Introduction

The assessment of rock mass quality is one of the main issues that affect the support design and operation cost of tunnel engineering (Lowson and Bieniawski 2013; Rehman et al. 2018). Inaccurate evaluation may cause the failure of support, and even lead to irreparable disasters such as water and mud gushing (Lu and Liu 2009; Li et al. 2017; Han et al. 2020) and sudden collapse of a tunnel (Shin et al. 1999). Accurate, effective, and objective rock mass quality assessment can reduce the cost and improve safety for tunnel engineering (Wang et al. 2020b).

In recent decades, the most commonly employed rock mass quality assessment system is the rock quality designation (RQD) system (Deere 1964), Norwegian geotechnical institute Q-system (Q) (Barton et al. 1974), rock mass rating (RMR) system (Bieniawski 1973), and geological strength index (GSI) system (Hoek and Brown 1997). In Japan, the Japan Highway Public Corporation (JH) system, which is based on the RMR system, is commonly used to quantitatively evaluate the rock mass quality (Masahiro et al. 1999; Akagi et al. 2001; Yuji et al. 2006). Similar to other assessment systems (e.g., Q-system, RMR system, and GSI system), when the JH system is utilized, the rock mass quality score (RQS), the scores of tunnel face observation items (e.g., compressive strength, weathering, and spacing of joints), is utilized to evaluate and grade the rock mass quality. Although these proposed assessment systems are extensively employed in rock engineering and tunnel engineering, the objectivity of the evaluation is insufficient due to subjective judgments that are based on the engineer’s experience observation items (Palmstrom 2005; Rahmati et al. 2014; Zolfaghari et al. 2015; Wang et al. 2020a).

Over the last decades, with the development of measure-while-drilling (MWD) technology, this technology can be used to predict and evaluate rock mass quality in broad tunneling projects (Lear and Dareing 1990; Nilsen 2015; Navarro et al. 2018). Aoki et al. (1999) proposed to use MWD data to evaluate the geological conditions of different drilling depths. Yue et al. (2004) used the MWD data to study decomposition grades in the ground, and confirmed that there is a certain relationship between the drilling speed and decomposition. Zhou et al. (2011) developed an unsupervised method to predict rock types based on MWD data, who verified that the method is effective in rock type recognition. Leung and Scheding (2015) adopted a modulation-specific energy to overcome the low specificity and high variability of existing MWD methods. Considering hundreds of thousands or more MWD data, the more efficient prediction method of fitting these MWD data to a rock mass quality index should be further developed.

With the development of artificial neural network (ANN) technology, the application of ANN in rock mass quality prediction has been successful (Sousa et al. 2012). Xu et al. (2007) used a back-propagation neural network to assess the rock mass quality. The RMR values were estimated by Hussain et al. (2016), who compared ANN technology with multiple regression technology. Karlaftis (2018) proposed an ANN model to classify rock masses using data from tunnels in Greece. The results demonstrated that the ANN can place a rock mass in the classification ratings very quickly and with very high accuracy with a smaller number of input variables. The applicability of ANN technology for an automatic online classification of rock mass was explored by Erharter et al. (2019). This research obtained a final classification accuracy of 74.4%.

ANN is one of the most innovative research fields in the field of science and engineering, with strong nonlinear mapping ability (Mohamad et al. 2012; Momeni et al. 2015). However, slow learning speed and easy to fall into local minima are some defects of ANN (Armaghani et al. 2018, 2019). Applications of population-based evolutionary algorithms, such as genetic algorithm (GA), particle swarm optimization (PSO), and imperialist competitive algorithm (ICA), are helpful to overcome these shortcomings. Using these algorithms to optimize artificial neural network can solve complex engineering problems and become a research hotspot (Hasanipanah et al. 2017; Khandelwal et al. 2017; Liu et al. 2020).

Although the feasibility of traditional rock mass quality assessment system has been identified, research on the development of more objective, intelligent, and efficient evaluation methods require further study. The purpose of this research is to introduce hybrid ANN technologies, including GA-ANN, PSO-ANN, and ICA-ANN, to predict the RQS (a rock mass quality index used in Japan) using the MWD data obtained from the new Nagasaki tunnel (east) of the West Kyushu line of the high-speed railway project in Japan. A conventional regression model and simple ANN model are developed. Subsequently, three hybrid ANNs are established and compared with the developed regression model and ANN model. Through comparative analysis, the best model to predict RQS value is selected. This method can evaluate rock mass quality accurately, effectively, and objectively.

