.34	0.95

The times for the different analytical methods were obtained, giving the following results: The SM method was the fastest, with times in the order of tens of microseconds, demonstrating the method’s near-zero complexity. Both the FFT and MEM methods showed acceptable performance, with execution times in the order of milliseconds, making them almost suitable for real-time analysis. Finally, the ST method had the highest computational cost, with execution times of several seconds, consistent with the time-frequency resolution and much more robust in terms of the aforementioned aspects.

For the reasons mentioned above, the impact of hardware can be considered negligible for SM, FFT, and MEM, but for ST it must be considered in accordance with the processor architecture, cache, and memory bandwidth. The considerably longer times in PC2 and PC3 confirm the high computational cost of ST on resource-constrained systems.

Generally speaking, the two methods that offer the best performance (MEM and ST) also have the highest computational demands. For ST, optimization might be necessary for real-time applications, or alternatives such as the Discrete Orthonormal Stockwell Transform (DOST) could be used to reduce execution times [42,43]. On the other hand, SM and FFT can be considered highly viable from an execution-time perspective, but with the drawback of reduced robustness.

The benchmarking was performed on three systems: PC1–Dell G15 laptop computer (Dell Technologies Inc., Round Rock, TX, USA), equipped with an Intel Core i7-13650HX processor (Intel Corporation, Santa Clara, CA, USA), 16 GB RAM, and MATLAB R2025a; PC2–Dell G7 laptop computer (Dell Technologies Inc., Round Rock, TX, USA), equipped with an Intel Core i7-8750H processor (Intel Corporation, Santa Clara, CA, USA), 16 GB RAM, and MATLAB R2020b (The MathWorks, Inc., Natick, MA, USA); and PC3–Huawei MateBook D14 laptop computer (Huawei Technologies Co., Ltd., Shenzhen, China), equipped with an AMD Ryzen 7 5700U processor (Advanced Micro Devices, Inc., Santa Clara, CA, USA), 16 GB RAM, and MATLAB R2025a. These configurations provide a representative spectrum of computational capabilities for laboratory and embedded contexts.

Therefore, this work allows us to evaluate the desirable characteristics or requirements of the expected results in order to achieve an appropriate balance between the desired stability and the available computational efficiency.

4.8. Integrated Comparison Using Normalized Performance Indicators

To obtain a suitable and easily interpretable representation of the analytical methods used (SM, FFT, MEM, and ST) to determine the $C_r$ of EN signals, five normalized indicators between 0 and 1 were calculated to ensure balanced comparisons, as shown in Table 11. These indicators were global stability, intra-medium robustness, normalized range, computational efficiency, and noise tolerance.

Table 11. Normalized performance indicators for each analytical method.

Method	Stability	Robustness	Range	Time	Noise
FFT	0.32	0.32	0.30	0.08	0.29
MEM	1.00	1.00	1.00	0.00	0.64
SM	0.00	0.00	0.00	1.00	0.00
ST	0.49	0.49	0.42	0.00	1.00

As can be seen in Figure 7, which displays the five aforementioned indicators in a radar chart, clear comparisons can be made between the four different methods used in the analysis. In this radar chart, it can be seen that MEM has the highest scores in three of the five indicators: stability, robustness, and range. Therefore, it is concluded that its only limitations in applications could be computational efficiency or noise tolerance. On the other hand, the ST method, which has good performance, is slightly below the MEM method in terms of stability, robustness, and range, but surpasses the other methods in terms of noise tolerance. The FFT analytical method could be considered insufficient compared to the MEM in virtually all aspects, and while it may be a cost-effective alternative, it has limitations. Finally, the SM method, which has the lowest computational cost, obtained the lowest scores in the remaining indicators, highlighting the trade-off between computational simplicity and analytical performance.

Figure 7. Radar plot visualization of normalized performance indicators for each method.

The images in Figure 8 clearly show that the steel specimens exhibit the most severe damage in the most corrosive media (marine environments and NaCl), which is consistent with the higher $C_r$ values obtained from the EN analysis. Conversely, the samples exposed to river water and RO conditions showed the lowest levels of visible degradation, consistent with the lower $C_r$ estimates.

