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Bassel Habeeb, Emilio Bastidas-Arteaga, Helena Gervásio, Maria Nogal
To cite this version:
Bassel Habeeb, Emilio Bastidas-Arteaga, Helena Gervásio, Maria Nogal. STOCHASTIC CAR-
18th International Probabilistic Workshop, Lecture Notes in Civil Engineering 153, pp.753-765, 2021,
�10.1007/978-3-030-73616-3_58�. �hal-03227811�
18th International Probabilistic Workshop
May 12-14, 2021, Guimarães, Portugal
Bassel Habeeb1, Emilio Bastidas-Arteaga1,2, Helena Gervásio3 and Maria Nogal4
1 Institute for Research in Civil and Mechanical Engineering UMR CNRS 6183, University of Nantes,
2 Laboratory of Engineering Sciences for Environment UMR CNRS 7356, La Rochelle University, France.
3 Institute for Sustainability and Innovation in Structural Engineering, University of Coimbra, Portugal.
4 Faculty of Civil Engineering and Geosciences, TU Delft, Netherlands.
Keywords: Seasonal Stochastic Markovian Autoregressive Integrated Moving Average model, In-
frastructure reliability, Carbon dioxide forecasting, Concrete carbonation, Climate change.
Over the Earth’s history, the climate has changed considerably due to natural processes affecting
directly the earth. In the last century, these changes have perpetrated global warming. Carbon
dioxide is the main trigger for climate change as it represents approximately up to 80 percent of
the total greenhouse gas emissions. Climate change and concrete carbonation accelerate the cor-
rosion process increasing the infrastructure maintenance and repair costs of hundreds of billions
of dollars annually. The concrete carbonation process is based on the presence of carbon dioxide
and moisture, which lowers the pH value to around 9, in which the protective oxide layer surround-
ing the reinforcing steel bars is penetrated and corrosion takes place. Predicting the effective re-
tained service life and the need for repairs of the concrete structure subjected to carbonation
requires carbon dioxide forecasting in order to increase the lifespan of the bridge. In this paper,
short term memory process models were used to analyze a historical carbon dioxide database, and
specifically to fill in the missing database values and perform predictions. Various models were
used and the accuracy of the models was compared. We found that the proposed Stochastic Mar-
kovian Seasonal Autoregressive Integrated Moving Average (MSARIMA) model provides
of 98.8%, accuracy in forecasting value of 89.7% and a variance in the value of the individual
errors of 0.12. When compared with the CO2 database values, the proposed MSARIMA model pro-
vides a variance value of -0.1 and a coefficient of variation value of -8.0
Habeeb et al.
IPW202018th International Probabilistic Workshop
Civil infrastructure investment in the European Union has been in a steady decline since the
outbreak of the economic and financial crisis. Although the decrease appears to gradually level
off from 2015 onwards with an increase of 5% [1]. The increase in the infrastructure investment
from 2015 onwards was illustrated as an action by the European Union for the sake of designing
and maintaining these systems for a certain service lifetime, which was recognized as critical
issues worldwide.
Decision making in the civil infrastructure investment in the European Union utilizing the qual-
ity control plan is involved in the case of repairing or demolition of the reinforced concrete
bridges, depending on the recent key performance indicators (KPI). The KPI are specified by
engineering consultants regarding the current condition of the bridge and the strategies to be
followed (Reference strategy / Representative strategy) taking into consideration the reliability,
the cost and the availability of the bridge.
Reinforced concrete bridges are characterized by high durability, despite that, they are also
vulnerable to natural hazards, as well as extreme events that affect their performance and ser-
viceability. Statistics on bridge collapses worldwide reveal that natural hazards are the predom-
inant cause of failure. French government revealed that among the 12,000 maintained bridges
after the collapse of the motorway bridge located in Genoa, 840 are at risk of collapsing. This
issue is common across Europe [2].