Case study and dataset collection

Case study

A case study of the new Nagasaki tunnel in Japan was carried out. Excavation of the tunnel, which connects Nagasaki City and Isahaya City in southeastern Nagasaki province and central Nagasaki province, respectively, commenced in March 2013. As shown in Fig. 1, the MWD data of this study were obtained from the Nagasaki tunnel (east), with a length of 3.885 km, and adjacent to it is the new Nagasaki tunnel (west), with a length of 3.575 km. The New Austrian Tunneling Method was employed. The stratum exposed by this tunnel is mainly volcanic rock. The type of rock passing from west to east is propylite andesite, hornblende andesite, tuff breccia, pyroxene andesite, and tuff breccia. Among them, propylite andesite accounts for the largest proportion, about 3 km. The geological condition of the surrounding rock is poor, and the measured compressional wave velocity (P wave) ranges from 2.5 to 3.5 km/s.

Fig. 1
figure 1

Locations of the new Nagasaki tunnel (east) high-speed railway

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During tunnel excavation, the JH method based on the observation report of tunnel face was adopted to evaluate the rock mass exposed by the tunnel. This method uses ten rock parameters of total state, self-stability, intact rock strength, weathering, joint proportion, spacing of joints, joint aperture, distribution of joints, ground water inflow, and ground water deterioration to evaluate the rock mass quality. As shown in Table 1, when using the JH method for rock mass classification, all observation items are configured with four values from 1 to 4. The sum of each item value is the final rock rating score, which is called the RQS value. The larger RQS value represents the worse geological conditions of the tunnel and the higher support strength is required.

Table 1 Grading configuration of each item of rock masses
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At the same time of tunnel excavation, the MWD data of drilling in advance working face were recorded. Figure 2 shows the drilling equipment and data collection equipment. The parameters of MWD data include the penetration rate (PR), hammer pressure (HP), rotation pressure (RP), feed pressure (FP), hammer frequency (HF), and specific energy (SE). The MWD data and the corresponding RQS constitute one dataset.

Fig. 2
figure 2

Drilling rig and data-recording device

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Dataset collection and analysis

In the process of tunneling, a total of 1270 datasets were collected in the new Nagasaki tunnel (east). To model and predict the RQS value, six parameters of MWD data were considered the input parameters. The RQS value was calculated form the observation report of the tunnel face. As shown in Table 2, descriptive statistics of parameters in the datasets were carried out. Their visual statistic distribution is provided in Fig. 3. The statistical results show that the values are extensively distributed. In addition, the correlation between each MWD parameter and their corresponding RQS was investigated. Figure 4 shows the distribution trend of each MWD parameter and RQS value along the tunnel chainage. The coefficient of determination (R2) of each MWD parameter and the corresponding RQS were calculated, as shown in Fig. 5. As shown in Figs. 4 and 5, a positive correlation exists among RQS and PR, HP, HF, and FE, while a negative correlation exists for RP and FP. The correlation between RQS and FP is better than that of other parameters; however, a low correlation factor (R2 = 0.395) is obtained. These results show a low correlation between MWD data and the RQS. The proportion of the R2 value obtained by calculating each MWD parameter in the total R2 value obtained by all parameters is taken as the influence weight for each MWD parameter on the estimation of RQS. Through calculation, parameters PR, HP, RP, FP, HF, and SE respectively obtain weight values of 0.26, 0.01, 0.11, 0.36, 0.05, and 0.20. The results show that parameter FP has the greatest influence on the estimation of RQS, while parameter HP has the least. At the same time, a correlation analysis between input parameters was carried out. The correlation results of the parameters evaluated by power, exponential, and linear equations are shown in Fig. 6. Considering the evaluation index of R2, the equation is evaluated. Figure 6 shows a high correlation between PR and SE; however, not too much R2 value (R2 = 0.739) is obtained. These results show that the correlation between the parameters is low. Therefore, using all of the MWD parameters to develop prediction model requires further study. The above analysis shows that the prediction ability of the model developed with a single MWD parameter is weak, and the modeling analysis of multiple MWD parameters needs to be carried out. In the following sections, the attempt to apply an advanced hybrid ANN technology to develop the RQS prediction models by multiple MWD parameters will be carried out.

