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Abstract By depletion of minerals at shallow depths, there is a notable growing trend towards mining operations in deeper grounds whole the world. However, as the depth of mining and underground constructions increases, the occurrence of stress-induced failure processes, such as rockburst, both inside the rock masses, away from the mined-out areas, and near excavations is inevitable. Rockburst is defined as the sudden and violent failure of a large volume of overstressed rock, which can damage structures and workers, and considerably affect the economic viability of the projects. The propensity of rocks to bursting behaviour can be aggravated by the seismic disturbances induced by different sources in deep underground openings. Therefore, the in-depth understanding of the rockburst mechanism and its prediction and treatment is of paramount significance. Due to the high-complex and non-linear nature of this hazard and the vague relationship between its influential parameters, the common conventional criteria available in the literature, cannot predict rockburst occurrence and its risk level with sufficient accuracy. However, the machine learning (ML) algorithms, which benefit from an inherent intelligence procedure, can be utilised to overcome this problem. During the last decade, significant progress has been made in implementing ML techniques to predict the propensity of rocks to bursting behaviour; however, the proposed models have complex internal structure and are difficult to use in practice. On the other hand, the experimental studies in this field are limited to measuring the bursting intensity of rocks under true-triaxial loading/unloading conditions. However, the complete stress-strain relation of rocks (i.e. the pre-peak and the post-peak regimes) subjected to different cyclic loading histories can open new insights into the rockburst/brittle failure mechanism and the long-term stability of the underground structures. The common load control techniques (i.e. the axial load-controlled and displacement-controlled techniques) cannot be employed directly to conduct the systematic cyclic loading tests and capture the failure behaviour of rocks, specifically for rocks showing class II/self-sustaining behaviour in the post-peak regime. Therefore, most current rock fatigue studies have focused on characterising the evolution of mechanical rock properties and damage parameters in the pre-peak regime. Given the above, the main focus of this thesis was on developing practical and accurate models to predict rockburst-related parameters as well as better understanding the effect of seismic disturbances on the failure mechanism of rocks using data-driven and experimental approaches. i

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The robust ML algorithms, such as gene expression programming (GEP), GEP-based logistic regression (GEP-LR), classification and regression tree (CART) etc., were programmed and employed for the following tasks: (a) Providing a mathematical binary model to estimate the occurrence/non-occurrence of rockburst hazard; (b) developing a model to cluster the rockburst events based on their risk levels; (c) proposing a novel and practical multi-class classifier to distinguish three most common failure mechanisms of squeezing, slabbing and rockburst in underground mines based on intact rock properties; (d) quantifying the rockburst maximum stress (i.e. the stress level that bursting occurs) and bursting risk level based on the comprehensive database compiled from the true-triaxial unloading tests for different rock types and (e) predicting the peak strength variation of rocks subjected to cyclic loading histories. The obtained results from the above studies proved the high performance and capability of the used ML techniques in dealing with high-complex problems in mining projects, such as rockburst hazards. The newly proposed models in this research project outperformed the conventional rockburst criteria in terms of prediction accuracy and can be used efficiently in underground mining projects. A new testing methodology namely “Double-Criteria Damage-Controlled Test Method” was developed in this research project to measure the complete stress-strain relation of rocks under different cyclic loading histories. This methodology, unlike the common testing methods, benefits from two controlling criteria, including the maximum stress level that can be achieved and the maximum lateral strain amplitude that the specimen is allowed to experience in a cycle during loading. The conducted uniaxial multi-level systematic cyclic loading tests on Tuffeau limestone proved the capability of this testing method in capturing the post-failure behaviour of rocks. The preliminary results also showed that rocks tend to behave more brittle by experiencing more cycles. Furthermore, a quasi-elastic behaviour dominated over the pre-peak regime during cyclic loading, which finally, resulted in strength hardening. In another comprehensive experimental study, 23 uniaxial single-level systematic cyclic loading tests were undertaken on Gosford sandstone specimens at different stress levels to unveil the failure mechanism of rocks subjected to seismic events. It was found that there exists a fatigue threshold (FTS) that lies between 86-87.5% so that below this threshold, no macroscopic damage is created in the specimen; rather, strength hardening induced by rock compaction occurs. Moreover, according to the evolution of damage parameters and brittleness index, the pre-peak and post-peak behaviour of rocks below the FTS was found to be independent of the cycle number. However, for the cyclic tests beyond the FTS, the instability of rocks increased ii

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with the applied stress level, representing the propensity of rocks to brittle failures like rockburst. To better replicate the rock stress conditions in deep underground mines and understand more about the evolution of some specific rock fatigue characteristics, such as strength hardening, FTS and post-peak instability with confining pressure, a comprehensive cyclic loading study was carried out on Gosford sandstone in triaxial loading conditions under seven confinement levels (𝜎 /𝑈𝐶𝑆 ). It was found that by an increase in 𝜎 /𝑈𝐶𝑆 from 10% to 100%, FTS 3 𝑎𝑣𝑔 3 𝑎𝑣𝑔 decreases from 97% to 80%. An unconventional trend was observed for the stress-strain relations of rocks by varying 𝜎 /𝑈𝐶𝑆 . A transition brittle to the ductile point was identified 3 𝑎𝑣𝑔 at 𝜎 /𝑈𝐶𝑆 = 65%. Therefore, it can be inferred that with an increase in depth in rock 3 𝑎𝑣𝑔 engineering projects, the propensity of rock structures to brittle failures such as rock bursting at stress levels lower than the determined average peak strength can be aggravated. Also, it was observed that below the transition point, cyclic loading has a negligible effect on rock brittleness; while for 𝜎 /𝑈𝐶𝑆 = 80% and 100%, the weakening effect of cyclic loading 3 𝑎𝑣𝑔 history was visible. According to the results of acoustic emission (AE), tangent Young’s modulus (𝐸 ), cumulative irreversible axial strain (𝜔𝑖𝑟𝑟) and axial strain at failure point (𝜀 ), 𝑡𝑎𝑛 𝑎 𝑎𝑓 it was found that for the hardening cyclic loading tests (with positive peak strength variation), the quasi-elastic behaviour was dominant during the pre-peak rock deformation. However, for the weakening cyclic loading tests (with negative peak strength variation), more plastic strains were accumulated within the rock specimens, which resulted in gradual damage evolution and stiffness degradation during cyclic loading before applying final monotonic loading. The peak deviator stress of Gosford sandstone under different confining pressures varied between - 13.18% and 7.82%. An empirical model was developed using the CART algorithm as a function of confining pressure and the applied stress level. This model is helpful in predict peak strength variations of Gosford sandstone. Keywords: Rockburst; Machine learning algorithm; Gene expression programming (GEP); Classification and regression tree (CART); Multi-class classification; True-triaxial unloading test; Failure mechanism; Systematic cyclic loading; Fatigue; Uniaxial cyclic loading test; Triaxial cyclic loading test; Acoustic emission; Brittleness; Strain energy; Pre-peak and post- peak behaviour; Brittleness; Damage; Irreversible Strain iii

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Statement of Originality I certify that this work contains no material which has been accepted for the award of any other degree or diploma in my name in any university or other tertiary institution and, to the best of my knowledge and belief, contains no material previously published or written by another person, except where due reference has been made in the text. In addition, I certify that no part of this work will, in the future, be used in a submission in my name for any other degree or diploma in any university or other tertiary institution without the prior approval of the University of Adelaide and where applicable, any partner institution responsible for the joint award of this degree. The author acknowledges that copyright of published works contained within this thesis resides with the copyright holder(s) of those works. I also give permission for the digital version of my thesis to be made available on the web, via the University’s digital research repository, the Library Search and also through web search engines, unless permission has been granted by the University to restrict access for a period of time. Roohollah Shirani Faradonbeh Signature: Date: 26 August 2021 iv

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Chapter 1 Thesis Overview 1.1. Introductory Background With an increase in depth of mining and underground constructions, due to the complex stress state induced by different loading conditions (i.e. static, quasi-static and dynamic loadings), the occurrence of some destructive phenomena such as rockburst in the confined rock mass and/or near excavation is inevitable. Although there is no international consensus on the definition of a rockburst, it can be defined as a sudden and violent expulsion of overstressed rocks from the surrounding rock mass, resulting in the instantaneous release of accumulated strain energy. This phenomenon may cause injury to workers, damage to mine infrastructure and equipment, and possibly endanger the economic viability of the project (Cai and Kaiser 2018). From the viewpoint of the triggering mechanisms and physical modelling approaches, rockburst can be categorised into two main groups of strainburst and impact-induced rockburst (He et al. 2012). Strainburst, as a self-initiated rockburst, frequently occurs by local stress concentration at the edge of underground openings (brittle rocks) in the form of the sudden release of stored energy and is usually associated with the development of drifts, shafts, stope faces, and mining pillars. However, rockburst occurrence is not only associated with the strain energy accumulation in rocks during excavation but also with the human- (e.g. drilling and blasting operation, haulage system vibration, mechanical excavation, backfilling etc.) and/or environmental-induced (e.g. earthquake, volcanic activities, fault slip etc.) seismic disturbances (He et al. 2018). This type of rockburst is known as impact-induced rockburst. The deformation and failure characteristics of rocks subjected to seismic disturbances are completely different from those under conventional loading conditions (Taheri et al. 2016). Many factors affect the rockburst triggering, including the mechanical rock properties, excavation geometry, discontinuities, in-situ and mining-induced stresses and construction method, which have complicated rockburst mechanism (He et al. 2015). A considerable number of studies have been carried out by different researchers on rockburst hazard using theoretical and experimental approaches. However, due to the complex nature of rockburst and 1

