#### Abstract

It is currently known that using stress wave and drilling resistance to detect the internal damage in the ancient timber structure is not a highly precise process. To improve the detection precision of this process, a simulation test was used to detect the internal damage of poplar and elm in ancient buildings. In this empirical study, we compared the detection precision of these two detection methods. Based on the idea of variable weight, we introduced three combined forecasting models based on the IOWA operator, IOWGA operator, and IOWHA operator to predict the internal damage in the ancient timber structure. The results show that the combined forecasting model based on the IOWA operator is more effective in predicting compared to a single detection method and other combined forecasting models. To be more specific, the results show that the detection precision of the combined model is increased by 25.8% and 4.7%, respectively, compared to the precision of the stress wave and drilling resistance tests. The error indicators of the combined forecasting model based on the IOWA operator are better than those of the other combined forecasting models. In addition, the analysis results based upon cross-validation theory show the combined forecasting model based on the IOWA operator has the best applicability, which provides a new practical method for evaluating internal damage of timber components in ancient buildings.

#### 1. Introduction

Ancient timber structures have high historical value, artistic value, and cultural value. Because wood is a natural material, it is easily affected by environmental factors during its life cycle. Split, insect attacks, hollow, decay, and other damages (see Figure 1) are easily found in the wooden components of ancient buildings. These may often result in some structural abnormalities of timber components [1]. Therefore, appropriately and precisely predicting the internal damage of timber components in ancient buildings is of great importance for sustaining the health and safety of ancient timber structures.

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In recent years, national and international scholars have conducted many studies in detecting the internal damage of timber components in ancient buildings and have achieved good results. Commonly used nondestructive testing methods [2] for detecting the internal damage of timber components mainly include stress wave [3–5], X-ray scanning [6–8], drilling resistance [9–11], and ultrasonic wave [12–14]. However, it is difficult to achieve precise detection by using only one single detection method because each of these methods has its own advantages and disadvantages [15].

The basic principle of stress wave nondestructive testing is that one end of the wood is subjected to the impact force, and the stress wave propagates inside. The properties of the wood material are determined by measuring the change of the stress wave propagation velocity. Riggio et al. [8] stated stress waves could be used on-site for the identification of internal macroscopic defects in wooden structures. However, the described technique permits only qualitative and large-scale analysis. Other previous researches, however, have also shown the stress wave detection method is simple, convenient, fast, accurate, and of low cost, and it is not easily affected by the detection environment and suitable for field detection [16, 17]. In contrast, other studies found that the stress wave detection method should be improved. For instance, Li et al. [18] presented the stress wave velocity model to diagnosis the internal defects in urban trees. According to Guntekin et al. [19], the imaging results and detection precision still need to be improved, although many stress wave detection devices are capable of generating two-dimensional or three-dimensional tomographic images of wood cross sections. Sun and Wang [20] found that the two-dimensional image detected by the stress wave was not accurate enough to show the shape of the decayed area.

For X-ray scanning, since wood parts and wood defects have different X-ray absorption capacities, the image formed by the X-ray scanning technique is also different, and the internal defects of the wood can be measured according to a certain process identification analysis. Lechner et al. [6] found that X-rays allowed a view into the structural member or the connections. Riggio et al. [2] used X-rays to test the original roof timbers of the Saint Anne’s Church in Prague, Czech Republic. The results showed when the loss of wood could not be determined visually, it is possible to estimate the extent of the void by measuring the dark area on the radiograph. Wei et al. [21], however, pointed out that X-rays had a low spatial resolution of the images when detecting wood defects. In addition, because of the high costs, equipment mobility issues, and health risks, this method is usually employed only in industrial environments [22]. For instance, Yu et al. [23] reported that the use of X-rays was neither portable nor practical in field assessment, and this approach posed a radiation hazard toward the users.

The drilling resistance method can quickly obtain the defect results inside the wood according to the resistance curve. Nowak et al. [11] used the drilling resistance method in in situ assessment of structural timber to assess the extent of wood damage in the tested elements. Nowak et al. also found that the drilling resistance graph was influenced by many factors, including angle and direction of drilling, interwood moisture, and drill bit sharpness. Chang et al. [24] and An et al. [25] found that the result detected by drilling resistance could only reflect the wood damage condition on the probe path; thus, it was difficult to give a specific and intuitive three-dimensional image. In order to obtain more precise information, more drilling resistance paths should be provided [26].

