Bayesian assessment of surface recession patterns in brick buildings with critical factors identification

b o l e t í n d e l a s o c i e d a d e s p a ñ o l a d e c e r á m i c a y v i d r i o 6 1 (2 0 2 2) 357–373 www.elsevier.es/bsecv Bayesian assessment of surface recession patterns in brick buildings with critical factors identification E. Menéndeza,∗, L.M. Gil Martínb, Y. Salema, L. Jalónb, E. Hernández-Montesb, M.C. Alonsoa aCSIC - Instituto de Ciencias de la Construcción, Eduardo Torroja (IETCC), Madrid, Spain bDepartment of Structural Mechanics and Hydraulic Engineering, University of Granada, Spain a r t i c l e i n f o Article history: Received 22 February 2022 Accepted 1 April 2022 Available online 7 May 2022 Keywords: Bayesian system identification Cultural heritage brick building Surface recession assessment Photogrammetric point cloud Degradation factors a b s t r a c t The deterioration of built heritage is a major concern for many countries. In the specific case of brick heritage buildings, degradation depends on several factors such as the chemical and mineralogical composition and the porosity of the bricks, floor insulation and exposure to environmental conditions. This paper applies a probabilistic Bayesian approach to identifying the pattern of recession of the brick walls using digital photogrammetry data. Two cases studies are presented corresponding to two buildings classified as Assets of Cultural Interest by the Spanish Ministry of Culture. The analysis of physico-chemical factors and photogrammetric descriptions allow a holistic understanding of the deterioration process to be developed. The degradation patterns and degradation velocities obtained are used for the structural assessment of the buildings. The comparison of the degradation patterns of both buildings considering their individual features allows the most critical factors in the degradation process of brick walls to be identified. © 2022 The Author(s). Published by Elsevier Espa ˜ na, S.L.U. on behalf of SECV. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/ by-nc-nd/4.0/). Evaluación bayesiana de patrones de recesión superficial mediante la identificación de factores críticos en edificios de ladrillo Palabras clave: Sistema de identificación bayesiano Edificios de ladrillo-patrimonio cultural Evaluación de la recesión de superficies Factores de degradación r e s u m e n El deterioro del patrimonio construido es una gran preocupación para muchos países. En el caso específico de los edificios patrimoniales de ladrillo, la degradación depende de varios factores como la composición química y mineralógica y la porosidad de los ladrillos, el aislamiento del suelo y la exposición a las condiciones ambientales. Este artículo aplica un enfoque probabilístico bayesiano para identificar el patrón de recesión de las paredes de ladrillo utilizando datos de fotogrametría digital. Se presentan dos casos de estudio correspondientes a dos edificios catalogados como Bienes de Interés Cultural por el Ministerio de Cultura espa ˜ nol. El análisis de los factores físico-químicos y las descripciones fotogramétricas permiten desarrollar una comprensión holística del proceso de deterioro. Los patrones de ∗Corresponding author. https://doi.org/10.1016/j.bsecv.2022.04.002 0366-3175/© 2022 The Author(s). Published by Elsevier Espa ˜ na, S.L.U. on behalf of SECV. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). 358 b o l e t í n d e l a s o c i e d a d e s p a ñ o l a d e c e r á m i c a y v i d r i o 6 1 (2 0 2 2) 357–373 degradación y las velocidades de degradación obtenidos se utilizan para la evaluación estructural de los edificios. La comparación de los patrones de degradación de ambos edificios considerando sus características individuales permite identificar los factores más críticos en el proceso de degradación de las paredes de ladrillo. © 2022 El Autor(s). Publicado por Elsevier Espa ˜ na, S.L.U. en nombre de SECV. Este es un art´ ıculo Open Access bajo la licencia CC BY-NC-ND (http://creativecommons.org/licenses/ by-nc-nd/4.0/). Introduction Brick has been used for building purposes for thousands of years. The oldest bricks were made with mud dried in the sun to harden it, but it was the invention of fired brick around 3500 BC which allowed the use of bricks even in cool areas. The Romans spread the use of construction with fired bricks by constructing public and