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many influential parameters, its mechanism is still unclear. Therefore, there exists a remarkable theoretical significance and engineering value to deeply understand the rockburst mechanism and find solutions for its prediction and treatment. 1.2. Literature Review and Research Gaps 1.2.1. Rockburst Occurrence and its Risk Level The main focus of researchers during the last decade was on the prediction and control methods. From the viewpoint of prediction, rockburst can be assessed in the short term and long term. Short-term prediction of rockburst refers to the in-situ measurement techniques, including micro-seismic monitoring, microgravity, acoustic emission (AE), geological radar and so forth, which can be employed to determine the time and location of bursting. These techniques, however, are very costly and time-consuming. On the other hand, long-term prediction of rockburst is based on empirical criteria, numerical analyses, rockburst charts and data-driven techniques (soft-computing algorithms), which are usually used at the design stage of the projects to evaluate the propensity of different areas to bursting. These techniques are relatively quick, easy to use, and accurate, which can be implemented straightforwardly by engineers in practice. According to the state-of-the-art literature review conducted by (Zhou et al. 2018), approximately 100 rockburst empirical criteria have been proposed by different researchers from 1996 to the present, mostly based on strength/stress, strain and strain energy parameters. These criteria classify rockburst risk level (intensity) into four main classes of “None”, “Light”, “Moderate,” and “Strong,” based on the compiled information from the bursting location such as failure pattern, the scale of damage, and the sound of rockburst. The simplicity and operability are the most prominent advantages of empirical criteria. However, the empirical criteria suffer from some critical drawbacks. Firstly, as mentioned above, rockburst is affected by many geological, rock mechanical and operational factors, whilst the empirical criteria only consider single or few parameters and cannot reflect the mutual effects of the influential factors for rockburst assessment. Secondly, the thresholds defined by the researchers for the empirical criteria are not unique, even for those having similar expressions. This is mainly due to the case study-based nature of these criteria and also the limited number of datasets used for their development by scholars. Thirdly, in several studies (Jian et al. 2012; Liu et al. 2013; Li et al. 2017), these criteria have shown low prediction accuracy, which raises doubts concerning their efficiency. Fourthly, some engineering assumptions have been applied to the empirical criteria which may affect their reliability. Given 2

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such essential limitations and the complex non-linear nature of rockburst hazard, recently, the application of data-driven approaches such as machine learning (ML) algorithms have been increased in this field. The ML techniques (supervised and unsupervised algorithms) are capable of including more input parameters/predictors, dealing with noisy data, finding the latent non-linear relationships between inputs and the corresponding output and selecting the most influential parameters on rockburst occurrence using a smart feature selection procedure. As such, the ML algorithms do not need any prior knowledge concerning the mechanism of the problem and interrelationship of parameters, which is a significant benefit over the common criteria and statistical methods. A considerable number of ML techniques, including artificial neural network (ANN), Bayesian network (BN), support vector machine (SVM), and logistic regression (LR), has been used extensively during the last decade by researchers to predict either rockburst occurrence/non- occurrence (a binary problem) or rockburst risk level (a multi-class problem) (Pu et al. 2019). In most of these studies, the uniaxial compressive strength (𝜎 ), uniaxial tensile strength (𝜎 ), 𝑐 𝑡 maximum tangential stress (𝜎 ), elastic strain energy index (𝑊 ) and their combinations have 𝜃 𝑒𝑡 been used as input parameters. The results prove the high performance of such algorithms in rockburst assessment. However, the ML techniques still have the following limitations: a) most of these algorithms are known as black-box techniques and have a complex internal computational procedure which is very difficult to understand by human, b) some of these techniques are prone to the over-fitting problem and may get stuck in local minima (solutions), and c) more importantly, most of the used techniques in the literature are not very practical since they cannot offer any mathematical or visual output to let the engineers and researchers apply them without using a code. Therefore, to overcome the above problems and provide practical and user-friendly models for the prediction of rockburst occurrence and its risk level (intensity), it is required to perform a comprehensive statistical analysis on the compiled database and utilise robust white-box techniques for modelling. Furthermore, by developing practical models that have an apparent internal structure, it will be possible to perform different statistical analyses, evaluate the rockburst vulnerability in associations with different input parameters, and finally propose an appropriate controlling technique. From the viewpoint of rockburst control, several techniques have been proposed as potential solutions to mitigate this hazard (Saharan and Mitri 2011; Feng 2017; He et al. 2018): (1) Application of energy-absorbing bolts/cables which have a constant resistance under static and dynamic loadings and benefit from a large elongation capacity. These bolts/cables compared 3

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with the ordinary ones, have higher resistance against dynamic loads and are capable of absorbing energy from multiple impacts, and finally, can maintain the large deformation of rock masses; (2) Application of ground preconditioning techniques such as destressing and water infusion (hydrofracturing). Destressing can be conducted using destress blasting and destress drilling (i.e. boreholes without explosives or pilot tunnels in civil tunnels excavated by TBMs) methods. The argument for destressing using blasting operation is that if destressing is carried out ahead of an advancing underground opening, the high-stress concentration zone would be transferred farther away from the working face into the solid rock mass. Therefore, a protective barrier (buffer zone) is created between the working face and the highly-stressed zone for the next mining operation. Hydrofracturing, as another rockburst control technique, changes the rock properties and decreases the ability of the rock masses in absorbing the strain energy (source of bursting). This method is mostly used for coal seams. (3) Application of alternative mining methods such as pillarless mining and mining with protective seams/veins or sacrifice galleries. This technique can be used in longwall mining of coal seams and can reduce the risk of spontaneous failures. 1.2.2. Rockburst and other Failure Mechanisms In deep underground conditions, the rockburst is not the only failure mechanism. Different types of failure, such as high-stress slabbing and squeezing, may be observed based on the stress distribution around the excavation and the influential uncertain factors. However, to the author’s knowledge, there has been no attempt to develop a practical model to distinguish different failure mechanisms for over-stressed rock masses in the deep underground. This is while the proper measurement of this issue at the initial stages of the projects can help engineers to optimise mining layout, apply the adequate supporting system and reduce high costs. According to the robustness and the approved capabilities of the ML techniques in dealing with high complex non-linear problems, this gap can be addressed properly by incorporating the most influential parameters on different failure modes and designing novel hybrid models (multi-class classifiers). 1.2.3. Experimental Studies and Rockburst Maximum Stress As mentioned earlier, rockburst can also be investigated using experimental methods. In other words, the stress state around the excavations can be simulated using laboratory tests, and subsequently, study the failure mechanism/characteristics of rocks under different loading histories and loading conditions. Furthermore, the obtained results from these tests can be used 4

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to calibrate the numerical models as well as to identify the critical stress conditions leading to dynamic failures. These experimental tests include uniaxial compression/tensile tests (Gong et al. 2019), conventional triaxial unloading tests (Huang et al. 2001), combined uniaxial and biaxial static-dynamic (cyclic) tests and true-triaxial loading/unloading tests (Bagde and Petroš 2005; He et al. 2010; Su et al. 2018). The conventional uniaxial compression and tensile tests usually have been used by the researchers to measure the mechanical rock properties (e.g. 𝜎 , 𝑐 𝜎 , elastic modulus and so on), perform the energy analysis based on the obtained stress-strain 𝑡 curves and finally, to develop the strength- and strain/energy-based rockburst empirical criteria (e.g. rock brittleness index, 𝐵𝐼 = 𝜎 /𝜎 ). The combined static-dynamic (cyclic) tests in 𝑐 𝑡 uniaxial, bi-axial and true-triaxial conditions are also significant to reproduce the stress state affecting on underground structures (e.g. mining pillars) which are exposed to in-situ stress and cyclic loading induced by different seismic sources (e.g. blasting waves). However, among the foregoing experimental methods, the true-triaxial unloading test can better simulate physically the rockburst process in deep underground conditions. The true-triaxial unloading apparatus is capable of applying the in-situ stresses to the specimen simultaneously and independently, and by unloading the pressure on one or more surfaces of the specimen, can simulate the strain bursting at different locations of underground excavations. In studies undertaken using the true-triaxial testing system, the bursting propensity of rocks has been investigated based on the evolution of acoustic emission (AE) parameters (e.g. AE energy, hits, frequency, b-value), the kinetic energy of ejected rock fragments from the free face of the tested rock specimens, ejection velocity parameter, size of the rock fragments, the evolution of strain energy components and failure mode. Also, in some of these studies, the influence of different parameters such as temperature, moisture content, aspect ratio, loading and unloading rate, deviator stress, tunnel axial stress and radial stress gradient on rockbursting have been evaluated. Although considerable studies have been conducted using the true-triaxial test method for rockburst assessment by different researchers, most of them are limited to some specific rock types and loading histories, and there is no holistic and convenient approach to quantify the bursting potential of rocks. Rock specimens subjected to true-triaxial unloading conditions usually experience an explosion-like failure at a specific stress level, known as rockburst maximum stress (𝜎 ). The proper estimation of this 𝑅𝐵 stress level for different rock types can help engineers to identify rockburst hazards in different in-situ stress conditions, to increase the long-term stability of the underground openings as well as for numerical studies. This task can be accomplished by compiling a comprehensive 5