Carbonation of concrete is one of the main causes of corrosion and occurs by the reaction given
in Equation (1) between atmospheric CO2 and the hydrated phases of concrete. This reaction
generates calcium carbonate, leading to a drop in the pH value, in which the protective oxide
layer of the reinforcing steel bars is broken and corrosion starts. Therefore, the life span of the
concrete infrastructure is affected by the enhanced risk of carbonation induced corrosion [3].
The temperature significantly affects the diffusion coefficient of CO2 into concrete, the rate of
reaction between CO2 and Calcium Hydroxide (Ca(OH)2), and their rate of dissolution in pore
water [4]. The optimum relative humidity condition for the carbonation process is between 50%
and 70%, including wetting and drying cycles that enhance the reaction [5].
The carbonation process is very sensitive to the local climate depending on the environmental
conditions [6]. Climate change impacts the infrastructure as the increase in CO2 levels associ-
ated with global warming will increase the carbonation-induced corrosion. Moreover, changes
in humidity and temperature significantly affect the initiation time of corrosion [7]. Since stud-
ies on global warming have predicted several changes in climate, the impact of climate change
on structural reliability should be considered. For example, Bastidas-Arteaga has calculated
numerically in the oceanic environment a reduction in the lifetime of failure that ranges between
1.4% and 2.3% and up to 7% when cyclic loading is considered [8].
A carbon dioxide database is essential to study the influence of realistic exposure conditions on
concrete carbonation. Databases could be also used to establish probabilistic prediction models.
Therefore, this study proposes a prediction model that is established based on the time-domain
analysis of the database and evaluated with a short memory process. The model is also com-
pared with other autoregressive models. The proposed Stochastic Markovian Seasonal Auto-
regressive Integrated Moving Average model (MSARIMA) is also used to fill the missing
Stochastic carbon dioxide forecasting model for concrete durability applications
IPW2020 18th International Probabilistic Workshop
database and to perform predictions, taking into account the statistical analysis on the previ-
ously existing historical database and seasonality.
Climate models are based on well-notarized physical processes that simulate the transfer of
energy and materials through the climate system. Climate models, also known as general cir-
culation models, use mathematical equations to characterize how energy and matter interact in
different parts of the ocean, atmosphere and land [9]. Climate models are operated using varia-
bility that is driving the climate and predicting the climate change in the future. External factors
are the main inputs into the climate models that affect the amount of the solar energy absorbed
by the Earth or the amount trapped by the atmosphere, these external factors are called “forcing”.
They include variations in the sun’s output, greenhouse gases and tiny particles called aerosols
that are emitted from burning fossil fuels, forest fires and volcanic eruptions. The aerosols re-
flect incoming sunlight and influence cloud formation except the black carbon.
Climate models provide results that vary with respect to the actual historical database; those
variations are at the expense of each model differences in: (ensemble, data source, forcing, the
initial state of run, driving model, aerosols influence and jet stream impact). However, the pro-
posed model is based on stochastic time series analysis that avoids the climate models variations
and provides database that is statistically related to the existing historical database.
Time series forecasting is a quantitative approach that uses information based on historical val-
ues and associated patterns to predict future observations. Time series analysis comprises meth-
ods for analyzing time-series data to extract meaningful statistics and other characteristics of
the data. The analysis includes trend, seasonality and irregular components. A time-series anal-
ysis quantifies the main features in data and random variation. These reasons, combined with
improved computing power, have made time series methods widely applicable.
2.1 Methodology
2.1.1 Time series analysis
Time series analysis for carbon dioxide database is based on the time-domain analysis (auto-
correlation analysis and cross-correlation analysis), in which the type of the process deduced is
a short-term memory process with short-range dependence that is characterized by an exponen-
tial decay of the autocorrelation function (Acf) for the historical database.
2.1.2 Decomposition
Time series consists of two systematic components: trend and seasonality, and a non-systematic
component called noise. A multiplicative nonlinear model is used as the seasonality increases
with the increase in the trend. The autocorrelation function of the non-systematic component
demonstrates the characteristics of the autoregressive model in terms of damaged cosine shape.