The ultrasonic spectrum technology primarily detects wood defects based on changes in the velocity of the ultrasonic waves in the wood. The last commonly used method to be examined is the ultrasonic wave method. Dackermann et al. [5] adopted the ultrasonic echo technique to obtain the direct localization of a reflector such as a backwall or any inhomogeneity or damage in the wood element. But it is difficult to locate the exact position of damage within the specimen and distinguish between one large irregularity such as a knot and a cluster of small ones. Perlin et al. [22] also used ultrasound to determine an accurate pith location. When applied to a given wood structure, this method can improve its structural assessment given by other nondestructive methods such as drilling resistance, stress wave transmission, and stress wave tomography. Due to the complexity of the wood structure, there are still problems to be solved by applying the ultrasonic technology. For example, the air gap between the ultrasonic probe and wood requires a good coupling agent [16].

Although many methods provide nondestructive assessment of the internal damage of old wood in timber structures, no single method can provide a complete data set for analysis [11]. In recent years, many scholars have used a combination of detection methods to detect internal damage of wood components [8, 15, 20, 24–31]. However, currently, there are not many methods to quantify uniformly the detection results detected by different methods. In order to overcome the loss of information caused by a single detection method and quantify uniformly the detection results, combined forecasting methods [32] are introduced to detect and predict the internal defects of wooden components. Chang et al. [24] proposed a combined forecasting method based on the Shapley value to predict the internal defects of wooden components. The weight coefficients of the stress wave and drilling resistance tests were fixed under each working condition, without considering the dynamics of the detected effects. However, the detected precision of this detection method is not the same under different working conditions. The detected precision is high under a certain working condition, but when it is under another different working condition, it may be low. Therefore, this combined forecasting method could not provide a consistent precise prediction and still needs to be improved.

In order to overcome the disadvantage of assigning weight in the previous combined forecasting model, this paper introduced the idea of variable weight. According to the detected precision of a single detection method under various working conditions, the weight coefficients were given in an order from high to low. This greatly reduced the sensitivity of the result to a poor detection method. Additionally, it effectively improved the forecasting precision of the internal damage of wooden components in ancient buildings. The following four steps are used for evaluating the internal defect of wood components in ancient buildings based on the optimal combined forecasting model: Step 1: considering the relative lower costs of equipment and simpler execution in field applications, test methods based on stress wave and drilling resistance were used to detect the internal defects of poplar and elm based on the idea of reverse simulation. Step 2: based on the idea of variable weight and taking the square sum of error, square sum of logarithmic error, and square sum of reciprocal error as a guideline, three combined forecasting models were established based on the IOWA (induced ordered weighted average) operator [33–35], IOWGA (induced ordered weighted geometric average) operator [36, 37], and IOWHA (induced ordered weighted harmonic average) operator [38, 39]. Additionally, the combined forecasting models based on the entropy value [40, 41] and Shapley value [24–42] were used to compare with the proposed methods. Step 3: based on the five indicators, a comprehensive evaluation index was developed to select the optimal combined forecasting model. Step 4: according to the cross-validation theory [43], the optimal combined forecasting method was generalized into a model.

#### 2. Nondestructive Tests

##### 2.1. Specimen Fabrication

Poplar and elm, commonly used in ancient building timber components (e.g., beams and columns in the Guanyin Temple, Changzhi City, Shanxi Province), were selected as test specimens to simulate the hollow and insect attacks in the wooden structure. The raw materials were sawed into a cylindrical shape of 100 mm height (see Figure 2(a)), and the sawing plane of the test piece was required to be flat. Based on the cross-sectional area (*S*) of the test piece, there are five simulated damage ratios, which are, respectively, 1/32, 1/16, 1/8, 1/4, and 1/2 of the cross-sectional area (*S*) of the test piece (see Figure 2(b)). According to the method of reverse simulation, the internal hollow and insect attacks were simulated by manual digging (see Figure 2(c)) and drilling (see Figure 2(d)) in the cross sections of the wood component.