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database from rockburst tests and the application of robust ML techniques. By doing so, the developed model can be used conveniently in practice to predict bursting stress when the testing apparatus is not available. 1.2.4. Seismic Events and Rock Failure Behaviour As stated earlier, rockburst can also be triggered by seismic disturbances induced by different sources in deep underground mines (i.e. impact-induced rockbursts). These seismic events can be replicated as time-dependent loads, i.e. cyclic and dynamic loadings, on a laboratory scale. Almost sixteen types of stress waves (waveforms) including ramp wave, sinusoidal wave, square wave, sawtooth wave and so forth can be generated in the laboratory to simulate rockburst with different magnitudes (He et al. 2018). The literature review (Bagde and Petroš 2005; Cerfontaine and Collin 2018) shows that different researchers have made tremendous efforts during the last decades to unveil the rock fatigue mechanism under different loading histories and loading conditions (i.e. uniaxial tests, triaxial tests, flexion tests, freeze-thaw tests and wetting and drying tests). Generally, prior studies can be classified into two main groups of systematic cyclic loading tests with a constant loading amplitude and damage-controlled cyclic loading tests with an incremental loading amplitude. However, systematic cyclic loadings having the ramp or sinusoidal waveforms can better represent the seismic events that are common during the mining operation. In rock fatigue studies, the results are usually analysed based on the information withdrawn from the measured stress-strain relations. Indeed, the complete stress-strain relation (i.e. the pre-peak and the post-peak regimes) is an efficient tool to manifest the evolution of strain energy (source of rockbursting) during the loading process as well as determining rock failure behaviour. However, the majority of prior studies have focused on the effect of cyclic loading effect on the pre-peak characteristics of rocks (i.e. damage evolution, variation of peak strength and deformability parameters and determination of fatigue life and fatigue threshold stress), and, no significant progress has been made regarding the post-failure behaviour of rocks under cyclic loading. This is while in practical engineering, due to the release of in-situ rock stresses in the field, the surrounding rocks experience damage and instabilities in the post-peak state. In this regard, the rock brittleness showing the release mode of stored strain energy during loading is a very significant parameter in the process of rockburst assessment. However, the common method of brittleness measurement, i.e. 𝐵𝐼 = 𝜎 /𝜎 , cannot represent the brittleness 𝑐 𝑡 of rocks properly as the physical meaning of this index does not reflect the rock fracturing 6

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process as well as 𝜎 and 𝜎 can be obtained from each other. Moreover, previous studies show 𝑐 𝑡 that rocks with different 𝜎 and 𝜎 may have similar 𝐵𝐼 values representing the narrow range 𝑐 𝑡 of variation of this index (Munoz et al. 2016; Meng et al. 2020). Hence, the rock brittleness can be measured in a more reliable manner based on the energy evolution in both the pre-peak and the post-peak regimes of rocks. On the other hand, rockburst usually occurs in rocks showing Class II behaviour during the failure stage accompanied by the release of excess energy and rock ejection (Li 2021). Therefore, the proper measurement of post-peak behaviour of rocks under cyclic loading is of paramount significance to quantify the post-peak fracture energy, determine the rock brittleness, and consequently, understand more about the mechanism of severe geotechnical hazards like rockburst. However, as mentioned above, the current testing methods are not capable of capturing the post-peak stress-strain curve of rocks under cyclic loading adequately, specifically for brittle rocks which show a snap-back/self-sustaining failure behaviour in the post-peak regime. This is relevant to difficulties in controlling the axial load and damage extension in the post-peak regime for such rocks. The post-peak behaviour of rocks usually is characterised by either Class II (positive post-peak modulus representing an unstable fracture propagation) or a combination of Class I (negative post-peak modulus representing stable fracture propagation) and Class II behaviour. As it is discussed in detail in Chapters 6 and 7, the current axial load-controlled, axial displacement-controlled and lateral displacement- controlled loading techniques have significant limitations in controlling the axial load in the post-peak regime of rocks subjected to systematic cyclic loading. Thus, applying the current loading techniques results in a sudden failure without capturing the post-peak response properly. Therefore, a new testing methodology having the capability of performing different cyclic loading histories and measuring the complete stress-strain relations of rocks in both uniaxial and triaxial loading conditions is required. 1.2.5. Evolution of Rock Fatigue Characteristics In prior rock fatigue studies, little attention has been made to some specific phenomena/parameters, including cyclic loading-induced strength hardening, fatigue threshold stress and post-peak instability of rocks and their variations at different confining pressures (𝜎 ) and stress levels. This is while in rock engineering projects, depending on the depth and 3 geometry of excavations, surrounding rocks usually experience systematic cyclic loading at different confinement levels. Therefore, having an in-depth knowledge regarding the evolution 7

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of the foregoing parameters with confinement level can open new insights into the failure mechanism of rocks, long-term stability of openings and reinforcement design. This task, however, requires applying a triaxial testing method, capable of recording the large lateral deformations created in the post-failure stage. 1.3. Research Objectives and Thesis Layout Figure 1.1 represents the objectives, methodology and outcomes of this research schematically. According to the introductory background and the research gaps discussed in Sections 1.1 and 1.2, the present thesis addressed the following objectives: 1) To develop practical models to predict the occurrence or non-occurrence of rockburst hazard in deep underground mines through a binary expression and evaluate the effect of different parameters on rockbursting. 2) To assess rockburst risk levels (intensities) using robust ML techniques and evaluate the performance of the empirical criteria. 3) To measure the propensity of the over-stressed rock masses to different failure mechanisms in deep underground conditions. 4) To develop practical models for predicting both rockburst maximum stress (𝜎 ) and 𝑅𝐵 rockburst risk index (𝐼 ) based on the results obtained from the true-triaxial unloading 𝑅𝐵 tests. 5) To develop a new experimental methodology to capture the post-failure behaviour of rocks subjected to systematic cyclic loading in uniaxial loading conditions. 6) To investigate the effect of pre-peak systematic cyclic loading at different stress levels on damage evolution and failure characteristics of rocks in uniaxial conditions. 7) To investigate the effect of confining pressure on some specific rock fatigue characteristics, including fatigue threshold stress, post-peak instability, and strength hardening induced by cyclic loading. In this thesis, the data-driven approaches and rock mechanics laboratory tests were utilised as two main research tools to achieve the above objectives. According to the defined research objectives above, this thesis has been structured into eight chapters as follows: 8

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The current chapter, Chapter 1, provides an introductory background regarding this research and contains topics including problem statement, literature review and research gaps, research objectives and thesis layout and conclusions and recommendations. In Chapter 2, to address objective 1, a comprehensive study is carried out on the prediction of rockburst occurrence/non-occurrence based on a database containing 134 rockburst events, compiled from different underground mines. Several significant parameters, including uniaxial compressive strength (𝜎 ), uniaxial tensile strength (𝜎 ), maximum tangential stress (𝜎 ) and 𝑐 𝑡 𝜃 elastic energy index (𝑊 ) are chosen as input parameters, while a binary condition (i.e. “1” 𝑒𝑡 for occurrence and “0” for non-occurrence) is defined for rockburst as the output parameter. The homogeneity of the database is initially evaluated using different statistical tests. New models are then developed using three robust supervised ML techniques, including genetic algorithm-based emotional neural network (GA-ENN), decision tree-based C4.5 algorithm and gene expression programming (GEP) algorithm. Finally, the performance of the proposed new models, along with five empirical criteria, are evaluated, and the sensitivity analysis is performed on the best model to identify the most influential parameters on rockbursting. The results showed the high performance of the ML techniques in solving complex nonlinear geotechnical hazards like rockburst and their capability to improve practical models that can be used in the pre-design stages of an underground opening. The results of this study were published as a journal paper entitled “Long-term prediction of rockburst hazard in deep underground openings using three robust data mining techniques”. The details of this paper are as follows: Shirani Faradonbeh R, Taheri A (2019) Long-term prediction of rockburst hazard in deep underground openings using three robust data mining techniques. Engineering with Computers 35(2):659–675 (IF= 7.963, Q1) In Chapter 3, two robust unsupervised algorithms, self-organizing map (SOP) and Fuzzy C- Mean (FCM) are used to cluster and identify rockburst risk level (intensity) as a multi-class problem based on the collected database (i.e. objective 2). The input parameters in this study are the same used in Chapter 2. The output, however, is a qualitative parameter showing different degrees of bursting, i.e. “None”, “Light”, “Moderate” and “Strong”, which have been defined based on an empirical classification/visual inspection of rockburst location. These two applied unsupervised algorithms are capable of finding the latent relationships between the 9

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input parameters and the corresponding output during a smart procedure, and finally, link each observation (rockburst event) to an appropriate cluster (risk level). In addition to SOM and FCM techniques, five empirical criteria are also employed to assess their capability in clustering rockburst events. Five common performance measures comprising accuracy (%), precision (%), Recall (%), F1 score (%) and Kappa (%) are calculated for all models and results are compared. This study revealed the superiority of the unsupervised ML techniques in terms of accuracy over the conventional criteria in assessing rockburst intensity. The results of this study were published as a journal paper entitled “Application of self-organizing map and fuzzy c-mean techniques for rockburst clustering in deep underground projects”. The details of this paper are as follows: Shirani Faradonbeh R, Shaffiee Haghshenas S, Taheri A, Mikaeil R (2020) Application of self-organizing map and fuzzy c-mean techniques for rockburst clustering in deep underground projects. Neural Computing and Applications 32(12):8545–8559 (IF= 5.606, Q1) Chapter 4 aims to address objective 3, i.e. developing a practical and easy-to-use model for distinguishing different failure mechanisms of the over-stressed rock masses in deep underground conditions. For this aim, a database containing 35 failure events recorded from different underground projects is compiled. This database contains a wide range of rock types with compressive strength varying from 41 MPa to 335 MPa and includes three common types of failure, i.e. squeezing, strainbursting and slabbing. The intact rock properties, including uniaxial compressive strength (𝜎 ), Brazilian tensile strength (𝜎 ), elastic modulus (𝐸) and 𝑐 𝑡 Poisson’s ratio (𝜐), which can be measured straightforwardly in the laboratory and have a significant effect on failure mechanisms are chosen as the predictors, while the failure mode is selected as the output parameter. In this chapter, a novel hybrid data-driven approach, namely gene expression programming based-logistic regression (GEP-LR), is proposed and implemented as a multi-class classifier to estimate the failure mechanism based on the given intact properties. Three separate binary mathematical models are initially developed using the GEP algorithm to reveal the relationship between failure mode and input parameters. Then, a probabilistic approach (i.e. LR) is linked to the GEP models to determine the probability of occurrence of each failure mechanism with high accuracy. Finally, the failure type having the highest probability index is selected as the output. The developed model in this study is provided as MatLab codes which researchers and engineers can use in practice to identify the 10