Figure 8. Condition of reinforced-concrete samples after 220 days of exposure. Corrosion-product accumulation is qualitatively consistent with $C_r$ in aggressive media.

The combination of MEM and ST methods proved to be the most stable and robust under the evaluated conditions for EN-based corrosion assessment. MEM exhibited the highest levels of statistical stability and intra-medium robustness, whereas ST showed superior tolerance to noisy and transient signals. FFT provided intermediate performance and may represent a practical compromise between analytical capability and computational cost. Finally, the SM method represented the computationally most efficient approach, although it exhibited the lowest scores in the remaining indicators, highlighting the trade-off between computational simplicity and analytical performance. These observations are supported by the normalized performance indicators and are qualitatively consistent with the visual condition of the exposed specimens.

5. Conclusions

Once each analytical method was applied and evaluated for reinforced-concrete specimens exposed to six aqueous environments, and considering the statistical, spectral, and time–frequency analyses performed on the electrochemical noise (EN) signals, the following conclusions can be drawn:

• MEM exhibited the highest overall statistical stability among the evaluated methods. This method consistently provided the lowest and most stable $C_r$ estimates throughout the different analyses, achieving the best performance in 80% of the normalized indicators presented in Table 11. Its high spectral resolution allowed for the detection of subtle variations in the EN signals, albeit at a higher computational cost than the SM and FFT methods, all under the evaluated conditions.
• The ST method was classified as a high-performance complementary analysis system due to its ability to process noisy and non-stationary signals. According to normalized indicators, its performance was comparable to that of MEM in several aspects and superior to that of FFT and SM under highly variable exposure conditions. Therefore, ST represents an excellent alternative for corrosion monitoring in aggressive environments, although it has the highest computational cost.
• The analysis of the sensitivity of the autoregressive order of MEM showed that ($\mathit{M}=\mathbf{200}$) falls within a stable operating region. The variability in the estimated corrosion rates remained limited across a wide range of model orders, supporting the robustness of the selected value of M and confirming a suitable balance between spectral resolution and model stability.
• The different analytical methods showed a strong dependence of the corrosion rate on the exposure medium. Corrosion rates ranged from $0.0366$ mm/year (MEM, RO) to $0.3504$ mm/year (ST, 3.5% NaCl), demonstrating considerable variability among the different corrosive media. Chloride-rich media (CSW, ADSW, and NaCl) consistently produced the highest corrosion rates, while river water and reverse-osmosis water exhibited the lowest values.
• The corrosion rate trends obtained from the EN analysis were qualitatively consistent with the visual condition of the samples after exposure. Samples submerged in chloride-rich environments exhibited the most severe visible deterioration and, therefore, the highest corrosion rates, which is consistent with the ability of the analytical methods used to detect differences in environmental aggressiveness.
• All results obtained confirm that chloride-rich environments produced the highest estimated corrosion rates throughout the experimental period. These findings highlight the dominant role of chlorides in the deterioration of reinforced concrete and underscore the importance of considering local environmental conditions when evaluating structural durability and maintenance strategies.
• The statistics confirmed the significant effects of the analytical method, the corrosive medium, and their interaction on the estimated corrosion rates. Diagnostic analyses showed deviations from some classic ANOVA assumptions; these deviations are consistent with the temporal and non-stationary nature of long-term electrochemical noise measurements. Consequently, the ANOVA results were used as a comparative and exploratory framework, while the overall trends remained clear and statistically significant.
• Practical implications for its development in engineering. The comparative evaluation of SM, FFT, MEM, and ST establishes objective criteria for determining the most suitable analytical method for EN analysis in reinforced-concrete monitoring. In particular, MEM and ST showed the most favorable balance between sensitivity and robustness, supporting the future development of structural health monitoring systems aimed at optimizing service life and improving infrastructure safety.

Overall, the results demonstrated that EN-based corrosion studies are a viable tool for evaluating RC structures. The incorporation of statistical, spectral, and time-frequency approaches provides complementary information that can support future work in durability studies for monitoring corrosion resistance $C_r$ and condition assessment and durability monitoring.