Table 2 Distribution statistics of dataset parameters
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Fig. 3
figure 3

Distribution of the parameters of the input data and output data

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Fig. 4
figure 4

Distribution trend of among RQS and MWD parameters. a PR. b HP. c RP. d FP. e HF. f FE

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Fig. 5
figure 5

The correlation among RQS and MWD parameters. a PR. b HP. c RP. d FP. e HF. f FE

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Fig. 6
figure 6

Correlation analysis among the input parameters

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To reduce the influence of the order of magnitude, the datasets were normalized by following equation:

$$ {X}_{\mathrm{norm}}=\frac{x-{x}_{\mathrm{min}}}{x_{\mathrm{max}}-{x}_{\mathrm{min}}} $$
(1)

where Xnorm and x are normalized data and measured data, respectively. xmin and xmax are the minimum value and maximum value, respectively, of x. The established datasets were divided into training set and testing set, to establish and evaluate the created networks. In order to train these datasets, Swingler (1996), Looney (1996), and Nelson and Illingworth (1991) proposed 80%, 75%, and 70–80% of the whole datasets as training sets respectively. In this study, the ratio of training set and testing set was set to 8:2, which means that there are 1016 datasets for training prediction models and 254 datasets for testing the predictability of the developed prediction model.

Methods

MLR and MNR

Multiple regression analysis is a statistical analysis method that considers one variable a dependent variable and at least two other variable independent variables to establish a linear or nonlinear mathematical relationship among multiple variables (Knofczynski and Mundfrom 2008). Multiple regression techniques are commonly used to predict the dependent variable by the independent variables. The multiple linear regression (MLR) equation is , Y = b1x1 + b2x2 + ⋯ + bnxn + c where bi is a regression coefficient, xi is the independent variable, Y is the dependent variable, and c is the intercept. The form of multiple nonlinear regression (MNR) equation is generally determined by the relationship between each independent variable and dependent variable.

ANN

ANN was invented by McCulloch and Pitts (McCulloch and Pitts 1943), who show that, in principle, ANNs can calculate any arithmetic or logic function. The work of these researchers is often regarded as the origin of the field of ANN. Rosenblatt (1958) invented a perceptron network and associated learning rules and demonstrated its ability of pattern recognition, which marked the first practical application of ANN. The successful application of a neural network generated substantial interest in the research of the ANN. However, subsequent research showed that the performance of the basic perceptron network was limited. ANNs became a research hotspot in the late 1980s as information processing structures that were inspired by a biological neural organization structure and operation mode. An ANN is a system composed of several nodes (neurons). These basic nodes are connected and work in parallel to complete certain processing tasks. ANNs can automatically derive general rules by provided pairs of input and output signals. Using this rule, an ANN can generate prediction output for previously unused signals.

The traditional nonlinear regression technique is a highly fitting method for general nonlinear functions, which is based on the nonlinear structure of each element that is linearly combined. As a “universal approximation” technology, ANN is proved to be an “ Input-Hidden-Output “ hierarchical structure, which enables it to fit almost all functions (including nonlinear functions) (Hornik 1991; Back and Chen 2002; Monjezi et al. 2013). The nonlinear fitting ability of ANN is mainly attributed to the ability to express the properties of each element and the organizational structure between elements. In this research, a typical three-layer neural network is adopted to developed ANN models (Fig. 7). Figure 7 also shows the processing flow of ANN, including training stage and testing stage. In the training stage, the input and output of training dataset are input to the input layer of ANN in pairs. Then, the relationship between the training datasets was established by one training algorithm. The remainder of the dataset was applied as a testing dataset. The error back-propagation algorithm is employed as the training algorithm. This algorithm includes two passes: feed-forward pass and backward propagation pass. In the feed-forward pass, the output value of network and the weight and biases of network nodes are random. In the process of the backward propagation pass, the actual values of the outputs are subtracted from the calculated values in a previous pass to generate an error signal. The signal propagates backward to the input layer. According to this signal, the weight and biases are adjusted until the set condition is reached.