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most probable failure type in different locations and consequently apply an appropriate controlling technique. The results of this study were prepared as a journal paper entitled “Rockburst assessment in deep geotechnical conditions using true-triaxial tests and data-driven approaches”. The details of this paper are as follows: Shirani Faradonbeh R, Taheri A, Karakus M (2020) The propensity of the over-stressed rock masses to different failure mechanisms based on a hybrid probabilistic approach. Tunnelling and Underground Space Technology x(x): x-x. The revised format submitted on 15/06/2021 (Under review) (IF= 5.915, Q1) In Chapter 5, a comprehensive study is carried out by combining the results obtained from the true-triaxial unloading tests (rockburst tests) and two white-box machine learning (ML) algorithms to provide new models for estimating rockburst maximum stress (𝜎 ) and its risk 𝑅𝐵 index (𝐼 ) (objective 4). The information of rockburst laboratory tests conducted from 2004 𝑅𝐵 to 2012 are compiled in this study, and a series of statistical analyses are performed to provide a homogeneous database (i.e. removing missing values, identifying the outliers and natural groups in the original database). The prepared database contains different parameters including rock mass properties (i.e. 𝑈𝐶𝑆, 𝐸 and 𝜈), in-situ stresses, depth, rock density and horizontal pressure coefficient, which can be considered as input variables, and 𝜎 and 𝐼 , which are 𝑅𝐵 𝑅𝐵 defined as outputs. However, a systematic strategy, i.e. the stepwise selection and elimination (SSE) procedure, is followed to choose the most influential input parameters and subsequently decrease the complexity of the final models. The GEP algorithm that whose high performance in modelling complex problems was proved in previous chapters, is utilised along with the classification and regression tree (CART) algorithm to develop some explicit models (i.e. mathematical and graphical models) for estimating rockburst parameters. Validation of the developed models is completely verified using seven statistical indices and their corresponding thresholds. Parametric analysis is also performed in this study on the best models to evaluate the evolution of rockburst parameters by changing each input parameter within its range of values. The results point to the applicability of the proposed models for rockburst assessment with high reliability. These models can help researchers and engineers to estimate the stress level that rocks are prone to bursting and evaluate the rockburst risk level. The results of this study were published as a journal paper entitled “Rockburst assessment in deep geotechnical conditions using true-triaxial tests and data-driven approaches”. The details of this paper are as follows: 11

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Shirani Faradonbeh R, Taheri A, Ribeiro e Sousa L, Karakus M (2020) Rockburst assessment in deep geotechnical conditions using true-triaxial tests and data-driven approaches. International Journal of Rock Mechanics and Mining Sciences 128:104279 (IF= 7.135, Q1) In Chapter 6, by reviewing the prior rock fatigue studies, a holistic classification is proposed for cyclic loading tests based on the loading history and load control technique. Also, a new experimental methodology, namely “Double-criteria damage-controlled cyclic loading test” is introduced in this chapter to capture the complete stress-strain relation of rocks (i.e. the pre- peak and the post-peak regimes) subjected to systematic cyclic loading (objective 5). In this new testing method, two criteria including the maximum axial stress level that cyclic loading is applied and the maximum lateral strain amplitude, 𝐴𝑚𝑝.(𝜀 ), that a rock specimen is allowed 𝑙 to experience in a cycle during loading are adopted to control the axial load and damage extension before and after failure point. Tuffeau limestone is used in this study as a soft porous rock to evaluate the applicability of the proposed testing method in capturing the post-failure behaviour of rocks. A series of multi-level systematic cyclic loading tests are undertaken in this study by applying the axial load at approximately 81% of the determined average 𝑈𝐶𝑆, and the post-peak behaviour is captured in a controlled manner. Based on the obtained complete stress-strain relations, a preliminary evaluation is performed on post-peak behaviour as well as the evolution of fatigue damage parameters. Generally, the results represent the success of the proposed technique in measuring the full response of rocks under cyclic loading, which can open new insights regarding the rock failure mechanism. Also, a strength hardening induced by cyclic loading is observed for this rock type which needs to be further investigated. The results of this study were published as a journal paper entitled “Post-peak behaviour of rocks under cyclic loading using a double-criteria damage-controlled test method”. The details of this paper are as follows: Shirani Faradonbeh R, Taheri A, Karakus M (2021) Post-peak behaviour of rocks under cyclic loading using a double-criteria damage-controlled test method. Bulletin of Engineering Geology and the Environment 80(2):1713–1727 (IF= 4.298, Q1) In Chapter 7, a more comprehensive experimental study is undertaken using the developed test method in Chapter 6 to investigate the effect of pre-peak systematic cyclic loading applied at different stress levels on both pre-peak and post-peak characteristics of Gosford sandstone in 12

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uniaxial loading conditions (objective 6). This chapter also intends to examine some specific behaviours observed in the previous chapter (e.g. cyclic loading-induced strength hardening) in more depth. In this chapter, the uniformity of the testing material is initially evaluated based on the performed six 𝑈𝐶𝑆 tests and the measured mechanical rock properties. Seventeen (17) single-level systematic cyclic loading tests are then designed at different stress levels ranging from 80% to 96% of the average monotonic strength (i.e. in the unstable crack propagation stage). This study defines two types of cyclic loading tests: hardening cyclic loading tests (the specimens that do not fail during 1500 cycles) and fatigue cyclic loading tests (the specimens that fail in the cycle). For both types of tests, the double-criteria damage-controlled cyclic loading test method is adjusted in such a way that the post-peak behaviour of rocks is captured in a controlled manner, and based on the measured complete stress-strain relations, the damage evolution, post-peak instability of rocks (rock brittleness) and strength hardening phenomenon is investigated comprehensively. The results of this study were published as a journal paper entitled “Failure behaviour of a sandstone subjected to the systematic cyclic loading: Insights from the double-criteria damage-controlled test method”. The details of this paper are as follows: Shirani Faradonbeh R, Taheri A, Karakus M (2021) Failure behaviour of a sandstone subjected to the systematic cyclic Loading: Insights from the double-criteria damage- controlled test method. Rock Mechanics Rock Engineering x(x): x-x (IF= 6.730, Q1) In Chapter 8, for the first time, a comprehensive study is carried out in triaxial conditions to better replicate the stress state in deep underground openings and consequently understand more about the failure mechanism of rocks subjected to cyclic loading under different confining pressures. A modified triaxial testing system (by mounting four strain gauges on the Hoek cell membrane and connecting them to a half-bridge circuit) is utilised to provide a single lateral strain-based feedback signal. With this arrangement, failure behaviour was accurately investigated. Seven confinement levels (i.e. 𝜎 /𝑈𝐶𝑆 = 10-100%) are defined to evaluate the 3 𝑎𝑣𝑔 effect of both confining pressure and systematic cyclic loading history on the evolution of some specific rock fatigue characteristics, including post-peak brittleness, fatigue threshold stress and strength hardening. At each confinement level, the specimens experience 1000 loading and unloading cycles at different stress levels. Should the specimen did not fail in cycles, a final monotonic loading is applied under lateral strain-controlled loading conditions to capture the failure behaviour. The non-destructive AE technique also is employed to analyse damage 13

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Prediction and control of rock burst phenomenon in deep underground mining based on rock behaviour Objective 1: A practical model for rockburst Objective 5: New testing method to capture post- occurrence prediction peak behaviour of rocks under cyclic loading Objective 2: Assessing rockburst risk levels accurately Objective 6: Investigate the effect of the pre-peak Objective 3: Practical model for predicting the failure systematic cyclic loading on the failure behaviour mechanism of the over-stressed rock masses of rocks in uniaxial conditions Objective 4: Practical models for predicting rockburst Objective 7: Investigate the effect of confining maximum stress and its risk index pressure on rock fatigue characteristics Machine Learning Analysis Experimental Analysis 1- Database Preparation 1- Developing a new testing methodology 2- Statistical analysis of the database 2- Conducting UCS tests 3- Model development using robust ML methods 3- Performing uniaxial single-level 4- Validation verification of the models and multi-level cyclic loading tests 5- Comparing the results 4- Performing triaxial cyclic loading tests 6- Sensitivity/parametric analysis 5- Comprehensive analysis of the test results Paper #1 Paper #2 Paper #3 Paper #4 Paper #5 Paper #6 Paper #7 (Chapter 2) (Chapter 3) (Chapter 4) (Chapter 5) (Chapter 6) (Chapter 7) (Chapter 8) Conclusions & Recommendations (Chapter 9) Figure 1.1 The objectives, methodologies and outcomes of the present thesis References Bagde MN, Petroš V (2005) Fatigue properties of intact sandstone samples subjected to dynamic uniaxial cyclical loading. International Journal of Rock Mechanics and Mining Sciences 42(2):237–250 Cai M, Kaiser P (2018) Rockburst support reference book—volume I: rockburst phenomenon and support characteristics. Laurentian University. 284 15

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Cerfontaine B, Collin F (2018) Cyclic and fatigue behaviour of rock materials: review, interpretation and research perspectives. Rock Mechanics and Rock Engineering 51(2):391–414 Feng X (2017) Rockburst : mechanisms, monitoring, warning, and mitigation. Butterworth- Heinemann Gong F, Yan J, Li X, Luo S (2019) A peak-strength strain energy storage index for rock burst proneness of rock materials. International Journal of Rock Mechanics and Mining Sciences 117:76–89 He M, e Sousa LR, Miranda T, Zhu G (2015) Rockburst laboratory tests database - Application of data mining techniques. Engineering Geology 185:116–130 He M, Ren F, Liu D (2018) Rockburst mechanism research and its control. International Journal of Mining Science and Technology 28(5):829–837 He M, Xia H, Jia X, et al (2012) Studies on classification, criteria and control of rockbursts. Journal of Rock Mechanics and Geotechnical Engineering 4(2):97–114 He MC, Miao JL, Feng JL (2010) Rock burst process of limestone and its acoustic emission characteristics under true-triaxial unloading conditions. International Journal of Rock Mechanics and Mining Sciences 47(2):286–298 Huang RQ, Wang XN, Chan LS (2001) Triaxial unloading test of rocks and its implication for rock burst. Bulletin of Engineering Geology and the Environment 60(1):37–41 Jian Z, Xibing L, Xiuzhi S (2012) Long-term prediction model of rockburst in underground openings using heuristic algorithms and support vector machines. Safety Science 50(4):629–644 Li CC (2021) Principles and methods of rock support for rockburst control. Journal of Rock Mechanics and Geotechnical Engineering 13(1):46–59 Li N, Feng X, Jimenez R (2017) Predicting rock burst hazard with incomplete data using Bayesian networks. Tunnelling and Underground Space Technology 61:61–70 Liu Z, Shao J, Xu W, Meng Y (2013) Prediction of rock burst classification using the technique 16