2.1.3 Stationarity
Stationarity of the database is essential to maintain the statistical properties of the time series,
a stationarized series is relatively easy to predict, the stationarity is achieved through differenc-
ing and log transformation. The basic idea of stationarity is that the probability laws that govern
the behavior of the process do not change over time. In a sense, the process is in statistical
equilibrium. Specifically, a process is strictly stationary if the distribution of existed state is the
same as the distribution of the previous state for all choices of time points and all choices of
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IPW202018th International Probabilistic Workshop
time step lag. The stationarity of the time series is checked using Kwiatkowski–Phillips–
Schmidt–Shin (KPSS) test and augmented Dickey-Fuller (ADF) test [10].
2.1.4 Models
The statistical technique utilized for forecasting the carbon dioxide is Seasonal Stochastic Mar-
kovian Autoregressive Integrated Moving Average (MSARIMA) which provides high accuracy
and precise results. Moreover, other statistical techniques that include moving average based
methods, such as Autoregressive Moving Average (ARMA), Autoregressive Integrated Mov-
ing Average (ARIMA), Holt-Winters’ Triple Exponential Smoothing and Seasonal Autoregres-
sive Integrated Moving Average (SARIMA) were performed in order to compare the variations
in the accuracy of the models.
Lately, Autoregressive Integrated Moving Average (ARIMA) model has been used to study the
short time-varying processes. However, one limitation of ARIMA is its natural tendency to
concentrate on the mean values of the past series data. Therefore, it remains challenging to
capture a rapidly changing process, in which the proposed model (MSARIMA) solves this issue
by triggering a Markovian step when the value of the integration part is >1 and the probability
of occurrence is related to the previous seasonal events.
2.2 Models description
Models presented are divided into two categories: auto regression (AR) moving average (MA)
parameters and exponential smoothing parameters. The proposed MSARIMA model is based
on the AR and MA parameters. In addition, it accounts for seasonality and Markovian step
The autoregressive model of order p, which is denoted as AR(p), writes:
,%+-*./&,%"& *0%
&() #; 0%1#23&456*
is the state,
is a parameter of the model,
is constant,
is a random white noise
is the variance of the random white noise.
In this case, we denote by {
} 1 AR (p). In the same way, we can rewrite a process AR(p)
with a polynomial
8&;)#7+#+ :+#; 8&;)#+#<#=#8);#=8!;!#=#>>>=#8,;,
The moving average model of order q, which is denoted as MA(q), writes:
7++?+.@-:+"- *:+
is a parameter of the model and
is the expectation of
, often equals to zero.
Use the backshift operator B to rewrite Equation (4).
7++@&;)#:+#; @&;)#+ #<#*#@);*@!;!*#>>>*@.;.##
2.2.1 Autoregressive moving average model (ARMA)
The general ARMA model was described in the 1951 by Peter Whittle [11].
-() *.8-7+"- *:+/
The model could be written using the polynomials φ (B) and θ (B) in which the constant
are zero values:
Stochastic carbon dioxide forecasting model for concrete durability applications
IPW2020 18th International Probabilistic Workshop
-() +.@0:+"0
0() *:+
-() 7++&<*.#@0;0)
0() :+#
The ARMA model omits the integration part of its calculation leading to a non-stationary time
series model in which statistical parameters will vary with time. On the contrary, embedding
the integration part in the time series will control the stationarity in which the statistical prop-
erties such as mean, variance, autocorrelation, etc. are all constant over time.
2.2.2 Autoregressive integrated moving average model (ARIMA)
The ARIMA is an advanced ARMA model that solves the stationarity of the time series by
using difference operation, this value is up to the second-order of integration (
𝑑!"# = 2)
on the backshift operator Equation (9). Otherwise, it is solved using log transformation.