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##### 2.2. Test Hypothesis

Considering that wood is an anisotropic material, the physical properties of wood differ in different positions in the same tree. When the water content is the same, the wave propagation velocity increases linearly with the increase of density [44]. In the radial direction of the trunk, the change in wood density is divided into three cases: (1) increase from the bark to the pith, (2) first increase and then decrease from the bark to the pith, and (3) decrease from the bark to the pith [45]. To reduce the effect of this difference on the test results, we designed the damaged area of the test piece to be circular. Additionally, circular damaged areas of different sizes can better reflect the degree of internal damage of the wood. Therefore, we made two kinds of assumptions:(1)It is assumed that the circumference of the specimen is a complete circle, regardless of the special shape of the wood(2)It is assumed that the damage type of each specimen is a standard circle (see Figure 3)

##### 2.3. Test Conditions

The indoor temperature is 20°C and the air relative humidity is 65%, which meet the requirements of the “Standard for test methods of timber structures” (GB/T 50329-2012). Test equipment includes Fakopp (stress wave test equipment) made in Hungary, IML-RESI PD500 (drilling resistance test equipment) and GANN-HT85T (wood hygrometer) made in Germany, and electric percussion drill (BOSCH), which are shown in Figure 4.

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There are 18 working conditions in total. Each working condition simulates the detection of different defective areas of different tree species under different damage types. For example, the working condition 7 in Table 1 simulates a condition that the internal damage type of elm is hollow and the proportion of the damaged area is 1/4. Through visual inspection, surface percussion, and pressing, there are no obvious joints, splits, and other defects. Four specimens with an average moisture content of 9.18% are prepared for this test. They meet the requirements of the “Standard for design of timber structures” (GB 50005-2017) [46] and “Standard for test methods of timber structures” (GB/T 50329-2012) [47]. The specific parameters of the specimen are shown in Table 1.

##### 2.4. Stress Wave Detection

Fakopp 3D Acoustic Tomograph is able to nondestructively detect the size and location of the defective part in wood. It works based on sound velocity measurement between several sensors around the trunk. If there is a hole, the sound waves will have to pass around the hole. Thus, they require more time to reach the opposite sensors. In order to explain the complex velocity model, a two-dimensional image is constructed. Healthy wood is shown in green, decaying wood is shown in red, and hollow is shown in blue. This test selected 10 sensors to detect the internal damage of the specimen (see Figure 5). The specific test steps were as follows:(1)10 Sensors were placed around the specimen, connecting to the wood with steel nails(2)Sensors were connected to amplifier boxes(3)Bluetooth connection is established to PC(4)Each sensor is tapped 3 times by a hammer(5)The data are transmitted to a laptop to calculate the two-dimensional image

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##### 2.5. Drilling Resistance Tests

Drilling resistance tests are based on microdrilling of wood at a constant velocity by a standard drill. IML-RESI PD500 has a small needle driven by a motor to penetrate into the wood at a constant speed. When the drilling needle enters the interior of the wood, it encounters relative resistance in both directions which are the forward direction and the direction of rotation. The relative resistance value varies with the density of each tree species, and the instrument records the relative resistance during the test. The resistance image processing software (PD-Tools Pro) is applied to export the data information to Excel, which can be used to draw two-dimensional images of relative resistance. In the figure, the abscissa represents the path length and the ordinate represents the relative resistance that the drilling needle encounters. Based on the measured impedance curve, the width of the damaged area can be determined according to the changes in the peaks and troughs in the curve (see Figures 6(b) and 6(d)). The decay inside the wood can be judged [48, 49], and the test steps are as follows:(1)Three paths are selected for the test specimen.(2)The bit should be perpendicular to the direction of the rings (see Figures 6(a) and 6(c)).(3)The drilling needle rotation rate and advance rate parameters of the test equipment are set, respectively. The stability of the drilling resistance test equipment should be ensured.

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#### 3. Discussion and Analysis of Test Results

##### 3.1. Two-Dimensional Images

For example, in specimen 2, the tree species is elm and the simulated defect type is hollow. Because of limited pages, Figure 7 only shows the relative impedance curve in one path direction.

When there is no internal damage in the specimen (see Figure 7(a)), the two-dimensional image detected by stress wave tests is green, and the relative impedance curve detected by drilling resistance tests is continuous. Two detection methods indicate that the specimen is healthy wood. While the internal hollow is small (when the damaged proportion is less than 1/8), pale yellow (see Figure 7(b)) and red (see Figure 7(c)) colors are presented in the center of the two-dimensional image detected by stress wave tests. However, the relative impedance curve image detected by the drilling resistance tests starts to appear “blank” reflecting the approximate width of the hollow area. With the expansion of the internal hollow area, the center part of the two-dimensional image detected by stress wave tests shows a bright blue color. The stress wave tests are more accurate in identifying the size and location of internal hollow (see Figures 7(d)–7(f)). The stress wave tests visually express the location and size of internal hollow through colors, but the boundary of the hollow is relatively fuzzy. For drilling resistance tests, the length of the “blank” on the relative impedance curve increases with the expansion of the internal hollow area, which is basically similar to the test results detected by stress wave tests.

