Keywords

1 Introduction

Heritage constructions stimulating the interest of the scientific community for the development of non-destructive strategies and methodologies to support the maintenance of structural safety. Especially due to the damage mechanisms, constructive issues and its own heterogeneity, to assess masonries by non-destructives tests (NDT) has been a complex task. While since 60-decade several advances have been done in most isotropic materials, as steel and concrete, application on anisotropic materials as masonry are missed in the literature [1].

According to ICOMOS recommendations [2], ancient structures must be analyzed with less invasive investigation techniques and the rehabilitation measures must be carried out with the maximum efficiency and careful way, in order of do not introduces new damages to the structure. Among the usual tools for evaluating the structural characteristics, due to repeatability without introduce damages as well the necessity of sample extraction, NDT appears as a promising tool to analyze the properties of the historic buildings. With satisfactory results, as seen in the mechanical characterization of the Historic Center of Bragança (Portugal), NDT were performed with recurrence to the application of flat monkey test presented by [3]. In another hand, the work developed by [4] in a controlled environment showed that the employment of thermography can provide a significant contribution to structural assessment, but needs more developments on this technique before be largely use by the technical community.

According to [5], non-destructive tests can be also very useful for determining the characteristics of historic masonry. By correlation between sonic waves velocities and mechanical properties from samples experimentally tested in the laboratory, it is possible estimating mechanical parameters of the structure in situ. However, this experimental was limited to direct ultrasonic measurements and was baes on ultrasonic velocities values interpretation. But, even with these limitations, ultrasonic tests stand out as quite promising NDT.

In addition, ultrasonic data has been extensively used for masonries characterization, as the works developed by [6,7,8], but also limited on interpretation of ultrasonic velocities pulse. Although promising, the use of more sophisticated statistical methods in the analysis of ultrasonic data to assess the condition of a building has not been widely explored, as well the interpretation of the pattern distribution of ultrasonic waves over an area are not commonly reported on the literature.

The employment of sonic tests as a characterization tool in traditional buildings has been each more reported in the literature especially since from 90 years. Although technical reports show that different types of masonries were tested, the methodology of ultrasonic data acquisition or processing has not been significantly changed since then [9,10,11,12,13,14,15]. Basically, the common recommendations noted of the literature consists in the necessity to perform NDT in each analyzed material and the interpretation of the results are limited to the tested sample. Basically, since from 90 decade, the waves propagation velocity values have been used as a link to correlate the properties of the assessed material.

In a report, [16] performed sonic test on the existing walls of the Cathedral of Noto. A relevant portion of this structure was damaged in 1996 due to a collapse. The intention of these investigations was to verify the state of conservation of the walls and pillars in view of the reconstruction of the damaged part of the Cathedral. The results show that sonic tests are sensible and can be successfully applied for the diagnosis and control of the effectiveness of repair techniques.

Ultrasonic test is based on the emission and reception of ultrasonic waves. The characteristic parameters of sonic waves are velocity, amplitude, phase, frequency and attenuation. The stiffness of the structure and the material density interferes with velocity, while the wave attenuation parameter is an indicator of fractures and compaction conditions [17, 18]. The levels of material deformation associated with sonic tests are sufficiently small, assuming material behavior according to the theory of elasticity as a theoretical basis for the interpretation of the results. These tests result in specific values for the velocity of wave propagation, from which the elastic modulus can be derived. Thus, making possible to characterize the material or identify intrinsic anomalies. According to [19], this method is widely used and consolidated for concrete and steel samples and consisting in emitting an ultrasonic wave through the element and measuring its propagation time.