The general equation taking into account the constant
as a non-zero value, in which
is the mean of
, is as follows:
2.2.3 Seasonal Autoregressive integrated moving average model (SARIMA)
The seasonality of a model is detected using an autocorrelation function in which the peaks
evolve over the lag values of a defined time series with a scale value >24. The monthly seasonal
stationarity of a model is based on a lag value of
= 12 and is known as the seasonal monthly
differencing operator in Equation (11).
I&;4)8&;)&#7+=?)+ J&;4)@&;):+
The SARIMA model without the differencing operations is mentioned in Equation (12) and the
terms are illustrated below:
I&;4) + <=I);4=I!;!4#=#>>>=#I,;,4
is the seasonal AR parameter,
is the AR parameter,
is the seasonal MA parameter
is the MA parameter.
2.2.4 Markovian Seasonal Autoregressive Integrated Moving Average model (MSARIMA)
The proposed model is based on the SARIMA model. The MSARIMA solves the SARIMA
only limitation with its tendency to concentrate on the mean values of the past series data by
working on a sequence of time intervals changing their mean value in each time and by trigger-
ing a Markovian step Equation (17).
L + MN#O5PO5")! +#Q5")! R#
+ST&:6-")!)U <######57+V5"- *W#####
TX:6-")!YZ <#####57+V5"- *#<#####
Habeeb et al.
IPW202018th International Probabilistic Workshop
is the state,
is the mean value of the monthly seasonal errors of the value
the Markovian step value.
The MSARIMA model is developed based on the SARIMA model with a triggering condition
when the integration value >1, the model works on increasing the accuracy of the prediction
regarding the seasonal errors for the current state.
The step process depends on the most recent past event and the Markovian step is a renewable
process because it presents only positive values. This model neglects the
values in the pre-
vious equations and presents the Markovian step process value
for more accurate results. The
equation is as follows:
2.2.5 Holt-Winters’ multiplicative seasonal model
Winters (1960) extended Holt’s method to capture seasonality [12]. The Holt-Winters’ seasonal
method comprises the forecast equation and three smoothing equations. The multiplicative
method is used when the seasonal variations are changing proportionally to the trend of the
series. The seasonal component is expressed in relative terms and the series is seasonally ad-
justed by dividing through by the seasonal component. Within each year, the seasonal compo-
nent will sum up to approximately the seasonal frequency value.
+789+ +#&\+*]>^+)_+7:";<=7)>
\++` ?!
@!"# *&<=`)#&\+") *^+") )
_b#+c A%
B!"$"C!"$ *&<=c)_+"D
is the level,
is the trend,
is the seasonal component, m is the seasonal frequency,
are the model smoothing parameters.
The main objective of this section is to estimate the ability of the proposed approach in fore-
casting carbon dioxide concentration using an incomplete database. The forecasting and pre-
diction of the missing values are performed using the following mathematical and stochastic
models: ARMA, ARIMA, SARIMA, Holt-Winters’ and MSARIMA.
3.1 Database description
The concentration of greenhouse gases in Portugal is measured on the island of Terceira, which
is one of the nine islands in the archipelago of the Azores, located in the middle of the Atlantic
Ocean. The database is available since 1979 for different greenhouse gases. In particular, for
the three main gases, carbon monoxide (CO) since 1990, carbon dioxide (CO2) since 1979, and
methane (CH4) since 1983. However, the carbon dioxide database includes missing values. The
samples are collected on the island of Terceira and the analysis is carried out in NOAA lab,
Hawaii, in the scope of the Cooperative Global Air Sampling Network.
Stochastic carbon dioxide forecasting model for concrete durability applications
IPW2020 18th International Probabilistic Workshop
3.2 MSARIMA CO2 database prediction
The database offered by NOAA lab, Hawaii, in the scope of the Cooperative Global Air Sam-
pling Network includes missing values. Therefore, a stochastic MSARIMA model presents ac-
curate results in filling the database shown in Figure 1 and can be used in for forecasting
Figure 1: Example of CO2 assessment based on previous database
3.3 Stochastic models analysis
3.3.1 Stochastic models predictions
In this section, the prediction of the MSARIMA model is compared with SARIMA and Holt-
Winters’ models as both include seasonal components. This is implemented by forecasting a
historical CO2 starting from 2010 through 2018 Figure 2. The prediction of the MSARIMA
model seems to provide the best fitting results to the original database compared to the other
models. The errors associated with the predictions will be further studied in the next section.