Fig. 7
figure 7

Flowchart of the artificial neural network

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GA

The GA was developed by Holland (1992) to find the optimal solution based on the theory of natural evolution. The genetic operation of a GA consists of selection, crossover, and mutation. The GA has many advantages, such as no limit of derivation and function continuity, and better global optimization ability. Despite the advantages of the GA, it has numerous problems. These problems include difficult operator parameter selection, slow search speed, and dependence on initial population selection. However, GAs continue to be extensively employed in approximating nonlinear optimization (Yang 2010). In the first stage of the general process of GA, the problem solutions are encoded (e.g., binary encoding or real encoding). In the second stage, the loop process is executed by randomly generating chromosomes. In the final stage, the fitness of each chromosome is calculated. Two individuals are selected from the population for crossover and mutation based on the fitness and selection probability. A crossover operator is used to randomly select two chromosomes of the selected individuals to produce the next generation. A mutation operator is utilized to randomly select the chromosomes of the new generations for the mutation operation. The cycle is repeated until the stop condition is satisfied (Dybowski et al. 1996; Nasseri et al. 2008).

PSO

PSO, which is also a heuristic algorithm, is an evolutionary computing technology that was proposed by Kennedy and Eberhart (1995). Particle position and velocity are the core operators of PSO. The motion of particle is the process of individual search. The movement velocity of each particle is adjusted according to the historical optimal position of all particles and the historical optimal position of each particle. The optimal solution that each particle individually searches is referred to as the personal best (pbest), and the optimal individual extremum of all particles is the current global best (gbest). The velocity and particle position were updated (Eqs. (2) and (3)). The optimal solution is obtained by cyclic iteration.

$$ {v}_{\mathrm{new}}= wv+{c}_1{r}_1\left({p}_{\mathrm{best}}-p\right)+{c}_2{r}_2\left({\mathrm{g}}_{\mathrm{best}}-p\right) $$
(2)
$$ {p}_{\mathrm{new}}=p+{v}_{\mathrm{new}} $$
(3)

where vnew, v, pnew, p, and w are the new velocity, current velocity, new position, current position, inertia weight, respectively. r1 and r2 are random numbers that are usually chosen between [0, 1]. c1 is a positive constant, which is referred to as the coefficient of self-adjustment, and c2 is a positive constant, which is referred to as the coefficient of the social component. pbest and gbest are the best locations for individuals and the best locations for all particles. PSO has been widely used in civil engineering, traffic engineering, and other engineering fields. A detailed description of PSO and its application in different subjects are provided in the literature (Shi and Eberhart 1998; Gandomi et al. 2013; Nouiri et al. 2018).

ICA

Inspired by the colonial competition mechanism of imperialism, Atashpaz-Gargari and Lucas (2007) proposed a new intelligent optimization algorithm, imperialist competitive algorithm (ICA), in 2007. Different from GA, PSO, and other swarm intelligence algorithms inspired by biological behavior, ICA is an optimization method inspired by social behavior. ICA is also a population-based optimization algorithm. ICA divided the country into several empires. Within each empire, ICA brought the colonies closer to the imperialist countries through assimilation mechanism. Empire competition mechanism is an important operator of ICA algorithm. One or more weakest empires are annexed by the strongest empires through empire competition mechanism. Through the annexation of the empire, the exchange of information between the empires is completed. A more detailed description of the ICA and its application in different subjects is provided in the literature (Gazafroudi et al. 2014; Moayedi and Jahed Armaghani 2018).

Predictive models

MLR and MNR models

MLR and MNR prediction models were developed to correlate the RQS by the input parameters of PR, HP, RP, FP, HF, and SE of the training datasets. Equations (4) and (5) show the developed MLR and MNR model, respectively. The evaluations of these two models with testing datasets will be carried out in the “Results and discussion” section.

$$ \mathrm{RQS}=13.082+1.845\mathrm{PR}+1.034\mathrm{HP}-0.142\mathrm{RP}-1.717\mathrm{FP}+0.032\mathrm{HF}-0.001\mathrm{SE} $$
(4)
$$ \mathrm{RQS}=112.553-0.983\ \mathit{\ln}\left(\mathrm{PR}\right)+0.004{e}^{0.471\mathrm{HP}}-2.165\mathit{\ln}(RP)-1.514\mathrm{FP}+0.030\mathrm{HF}-69.971{\mathrm{SE}}^{0.037} $$
(5)

