Introduction

The study of geotechnical and mechanical properties of rocks would be critical keys for the construction, control, and maintenance of high seismic intensity regions such as dam areas. Seydoon dam (Khoozestan Province, southwest of Iran) has been constructed over a series of marls, shale and sandstones. Among these rocks, marls are particularly important due to their specific properties. Marls are composed of clay and carbonate minerals in different proportions and their characters mainly depend on the type and percentage of carbonate and clay minerals (Bellair and Pomerol 1980; El Amrani et al. 1998). It is well accepted that the force load in dam areas has a large effect on the mechanical properties of rocks, such as elasticity and strength (Abrams 1917; Watstein 1953; Malvar and Ross 1998; Yan and Lin 2006). Uniaxial compressive strength (UCS) and Young’s modulus (E) can be used to determine the durability of rocks against weathering agents and their fabrics. UCS and E are also can be used to determine their deformation and bearing capacity (Dehghan et al. 2010; Matin et al. 2017). Therefore, the determination of UCS and E for marls from the ground of the dam can play an essential role in understanding their mechanical properties and help to do appropriate maintenance for the site.

American Standards for Testing Materials (ASTM) and International Society for Rock Mechanics (ISRM) have been introduced as standard procedures for the determination of UCS and E. However, direct determinations of these indexes based on ASTM and ISRM approaches in the laboratory have few drawbacks (complex sample preparation, expensive and time consuming process) (Sousa 2014; Jamshidi et al. 2016a, b; Armaghani et al. 2016a, b). For solving limitations associated with laboratory tests, various investigations were performed for the indirect determination of UCS and E, based on non-destructive index experiments. In those studies, mineral properties (composition, porosity and density), ultrasonic tests [P-wave velocity (Vp) and S-wave velocity (Vp)] and other standard indexes were used to predict UCS and E by regression or other intelligent computing methods [i.e., artificial neural networks (ANNs) or the adaptive neuro-fuzzy inference system (ANFIS)] (Demirdag et al. 2010; Ersoy and Kanik 2012; Armaghani et al. 2016a, b). For generating a robust predictive model, it is essential to build a highly accurate system based on the most relevant parameters (inputs). Development of a model which can explore inter-correlations through various variables, detect and select the most effective ones [variable importance measurement (VIM)], and use them as inputs of a precise predictive model has several advantages: as irrelevant variables make noises in modeling, by VIM redundant variables can be removed (save time and reduce cost of analyses) and outliers (influential points) identified (Chehreh Chelgani et al. 2016a, b; Matin and Chelgani 2016; Matin et al. 2016; Shahbazi et al. 2017).

Mutual information (MI) is an intelligent computer method which can explore both the linear and nonlinear relationship between wide ranges of inputs and rank them based on their influences. In other words, MI provides a decision-making system that can be used to select the most effective inputs (variables which can represent the influence of other ones) and reduce the noises for the development of a model (Chelgani et al. 2018). In a predictive modeling problem, various researches indicated that combination of intelligent predictor models and development of an ensemble of predictors (experts) can construct an accurate model to deal with complicated problems (Masoudnia et al. 2012; Hadavandi et al. 2015, 2016). One of the popular ensemble methods is the neural network ensemble (NNE) (Hansen and Salamon 1990) and an efficient approach for creating an NNE model is Adaptive Boosting (Adaboost) that can adaptively improve the probability of sampling cases for accurate training experts for the NNE model. This approach can develop a model by using a wide distribution of inputs and reduce the prediction errors by considering the information of previous experts (Hansen and Salamon 1990; Freund and Schapire 1996; Solomatine and Shrestha 2004; Masoudnia et al. 2012; Tian et al. 2012; Zhai et al. 2012). Although the last decade has witnessed increasing applications of MI and Adaboost–NNE models, they have not yet been used in the exploration and prediction of earth science sectors. This work explores the relationships between the mechanical properties of marls (from the cores of Seydoon dam) to predict UCS and E based on various analyses (depth, density, ultrasonic tests, Brazilian test, point load test, etc.). The interpretation of variables was evaluated by using MI and Adaboost–NNE predictive models. The results of this investigation would be useful for the maintenance of the dam and could introduce a unique model for the prediction of other rock and geomechanical indexes.