Figure 2: Stochastic models comparison
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IPW202018th International Probabilistic Workshop
3.3.2 Stochastic models accuracy
A statistical study was performed to derive the variations between the mathematical stochastic
models and the meteorological station’s database. The difference of the relative frequencies for
CO2 presented in Figure 3 was performed for a time series starting from 01/2010 to 01/2018 to
describe the variations in the models. The proposed MSARIMA model presents the lowest var-
iations. Moreover, ARIMA and ARMA models show higher variations than the other models
as seasonality is not considered.
Figure 3: Difference in relative frequency
The stochastic models' statistical study in Table 1 illustrates the variation of the models with
the original database in terms of mean value, variance and coefficient of variation, in which the
MSARIMA model shows the lowest variation with the meteorological station’s CO2 database.
On the contrary, the others present higher variations in the results.
Table 1: Statistical differences with meteorological station's CO2 database
Mean value
Coefficient of variation (%)
The accuracy of the stochastic models is finally demonstrated by comparing SARIMA and
MSARIMA models with the original database for the data given in Figure 2. This study will be
carried out in terms of the error indicators in Table 2. In this table ME is the mean error, RMSE
is the square root of the average of the square errors, MAE is the mean absolute error, MAPE
is the mean absolute percentage error and
is the proportion of the fitted model variation with
the original database.
Table 2: Accuracy between MSARIMA and SARIMA models
Stochastic carbon dioxide forecasting model for concrete durability applications
IPW2020 18th International Probabilistic Workshop
The MSARIMA model presents the highest
value in which 98.8% of the CO2 database var-
iation is explained by the fitted model. The mean error refers to the average of all errors, it is
also described as the uncertainty in measurements, the proposed MSARIMA model provides
the lowest value in errors. The variation in the errors in the set of forecasts is diagnosed by the
difference between RMSE and MAE, in which lower values in RMSE-MAE show lower vari-
ance in the individual errors, as shown in Table 2 the MSARIMA model has the lowest RMSE-
MAE values. The accuracy of a model prediction is presented by the 1-MAPE value, as it cal-
culates the relation between forecasted values and original values, in which the MSARIMA
model has the highest accuracy in forecasting.
The prediction of the proposed Stochastic Markovian Seasonal Autoregressive Integrated Mov-
ing Average (MSARIMA) model seems to provide the best fitting results to the original CO2
database compared to the other models.
The proposed MSARIMA model provides
value of 98.8%, accuracy in forecasting value of
89.7% higher than all the other models and variance in the individual errors value of 0.12. When
compared with the CO2 database values, the proposed MSARIMA model provides a mean value
of -0.32, a variance value of -0.1 and a coefficient of variation value of -8.0
The provided results demonstrate that there is no overestimation in the predictions using the
proposed MSARIMA model, which might be an obstacle due to the proposed step methodology.
On the contrary, the MSARIMA model provided the best fit in predictions when compared with
the original CO2 database.
This paper was carried out in the framework of the Strengthening the Territory’s Resilience to
Risks of Natural, Climate and Human Origin (SIRMA) project, which is co-financed by the
European Regional Development Fund (ERDF) through INTERREG Atlantic Area Program
with application code: EAPA_826/2018. The sole responsibility for the content of this publica-
tion lies with the author. It does not necessarily reflect the opinion of the European Union.
Neither the INTERREG Europe program authorities are responsible for any use that may be
made of the information contained therein.
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ResearchGate has not been able to resolve any citations for this publication.