In this study, the coefficient of determination (R2), root mean square error (RMSE), and variance account for (VAF) indices were calculated to evaluate the prediction capacity of the developed models, as adopted by Yilmaz (2009) and Kayabasi (2012):

$$ {R}^2=1-\frac{\sum \limits_{i-1}^n{\left(y-y\prime \right)}^2}{\sum \limits_{i=1}^n{\left(y-\overline{y}\right)}^2} $$
(6)
$$ \mathrm{RMSE}=\sqrt{\frac{1}{n}\sum \limits_{i=1}^n{\left(y-y\prime \right)}^2} $$
(7)
$$ \mathrm{VAF}=\left[1-\frac{\operatorname{var}\left(y-y^{\prime}\right)}{\operatorname{var}(y)}\right] $$
(8)

where y is the ith measured value, y´ is the ith predicted value, \( \overline{y} \) is mean value of the y, and n is the number of datasets. High R2 and VAF values and low RMSE values indicate that the prediction performance is superior. If the R2 is 1, the RMSE is 0, and the VAF is 100, the model performance will be excellent.

ANN models

The prediction ANN model was proposed in this section. As previously mentioned, the Levenberg-Marquardt was employed as a training algorithm. Difficulty in establishing ANN model is encountered in selecting the number of hidden layer and nodes.

Many researchers reported that one hidden layer is usually enough to solve most problems. Therefore, the number of hidden layers in this study was set to one. The number of input and output layer nodes was set to 6 and 1, respectively. The prediction performance of ANN is mainly affected by the number of hidden nodes (Kanellopoulos and Wilkinson 1997; Gao 1998; Monjezi et al. 2011) to evaluate the influence of the number of hidden layer nodes on the performance of the ANN model. As shown in Table 3, 26 single hidden layer neural network models and 1–80 hidden layer nodes were constructed. Index R2 and RMSE were used to evaluate the prediction performance of the developed models. However, it is difficult to determine the optimal model. Zorlu et al. (2008) proposed a simple sorting selection method to deal with the above difficulties in selecting the optimal model. The principle of this ranking method is that the highest score means the best performance. For example, R2 values of 0.542, 0.621, 0.663, 0.724, 0.767, 0.759, 0.759, 0.760, 0.770, 0.796, 0.771, 0.815, 0.825, 0.790, 0.833, 0.819, 0.807, 0.766, 0.727, 0.788, 0.802, 0.803, 0.810, 0.797, 0.832, and 0.811 were obtained for the training datasets of models 1–26, as shown in Table 3. The ratings of the models were assigned 1, 2, 3, 4, 10, 5, 7, 8, 11, 18, 6, 20, 21, 12, 25, 24, 13, 14, 9, 23, 15, 17, 19, 22, 26, and 16, respectively. This method was also used to evaluate RMSE results. Table 3 showed the final sorting results. Results revealed that the network model no. 15, which had one hidden layer with the neural network architecture of 6-15-1, was considered the optimum model for RQS prediction. According to the results in this table, model no. 15 representing the network structure of 6-15-1 was selected as the best prediction mode. Additional discussions regarding the optimum developed model (among 5 runs) is given in the “Results and discussion” section. It is noted that different hybrid models of the neural network structure were developed based on 6-15-1.

Table 3 The performance index results of different number of hidden nodes for predicting RQS
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GA-ANN models

In this section, the hybrid model GA-ANN was established. Figure 8 shows the flowchart. The establishment process of hybrid GA-ANN prediction model for RQS is detailed in the following section.

Fig. 8
figure 8

Flowchart of the optimized artificial neural network by the genetic algorithm

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GA parameters

The main parameters of GA are population size (Spop), number of generations (Ngen-GA), mutation probability, and crossover probability. The main task of developing hybrid GA-ANN model is to determine these parameters. The mutation probability proposed by Momeni et al. (2014) was set to 25%, and the crossover probability was set to 70%. The roulette method was used as the selection method of crossover operation.

Value of the S pop

Different GA-ANN models with population sizes between 25 and 600 were established to determine the best Spop value, as shown in Table 4. It should be noted that these models were all developed with 6-15-1 neural network structure with the maximum generation of 100. Similar to the “ANN” model section, the simple sorting method was used to filter out the best Spop. The results show that the model no. 8 has the optimal prediction performance. As a result, the best Spop value was selected as 300.

Table 4 RQS prediction performance index results of the GA-ANN for different population size
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Value of the N gen-GA

A number of GA-ANN models were established to determine the most appropriate Ngen-GA with the maximum number of 1000 and Spop values of 25–600. Other parameters of GA-ANN used in these models were set as the parameters determined in the previous steps. As shown in Fig. 9, after 700 generations, the RMSE value no longer continues to decline and remains constant. Thus, a value of 700 was chosen as the optimum Ngen-GA.