To sum up, the stress wave tests can quickly make an intuitive judgment on the general position and degree of damage, but the judgment on the damage type is weak and the boundary division of internal defects is fuzzy. However, the drilling resistance tests only reflect the internal damage of the wooden components under one path according to the relative impedance curve. It is not possible to detect every position of a cross section, and there is no great reference value when used alone. If enough information is needed, more drilling resistance paths should be provided. Through analysis, it is found that the stress wave image and the resistance curve have a good correspondence relationship in this test. Putting the results of the two together for comparative analysis can make up for their respective shortcomings.

##### 3.2. Detection Data

The detection data listed in Table 2 show that the detected precision of the same detection method is different while it is working in various working conditions. The mean error and mean precision obtained by stress wave tests are 11.13 cm^{2} and 72.9%, respectively, while the mean error and mean precision obtained by drilling resistance tests are 8.47 cm^{2} and 87.6%, respectively. The correlation coefficients between the detected data obtained by stress wave and drilling resistance tests and the actual value are 0.9894 and 0.9989. The overall detection effect of drilling resistance tests is more precise compared to that of stress wave tests.

Although the correlation coefficients between the two sets of test data and the real value are very high, the detected precision is still low under some conditions, especially when the stress wave detection is used. We find that the detected precision of the stress wave tests under working condition 1, working condition 4, working condition 9, working condition 14, and working condition 15 is relatively low. The proportion of damage simulated under the five working conditions is also small. Therefore, it is of great engineering value to study the precision of stress wave detection with such a small internal damaged proportion.

When we examined the curves of detected precision under different working conditions (see Figure 8), we found that the detected precision obtained by stress wave tests increases with the increase of the internal damaged area in the wood.

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As far as drilling resistance tests are concerned, the detected precision increases with the increase of the internal defects in the wood when the internal damage type is hollow (see Figures 8(a) and 8(b)). While the internal damage type is insect attacks, the detected precision of specimen 3 does not change much with the increase of insect attack area (see Figures 8(c) and 8(d)).

In addition, when the internal defects are small, the detected precision of drilling resistance tests is higher than that of stress wave tests. With the further increase of simulated damaged area, the detected precision of both detection methods tends to be close to each other. When the internal damaged proportion of wood exceeds 1/4, the detected precision of stress wave tests is higher than that of drilling resistance tests (see Figures 8(b)–8(d)).

To sum up, comprehensive use of stress wave and drilling resistance tests can screen the type, position, and size of internal damage of timber components. However, the detection results of the two detection methods are quite different with low detected precision. In order to comprehensively use the information provided by the two detection methods, this paper introduces several new combined forecasting models which are different from the literature [24], in order to improve the prediction precision of the internal damage in ancient building wood components.

#### 4. Combined Forecasting Model

##### 4.1. Model Building

Based on the OWA operator [50], OWGA operator [51], and OWHA operator [38], great deals of extensions have been developed. These extensions are the IOWA operator, IOWGA operator, and IOWHA operator. In this study, we reordered the arguments by an inducing variable.

If there are *m* feasible single detection methods to detect internal defects of the timber components in ancient buildings under a certain working condition, the detection value of the *i*-th detection method in the *t*-th working condition is *x*_{it}, where *i* = 1, 2, …, *m* and *t* = 1, 2, …, *N*.