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of cloud models with attribution weight. Natural Hazards 68:549–568 Meng F, Wong LNY, Zhou H (2020) Rock brittleness indices and their applications to different fields of rock engineering: A review. Journal of Rock Mechanics and Geotechnical Engineering, 68(2):549-568 Munoz H, Taheri A, Chanda EK (2016) Fracture Energy-Based Brittleness Index Development and Brittleness Quantification by Pre-peak Strength Parameters in Rock Uniaxial Compression. Rock Mechanics and Rock Engineering 49(12):4587–4606 Pu Y, Apel DB, Liu V, Mitri H (2019) Machine learning methods for rockburst prediction- state-of-the-art review. International Journal of Mining Science and Technology 29(4):565–570 Saharan MR, Mitri H (2011) Destress blasting as a mines safety tool: Some fundamental challenges for successful applications. In: Procedia Engineering. Elsevier, pp 37–47 Su G, Hu L, Feng X, et al (2018) True triaxial experimental study of rockbursts induced by ramp and cyclic dynamic disturbances. Rock Mechanics and Rock Engineering 51(4):1027–1045 Taheri A, Yfantidis N, L. Olivares C, et al (2016) Experimental Study on Degradation of Mechanical Properties of Sandstone Under Different Cyclic Loadings. Geotechnical Testing Journal 39(4):673-687 Zhou J, Li X, Mitri HS (2018) Evaluation method of rockburst: State-of-the-art literature review. Tunnelling and Underground Space Technology 81:632–659 17

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Statement of Authorship Title of Paper Long-term prediction of rockburst hazard in deep underground openings using three robust data mining techniques Publication Status Published Accepted for Publication Submitted for Publication Unpublished and Unsubmitted work written in manuscript style Publication Details Shirani Faradonbeh R, Taheri A (2019) Long-term prediction of rockburst hazard in deep underground openings using three robust data mining techniques. Engineering with Computers 35(2):659–675 Principal Author Name of Principal Author (Candidate) Roohollah Shirani Faradonbeh Contribution to the Paper Literature review and database preparation, statistical analysis, development of models and preparation of the manuscript Overall percentage (%) 80% Certification: This paper reports on original research I conducted during the period of my Higher Degree by Research candidature and is not subject to any obligations or contractual agreements with a third party that would constrain its inclusion in this thesis. I am the primary author of this paper. Signature Date 17 June 2021 Co-Author Contributions By signing the Statement of Authorship, each author certifies that: i. the candidate’s stated contribution to the publication is accurate (as detailed above); ii. permission is granted for the candidate in include the publication in the thesis; and iii. the sum of all co-author contributions is equal to 100% less the candidate’s stated contribution. Name of Co-Author Abbas Taheri Contribution to the Paper Research supervision, review and revision of the manuscript Signature Date 21 June 2021 18

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Chapter 2 Long-term Prediction of Rockburst Hazard in Deep Underground Openings using Three Robust Data Mining Techniques Abstract Rockburst phenomenon is the extreme release of strain energy stored in surrounding rock mass which could lead to casualties, damage to underground structures and equipment and finally endanger the economic viability of the project. Considering the complex mechanism of rockburst and a large number of factors affecting it, the conventional criteria cannot be used generally and with high reliability. Hence, there is a need to develop new models with high accuracy and easy to use in practice. This study focuses on the applicability of three novel data mining techniques including emotional neural network (ENN), gene expression programming (GEP), and decision tree-based C4.5 algorithm along with five conventional criteria to predict the occurrence of rockburst in a binary condition. To do so, a total of 134 rockburst events were compiled from various case studies and the models were established based on training datasets and input parameters of maximum tangential stress, uniaxial tensile strength, uniaxial compressive strength, and elastic energy index. The prediction strength of the constructed models was evaluated by feeding the testing datasets to the models and measuring the indices of root mean squared error (RMSE) and percentage of the successful prediction (PSP). The results showed the high accuracy and applicability of all three new models, however, the GA- ENN and the GEP methods outperformed the C4.5 method. Besides, it was found that the criterion of elastic energy index (EEI) is more accurate among other conventional criteria and with the results similar to the C4.5 model, can be used easily in practical applications. Finally, a sensitivity analysis was carried out and the maximum tangential stress was identified as the most influential parameter, which could be a guide for rockburst prediction. Keywords: Rockburst occurrence, Data mining techniques, Emotional neural network, Gene expression programming, C4.5 algorithm, Conventional criteria 19

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2.1. Introduction One of the most important concerns in deep underground activities such as mining and civil projects is the occurrence of rockburst phenomenon. Rockburst is an unexpected and severe failure of a large volume of over-stressed rock caused by the instantaneous release of accumulated strain energy. This phenomenon usually is accompanied by other events such as spalling, slabbing, and throwing of rock fragments which could be led to injuries, deformation of supporting system, damage to equipment or even collapse of a large area of the underground excavation and finally cease the operation (Dong et al. 2013; Adoko et al. 2013; Li et al. 2017; Weng et al. 2018) . In deep underground activities, the induced seismicity has a great role in rockburst occurrence, therefore, the identification and localization of seismic events are essential in rockburst assessment (Dong et al. 2016a, b, 2017a, b). Great number of theoretical and experimental studies have been performed since 1930 by many researchers on the mechanism, prediction, and control of rockburst (Weng et al. 2017; Akdag et al. 2018). However, rockburst still remains an unsolved problem in deep mining (He et al. 2015). Rockbursts can be classified using various criteria comprising potential damage, failure pattern, scale, and severity. From the viewpoint of damage, it classifies into four classes of none, light, moderate, and strong. Based on the failure pattern, there are four types of failures including slabby spalling, dome failure, in-cave collapse, and bending failure. In terms of scale, rockbursts can be introduced as sparse with the rockburst length lower than 10 m, large-area with the rockburst length between 10-20 m and continuous rockburst with the length higher than 20 m. The severity of rockbursts can be assessed as a function of failure depth (He et al. 2012; Wang et al. 2012). According to the influence diagram developed by Sousa and Einstein (2007), many factors affect the occurrence of rockburst such as mechanical properties of rock, geological circumstances, construction method, and in-situ stress state. Considering the great number of effective parameters and the vague mechanism of rockburst, prediction, and control of this hazardous phenomenon is a very difficult task. Rockburst can be predicted in short-term and long-term. In-situ measurement techniques such as microseismic monitoring system and acoustic emission can be used to acquire the exact location and the specific time of rockburst occurrence at each stage of the project (i.e. in short-term). However, these techniques are time- consuming, costly, and require precise surveying strategies. On the other hand, rockburst prediction in long-term is mainly based on conventional criteria, numerical models, and data mining techniques. Compared to the short-term prediction technique, the long-term one can be served as a quick guide for engineers during the initial stages of the project and consequently, 20

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enable them to decide about the excavating and controlling methods (Adoko et al. 2013; Li et al. 2017). During the last three decades, various rockburst proneness indices have been developed based on strength parameters and rock strain energy (see Table 2.1) [15]. Table 2.1 A summary of conventional criteria for rockburst prediction Criterion Equation Input parameters Rockburst discrimination 𝜎 Russenes criterion (Russenes 1974) 𝜃 𝜎 ,𝜎 ≥0.25 𝜃 𝑐 𝜎 𝑐 𝜎 Hoek criterion (Hoek and Brown 1980) 𝑐 𝜎 ,𝜎 ≤3.5 𝜃 𝑐 𝜎 𝜃 𝜎 Stress coefficient (Wang et al. 1998) 𝜃 𝜎 ,𝜎 ≥0.3 𝜃 𝑐 𝜎 𝑐 𝜎 Rock brittleness coefficient (Wang et al. 1998) 𝑐 𝜎 ,𝜎 ≤40 𝑡 𝑐 𝜎 𝑡 Elastic energy index (Wang et al. 1998) 𝐸 𝑅 𝐸 𝑅,𝐸 𝐷 ≥2.0 𝐸 𝐷 𝜎 is the maximum tangential stress, 𝜎 is the uniaxial compressive stress, 𝜎 is the uniaxial tensile 𝜃 𝑐 𝑡 stress, 𝐸 is the retained energy, 𝐸 is the dissipated energy 𝑅 𝐷 According to Table 2.1, the conventional criteria only consider very few input parameters, therefore, cannot take into account a wide range of parameters that may influence rock- bursting. Data mining is a relatively new computational method with the aim of discovering latent patterns and relationships between raw datasets which combines different areas such as statistics, machine learning, and so on. Data mining techniques have the capability to deal with the datasets containing multiple input and output variables (Berthold and Hand 2003; Jian et al. 2012). Hence, they have been used extensively in geosciences (Khandelwal et al. 2017a, b; Aryafar et al. 2018; Mikaeil et al. 2018a). As a first attempt, Feng and Wang (1994) developed two artificial neural networks (ANNs) to predict and control the probable rockbursts. Their successful experience encouraged other scholars to investigate the applicability of novel data mining techniques in rockburst assessment (Zhao 2005; Gong and Li 2007; Shi et al. 2010; Zhou et al. 2010). Although the methods used by the scholars could consider more input parameters, most of them are black-box, i.e. they cannot provide a clear and comprehensible relationship between the input and output parameters. Consequently, the developed models using such opaque techniques cannot easily be used in practice. On the other hand, the conventional criteria as reported in many studies, could not predict rockburst with high accuracy. Therefore, there is still a need to develop transparent and easy to use rockburst 21