Fig. 9
figure 9

Performance comparison of the GA-ANN models with the different number of generations

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Network modeling

Using the best parameters obtained in the above steps, the final hybrid GA-ANN model was established, and 5 times of training were carried out. The performance index values obtained of the development models are shown in Table 8. Further comparative analysis will be performed in the “Results and discussion” section.

PSO-ANN models

Similar to the GA, several researchers have successfully employed the PSO to optimize ANN (Hoballah and Erlich 2009; Vasumathi and Moorthi 2012; Moayedi et al. 2019). In this section, the optimal parameters of the hybrid PSO-ANN model will be determined. Figure 10 shows the flowchart of the hybrid PSO-ANN algorithm.

Fig. 10
figure 10

Flowchart of the particle swarm optimization algorithm

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PSO parameters

For PSO algorithm, the main parameters are the coefficient of velocity equation, number of particles (Npar), number of generations (Npar-PSO), c1, c2, and w. It should be emphasized that in the establishment of all PSO-ANN models, the values of c1, c2, and w were all set to c1 = c2 = 2 and w 0.25 recommended by Kennedy and Eberhart (1997) and Clerc and Kennedy (2002).

Value of the N par

Several PSO-ANN models were established with a range of Npar from 25 to 600 to select the optimal Npar value, as presented in Table 5. The architecture of 6-15-1 and maximum generation of 100 were utilized, and the performance indices of R2 and RMSE were employed to assess the developed models. As discussed in the previous sections, the performance indexes of the established models were sorted. According to Table 5, model no. 11 with 450 particles obtained the highest sorting value. As a result, 450 was determined to be the best Npar value.

Table 5 Effect of the swarm size on the hybrid PSO-ANN in predicting RQS
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Value of the N par-PSO

Different models with a fixed number of generations of 1000 and Npar in the range of 25–600 were developed to choose the optimum Npar-PSO. Figure 11 shows that the values of the RMSE do not change after the Npar-PSO value of 800. Therefore, the optimum Npar-PSO was set to 800 in this study.

Fig. 11
figure 11

Performance comparison of the PSO-ANN models with the different number of generations

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Network modeling

Similarly, the PSO-ANN model was developed with ANN architecture of 6-15-1 and the obtained optimum parameters of PSO and was also trained five times. The performance index values of the final model are recorded in Table 8. The developed PSO-ANN model will be evaluated further in the “Results and discussion” section.

ICA-ANN models

The hybrid ICA-ANN model was also established to predict RQS. The hybrid ICA-ANN algorithm flowchart is shown in Fig. 12. The establishment process of the hybrid PSO-ANN prediction model will be discussed in the following sections.

Fig. 12
figure 12

Flowchart of the imperialist competitive algorithm

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ICA parameters

The main parameters of the ICA are the number of countries (Ncou), number of imperialists (Nimp), number of decades (Ndec), β (number greater than 1), θ (random number), and ξ (positive number less than 1). As previously mentioned, before establishing the hybrid model, the parameters β, θ, and ξ were selected as 1.5, π/4, and 0.2, respectively, according to the recommendations in the literature (Ahmadi et al. 2013; Marto et al. 2014). The remaining parameters of the Ncou, the Nimp, and the Ndec are determined in the following sections.

Value of N cou

To select the optimal Ncou, different ICA-ANN models were developed with a range of Ncou from 50 to 600, as presented in Table 6. The ANN architecture of 6-15-1 and maximum decade of 100 were employed. Table 6 records the ranking results of the prediction performance indexes of the established prediction models. The results show that model no. 10, which represents 450 countries, has the largest ranking value. Thus, 450 was selected as the best Ncou value.

Table 6 RQS prediction performance index results of the hybrid ICA-ANN for the different number of countries
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Value of Nimp

In this section, different ICA-ANN models were constructed with Nimp ranging from 50 to 200 and Ncou of 450. Table 7 shows that the no. 5 model representing Nimp = 90 has the highest sorting value. Therefore, 90 was set as the best Nimp value.

Table 7 RQS prediction performance index results of the hybrid ICA-ANN for the different number of imperialists
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Value of the N dec

As shown in Fig. 13, different models were developed to determine the best Ndec value with the fixed Ndec values of 1000 and the Ncou values between 50 and 600. The results show that the RMSE values of all models do not continue to decline after Ndec = 800. Therefore, 800 was determined as the best Ndec value of ICA-ANN prediction model.