private buildings all over the entire empire (see the Herculaneum gate in Pompeii in Fig. 1a). Between the 12th and 16th centuries, in the Baltic countries spectacular buildings were built mainly with fired red clay bricks (during the Brick Romanesque, Brick Gothic, and Brick Renaissance periods, see in Fig. 1b). During the Renaissance and Baroque periods, brick masonry fac¸ades were not common, and it was not until the mid-18th century that brick walls started appearing, a resurgence that was mainly due to the fact that the industrial revolution allowed for bricks to be produced on a massive scale, which meant they stopped being made by hand. In the 20th century, exposed brick walls started to be used again because of their durability, the fact that they needed less costly maintenance, and the influence of famous architects, such as Le Corbusier, who used them in their works (see Fig. 1c). Currently, some of these brick masonry buildings have been classified as Cultural Heritage (CH) assets and they have been the subject of major interest. However the degradation of our cultural brick heritage is happening faster today than at any time in the past [1]. Accordingly, the structural assessment of these kind of heritage constructions is essential. Recent works have applied the non-destructive testing (NDT [2–5]) and monitoring techniques available for the characterization of brick masonry elements [6–8]. In this paper, the degree of deterioration of CH brick masonry elements are studied, considering both physicochemical material deterioration and high-scale structurallevel degradation, which is directly related to the actual structural integrity of a building. As in [9], a probabilistic approach has been adopted for the surface recession assessment based on a simple non-destructive technique: digital photogrammetry. The method proposed is based on solid Bayesian system identification principles [10], allowing for the identification of the surface corresponding to the recession pattern for a given photogrammetric dataset. A suitable quantification of the level of degradation and its magnitude requires taking into account several factors that cause deterioration or erosion in bricks. This degradation impairs the aesthetics of fac¸ades and poses threats not only at a cultural level but also in terms of safety and cost. In this paper, how exposure to degrading agents impacts brick walls service life is analyzed. The measurement of the degree of deterioration of CH brick buildings is a useful tool for supporting decision making about preventive maintenance [11], restoration [12], and resilience to climate change [13]. Cycles of moisture exposure (i.e. first contact with water, wetting, and drying) [14] and freezing while wet are the major causes of damage to brick buildings [15]. The effect of salt contamination accumulated over time on CH brick buildings is another cause of damage. The damage associated with salt implies the presence of moisture that enables dissolution-crystallization cycles and the transportation of the salts [16,17]. In this work, several degradation factors are considered: the level of exposure to the direct action of water [18] and/or wind [19], the level of insulation of the bricks from the ground, the chemical and mineralogical composition of the bricks and the influence of hazard events such as freeze–thaw cycles [20], moisture exposure [14], and environmental pollution. Methodology Characterization of the material In this study, representative samples of bricks from two CH buildings have been used to characterize the material. Chemical, mineralogical, and physical characterizations have been conducted. Regarding the mineralogical composition of the bricks, in order to characterize the crystalline phases present in the bricks, both the majority and minority crystalline compounds were determined by X-ray diffraction (XRD). Samples are ground and sieved until a grain size smaller than 80 ␮m was obtained. The powder samples were analyzed using a D8 ADVANCE Theta-Theta Equipment diffractometer whose X-ray tube is a 2.2 kW copper anode. The chemical composition was determined from an elemental chemical analysis of the brick samples performed using the X-ray fluorescence (XRF) technique in melted samples with a wavelength dispersion X-ray spectrometer, Bruker’s S8 Tiger. The percentage of CO2was determined by loss to fire in accordance