If *l*_{m} is the weight of the *m*-th single detection in the combined forecasting model, the weight satisfy the normalization and nonnegativity, such thatwhere *a*_{it} represents the detected precision of the *i*-th detection method under the *t*-th working condition, *a*_{it} ∈ [0, 1]. When *a*_{it} is regarded as the inducement value of *x*_{it}, it can form *m* two-dimensional arrays which are (*a*_{1t}, *x*_{1t}), (*a*_{2t}, *x*_{2t}), …, (*a*_{mt}, *x*_{mt}). The detected precision sequence (*a*_{1t}, *a*_{2t}, …, *a*_{mt}) of *m* detection methods under the *t*-th working condition is arranged from high to low. Let us hypothesize that *a*-index(*it*) is a subscript of the *i*-th largest value among the detection sequence.(1)Model based on the IOWA operator [34]: the square sum of error is taken as the criterion to establish the combined forecasting model. According to the detected precision sequence, the combined forecasting value based on the IOWA operator can be obtained byThe optimal combined forecasting model based on the IOWA operator with the square sum of error as the criterion can be expressed as follows:(2)Model based on the IOWGA operator [36]: the square sum of logarithmic error is taken as the criterion to establish the combined forecasting model. According to the detected precision sequence, the combined forecasting value based on the IOWGA operator can be obtained byThe optimal combined forecasting model based on the IOWGA operator with the square sum of logarithmic error as the criterion can be expressed as follows:(3)Model based on the IOWHA operator [38]: the square sum of reciprocal error is taken as the criterion to establish the combined forecasting model. According to the detected precision sequence, the combined forecasting value based on the IOWHA operator can be obtained byThe optimal combined forecasting model based on the IOWHA operator with the square sum of reciprocal error as the criterion can be expressed as follows:

##### 4.2. Solving Model

Taking the simulated hollow test of specimen 1 as an example, we can list the two-dimensional array of detection values and its detected precision under the *t*-th working condition as follows:

The prediction value based on the IOWA operator is calculated according to equation (2). The solving process is shown as follows:

By substituting them into equation (3), the optimal combined forecasting model based on the IOWA operator is arranged as follows:

The MATLAB software is used to solve the optimal weight (*l*_{i}). Some parameters of the genetic algorithm are set as follows: Generations = 50 StallGenLimi = 100 PopInitRange = [zeros(1, *m*); ones(1, *m*)] PopulationSize = 10000

After 50 genetic iterations, the MATLAB software shows that (*l*_{1}, *l*_{2}) is (0.69, 0.31), respectively. The black and blue points in Figure 9 are the best fitness value and the mean fitness value, respectively. It is found that the mean fitness value of the population represents a smooth downward trend with the increase of the number of iterations and gradually moves towards the best fitness value (see Figure 9).

Similarly, the solving processes of the models based on the IOWGA operator and IOWHA operator are the same as that of the model based on the IOWA operator. In order to select the optimal model, the traditional combined forecasting models based on the entropy value and Shapley value are introduced in this paper for comparison.

##### 4.3. Analyzing Precision of Different Combined Forecasting Models

In Table 3, the mean precision from large to small is , , , , and . Compared to detected precision of stress wave tests, the precision is improved by 25.8%, 25.4%, 25.2%, 21.5%, and 17.6%, respectively. While compared to detected precision of drilling resistance tests, the precision is improved by 4.7%, 4.3%, 4.2%, 1.1%, and 2.2%, respectively. Additionally, the mean absolute error from small to large is *e*_{1}, *e*_{2}, *e*_{3}, *e*_{5}, and *e*_{4}. So, we find that the models based on the IOWA operator, IOWGA operator, and IOWHA operator have better forecasting effects compared to others (see Figure 10).

Other than that, through statistical analysis of working condition 1, working condition 4, working condition 9, working condition 14, and working condition 15 (see Table 4), we also find that the combined forecasting models based on the IOWA operator, IOWGA operator, and IOWHA operator are more effective compared to the combined forecasting models based on the entropy value and Shapley value in improving the detected precision of stress wave tests in the case of small defects inside the wood.

##### 4.4. Forecasting Effect Evaluation

According to the evaluation principle of the forecasting effect, SSE, MSE, MAE, MAPE, and MSPE are selected as evaluation indexes to reflect the effectiveness of the combined forecasting models. The calculation results are shown in Table 5:

In Table 5, it is found that the first four indexes of the combined forecasting method based on the IOWA operator are significantly lower than those of others. Although MSPE of the combined forecasting method based on the IOWA operator is not the least, it is close to MSPE of combined forecasting methods based on the IOWGA operator and IOWHA operator. Therefore, the combined forecasting model based on the IOWA operator has the best effect. Meanwhile, by normalizing the above five indexes, the expression of the comprehensive evaluation index *C* is obtained as follows:where *C*_{i} is the comprehensive evaluation index of the *i*-th method, *i* = 1, 2, …, *M*; *E*_{ij} is the *j*-th index of the *i*-th method, *j* = 1, 2, …, *n*; and min(*E*_{j}) is the minimum value among the *j*-th indexes of *m* methods. The higher the *C* is, the better the corresponding combined forecasting model is. The index *C* of each method is developed by bringing the five index values in Table 5 into equation (12).