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increase of 𝑊 , the probability of rockburst occurrence and its intensity will increase 𝑒𝑡 (Palmstrom 1995; Jian et al. 2012; Liu et al. 2013; Li et al. 2017). Therefore, in the current study, four parameters of 𝜎 , 𝜎 , 𝜎 , and 𝑊 were adopted as the input parameters. Table 2.2 𝜃 𝑡 𝑐 𝑒𝑡 shows the descriptive statistics of the relevant input parameters that are used to develop rockburst models. For convenience, the abbreviations of input parameters were considered for modelling instead of their symbols; they are characterized by MTS, UTS, UCS, and EEI for 𝜎 , 𝜎 , 𝜎 , and 𝑊 , respectively. To understand more about the relationship between the input 𝜃 𝑡 𝑐 𝑒𝑡 parameters, Pearson correlation coefficients were computed which the results are listed in Table 2.3. According to this table, there are moderate correlations for the UTS-UCS and EEI- UCS if the categorizations proposed by Dancy and Reidy (2004) are followed. Table 2.2 Descriptive statistics of the input parameters within the database Parameter Abbreviation Unit Minimum Maximum Mean Std. deviation Variance 𝜎 MTS MPa 2.6 108.4 51.354 28.567 816.055 𝜃 𝜎 UTS MPa 1.3 22.6 7.519 4.926 24.268 𝑡 𝜎 UCS MPa 20.0 306.6 127.957 59.417 3530.415 𝑐 𝑊 EEI Dimensionless 0.85 10.6 4.726 2.196 4.824 𝑒𝑡 Table 2.3 Correlation coefficients between the input parameters Variables T UTS UCS EEI T 1 0.569 0.589 0.508 UTS 0.569 1 0.650 0.443 UCS 0.589 0.650 1 0.636 EEI 0.508 0.443 0.636 1 Prior to any modelling, the statistical analysis of original database has high importance. The presence of outliers in the database negatively affects the ability of algorithms to find a precise relationship between input and output parameters and consequently, decreases the reliability of the developed model. Additionally, outliers may create some natural groups with different behaviours in a single dataset and if this is the case, it is necessary to identify them and develop separate models (Middleton 2000; Tiryaki 2008). The box-plot is a common and standardized method to display the distribution of data based on minimum, first quartile (𝑄1), median (𝑄2), third quartile (𝑄3), and maximum values. The measurements outside the range of (𝑄1− 23

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3(𝑄3−𝑄1),𝑄3+3(𝑄3−𝑄1)) are defined as extreme outliers and should to be omitted from the database, while those which are in the range of (𝑄1−1.5(𝑄3−𝑄1),𝑄3+1.5(𝑄3−𝑄1)) are known as suspected outliers which are common in a big database and could be considered in modelling (Middleton 2000). Fig. 2.1 shows the box-plots of input parameters. According to this figure, the median line is not in the centre of boxes which indicates that the input parameters do not have a symmetric distribution. Besides, with the exception of MTS, other input parameters have few suspected outliers. MTS (MPa) UTS (MPa) UCS (MPa) EEI 120 25 350 12 300 100 10 20 250 80 8 15 200 60 6 150 10 40 4 100 5 20 2 50 0 0 0 0 Figure 2.1 Box plots of input parameters As a second effort, a principal component analysis (PCA) was conducted to check the existence or non-existence of natural groups in the database. PCA is a dimension reduction technique that enables the user to transform the initial correlated variables from an 𝑚-dimensional space to an 𝑛-dimensional one where 𝑛 < 𝑚. The new uncorrelated variables are nominated as principal components (PCs) which are the linear combination of initial variables (Sayadi et al. 2012; Faradonbeh and Monjezi 2017). To perform this analysis, firstly, the datasets were normalized between 0 and 1 using the Min-Max method to eliminate the effect of range. In the second step, the correlation matrix was created for input parameters. Then, the eigenvalues and eigenvectors corresponding to the previous correlation matrix were calculated for each PC as follows: 𝜆 ,𝜆 ,…,𝜆 𝑋𝑉 = 𝜆𝑉 → (𝑋−𝜆𝐼)𝑉 = 0 → 𝑑𝑒𝑡𝑒𝑟𝑚𝑖𝑛𝑎𝑛𝑡(𝑋−𝜆𝐼) = 0 → { 1 2 𝑛 (2.1) 𝑉 ,𝑉 ,…,𝑉 1 2 𝑛 where 𝑋, 𝜆, and 𝑉 are the matrix of datasets, eigenvalue, and eigenvectors, respectively. 24

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Eventually, the PCs were obtained by multiplying the input parameters in related eigenvectors. Fig. 2.2 shows the scree plot of eigenvalues against the number of components. According to this figure, 92.872 % of the database variations can be explained just with three first PCs by projecting the observations on these axes (i.e. PC1, PC2, and PC3). The scatter plots of PC1- PC2 and PC1-PC3 are shown in Fig. 2.3. As can be seen, there is not any natural group, i.e. the concentration of observations in specific areas in the database. Besides, few suspected outliers mentioned in the previous analysis can be seen in this figure as well. As a result, it can be said that the prepared database is suitable for further analysis. The output parameter is the rockburst occurrence, if any, was nominated as “Yes” otherwise, was nominated as “No”. Since the output is a qualitative parameter, we transferred it to a binary parameter, i.e. 0 (No) and 1(Yes). 3 100 2.5 80 ) % ( y e 2 60 tilib u a la v n 1.5 ir a v e e g 40 v iE 1 ita lu m 0.5 20 u C 0 0 PC1 PC2 PC3 PC4 Eigenvalue 2.702 0.567 0.446 0.285 Cumulative 67.559 81.734 92.872 100.000 Figure 2.2 Scree plot of PCA analysis 25

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and output layers that finally lead to high computational complexity (CC). Recently, a limbic- based emotional neural network (ENN) is developed by Lotfi and Akbarzadeh-T (2014) based on the emotional process of the brain with a single layer structure. Unlike ANNs that is formed based on a biological neuron, ENNs are based on the interaction of four neural areas of the emotional brain comprising thalamus, sensory cortex, orbitofrontal cortex (OFC), and amygdala. These four areas using the features of expanding, comparing, inhibiting, and exciting, overcome the shortages related to the common ANNs and provide more precise solutions. Initial ENNs have a low CC during the learning process, but the number of patterns which can be stored is limited that makes a low information capacity (IC) for this method. Lotfi and Akbarzadeh-T (2016), thanks to a winner-take-all approach (WTA), introduced a new version of ENN with the name of WTAENN which is able to increase the IC of the algorithm. The structure of WTAENN with 𝑛 input, one output, and 𝑚 = 1 competitive part is shown in Fig. 2.4. According to this figure, original input data (i.e. 𝑝⃗ = [𝑝 ,𝑝 ,…,𝑝 ]) first enter to 1 2 𝑛 thalamus part. In the thalamus, input data will expand by the following equation: [𝑝 ,…,𝑝 ] = 𝐹𝐸 (𝑝 ) (2.2) 𝑛+1 𝑛+𝑘 𝑗=1,…,𝑛 𝑗 where 𝐹𝐸 is an expander function which can be a Gaussian or Sinusoidal function or in general can be defined as: 𝐹𝐸 (𝑝 ) = max (𝑝 ) (2.3) 𝑗=1,…,𝑛 𝑗 𝑗 𝑗=1,…,𝑛 Then, the expanded signals are sent to winner sensory cortex 𝑖∗which is selected if only and only if: ∀𝑖 ‖[𝑝 ,𝑝 ,…,𝑝 ]−[𝑐 ,𝑐 ,…,𝑐 ]‖ ≤ ‖[𝑝 ,𝑝 ,…,𝑝 ]−[𝑐 ,𝑐 ,…,𝑐 ]‖, 1 ≤ 𝑖 ≤ 1 2 𝑛 1,𝑖∗ 2,𝑖∗ 𝑛,𝑖∗ 1 2 𝑛 1,𝑖 2,𝑖 𝑛,𝑖 𝑚 (2.4) where 𝑐 ,𝑐 ,…,𝑐 are the learning weights. 1 2 𝑛 Afterwards, the signals propagate to the related OFC and amygdala and the weights of 𝑤 ,𝑤 ,…,𝑤 from the 𝑖th OFC and the weights of 𝑣 ,𝑣 ,…,𝑣 from 𝑖th amygdala are 1,𝑖 2,𝑖 𝑛,𝑖 1,𝑖 2,𝑖 𝑛,𝑖 used during the learning process to determine the final output. During the learning process, amygdala receives the imprecise input of 𝑝 from the thalamus to determine the output signal 𝑛+1 of 𝐸 . After that, amygdala receives an inhibiting signal from OFC (𝐸 ) which with applying 𝑎 𝑜 the activation function (e.g. 𝑝𝑢𝑟𝑒𝑙𝑖𝑛, 𝑡𝑎𝑛𝑠𝑖𝑔, ℎ𝑎𝑟𝑑𝑙𝑖𝑚 and 𝑙𝑜𝑔𝑠𝑖𝑔 functions), the final emotional signal (predicted value) will be achieved. The final output can be calculated by the following equation: 𝐸 (𝑝⃗) = 𝑓(𝐸 −𝐸 ) = 𝑓(∑𝑛+1(𝑣 𝑝 )−∑𝑛 (𝑤 𝑝 )−𝑏 ) (2.5) 𝑖 𝑎𝑖 𝑜𝑖 𝑗=1 𝑗,𝑖 𝑗 𝑗=1 𝑗,𝑖 𝑗 𝑖 27