with the ASTM C114-18 standard [21]. The alkalis were determined by using the inductively coupled plasma technique (ICP) with Varian 725-ES equipment. Chlorides were analyzed with a Titrino 702 SM potentiometric validator, with a chloride selective electrode. Nitrites, nitrates, and phosphates were determined by using Ionic Chromatography with Metrohm 930 Compact IC Flex equipment. b o l e t í n d e l a s o c i e d a d e s p a ñ o l a d e c e r á m i c a y v i d r i o 6 1 (2 0 2 2) 357–373 359 Fig. 1 – Construction works made of bricks. (a) The Herculaneum gate of Pompeii. (b) Frederiksborg Palace in Denmark, built during the first years of the 17th century. (c) Maisons Jaoul of Le Corbusier in France. A chemical analysis of soluble extracted salts was carried out in order to assess the possible contribution of easily extractable salts to brick deterioration. In order to evaluate both the potential extraction of salts and the alkalis, a 7-day leaching test at room temperature was conducted based on the test procedure for alkali extraction in concrete [22]. Once the established period had ended, the samples were filtered with a 25 mm millipore filter by using ionic chromatography (IC) to obtain the nitrites and nitrates. The extracted salts were analyzed and the total alkali content (Na2O and K2O) was determined by using the inductively coupled plasma technique (ICP). Sulphates and chlorides were determined using gravimetry and potentiometry methods, respectively. The open porosity of the bricks was also determined. Both the porosity accessible to water and the apparent density of the representative samples of the bricks were determined in accordance with the test method of the UNE EN 83980: 2014 standard [23]. The water absorption, density and porosity accessible to water were determined by the method of water absorption by using vacuum and hydrostatic weighing of the samples. Once the brick samples had been saturated through water immersion, the specimens were dried until they reached a constant weight at low humidity. Characterization of the climate conditions Historical weather data in Granada and Madrid are shown in Tables A1 and A2, respectively (see Appendix). The climatological data from the last 40 years summarized in Tables A1 and A2 have been obtained from official Spanish meteorological sites [24,25]. The climatological parameters are average and absolute maximum and minimum temperatures in degrees centigrade, absolute maximum gust in km/h, and average pluviosity in mm. Identification of surface recession The surface recession defines the loss of volume of the fac¸ade walls of a building due to the action of degradation factors (chemical, climatic, . . .). A Bayesian inference framework has been adopted to identify a surface recession geometrical pattern using digital photogrammetry data. The methodology for both the photogrammetry survey and the Bayesian inference was previously published in [9], in which it was applied to degraded stone facades, but here it has been applied to brick walls containing different deterioration areas in the same section. Among all the possible candidate regression profiles (parabolic, triangular, bilinear, rectangular, . . .) [9], it has been proved that the one that best fit to the two cases studied of brick wall is the rectangular one. So, the parameterized degradation pattern depicted in Fig. 2 has been assumed. In Fig. 2, two different deteriorated areas can be seen being B1 and B2the wall’s thickness losses and H the height of the wall most affected by degradation, respectively. However, the 360 b o l e t í n d e l a s o c i e d a d e s p a ñ o l a d e c e r á m i c a y v i d r i o 6 1 (2 0 2 2) 357–373 Fig. 2 – Candidate degradation pattern. methodology can be applied to n steps of degradation by using Bayesian inference. The plausible values of the degradation parameters  = {B, H} defining the geometry of the degraded area were obtained in the form of Probability Density Function (PDF) p(|zD), where zDrepresents the data from photogrammetry. In this research, only one candidate degradation pattern (Fig. 2) has been considered in order to compare the basic parameters of the degradation  = {B, H} obtained in the buildings studied, that is: the height and the depth of the degradation. This candidate degradation pattern is described in a probabilistic way by introducing two error terms e1, e2which measure