Recent advances on employment of ultrasonic techniques for masonry characterization have point to:

  • Sonic and ultrasonic tests can be used to assess the effect of retrofitting techniques in masonries, as showed by [16];

  • Sonic and ultrasonic tests can used in a combined way with others NDT and semi-destructives strategies for estimating anisotropic material mechanical properties, as presented by [5, 10];

  • In sonic and ultrasonic tests, the masonry joints can influence the waves behavior [20, 21];

  • Ultrasonic tests can be used to identify heterogeneities in ancient masonries and to identify if a join of walls presents similar constructive properties, as well the work also shows that the loading and opening sections can influence the ultrasonic waves behavior in masonries [6];

  • The presence of a mortar recovery layer (15 mm of thickness) does not presents significant influences on the ultrasonic wave behavior, as suggested by [21, 22];

In 2020, an important contribution to employment of ultrasonic test to characterize ancient masonries [23], especially concerning clay bricks masonries was the publication of the ABNT NBR 16805 [24]. In this code, a methodology for collect ultrasonic data is presented and can be used to characterize the wall homogeneity, detection of anomalies, monitoring materials changes, as well to assess retrofitting measures impact. Nonetheless, in order to enlarge the application of ultrasonic tests, it is needed to proceed with scientific investigation on how different situations or materials conditions can affect the behavior of the waves inside a material.

This way, in order to give a contribution to the advance for the employment of non-destructive tests for masonries characterization, the present work has as the main aim to propose a novelty approach for detection of voids in ancient masonries based on ultrasonic data collected by indirect measurements.

2 Statistical Approaches for Ultrasonic Data Exploration

In this section, the methodology for ultrasonic data analysis proposed in this work based on statistical tools is presented.

The frequency of a wave propagation is related with Dynamic Modulus of Elasticity. Thus, ultrasonic waves can be used to analyze material properties. In the literature, several examples of the application of ultrasonic waves for analysis and properties estimating of steel, wood, stone and concrete are often reported. According to ABNT NBR 8802 [25], ultrasonic tests performed in concrete specimens can be used to analyze the material homogeneity, estimating the compressive strength, detection of anomalies or crack depth. So, if the properties of materials are knowing and the behavior of the wave propagation across the material sections as well, this kind of testing can detect heterogeneities and be useful for estimating mechanical characteristic. Thus, characterize and understand how some anomalies interferes in the wave behavior in a determined material is the way for advances the employment of ultrasonic tests to most heterogeneous materials, as masonries, for instance.

The method proposed in this work for ultrasonic data processing initially consists into plot the ultrasonic waves data of each one frame as function of the time of wave propagation versus distance. The Fig. 1 shows the 24 curves (12 curves from P1 and 12 curves from P2), here so-called “profile” of the wave behavior. In statistic, these types of curves are often used to analyze the behavior of a data collected by repeated measures at the time.

Fig. 1
figure 1

Profiles of the walls frame plotted in time of wave propagation versus distance

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In this work, the repeated measure was also performed. The data refer to the time that an ultrasonic wave takes to go from a point, x0, till the positions x1, x2,… x6 in each frame. Due to the methodology of data collection and considering that the material presents similar properties, it is expected a linear tendency for each one of the profiles. Based on that, some issues were highlighted in this work:

  1. 1.

    Is it possible to observe any change in the linear tendency of the ultrasonic walls profiles?

  2. 2.

    Assuming that is possible to observe changes in the linear tendency of the ultrasonic wall profiles, can the profiles be used to identify an anomaly in a masonry wall?

2.1 Mixed Model Theoretical Background

A regression analysis with mixed-effect is a methodology where the effect of each one of the frames are considered in the analysis by including a vector of random variable in a regression model, here assigned as by b. This model is called in the statistical literature as a mixed model, for details about this kind of statistical model see for example [26,27,28].

For a brief explanation of the mixed model, consider a sequence (Tij, Dij) (i = 1, 2, …, N e j = 1, 2, …, 6) where Tij represents the time that the wave takes to range the j-th distance, Dij, in the i-th frame, and N the total of aniseed frame. Each Dij is determined by the distance between the point x0 and each point xi, for i = 1, …, 6, as described before, all points aligned in the same horizontal line.