Fig. 13
figure 13

Performance comparison of the ICA-ANN models with the different numbers of decades

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Network modeling

As before with the developments of the GA-ANN and PSO-ANN models, the final ICA-ANN model was developed with the ANN architecture of 6-15-1 and determined optimal ICA parameters. The final ICA-ANN model was also trained five times. The performance index results of development models are shown in Table 8. All final prediction models established above will be discussed in next section.

Table 8 RQS prediction performance index results of the MLR, ANN, GA-ANN, PSO-ANN, and ICA-ANN models
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Results and discussion

In this section, the final model previously developed will be compared to select the model with the highest predictive performance. In the last stage of the developing models, except MLR and MNR models, other models were trained five times to predict the RQS. The indices of R2, RMSE, and VAF are utilized to assess the performance of these developed models.

Table 8 displays the obtained results of the developed models. Determining the optimal model is not easy because the results are similar. The method of ranking, as previously mentioned, was employed to select the optimal model in the same way. The total rank of all developed models is indicated in Table 9. The results show that ANN model no. 5, GA-ANN model no. 3, PSO-ANN model no. 2, and ICA-ANN model no. 1 have a total rank of 27, 27, 30, and 27, respectively, which indicates the optimal performance. For a certain training dataset, note that only one prediction formula can be fitted using the MLR method. Therefore, only one prediction model of the MLR was developed in this study.

Table 9 Results of total rank values of all stages of the developed final models
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Table 10 shows the best performance index results of the 6 final prediction models of MLR, MNR, GA-ANN, PSO-ANN, and ICA-ANN. The results revealed that the performance level of the regression models can be increased from approximately 0.60 (for MLR and MNR models) to approximately 0.83 (for ANN models) based on R2 by developing the ANN model. The performance level of the ANN model can be increased based on R2 by developing a hybrid model, i.e., PSO-ANN and ICA-ANN from approximately 0.83 (for ANN models) to approximately 0.86 (for PSO-ANN and ICA-ANN models). The measured and predicted RQS values of RQS obtained by the five optimal models are shown in Fig. 14. The comparison between measured RQS and predicted RQS using all five models with testing datasets are shown in Fig. 15. In addition, the weight of prediction effect of each model (Table 10) is obtained by calculating the proportion of R2 value obtained by each model at the test stage in the total R2 value of all models. It should be noted that the test stage is an important part of the evaluation of the prediction performance, and it should be emphasized that the test stage is an important part of the evaluation of the prediction effect and the change trend of the three evaluation indexes (R2, RMSE, and VAF) is basically the same. Therefore, only the R2 value of the test stage is considered when calculating the weight value of each model. Table 10 shows that MLR, MNR, ANN, GA-ANN, PSO-ANN, and ICA-ANN obtain weight values of 0.133, 0.141, 0.178, 0.180, 0.185, and 0.183 respectively. The results reveal that the hybrid models of PSO-ANN and ICA-ANN are better than the MLR, MNR, ANN, and GA-ANN models. When both the training dataset and the testing dataset are considered, R2 values of 0.875 and 0.862 and R2 values of 0.873 and 0.857 for the PSO-ANN technique and ICA-ANN technique, respectively, indicate that the PSO-ANN model has slightly higher performance compared with other models.

Table 10 Results of the best performance indices of the developed final models
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Fig. 14
figure 14

The coefficient of determination of measured and predicted RQS values of five developed final models for training and testing datasets. a MLR. b MNR. c ANN. d The GA-ANN. e PSO-ANN. f ICA-ANN

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Fig. 15
figure 15

Comparison of measured and predicted RQS values of five developed final models for training and testing datasets. a MLR. b MNR. c ANN. d The GA-ANN. e PSO-ANN. f ICA-ANN

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The comparison results of learning rates of ANN, GA-ANN, PSO-ANN, and ICA-ANN algorithms were shown in Fig. 16. ANN obtains the lowest RMSE value of 0.292 when epochs are about 160 steps. However, GA-ANN, PSO-ANN, and ICA-ANN got the lowest RMSE values when epochs are 80 steps, which are 0.212, 0.193, and 0.201, respectively. The results show that GA, PSO, and ICA can improve the learning rate of ANN and get better training effect.