the discrepancy between the degradation pattern zM() and the data in the two different degradation areas of the wall zD: zD=zM()+ e1, zM()≥ H zM()+ e2, zM()< H (1) Using the Principle of Maximum Information Entropy [26], a zero mean Gaussian distribution was conservatively assumed to model error variables e1, e2. Thus, the degradation profile of the wall zDwas represented using the following probability density function: p(zD|)=⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ 22 e1 −Ns 2exp −1 2F() e12, zM()≥ H 22 e2 −Ns 2exp −1 2F() e22, zM()< H (2) where F() is a goodness-of-fit function which is defined as the l2-norm of the measured and modeled data as: F()= ⎛ ⎝ Ns  j=1z−1 j,M ()− z−1 j,D ()⎞ ⎠ 1/2 , with z−1 j,M ()and z−1 j,D ()being the abscissa image of the jth element of the vectors zMand zD, respectively, and eiis the standard deviation of the jth component of the model error, with j = 1,. . ., Ns. Following the Bayesian approach, the posterior distribution of the model parameters p(|zD)can be obtained using Bayes’ Theorem, as follows: p(|zD)=p(zD|)p() p(zD)(3) where p(zD|)is the likelihood function represented by Eq. (1), p()is the prior distribution of the uncertain parameters, and p(zD)is the evidence of the model in representing the data zD. To solve Eq. (3), the Metropolis–Hastings algorithm [27,28] has been adopted for its simplicity and efficiency. Results Real case studies The influence of several factors, such as the chemical and mineralogical composition, the porosity of the bricks, moisture, freeze–thaw cycles, salts and leached alkalis, ground isolation, and pollution in the degradation patterns is illustrated here for cultural heritage brick buildings. In particular, two different Spanish CH buildings Puerta Elvira (Granada) and the Residencia de Estudiantes (Madrid) were investigated. Puerta Elvira (Bab Ilvira, see Fig. 3), is a fortress gate constructed with several materials (bricks, rammed earth and stone). Puerta Elvira was built in the 11th century and has undergone several transformations throughout its history. In 1612, the esplanade in front of the gate was leveled, and houses that were attached to the wall were built, which have remained almost unchanged. Currently Puerta Elvira is formed by the exterior arch (from the 14th century), two rammed earth towers that flanked the arch, and by three brick niches [29]. Throughout the 20th century it underwent depth restoration and consolidation works. Numerous repairs are visible on the main facade. The most important intervention was the inlay of handmade brick pieces of different thicknesses to fill gaps produced by the loss of the original rammed earth in 1957 [29,30]. Puerta Elvira was classified as a “National Historical and Artistic Monument” in 1896, considered valuable to society, and therefore worthy of preservation to future generations. b o l e t í n d e l a s o c i e d a d e s p a ñ o l a d e c e r á m i c a y v i d r i o 6 1 (2 0 2 2) 357–373 361 Fig. 3 – Puerta Elvira, Granada (Spain). Detail of the bricks. Regarding the soil, Puerta Elvira is located over the alluvial fans called Alhambra conglomerate, mostly constituted of rounded stones with an average size of 10 cm that generally has a silty-sandy matrix, that is sometimes clayey [31]. In this research, the brick strip located on the left side of the main facade of PE is analyzed in detail. The Residencia de Estudiantes is an architectural complex made up of four buildings located in the center of Madrid. In 1913, the Pabellones Gemelos were constructed, two linear blocks with an east–west orientation. In 1915, two new buildings were added: the Transatlántico and the Pabellón Central (see Fig. 4). The four buildings are brick facade. In the 1940s an additional floor was added to the Pabellones Gemelos. The complex suffered major deteriorations because of years of neglect until being rehabilitated in the period 1991–2001 (in 1991 Transatlántico, in 1994 Central and in 1998 the Pabellones Gemelos). In 1978, the architectural complex was declared as an Asset of Cultural Interest (BIC), a legal frameword for the protection of Spanish historical heritage, from the Spanish Ministry of Culture. In 1997, both the gardens and pavilions were included in the Protected Buildings Catalog of Madrid, and in 2006, the Residencia de Estudiantes received the title of Spanish European Heritage Site from the European