Materials and methods

Database

Thirty-nine core marl samples were collected from the Sydoom dam area by drilling exploration boreholes to different depths during geotechnical studies. The density of samples was determined by the weighting of cores. All sample analyses were based on ISRM and ASTM procedures. Core marls were highly weathered and weak against water and cutting blade; therefore, a high-speed thin cutting blade was used for sample preparation. Dynamic Poisson ratio (νd), S-wave (Vs) and P-wave (Vp) velocities were measured by 200 and 1000 kHz ultrasonic transducers through core samples. Brazilian test σt (MPa) was performed by using a 15 ton jack and a pump to generate forces to a cylindrical specimen between Brazilian frames. Point load test (Is(50)) core sample was subjected to a comprehensive load between two conical platens and, as a result of tension, the broke point was recorded. UCS and triaxial compression test [cohesion of rock material (C (MPa)) and friction angle of rock material (ϕ (°))] of samples were performed by an MTS machine. These tests were done using different confining pressures (1–6 MPa). The results of various experiments and their representative UCS and E are reported in Table 1.

Table 1 Various experiments on core of marls and their representative UCS and E
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Mutual information

Mutual information (MI) as a powerful VIM tool can quantify the inter-dependency between random variables. In other words, MI is the amount of shared information between model inputs. The MI (0 « I(X;Y) « 1) between two variables (X and Y) is defined based on the joint probability distribution p(x,y) and the product distribution p(x)p(y):

$$ I\left( {X;Y} \right) = E_{{p\left( {x,y} \right)}} \left[ {\log \frac{{p\left( {x,y} \right)}}{p\left( x \right)p\left( y \right)}} \right] = \mathop \sum \limits_{x\smallint X} \mathop \sum \limits_{y\smallint Y} p\left( {x,y} \right)\log \frac{{p\left( {x,y} \right)}}{p\left( x \right)p\left( y \right)}, $$
(1)

where Ep is the mathematical expectation. MI can reduce the prediction error by the maximization scheme between input variables and targets. In other words, MI can consider the information of more than single input to predict an output. This system detects the most relative variables, ranked them based on VIM and feed them to the predictive model. Variable selection by MI reduces learning algorithm time, increases the size of the search space and prevents overfitting in the predictive model (Kerroum et al. 2010; Lee and Kim 2013; Han et al. 2015; Hansen and Salamon 1990; Freund and Schapire 1996; Solomatine and Shrestha 2004; Masoudnia et al. 2012; Tian et al. 2012; Zhai et al. 2012).

Adaptive boosting

One approach for modeling complicated relationships is generating an ensemble system by combining the single prediction models (based on components) and exploiting the different local behavior of these base models to improve the performance of the overall prediction model (Masoudnia et al. 2012). The neural network ensemble (NNE) is a popular ensemble model that is developed based on a combination of neural network experts (Masoudnia et al. 2012; Tian et al. 2012; Zhai et al. 2012). The sequential manipulating of instances to train an individual neural network is one of the typical methods for the construction of NNEs that is called boosting method (Freund and Schapire 1996). For the last two decades, boosting as one of the most powerful ensemble methods was generated with a high learning capability. Adaptive boosting (Adaboost) changes the distribution of training set based on the performance of the previous NN components which is added in an ensemble model (Adaboost–NNE) (Tian et al. 2012; Zhai et al. 2012). Adaboost–NNE adaptively increases the probability of instances, which have higher prediction errors, by the previous components. The main idea in an Adaboost–NNE model is filtering out examples with the relative prediction error higher than the pre-set threshold value, and then following the Adaboost procedure (Hansen and Salamon 1990; Solomatine and Shrestha 2004). In this study, for prediction of rock mechanic indexes, an Adaboost–NNE model was developed in T iterations (T is the number of multi-layered perceptron neural network experts in the ensemble model). The training model is presented in algorithm 1:

figure a

Results and discussions

Variable selection

For making a robust system, before generating Adaboost–NNE model for the estimation of UCS and E, VIMs by MI (MI-VIM) is applied through all measured variables (νd, Vp, Vs, Is(50), σt, depth, density, C and ϕ) to evaluate their importance for the prediction, and as a result select the most effective inputs. MI-VIM results (Fig. 1) indicated that there are complicated interactions among variables, although σt and Vs showed a direct correlation with the outputs. On the other hand, MI ranked variables based on their importance and results illustrated that Vs and σt have the highest effectiveness for the prediction of UCS and E, respectively (Fig. 2). These outcomes demonstrated that Vs and σt can represent the correlation of other variables for the generation of predictive models and can be selected as input variables. There is a good agreement with these VIMs and theoretical studies where Castagna et al. (1985) and Pickett (1963) indicated that increasing the percentage of porosity would decrease the strength of rock. Therefore, the velocity of sonic waves (P or S) would be lower during passing through voids; therefore, results of ultrasonic tests potentially could be a good indicator of the mechanical properties of rocks. Moreover, several investigations demonstrated that sonic waves (Vp or Vs) can be strong predictors of UCS and E (Yasar and Erdogan, 2004a, b). On the other hand, the dependency evaluation of these indexes with Brazilian tensile strength (σt) (as an independent variable) showed that σt can be an appropriate predictor for UCS and E modeling (Jamshidi et al. 2016a, b; Fereidooni 2016).

Fig. 1
figure 1

Dependency measurement among input variables by MI for the prediction of UCS and E

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Fig. 2
figure 2

Variable importance measurements by MI and ranking inputs for the prediction of UCS and E

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The complex interactions of selected variables with the outputs are illustrated in Fig. 3. Linear correlations (Pearson correlation) between the MI selected variables and outputs (Table 2) indicate that there are significant positive correlations between them. To develop comprehensive Adaboost–NNE models (with 6 MLP experts) for the prediction of UCS and E based on MI-VIM selected variables, approximately 75% of records from the dataset were randomly applied for the training stage and the remaining 25% of the samples for the testing stage of the models. After training, the model was tested and the results of the testing phase showed that the Adaboost–NNE models could accurately predict the outputs, with the correlation of determination values (R2) of 0.92 and 0.98 for the E and UCS, respectively. The differences (Fig. 4) between laboratory measured variables (actual values) and Adaboost–NNE predicated ones showed that models could provide high satisfaction in the prediction of E and UCS. These results indicated the potential of MI coupled with Adaboost–NNE in the prediction of geomechanical indexes, and that these systems can be used for the assessment of other complicated variables in rock mechanics and other related disciplines.

Fig. 3
figure 3

Topographic relationship between selected variables by MI with UCS and E

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Table 2 Pearson correlation between VIM selected variables and UCS-E
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Fig. 4
figure 4

Relationship between predicted and actual value of UCS and E, and their differences

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Conclusion

Uniaxial compressive strength (UCS) and Young’s modulus (E) indexes can play critical roles for the control and maintenance of high seismic intensity regions such as dam areas. This study has introduced a new method [mutual information (MI)] for feature selections through rock properties based on variable importance measurements (VIMs) to predict UCS and E of marls from the Sydoon dam (Iran) by a powerful ensemble method called Adaboost–NNE (adaptive boosting- neural network ensemble). MI-VIM can assess the impact of each rock index individually and also in multivariate interactions with other variables. Based on MI-VIM results, the most effective features could be detected and selected to generate an unbiased and broadly applicable Adaboost–NNE model. Various rock mechanic analyses were performed [depth, density, ultrasonic tests (Vp and Vs), Brazilian test (σt), and point load test], MI-VIM through provided variables indicated that Vs and σt have the highest importance for the prediction of both UCS and E among other measured parameters. These two variables were selected to generate predictive Adaboost–NNE models. Testing results of the developed models indicated that Adaboost–NNE could predict UCS and E quite accurately with the correlation of determination values of 0.98 and 0.92, respectively. These results demonstrated that variable selection by MI and prediction by Adaboost–NNE make a robust system which can be used for expanding the knowledge surrounding modeling of rock and geomechanical indexes, and powerful tools for control and maintenance of other embankments.