Figure 11 shows that the comprehensive evaluation index of each combined forecasting model is significantly higher than that of the two single detection methods, indicating that the combined forecasting model can improve the forecasting precision of the internal defects of the timber components. Furthermore, the *C* of the combined forecasting model based on the IOWA operator is 97.2%, higher than others. All the analysis shows that the combined forecasting model based on the IOWA operator is the optimal model, followed by the combined forecasting models based on the IOWGA operator and IOWHA operator.

#### 5. Applicability Assessment

An applicability assessment is performed for the combined forecasting models based on the IOWA operator, IOWGA operator, and IOWHA operator.

##### 5.1. Cross-Validation Theory

Since there are a total of 18 working conditions in this test, each working condition has a corresponding set of actual values and detected values obtained by stress wave and drilling resistance tests (see Table 2). Because of the small number of data samples, and each combined forecasting model is tested only once, the randomness is large, which does not prove that the above optimal models have good universality. In order to make full use of the data samples, a cross-validation method is applied to carry out the test. The research process is shown in Figure 12.

Firstly, by setting random samples, we randomly extracted *N* (*N* = 5, 6, …, 17) working conditions from 18 working conditions as a training set (*I*_{N}), and then the corresponding remaining 18 *−* *N* working conditions are considered as a testing set (). For the number (*N*) of randomly selected samples, there are subsets for both the training set and the testing set (see Table 6). For example, if *N* = 5, both the training set (*I*_{5}) and the testing set () will have 8568 subsets.

If *I*_{5} = {working condition 1, working condition 2, working condition 3, working condition 4, working condition 5}, then = {working condition 6, working condition 7, working condition 8, working condition 9, working condition 10, working condition 11, working condition 12, working condition 13, working condition 14, working condition 15, working condition 16, working condition 17}.

By analogy, we can get 8568 subsets for the training set (*I*_{5}) and the testing set ().

Secondly, based on random sample data in each training set (*I*_{N}), three combined forecasting models based on the IOWA operator, IOWGA operator, and IOWHA operator are established to solve three sets of corresponding optimal weights. Based on the cross-validation method, three sets of weights are substituted into the corresponding testing set (). For the training set and testing set, we can get the comprehensive evaluation index *C* corresponding to each combined forecasting model using equation (12).

Finally, we calculate the mean value (*E*_{C}) and variance (*V*_{C}) of the comprehensive evaluation index *C*. Through the statistical analysis of the change law of the mean value (*E*_{C}) and variance (*V*_{C}), the universality of the optimal combined forecasting model is judged.

##### 5.2. Cross-Validation Result Analysis

The mean value (*E*_{C}) and the variance (*V*_{C}) of the comprehensive evaluation indexes are evaluated by each combined forecasting model in the training sets (see Table 7).

In Table 7, we find that *E*_{C-IOWA} of each training set has a significantly higher value compared to *E*_{C-IOWGA} and *E*_{C-IOWHA}. The higher the *E*_{C} is, the better the overall forecasting precision is. The alignment of the variance of the comprehensive evaluation indexes is *V*_{C-IOWA} < *V*_{C-IOWGA} < *V*_{C-IOWHA}. The smaller the *V*_{C} is, the smaller the dispersion of the *C* value is and the more stable the data change is. This indicates that the combined forecasting model based on the IOWA operator is well applicable.

In Figure 13, the distribution maps of *E*_{C} and *V*_{C} corresponding to each training set show a roughly linear changing trend. When the sample data in the training set (*I*_{N}) increase, *E*_{C-IOWA} and *E*_{C-IOWH} increase and *E*_{C-IOWGA} decreases. With the increase of the number of sample data in the training sets, the *V*_{C} of the three combined forecasting models decreases. But the change of *V*_{C-IOWA} is smallest, indicating that the *C* does not fluctuate much and the data are very stable.

Based on the cross-validation theory, the optimal weights obtained by each training set are brought into the corresponding testing set. *E*_{C} and *V*_{C} of the testing sets are listed in Table 8, and the distribution maps of *E*_{C} and *V*_{C} calculated by each testing set are shown in Figure 14.