Heritage Committee. The soil in Madrid or Madrid facies are basin edge deposits consisting mainly of quartz-feldspathic sands and silty clays. In this detrital set, two different units can be distinguished, depending on the content of fines. The upper material (from the Terciary of Madrid) is constituted by the unit of Miocene “arenas de miga”, in its geotechnical characterization (detrial materials – arkose–) and “Toscoh” that are arkose with intercalations of brown clays – that define the facies of Madrid [33]. The above description of the soil is a general explanation that does not necessarily represent the characteristics of the ground where the buildings are located. In order to compare both CH buildings, Table 1 shows the main characteristics of the two cases studied (Puerta Elvira and Pabellón Gemelo of the Residencia de Estudiantes). Fig. 4 – Residencia de Estudiantes, Madrid (Spain). (a) Views of the Pabellones Gemelos, (b) General scheme adapted from [32]. 362 b o l e t í n d e l a s o c i e d a d e s p a ñ o l a d e c e r á m i c a y v i d r i o 6 1 (2 0 2 2) 357–373 Table 1 – Comparative table of the main characteristics of Puerta Elvira and Residencia de Estudiantes. Puerta Elvira Residencia de Estudiantes Location Granada (Spain) Madrid (Spain) Main construction material Rammed earth Brick Main material deterioration Brick masonry Brick masonry Construction/Restoration with brick masonry 1957 1913 Soil type Alhambra conglomerate Garden Floor insulation Yes No Hazard Earthquakes, freeze–thaw cycles, storms, erosion of material over time (for example, collision of particles moved by wind, etc.), pollution due to daily traffic. Soluble salts from the soil, fertilizer, freeze–thaw cycles, storms, erosion of material over time (for example, collision of particles moved by wind, etc.) pollution due to daily traffic. Orientation EW NS Fig. 5 – XRD pattern of brick from: (a) Puerta Elvira. (b) Residencia de Estudiantes. Characterization of the bricks Samples of pieces of detached brick have been collected at each structure, Puerta Elvira and the Residencia de Estudiantes. These were damaged bricks since in none of the structures there was no interaction with their walls. XRD results The results of the X-ray diffraction tests are represented in Fig. 5. Table 2 shows the major and minor components observed by X-ray diffraction. Table 2 – Majority and minority crystalline compounds present in the samples studied. Majority crystalline compounds Minority crystalline compounds Puerta Elvira Quartz Calcite and Phengite Residencia de Estudiantes Quartz Microcline and Orthoclase b o l e t í n d e l a s o c i e d a d e s p a ñ o l a d e c e r á m i c a y v i d r i o 6 1 (2 0 2 2) 357–373 363 Table 3 – Chemical composition of Puerta Elvira and Residencia de Estudiantes bricks. Technique Component Puerta Elvira (Granada) Residencia de Estudiantes (Pabellón Gemelo) FRX (melted) SiO258.62% 72.27% Al2O318.60% 15.39% Fe2O36.39% 2.59% CaO 3.47% 1.35% MgO 5.15% 0.00% Na2O 2.44% 2.24% K2O 2.64% 4.83% TiO20.87% 0.36% P2O50.17% 0.13% Gravimetry SO30.55% 0.00% Potentiometry Cl−0.11% 0.00% FRX (melted) MnO 0.09% 0.05% ZnO 0.03% 0.00% SrO 0.07% 0.01% ZrO20.03% 0.02% BaO 0.12% 0.00% Rb2O 0.00% 0.05% Loss on ignition CO20.54% 0.72% Extraction in water (Ion Chromatography) NO2−0.00% 0.00% NO3−0.12% 0.00% Fig. 5a and b, and Table 2 show that the mineralogical composition in both bricks is similar. Quartz is the majority component in both samples. On the other hand, micas and feldspars are the minor crystalline components. In addition, it should be noted that a significant amount of calcium carbonate can be observed in the form of calcite in brick from Puerta Elvira. This calcite cannot be associated with the composition of the brick. XRF results The chemical composition of the bricks, expressed in the form of the most stable oxide, was determined by XRF with a model S8 Tigger 4 kW instrument (Bruker, Billerica, MA, USA). Additionally, and in order to determine compounds associated with environmental pollution such as nitrites, nitrates and carbon dioxide, complementary chemical analysis methods such as gravimetry, potentiometry, loss on ignition and ion chromatography are used. The chemical composition of the bricks analyzed in term of the most stable oxides is shown in Table 3. Table 3 shows that the chemical composition of the two types of bricks analyzed is, in general