The measurements of a cross of the frames is assumed as independents, and a mixed model is used to take into a count just the correlation between the data measured in each frame. So, consider that the time can be described as a function of the distance ranged by the ultrasonic wave, as:

$${\text{T}}_{\text{ij}} = {\text{f}}\left( {{\text{D}}_{\text{ij}} ;{\varvec{\upbeta}};{\mathbf{b}}} \right) + \upepsilon_{ij} ,$$
(1)

where f(.) can be a linear or non-linear function both in the parameters vector β and random-effect b, as the distance that is used to explain the time denoted by D; and ϵij is the term of error, considered here following a normal distribution with mean zero and variance constant, it is commonly denoted by: ϵij ∼ N (0, σ2).

The function f(Dij, β, b) in Eq. (1) describes the relationship between the wave propagation time and the distance range by the ultrasonic wave for this time. This relationship can be described by a non-linear function in Dij, as for instance, a polynomial curve with degree G, G > 1. The quantity β is a parameter vector that represents the common effect to all frames, this is called fixed effect parameter, and the vector b represents the effect of each frame in the model, and can be different across all of them, this is called mixed-effect variable.

In order to complete the model described in Eq. (1) let consider a normal distribution for each bi in b = (b1, b2, …, bN) as follows:

$${\text{T}}_{\text{ij}} = {\text{f}}\left( {{\text{D}}_{\text{ij}} ;\upbeta;{\mathbf{b}}} \right) + {\upepsilon }_{\text{ij}}$$
$${\upepsilon }_{\text{ij}} \sim{\text{N}}\left( {0,\upsigma^{2} } \right)$$
$${\text{b}}_{\text{i}} \sim {\text{N}}(0;\varSigma_{\text{b}} ), \,{\text{for}}\,{\text{i}} = 1, \ldots ,{\text{N}},$$
(2)

where bi ∼ N(0; Σb) represents that the bi follows a normal distribution with mean (or mean vector) µij and variance (or variance matrix) Σb.

In this work a polynomial mixed effect model with G = 3 was choose in Eq. (2) for the function f(Dij;.) and the vector b was considered as a scalar, b = b0. Then, this model became:

$${\text{T}}_{\text{ij}} = {\text{b}}_{0} +\upbeta_{0} +\upbeta_{1} {\text{D}}_{\text{ij}} +\upbeta_{2} {\text{D}}_{\text{ij}}^{2} +\upbeta_{3} {\text{D}}_{\text{ij}}^{3} + {\upepsilon }_{\text{ij}}$$
$${\upepsilon }_{\text{ij}} \sim {\text{N}}\,\left( {0,\upsigma^{2} } \right)$$
$${\text{b}}_{\text{o}} \sim {\text{N}}(0;\varSigma_{\text{b}} ),$$
(3)

Note that a polynomial curve can be also considered for the random-effect, in this case, the function f(.) became \({\text{f}}(.) = {\text{b}}_{0} +\upbeta_{0} + \left( {\upbeta_{1} + {\text{b}}_{1} } \right){\text{D}}_{ij} + \left( {\upbeta_{2} + {\text{b}}_{2} } \right){\text{D}}^{2}_{\text{ij}} + \left( {\upbeta_{3} + {\text{b}}_{3} } \right){\text{D}}^{3}_{\text{ij}} + {\upepsilon }_{\text{ij}}\).

To avoid the unwanted correlation between the transformation of the distance used in the model D, D2 e D3 in Eq. (3), the orthogonal polynomial is used in the expression \({\text{T}}_{\text{ij}} = {\text{b}}_{0} +\upbeta_{0} + \,\upbeta_{1} D_{\text{ij}} +\upbeta_{2} {\text{D}}^{2}_{\text{ij}} +\upbeta_{3} {\text{D}}^{3}_{\text{ij}} +\upvarepsilon_{\text{ij}}\) instead the original polynomial.