In Table 8, *E*_{C-IOWA} of testing sets is significantly higher than *E*_{C-IOWGA} and *E*_{C-IOWHA}, while *V*_{C-IOWA} is significantly lower than *V*_{C-IOWGA} and *V*_{C-IOWHA}. However, it is noted the values of *E*_{C-IOWGA} and *E*_{C-IOWHA} or *V*_{C-IOWGA} and *V*_{C-IOWHA} are very close to each other. Meanwhile, it is found from Figure 14 that *E*_{C} corresponding to each testing set decreases with the decrease of the sample number in the testing sets. Contrarily, *V*_{C} increases with the decrease of the sample number in the testing sets. The distribution map of *V*_{C-IOWA} has little change in slope, meaning the corresponding *C* value is more stable. The results show that the combined forecasting model based on the IOWA operator has the highest overall forecasting precision and best level of applicability among the three models.

#### 6. Nondestructive Tests for Double-Cirold Longevity Pavilion

Double-Cirold Longevity Pavilion is located in the Beijing Tiantan Park. It was built in the Middle Qing Dynasty and has a history of 277 years. Double-Cirold Longevity Pavilion is a combination of two round pavilions with double eaves and spires. Its structure is peculiar and precise, and its shape is novel and well proportioned. This kind of pavilion has high scientific, artistic, and cultural value in the Chinese timber structure. Affected by the natural environment and human factors all year round, timber components are damaged. Eventually, it leads to the loss of the external protective layer for timber components and the acceleration of the internal and external damage of timber components.

Nondestructive tests of timber components of the double-ring marsupial pavilion were performed by stress wave and drilling resistance (see Figure 15). It was found that the beams and the columns had internal defects. The B2 column of Double-Cirold Longevity Pavilion is an example with a moisture content of 9.8%. The perimeter of the wooden column is 109.9 cm, and the detected section area is 961.63 cm^{2}. According to the two-dimensional image obtained by the stress wave tests, the internal defects were located. Drilling resistance tests were conducted pertinently. There were two detected paths in drilling resistance tests, and each path passed through the location of defects.

Through nondestructive tests, it was found that there was an uncompacted sound when knocking the position of the B2 column, 400 mm from the ground. A certain degree of defect is found in the interior wood by the stress wave tests (see Figure 16). The damaged area detected by the stress wave tests accounts for 18% of the detected section, and the damaged area is 173.09 cm^{2}. The drilling resistance tests show the damaged area is 49 cm^{2}. The damaged area of the B2 column calculated by the combined forecasting model based on the IOWA operator is 87.47 cm^{2}. It is found that the B2 column is defective and the wooden column needs to be repaired.

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#### 7. Conclusion

(1)When used alone, both the stress wave and drilling resistance tests have their own advantages and disadvantages. Through analysis, it is found that the stress wave image and the resistance curve have good correspondence in this test, which can make up for their respective shortcomings. Stress wave and drilling resistance tests can be used together to qualitatively analyze the internal damage of the wood structure.(2)Weighing test results of the stress wave and drilling resistance and establishing a combined forecasting model can quantify the test results. Compared with the combined forecasting models based on the entropy value and Shapley value, the combined forecasting models based on the IOWA operator, IOWGA operator, and IOWHA operator have better forecasting effects according to the idea of variable weight, not only greatly reducing the sensitivity of the results to poor detection methods but also effectively improving the forecasting precision of internal damage of timber components in ancient buildings. When the internal damage of the wood specimen is small, the method proposed in this paper is more effective in improving the precision of stress wave detection.(3)The mean precision and mean absolute error calculated by the combined forecasting model based on the IOWA operator are 91.7% and 3.42 cm^{2}. The mean precision is improved by 25.8% and 4.7% compared to the stress wave and drilling resistance tests. In addition, *C*_{IOWA} is 97.2%, and the overall forecasting effect of the combined forecasting model based on the IOWA operator is the best of all. The analysis results based on the cross-validation theory show that the combined forecasting model based on the IOWA operator has the optimal performance and good applicability. The model can quickly and accurately analyze and judge the internal damage of timber components in ancient buildings qualitatively and quantitatively.

#### Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.

#### Acknowledgments

This study was financially supported by the National Key R&D Program of China (grant no. 2018YFD1100902-01), the National Natural Science Foundation of China (grant nos. 51678017 and 51678005), Beijing Municipal Education Commission Science and Technology General Project (grant no. KM201810005021), Beijing Natural Science Foundation Project (8182008), and the Open Fund of Shanghai Key Laboratory of Engineering Structure Safety (no. 2017-KF03).