terms, relatively Table 5 – Porosity accessible to water results. P (%) app (g/cm3) real (g/cm3) Puerta Elvira 29.51 2.08 2.95 Pabellón Gemelo RE 36.99 1.86 2.96 similar. Silica, aluminium, iron, carbon, sodium, and potassium are the most abundant components in both samples. The difference in the chemical composition related to the SiO2content in the bricks studied can be explained by the greater quantity of soluble salts at Puerta Elvira. The chemical composition of the Puerta Elvira brick has been corrected by subtracting the calcite content determined by TGA. Leaching test results The leaching test was performed, as explained above, in order to evaluate the potential extraction of salts and leached alkalis from both brick samples. The results obtained are summarized in Table 4, expressed in percentages. The results consider the leachable salts that contain nitrates, nitrites, chlorides, and phosphates, as well as leached alkalis, in accordance with the criteria established in [34]. Sulfate leaching has also been determined but it is not considered in the total sum of extractable salts, since it can come from the composition of the actual brick itself. The results in Table 4 show that there is a much higher quantity of soluble salts at Puerta Elvira than at the Residencia de Estudiantes. This is associated with environmental exposure and the type of soil on which the two structures are based. The nitrites and nitrates observed at Puerta Elvira are associated with organic contamination. At Puerta Elvira, the chloride ion is associated with the use of de-icing salts in winter. The authors assume that the phosphates come from fertilizers, which is why they are only detected in the soil of the Residencia de Estudiantes, which is located on a meadow. Sodium and potassium ions are observed at both structures, and the values are much higher at Puerta Elvira, which may also be associated with the use of de-icing salts. Porosity accessible to water results Results of the porosity test are summarized in Table 5. Obtained through dry, saturated, and submerged weights, the porosity is expressed in percentages. The results of the apparent density and the real density are also expressed. Table 4 – Results of leaching test for the two samples studied. Leached salts Leached salts Puerta Elvira Residencia de estudiantes ICP K2O 4.9 × 10−1(%) 1.3 × 10−3(%) Na2O 2.0 × 10−1(%) 2.0 × 10−3(%) Potentiometry Cl−2.2 × 10−2(%) 1.1 × 10−3(%) Chromatography NO21.0 × 10−4(%) 1.3 × 10−6(%) NO31.8 × 10−2(%) 2.1 × 10−5(%) P2O50.0 (%) 1.7 × 10−6(%) Total 7.7 × 10−1(%) 3.5 × 10−2(%) 364 b o l e t í n d e l a s o c i e d a d e s p a ñ o l a d e c e r á m i c a y v i d r i o 6 1 (2 0 2 2) 357–373 Table 6 – Average environmental parameters in the last 40 years. Environmental parameters Puerta Elvira (Granada) Residencia de Estudiantes (Madrid) Maximum Temperature 39.7 ◦C 37.9 ◦C Minimum Temperature −5.7 ◦C −2.5 ◦C Temperature variation 56.8 ◦C 48.0 ◦C Wind, maximum wind gust 79.7 km/h 78.9 km/h Rain, average rainfall 362.5 mm 391.6 mm Fig. 6 – Photogrammetric point cloud representation with the degradation profiles (S1, S2, S3, S4, S5) of Puerta Elvira (a) and Pabellón Gemelo of the Residencia de Estudiantes (b). Environmental exposure The mean values of the different environmental exposure parameters analyzed and the maximum temperature variation have been calculated. The values are summarized in Table 6. Geometrical 3D data The photogrammetric point cloud (PPC) has been obtained by using the methodology described in [9], and it is shown with the selected sections of both CH buildings in Fig. 6. The photogrammetric data zDhas been obtained up to 2 m above ground level, since this measurement contains two different degradation areas in both buildings. In both cases, the same number of degradation profiles (S1, S2, S3, S4, S5) separated by 50 cm have been considered to have the same a priori information. The degradation profiles have been obtained from the PPC using the CloudCompare software [35]. Preliminary data processing was carried out, removing the pavement and making a projection on the tangent line to the smallest deteriorated area. For illustrative purposes, the data processing is represented in Fig. 7 for Puerta Elvira. Table 7 – Prior information of model parameters. CH building Prior PDF p(|M) Puerta Elvira Residencia de Estudiantes 1= B1U(1e−5, 6e−2) U(1e−5, 6e−2) 2= H