Some approaches can be used to obtain estimative of the β and bq. In this work, the Bayesian methodology is adopted for allow also to obtain estimative of the values of bi [29].

The disadvantage into adopt a polynomial function for modelling the global tendency of wave propagation is to impose a behavior that not necessarily occur. So, for an initial exploration of this behavior, be important to consider a most flexible method. This way, a powerful method for modelling tendencies without to impose an analytical function is presented in the next section.

2.2 LOWESS Method Theoretical Background

The LOWESS method was proposed by Cleveland et al. (2017), and initially was called Loess Method. In this approach, for each considered distance D a linear regression is fitted in the neighborhood of the D using the last square method. For an explanation of this method, consider for each considered distance D a linear regression model is fitted in a set of point in the neighborhood of D determined by a constant h as:

$${\text{T}}_{i} \left( {\text{D}} \right) =\upbeta_{0} \left( {\text{D}} \right) +\upbeta_{1} \left( {\text{D}} \right){\text{D}}_{i} +\upvarepsilon_{i} ;\,{\text{for}}\;{\text{D}}_{i} \in \left[ {{\text{D}} - {\text{h}};{\text{D}} + {\text{h}}} \right],$$
(4)

where T is the time expended by the wave to range the i-th distance Di and ϵi is the term of error in the regression assumed with mean zero and variance constant for all ϵi. The points surrounding to D in the data can be choose subjectively or in a subjective form from the observed data. Note that, in this approach, several linear regressions can be fitted in the data considering only two parameter (β0(D) e β1(D)). Thus, the set of the points considered in each regression can be seen as a window around each D, in the cartesian plan where each regression model is considered.

For reducing the weight of the contribution of a particular point Di far away from D, in comparison with the other points in the neighborhood, weights can be associated with local regression models, given by Eq. (4). With this inclusion, the methodology consists basically to estimate β0; β1 and σ2 in each linear regression, in order to minimize the sum:

$$\Sigma \left\{ {\upomega_{\text{D}} \left( {{\text{D}}_{\text{i}} } \right)\,\left[ {{\text{T}}_{i} {-}\upbeta_{0} \left( {\text{D}} \right){-}\upbeta1\left( {\text{D}} \right){\text{D}}_{\text{i}} } \right]2} \right\};\,{\text{for}}\;{\text{D}}_{\text{i}} \in \left[ {{\text{D}} - {\text{h}};{\text{D}} + {\text{h}}} \right],$$
(5)

For defining the weights ωD (Di) in Eq. (5), a function Kernel is assumed as follow

$${\text{K}}_{\text{D}} \left( {\text{u}} \right) = \left( {1{-}\left| {\text{u}} \right|3} \right)3,\;{\text{for}} - 1 \le {\text{u}} \le 1;$$
(6)

where the weights ωD (Di) computed as:

$$\upomega_{\text{D}} \left( {{\text{D}}_{\text{i}} } \right) = {\text{KD}}\left( {{\text{D}}_{\text{i}} /\left( {{\text{D}} + {\text{h}}} \right)} \right).$$
(7)

For use this model in ultrasonic data, the R package as used by the library “ggplot2” performed by [30].

3 Experimental Setup

Two specimens of typical ancient walls from Ceará State were built in the Laboratory of Rehabilitation and Buildings Durability of the Federal University of Ceará. The first wall, here assigned as P1, was built without any anomaly and take as the reference wall, while the second wall, here assigned as P2, was built with some anomalies in its interior, essentially voids. Figure 2 shows P1 and P2.

Fig. 2
figure 2

Ancient specimens of walls used in this experimental setup

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Each one of the walls were built in the dimensions of 1.50 m of high, 1.00 m of width, and 0.135 m of stiffness. To the walls building ancient clay bricks with dimensions of 4.60 cm × 24.70 cm × 12.30 cm and specific mass of 2000 kg/m3, joined by a lime mortar in the proportion 1:1 (lime: fine aggregate, in mass) was used. The walls had one of the surfaces recovered by a 15 mm of lime mortar in the proportion 1:6 (lime: fine aggregate, in mass). The walls were left 40 days before starting the experimental measurements.