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where 𝑌𝑘 is the output of the winner part for 𝑘th input pattern, 𝑇𝑘 is the related target and 𝑚 is the number of training pattern targets. By minimizing the cost function, the best learning weights for WTAENN can be obtained (Lotfi et al. 2014; Lotfi and Akbarzadeh-T 2014, 2016). 2.3.1.1. Rockburst Prediction Using GA-ENN In this study, for the first time, the applicability of ENNs was examined to predict rockburst occurrence as a geotechnical engineering problem. In GA-based ENN algorithm, it is necessary to determine the optimum values of its parameters, i.e. the number of competitive parts (𝑚), number of generations, and the population size. The MatLab code was used to develop this model. Since the input parameters have different units and range of values, in soft computing techniques, it is better to normalize datasets on account of speeding up the modelling process, reducing errors, and more importantly preventing the over-fitting phenomenon. So, the input parameters were normalized between 0 and 1 using the following equation: 𝑋 = 1− 𝑋𝑚𝑎𝑥−𝑋 𝑖 (2.11) 𝑛𝑜𝑟𝑚 𝑋𝑚𝑎𝑥−𝑋 𝑚𝑖𝑛 where 𝑋 , 𝑋 , 𝑋 , and 𝑋 are 𝑖th actual value, minimum value, maximum value and 𝑖 𝑚𝑖𝑛 𝑚𝑎𝑥 𝑛𝑜𝑟𝑚 the normalized value of an input parameter, respectively. In the following, the initial database was divided into three parts of training (70% of the database), validation (10% of the database), and testing (20% of the database) to conduct a series of sensitivity analysis and subsequently to find the best combination of parameters. In the first analysis, the parameters of 𝑚 and activation function were fixed on 1 and “𝐻𝑎𝑟𝑑− 𝑙𝑖𝑚𝑖𝑡”, and the values of population size and the number of generations increased from 20 to 300. Fig. 2.5 shows the variation of mean square error (MSE) as the fitness function in each run. According to this figure, after generation no. 100, the MSE value remained constant and no change was observed up to generation no. 300. So, the value of 100 was selected as the optimum value for the parameters of population size and generation number. An increase in MSE can be seen between generations 60 to 100, which may refer to the stochastic mechanism of ENN algorithm for searching and finding the best combination of training coefficients (i.e. 𝑐, 𝑣, and 𝑤 weights) among all the possible solutions. Similarly, the second analysis with the aim of finding the optimum value of 𝑚 was executed by varying its value from 1 to 40 and recording the corresponding MSE values. The 𝑚 = 1 provided the minimum value of MSE. Eventually, the algorithm was executed for several times based on the obtained optimum values for parameters and the best model was identified. Table 2.4 indicates the characteristics of the best GA-ENN model. 29

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0.4 0.35 0.3 0.25 E S 0.2 M 0.15 0.1 0.05 0 0 50 100 150 200 250 300 Generation number and population size Figure 2.5 Variation of fitness function for different values of generation number and population size Table 2.4 Characteristics of developed GA-ENN model Parameter Value Input variables MTS, UTS, UCS, EEI Output variable Rockburst occurrence Yes: 1 No: 0 Generation number 100 Population size 100 Number of competitive units (m) 1 Activation function Hard-limit 2.3.2. C4.5 Algorithm One of the best-renowned data mining techniques is decision tree (DT). The decision tree is a nonparametric technique which benefits from simple and interpretable structure, low computational cost and the ability to represent graphically. DTs have proven their efficiency for various purposes such as classification, decision making and as a tool to make a relationship between independent variables and the dependent one (Breiman et al. 1984; Salimi et al. 2016; Hasanipanah et al. 2017b; Khandelwal et al. 2017a). The most important characteristic of a DT as a “white box” technique is its simple graphical structure which enables the user to clarify the relations between variables easier, while other machine learning techniques such as ANNs 30

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have a vague internal computational procedure, which means the results are difficult to interpret. In the case of having a problem with many variables which act in reciprocally and non-linear ways, finding a comprehensive model may be very difficult. In these circumstances, DT can be a suitable alternative which is able to break down (sub-divide) the initial space into smaller parts so that the interactions are easier to manage. A decision tree is a collection of nodes (root node, internal nodes, and terminal or leaf nodes), arranged as a binary tree. The root node and internal nodes belong to decision stage and represent specific input variables which are connected together based on a smaller range of values. The terminal nodes, show the final classes (Coimbra et al. 2014; Jahed Armaghani et al. 2016; Liang et al. 2016; Hasanipanah et al. 2017a). There are various types of decision trees, including classification and regression tree (CART), Chi-squared automatic interaction detection (CHAID), C4.5, ID3, quick, unbiased, efficient statistical tree (QUEST). C4.5 proposed by Quinlan (1993), is a powerful classification algorithm which is derived from the development of ID3 algorithm and is able to handle numeric attributes, missing values, and noisy data (Ghasemi et al. 2017). C4.5 identifies decision tree classifiers and using a divide-and-conquer method grows the decision tree. The C4.5 algorithm acts in two main stages: tree constructing and pruning. Tree constructing starts by calling the training dataset. All of the datasets firstly are concentrated in the root node and then divided into homogeneous sub-nodes based on a modified splitting criterion, called gain ratio. The attribute with the highest normalized information gain is chosen to make the decision (Quinlan 1993). This splitting will continue till the stopping condition is met, i.e. all instances in a node belong to the same class and this node is identified as a leaf node. The generated DT by training dataset often is prone to the over-fitting problem because of having a great number of branches and such DTs fail to classify the new unused data. To overcome this problem, there is a need to prune the tree. Pruning is the process of reducing decision tree size by eliminating parts of the tree which have little power for classifying and this process finally led to increasing the accuracy of the classifier and its reliability (Quinlan 1993; Ture et al. 2009; Hssina et al. 2014). 2.3.2.1. Rockburst Prediction Using C4.5 In this study, the C4.5 algorithm was applied to the training dataset using WEKA (Waikato Environment for Knowledge Analysis) software. There are two main parameters which should be adjusted to develop a high-performance C4.5 classifier including confidence factor (CF) and the minimum number of instances (MNI) (data samples) per leaf. The CF is used to compute a 31

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pessimistic upper bound on the error rate at a leaf/node. The smaller this value, the more pessimistic the estimated error is and generally the heavier the pruning. If a CF greater than 0.5 is chosen, then the pruning will be done on the basis of unchanged classification error on the training dataset and this is equivalent to turning off the pruning. The MNI affect the volume (i.e. the complexity) of the developed tree (Bui et al. 2012). Hence, according to Bui et al. (2012) and Ghasemi et al. (2017), the CF and MNI varied from 0.1 to 0.5 and 1 to 20 respectively, and the corresponding accuracy values were recorded. Finally, the optimum values of 0.25 and 2 were determined for CF and MNI, respectively. After adjusting the C4.5 parameters in WEKA software, the model was executed and the corresponding tree was obtained. Fig. 2.6 displays the results obtained by this algorithm which contains a root node, 5 internal nodes, and 7 leaves. There are two numbers in the parentheses of leaf nodes, which the first number belongs to the number of instances in that node and the second number shows the number of misclassified instances. The process of rockburst prediction using the developed tree model is very simple. For example, taking into account the values of 4.6, 3, 20, and 1.39 for MTS, UTS, UCS, and EEI respectively, and passing through the path of 𝑀𝑇𝑆 ≤ 25.7, 𝑈𝑇𝑆 ≤ 4.55, 𝐸𝐸𝐼 ≤ 2.04 and 𝑈𝐶𝑆 ≤ 30, the leaf node Yes (2,0) can be achieved which shows the occurrence of rockburst. Root node MTS Internal node <=25.7 >25.7 Leaf node MTS UTS >38.2 <=4.55 >4.55 <=38.2 Yes (71,0) EEI No (12,0) EEI <=2.04 >2.04 >1.8 <=1.8 Yes (3,0) No (2,0) Yes (15,1) UCS <=30 >30 Yes (2,0) No (2,0) Figure 2.6 Developed C4.5 tree model based on training dataset 2.3.3. Gene Expression Programming (GEP) During the progress of evolutionary algorithms (EAs) since 1975, Ferreira (2002) introduced a new powerful population-based algorithm called gene expression programming (GEP) that 32

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takes advantage of basic GA and genetic programming (GP) methods. The main goal of the GEP is to find a rational mathematical relationship between the independent variables and the corresponding dependent in such a way that the defined fitness function reaches its minimal value. In GEP, possible solutions are in the form of fixed-length coded chromosomes consist of two groups of entities: terminals and functions. Terminals can be both of input variables and user-defined constant values. Functions are algebraic symbols e.g. +, −, ×, /, 𝐿𝑛, 𝐿𝑜𝑔 and so on. The chromosomes can consist of one or more genes, and each gene comprises two parts of the head and tail so that the genetic operators create effective changes in these areas to produce better solutions. In contrast to multiple non-linear regression techniques, there is no need to consider a pre-defined mathematical framework (e.g. exponential, power, logarithmic, etc.) for GEP to develop a model. As a matter of fact, the GEP algorithm during its intelligent search is capable to find the optimum combination of terminals and functions to provide a predictive equation with enough accuracy. As shown in Fig. 2.7, the process of GEP modelling starts with the random generation of chromosomes in Karva language (a symbolic expression of GEP chromosomes) which are then expressed and executed as the tree and mathematical structures, respectively. Then, the generated chromosomes are evaluated according to the pre-defined fitness function. Bests of the first population are copied into the next generation, and the others are influenced by genetic operators, including selection and reproduction (i.e. mutation, inversion, transposition, and recombination). Finally, the modified chromosomes are transferred to the next generation and this process will continue until the stopping criteria (maximum generation number or reach to pre-defined fitness) are met (Ferreira 2002; Güllü 2012; Armaghani et al. 2016; Faradonbeh et al. 2016, 2018). The detailed information concerning genetic operators and their mechanisms can be found in (Ferreira 2002). 33