Fig. 16
figure 16

Comparison of learning rates of ANN, GA-ANN, PSO-ANN, and ICA-ANN algorithms at the training stage

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Furthermore, a statistical analysis of tunnel faces with large prediction error was carried out to reveal the reasons for low prediction performance, as shown in Table 11. Measured and predicted RQS, prediction error, and the number of corresponding testing datasets in the top 10 with large prediction error of the five final optimal models are recorded in this table. Tunnel faces 1–6 of testing datasets 433, 27, 652, 152(155), 503, and 324 achieved low prediction performance in at least three prediction models. The detailed geological description site photos of tunnel faces 1–6 are shown in Fig. 17. As displayed in Fig. 17, these tunnel faces have the geological characteristics of relatively developed cracks and ground water heavy deterioration. For example, many large cracks exist in the tunnel faces 1–6; large sheeting zones exist in tunnel faces 1, 4, 5, and 6; and water inrush deterioration is relatively serious in tunnel faces 3–5. The findings conclude that the MWD data of a single borehole cannot sufficiently characterize the complex rock mass quality. Therefore, when the complex rock mass is continuously exposed in the tunnel excavation, the application of more than one borehole for advance detection is suggested to improve the prediction accuracy.

Table 11 Predicted large error values (top ten) of testing stages of the developed final models
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Fig. 17
figure 17

Tunnel faces with low prediction performance. a Tunnel face 1 (chainage of 58 K533.9). b Tunnel face 2 (chainage of 57K990.0). c Tunnel face 3 (chainage of 59 K142.4). d Tunnel face 4 (chainage of 58K227.0). e Tunnel face 5 (chainage of 58 K759.5). f Tunnel face 6 (chainage of 58K311.9)

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Finally, it should be noted that the best prediction model is determined as PSO-ANN hybrid model, and the corresponding ANN structure is 6-15-1. In this structure, all six variables were taken as input variables. According to the results of relative influence of each input variable on RQS in the “Dataset collection and analysis” section, the weight values of 0.26, 0.01, 0.11, 0.36, 0.05, and 0.20 were obtained for PR, HP, RP, FP, HF, and SE respectively. In practical application, the weight values can be used to determine the final required input variables.

Conclusion

The accurate/objective prediction of the rock mass quality score (RQS) utilizing measure-while-drilling (MWD) data is one of the greatest challenges in tunneling operations. An advantage of artificial neural network (ANN) methods is that they can address complex multivariate nonlinear mapping problems. However, it is required to overcome the shortcomings of slow training speed and minimum layout through optimization ANN, so as to produce more reliable results in the prediction of RQS. Therefore, three optimization algorithms of the genetic algorithm (GA), particle swarm optimization (PSO), and imperialist competition algorithm (ICA) were employed to develop hybrid models of GA-ANN, PSO-ANN, and ICA-ANN to predict the RQS value. For this purpose, 1270 datasets, including six measure-while-drilling parameters of penetration rate (PR), hammer pressure (HP), rotation pressure (RP), feed pressure (FP), hammer frequency (HF), and specific energy (SE), were acquired from the new Nagasaki tunnel (east) of the West Kyushu Line high-speed railway in Japan and considered input parameters, while their corresponding RQS were considered output parameters. To compare the performance of the hybrid models, MLR and ANN models were also developed to predict the RQS.

Three performance indexes R2, RMSE, and VAF are used to compare the developed prediction model. The results show that the developed PSO-ANN and ICA-ANN models have higher accuracy and efficiency than other models. However, among the two hybrid models, the PSO-ANN hybrid model has slightly higher performance in predicting RQS. The results of R2 = 0.875 and 0.862, RMSE = 1.584 and 1.782, and VAF = 0.875 and 0.861 for the training dataset and testing dataset, and the results of R2 = 0.873 and 0.857, RMSE = 1.592 and 1.812, and VAF = 0.873 and 0.855 for the training dataset and testing dataset were obtained for the PSO-ANN model and ICA-ANN model, respectively.

Note that the models established in this paper are specific to the West Kyushu Line of the high-speed railway region, and the parameters of the final prediction model need to be modified according to their conditions for other regions and other tunnel construction methods. Based on the hybrid ANN algorithm proposed in this paper, the research on the influence of different combinations of MWD parameters on prediction performance needs to be further explored.