To ultrasonic data collecting the equipment model Pundit® 200 PROCEQ, with transducers of 54 kHz was used. The indirect ultrasonic method was used to collect the ultrasonic data of the walls. In this method, both transductors, emitter (E) and receiver (R) are positioned aligned in the same surface, as shows the schemes of the Fig. 3. Indirect ultrasonic method is used in the situation that the direct ultrasonic measurements are not possible, as for instance, limitation by dimensions of the assessed element.

Fig. 3
figure 3

Scheme of the indirect ultrasonic measurements

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The ultrasonic measurements were performed in the recovered surface of the walls. In each one of the walls, ultrasonic collecting data were performed in the horizontal direction (axis X) and vertical direction (axis Y). For that, the walls were divided into 12 frames with 20 cm × 40 cm and the ultrasonic measures were performed for each one of the frames. In order of the be free from walls boundaries influences, a distance of 10 cm from the wall edges were not considered during frames division. Also, 20 cm from the top and the bottom region of the walls were left out. Figure 4 shows the frames assigned in the walls.

Fig. 4
figure 4

Wall with the ultrasonic frames assigned

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In each frame, 6 ultrasonic measurements distributed along of the frame central region were done. The emitter and receiver transductor were distant since point x0, till the points x1, x2, …, x6. The emitter (E) is static during the ultrasonic measurements, while the receiver (R) vary according to distances linearly increasing. So, for each distance between the point x0 and a point x1, x2, …, x6, the time of ultrasonic wave propagation was measured. Figure 5 shows the schematic view of ultrasonic indirect measures performed in the frames.

Fig. 5
figure 5

Schematic view of the indirect ultrasonic measurements performed inside the walls frame

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The method for indirect ultrasonic measurements adopted in this work was based in the recommendations of the ABNT NBR 16805 [24] and in the work presented by [6].

4 Results

Initially, by Fig. 1 a linear and growing tendency can be observed, once the time spends by a wave to travel from a point to other is proportional to the distance. However, by Fig. 1 analysis it is not possible to identify if all profiles are following the same expected behavior, namely, linear and growing tendency. Since a most appropriate analysis is performed, it is possible to observe that in some profiles, a disagreement with the global behavior is noted. For that, a mixed polynomial regression model was loaded and the effect of each frame to the global curve was modelled (Fig. 6).

Fig. 6
figure 6

Curves of the ultrasonic profiles plotted by logarithmic transform for time

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Considering the results obtained by the mixed model, the Table 1 shows the random effect estimated from this model, showing also a credibility interval that contain the value of b0 with 95% credibility. The credibility interval represents the influence of each frame in the global tendency of the profiles. In this table, five frames can be observed as having different behavior, with a credibility of 95%, because the random effect can be considered as different of zero with 95% of certain, since zero is included in the credibility interval. Among this frame, four of them have anomalies.

Table 1 Estimated effects of the frames to estimated degree polynomial curve, G = 3, and 95% credibility intervals
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Table 1 shows the estimative of the individual frame effects to the global behavior of the curve, as well shows the credibility intervals that makes possible to identify if the frames are affecting the global behavior of the curve or not. In the case that the random effect presents zero in its interval that means that the value is very near to zero, making the distance between the obtained curve since the frame considered in relation to global curve do not be significant. Based on that, it is possible to identify anomalies on the behavior of the ultrasonic waves in the region of Q2, Q6, Q8, Q9 and Q10 from the P2 wall (Fig. 7). In addition, an anomaly as also noted in Q2 region in P1 wall. The results presented by Table 1 suggest that even if a not flexible curve be imposed, it is possible to verify frames with anomalies influencing the global tendency of the curve.