U(1e−5, 1.5) U(1e−5, 6e−1) 3= B2U(1e−5, 8e−3) U(1e−5, 1e−2) 4= ␴e1 U(1e−3, 4e−2) U(1e−3, 4e−2) 5= ␴e2 U(5e−4, 1e−2) U(5e−4, 8e−3) The final degradation is represented in Fig. 8 for the two buildings studied. The difficulty in fitting the values of the geometry degradation parameters  = {B, H} can be seen simply by observing the data. So, the plausible values of the geometry degradation parameters ␪ = {B, H} have been obtained in the form of the Probability Density Function p(|zD) using the Bayesian methodology proposed. Surface recession patterns The prior information of the model parameters  used in Eq. (3) is represented by uniform distributions and summarized in Table 7. Note that the Standard Deviation of the prediction errors e1, e2is assumed to be part of the set of uncertain parameters . Samples from the posterior PDFs (p(|ZD)) of the model parameters were obtained by using the M–H algorithm with 30,000 realizations in both buildings. The prior and the posterior PDF results of each individual parameter (1= B1, 2= H, 3= B2, 4= e1, 5= e2) are represented in Fig. 9 for Puerta Elvira and Residencia de the Estudiantes. The reduction in uncertainty from the prior PDF (dashed line) to the posterior PDF (solid line) is worth noting. Fig. 10 represents the forward model simulation results using the posterior PDFs of the model parameters as inputs in Eq. (2) for the two buildings studied. It can be observed that the candidate degradation profile (see Fig. 2) represents the experimental data reasonably well in both cases. In addition, note that model output represents the variability and complexity in the data as two uncertainties, one in the non-deteriorated data where the uncertainty is less than the deteriorated data, which presents the highest variability and complexity. Finally, it can be concluded that the mean degradation depth is similar in both buildings, however the mean degradation height is higher in Puerta Elvira (H = 0.8 m) than in the Pabellón Gemelo of the Residencia de Estudiantes (H = 0.25 m). Fig. 10 shows that the posterior mean degradation of the brick walls is larger in the case of Puerta Elvira than the Residencia de Estudiantes. The most significant difference is the degradation height, which is four times greater at Puerta Elvira. By assuming uniform degradation over time, Fig. 10 leads to the velocities of mean degradation indicated in Table 8. It can be observed that the degradation is much greater in height than in depth, which indicates good brick behavior. The average degradation speed, both in height and thickness, is higher at Puerta Elvira. This seems to be related to the environmental exposure factors of the bricks. b o l e t í n d e l a s o c i e d a d e s p a ñ o l a d e c e r á m i c a y v i d r i o 6 1 (2 0 2 2) 357–373 365 Fig. 7 – Data processing of Puerta Elvira. Fig. 8 – Measured degradation profiles of (a) Puerta Elvira and (b) Pabellón Gemelo of Residencia de Estudiantes. Table 8 – Mean degradation velocity assumed as uniform. Uniform mean degradation velocity Puerta Elvira Residencia de Estudiantes Degraded height 13.67 mm/year 2.3 mm/year Degraded depth 0.33 mm/year 0.18 mm/year Discussion At present, the brick facades show evidence of deterioration, mainly in the lower part of the walls. The deleterious effects cause damage that can be seen in the rounded corners of the bricks as shown in Fig. 6. Different groups of parameters related to the degradation of both structures have been considered. The different groups of parameters to be correlated are grouped as: • Physico-chemical factors: Total of potentially soluble compounds, CO2, soluble salts, open porosity and density. • Environmental factors: Maximum temperature, minimum temperature, temperature variation, maximum wind gust and average rainfall. • Degradation factors: Maximum degradation height, maximum degradation depth, height at which the greatest degradation depth occurs and depth at which the greatest degradation height occurs. Table 9 contains a summary of the parameters measured, in group of factors, and their normalization to 100% with respect to the largest parameter. This normalization has been carried out in order to be able to represent each group of parameters comparatively. Figs. 11–13 show the relative influence of each parameter. In Fig. 11, the physico-chemical factors are represented. 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