Full-text available
Durability of reinforced concrete (RC) structures is affected by certain environmental conditions and operational actions which can reduce their lifetime significantly. Among these actions, this paper proposes a stochastic model that accounts for the combined effects of chloride-induced corrosion, climate change and cyclic loading. Separately, corrosion leads to cross-section reduction, climate change produces changes in temperature and humidity and fatigue induces nucleation and propagation of cracks in the rebars. When considered together, pitting corrosion nucleates cracks while environmental factors affect the kinematics of chloride ingress and corrosion propagation. The proposed approach is illustrated with the reliability analysis of a bridge girder subjected to cyclic loading under various environmental conditions. The overall results indicate that climate change effect induces lifetime reductions ranging between 1.4 and 2.3% if fatigue load is neglected. Under cyclic loading, total lifetime reduction increases up to 7%.
Full-text available
Carbonation affects the performance, serviceability and safety of reinforced concrete (RC) structures when they are placed in environments with important CO 2 concentrations. Since the kinetics of carbonation depends on parameters that could be affected by climate change (temperature, atmospheric CO 2 pressure and relative humidity (RH)), this study aims at quantifying the effect of climate change on the durability of RC structures subjected to carbonation risks. This work couples a carbonation finite element model with a comprehensive reliability approach to consider the uncertainties inherent to the deterioration process. The proposed methodology is applied to the probabilistic assessment of carbonation effects for several cities in France under various climate change scenarios. It was found that climate change and local RH have a significant impact on corrosion initiation risks.
Atmospheric CO2 is a major cause of reinforcement corrosion in bridges, buildings, wharves, and other concrete infrastructure in Australia, United States, United Kingdom and most other countries. The increase in CO2 levels associated with global warming will increase the likelihood of carbonation-induced corrosion. Moreover, temperature rises will increase corrosion rates. Clearly, the impact of climate change on existing and new infrastructure is considerable, as corrosion damage is disruptive to society and costly to repair. The paper describes a probabilistic and reliability-based approach that predicts the probability of corrosion initiation and damage (severe cracking) for concrete infrastructure subjected to carbonation and chloride-induced corrosion resulting from elevated CO2 levels and temperatures. The atmospheric CO2 concentration and local temperature and relative humidity changes with time over the next 100 years in the Australian cities of Sydney and Darwin are projected based on nine General Circulation Models (GCMs) under (i) high CO2 emission scenario, (ii) medium CO2 emission scenario, and (iii) CO2 emission reduction scenario based on policy intervention. The probabilistic analysis included the uncertainty of CO2 concentration, deterioration processes, material properties, dimensions, and predictive models. It was found that carbonation-induced damage risks can increase by over 400% over a time period to 2100 for some regions in Australia. Damage risks for chloride-induced corrosion increase by no more than 15% over the same time period due to temperature increase, but without consideration of ocean acidity change in marine exposure. Corrosion loss of reinforcement is not significant. The results were most sensitive to increases in atmospheric CO2.
We propose automatic generalizations of the KPSS-test for the null hypothesis of stationarity of a univariate time series. We can use these tests for the null hypotheses of trend stationarity, level stationarity and zero mean stationarity. We introduce the asymptotic null distributions and we determine consistency against relevant nonstationary alternatives. We compare the properties of the tests with those of other proposed tests for stationarity. Monte Carlo simulations support the relevance of the tests when an autoregressive process with large positive autocorrelations is likely under the null hypothesis.
Investment in infrastructure in the EU -Gaps, challenges, and opportunities
  • I Zachariadis
I. Zachariadis, "Investment in infrastructure in the EU -Gaps, challenges, and opportunities," no. October, 2018. Available:
Bridge Maintenance and
  • Le Journal Du Dimanche
Le Journal du Dimanche, "Bridge Maintenance and," 2018.
Carbonation of Reinforced Concrete -Ronacrete
  • Ronac
Ronac, "Carbonation of Reinforced Concrete -Ronacrete," vol. 44, no. 0. pp. 1-6, 2018. Available:
  • S Kotz
  • N L Johnson
S. Kotz and N. L. Johnson, Breakthroughs in Statistics, vol. 40, no. 2. 1997.