Fig. 7
figure 7

Some panel quadrants with anomalies

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Following, the Loess model was used to analyze the ultrasonic data. For the data processing, routines R Core Team [31] were used with recurrence to ggplot2 pack available by [30] to fit smoothed curves since ultrasonic data. Figure 6 shows the smoothed curves for each one of the frames with and without anomalies.

The results have shown that the data series are largely correlated, not only because they are from the same samples, but also due to the fact that were obtained by the same points. So, it is expected that a higher correlation is observed between points from the same frame, than between different frames. Between two points observed in different frames, the wave propagation can be different according to material density.

To analyze the global tendency of the profiles, the Lowess method was used to adjust the mean curve of the data from each one wall. shows two smoothed curves where can be noted two different behavior for both global tendencies. To wall sample without anomalies (P1), it can be noted that the tendency is more linear, while in the wall with anomalies (P2), a change in the curve behavior over around the third measurement point is noted. This difference between both curves is better represented by Fig. 8.

Fig. 8
figure 8

Smoothed curves plotted by Lowess method

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In order to identify the curves behavior and the frames that are disturbing the smoothed curves, Fig. 9 was plotted including profiles assumed with disturbing effect to the curves. By these profiles, Q8, Q5 and Q1 presents similar behavior and with potential to influence the global tendency of the curve. To evidence the disturbing effect promoted by anomalies presence, Fig. 10 presents 3 smoothed curves, the first to the wall without voids, the second to the wall with voids presence but without the disturbers profiles and the third curve with frames classifieds as disturbers by visual analysis. It is possible to note that when the disturbers are excluded from the smoothed curve of the wall with the anomaly, the new curve came back to presents a linear and growing tendency.

Fig. 9
figure 9

Disturbers profiles identified by Lowes approach

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Fig. 10
figure 10

Behavior of the curve profiles according with behavior

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Considering the data from ultrasonic measurements on clay bricks walls, the observation on the disturbers of the measured data shows that the voids can change the behavior of the waves in a localized way, allowing, thus, the local identification of a void even if the wall presents a cover layer with mortar, with 15 mm of thickness.

An additional advantage about the application of Mixed-effect and Lowess approaches to process ultrasonic data is the possibility of the data visualization in a panoramic way, being possible to identify how the individual measures from a frame are influencing the global data. This proceed makes possible the classification of the different behavior between the analyzed data.

5 Conclusion

Ultrasonic analysis can be a powerful tool to support materials characterization, even if this one presents a high level of anisotropy. The results presented show that the ultrasonic data can be used to analyze the presence of voids inside ancient wall structures. For anisotropic materials like masonries, the disturbances in the wave distribution profile can be used to better understand the anomalies presented as voids.

The results demonstrate how voids can influence disturb in the time ultrasonic wave propagation, making it possible to not only identify the anomalies presence, as well its location. Additionally, this method also can be used in walls with presence of a recovery layer of mortar till 15 mm of thickness.

The combined approaches employed in this work, namely Mixed effect and Lowess method, provided a novelty approach to ultrasonic data processing. While Mixed effect can be used to identify if the curves presents a data join with disturbing effect to the global behavior of the curve, Lowess method can be used to confirm the disturbers.

Thus, this work contributes to advances on ultrasonic characterization of ancient walls, introducing a new approach to ultrasonic data processing based on pattern distribution of ultrasonic data by indirect measurements. Once each masonry presents its own settlement pattern of ultrasonic data that can vary according with masonry materials composition, geometry and arrangement, to understanding how the ultrasonic waves by indirect measurements behaves is a fundamental way to analysis the voids and others anomalies presence, or even the heterogeneities between a join of walls from the same building.

Finally, this work also demonstrates that concerning ultrasonic masonries characterization, while the analysis of the values obtained by ultrasonic tests be used in several researches to characterize the masonries properties, as ultrasonic velocities, presents limitations on its application, the pattern distribution of ultrasonic waves can be an alternative way to characterize masonries, opening a new field for investigation.