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Head Tail Create initial population e.g. Q * + - a b c d a a d Express chromosomes - c Q × b + Execute each program a (𝑎+𝑏).(𝑐−𝑑) 𝑛 Evaluate fitness e.g. 𝑅𝑀𝑆𝐸= 1/𝑛 (𝑥 −𝑥 )2 𝑖𝑟𝑒𝑎𝑙 𝑖𝑝𝑟𝑒𝑑 𝑖=1 Yes Termination? Genetic operators for reprodcution: End No 1) Mutation (an element is changed to another) Q * + - a b c d Q * + a a b c d Best of Generation? 2 ) Inversion (a fragment is inverted in th e head) Q * + - a b c d Q * + c b a c d No 3 ) Transition (IS type: a fragment is cop ied to the head) Selection Q * + - a b c d Q * + c b a c d 4 ) Recombination (one-point type: Two chromosomes exchnage a fragment) Reproduction Q * + - a b c d a a Q * + b - Q a b a a + * - b - Q a b a a + * - - a b c d a a Generation+1 Figure 2.7 GEP flowchart 2.3.3.1. Rockburst Prediction using GEP The GeneXproTools 5.0, an exceedingly flexible modelling tool designed for function finding, classification, time series prediction, and logic synthesis, was implemented to classify and predict rockburst events. This software classifies the value returned by the evolved model as “1” or “0” via the 0/1 rounding threshold. If the returned value by the evolved model is equal to or greater than the rounding threshold, then the record is classified as "1", "0" otherwise. Similar to the GA-ENN and C4.5 modelling, 80% of the database was applied to the software as the training dataset to develop the model. In the first step, a fitness function for the algorithm should be defined. The sensitivity/specificity with the rounding threshold of 0.5 was used for this aim. The sensitivity/specificity (𝑆𝑆 ) of a chromosome as a solution can be calculated by 𝑖 the following equation: 𝑆𝑆 = 𝑆𝐸 .𝑆𝑃 (2.12) 𝑖 𝑖 𝑖 34

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where 𝑆𝐸 is the sensitivity and 𝑆𝑃 is the specificity of the chromosome 𝑖, and are given by 𝑖 𝑖 the following formulas: 𝑇𝑃 𝑆𝐸 = 𝑖 (2.13) 𝑖 𝑇𝑃 +𝐹𝑁 𝑖 𝑖 𝑇𝑁 𝑆𝑃 = 𝑖 (2.14) 𝑖 𝑇𝑁 +𝐹𝑃 𝑖 𝑖 where 𝑇𝑃, 𝑇𝑁, 𝐹𝑃, and 𝐹𝑁 represent, respectively, the number of true positives, true 𝑖 𝑖 𝑖 𝑖 negatives, false positives, and false negatives. 𝑇𝑃, 𝑇𝑁, 𝐹𝑃, and 𝐹𝑁 are the four different 𝑖 𝑖 𝑖 𝑖 possible outcomes of a single prediction for a two-class case with classes “1” (Yes) and “0” (No). A false positive is when the outcome is incorrectly classified as “Yes” (or positive) when it is in fact “No” (or negative). A false negative is when the outcome is incorrectly classified as “No” when it is in fact “Yes”. True positives and true negatives are obviously correct classifications. Keeping track of all these possible outcomes is such an error-prone activity, that they are usually shown in what is called a confusion matrix. Thus, the fitness value of chromosome 𝑖 is evaluated by the following equation: 𝑓 = 1000.𝑆𝑆 (2.15) 𝑖 𝑖 which obviously ranges from 0 to 1000, with 1000 corresponding to the maximum prediction accuracy. In the second step, terminals and functions which are kernels of generated chromosomes should be assigned. Terminals are input parameters (i.e. MTS, UTS, UCS, and EEI). The most common arithmetic functions were selected as follows: 𝐹 = {+,−,×,/,𝑆𝑞𝑟𝑡,𝐸𝑥𝑝,𝐿𝑛,^2,^3,3𝑅𝑡} (2.16) The goal of GEP modelling is to develop a rockburst index in the form of 𝑅𝐵𝐼 = 𝑓(𝑀𝑇𝑆,𝑈𝑇𝑆,𝑈𝐶𝑆,𝐸𝐸𝐼). The third step is to determine the structural parameters, i.e. the number of genes and head size. These two parameters affect the length of the generated chromosomes and subsequently the complexity of the proposed formula. By trial and error, the best values of 4 and 9 were obtained for the number of genes and head size, respectively. In the fourth step, the ratios of genetic operators (i.e. mutation, inversion, transposition, and recombination) as chromosomes modifiers should be determined. A set of values has been recommended by the researchers for genetic operators that their validity has been confirmed in many engineering problems (Ferreira 2006; Kayadelen 2011; Güllü 2012; Khandelwal et al. 2016). So, these values were set for the operators in the current study as well (see Table 2.5). As the final step, since we face multi-genic chromosomes, we need to define a linking function 35

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to link genes to each other. Addition (+) is a most common linking function which was used for this aim. After adjusting the GEP parameters (Table 2.5), the model was executed in training mode for 2000 generations and the results were recorded. Eq. 2.17 shows the developed rockburst index based on GEP algorithm. By feeding the input parameters to the Eq. 2.17 and comparing the calculated value with the Eq. 2.18, the rockburst occurrence can be determined. 𝑀𝑇𝑆 𝑅𝐵𝐼 = 𝐸𝑥𝑝(𝑀𝑇𝑆)−𝑈𝐶𝑆3 +2𝑇+ 𝐸𝑥𝑝( 𝐸𝐸𝐼) +𝐸𝐸𝐼 −𝐸𝐸𝐼9 (2.17) 𝐸𝐸𝐼 𝐸𝐸𝐼 (𝑈𝑇𝑆−𝐸𝑥𝑝(𝑈𝑇𝑆))×√ 𝑈𝑇𝑆 1 (𝑌𝑒𝑠) 𝑅𝐵𝐼 ≥ 0.5 𝑅𝐵𝐼∗ = { (2.18) 0 (𝑁𝑜) 𝑅𝐵𝐼 < 0.5 Table 2.5 Characteristics of developed GEP models Type of setting Parameter Terminal set MTS, UTS, UCS, EEI Function set +,−,×,/,𝑆𝑞𝑟𝑡,𝐸𝑥𝑝,𝐿𝑛,^2,^3,3𝑅𝑡 Fitness function Sensitivity/Specificity Population size 90 General setting Number of generations 2000 Head size 9 Number of genes 4 Linking function Addition (+) Mutation rate 0.044 Inversion rate 0.1 Transposition rate 0.1 Genetic operators One-point recombination rate 0.3 Two-point recombination rate 0.3 Gene recombination rate 0.1 2.4. Performance Evaluation of the Proposed Models In this section, the remaining testing datasets (27 cases) were applied to the developed models of GA-ENN, C4.5, and GEP to evaluate their prediction performance. For further evaluation, five conventional criteria mentioned in Table 2.1 were considered as well. Table 2.6 shows the obtained results from eight different models in testing stage. A confusion matrix is a useful tool to describe the performance of a classifier on a set of test data. Each row of the matrix 36

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represents the instances in an actual class while each column represents the instances in a predicted class (or vice versa). Table 2.7 shows the confusion matrices of the developed models. According to Tables 2.6 and 2.7, GA-ENN and GEP models have the equal number of misclassified cases (i.e. 4 cases), while this number is equal to 9 for stress coefficient and brittleness coefficient criteria. In the following, two indices of root mean squared error (RMSE) (an index to measure the deviation between the actual and predicted data) and the percentage of the successful prediction (PSP) (the percentile quotient of the number of correct predictions to the total number of testing data) were used to investigate the accuracy and capability of the models. Ideally, RMSE and PSP are equal to 0 and 100%, respectively. The results of performance indices are shown in Table 2.8. As can be seen in this table, all three new constructed models (i.e. GA-ENN, GEP, and C4.5) have higher accuracy and lower estimation error compared with five conventional criteria. Table 2.8 also shows that, two models of GA- ENN and GEP with the similar results outperformed the C4.5. On the other hand, EEI criterion acted just like the C4.5 model which shows that this criterion with its simple formula can be used effectively to predict rockburst occurrence in engineering projects. Fig. 2.8 compares the prediction performance of the developed models. Table 2.6 Results of validation of developed models with testing dataset No. Input parameters Developed models Actual Russenes Hoek GA- MTS UTS UCS EEI Output C4.5 GEP SC BC EEI ENN criterion criterion 1 45.7 3.2 69.1 4.1 1 1 1 1 1 1 1 1 1 2 62.4 9.5 235 9 1 1 1 1 1 0 0 1 1 3 55.6 18.9 256.5 9.1 1 1 1 1 0 0 0 1 1 4 41.6 2.7 67.6 3.7 1 1 1 1 1 1 1 1 1 5 30.3 3.1 88 3 1 1 1 1 1 1 1 1 1 6 28.6 12 122 2.5 1 1 1 1 0 0 0 1 1 7 4.6 3 20 1.39 0 0 1 0 0 0 0 1 0 8 2.6 3 20 1.39 0 0 1 0 0 0 0 1 0 9 33.6 10.8 156 5.2 1 1 1 1 0 0 0 1 1 10 23 3 80 0.85 1 1 0 0 1 1 0 1 0 11 80 6.7 180 5.5 0 1 1 1 1 1 1 1 1 12 19 4.48 153 2.11 1 0 1 0 0 0 0 1 1 13 38.2 3.9 53 1.6 0 1 0 0 1 1 1 1 0 14 73.2 5 120 5.1 1 1 1 1 1 1 1 1 1 15 3.8 3 20 1.39 0 0 1 0 0 0 0 1 0 16 89.56 17.13 190.3 3.97 1 1 1 1 1 1 1 1 1 37

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17 18.8 6.3 171.5 7 0 0 0 0 0 0 0 1 1 18 105.5 12.1 170 5.76 1 1 1 1 1 1 1 1 1 19 39 2.4 70.1 4.8 1 1 1 1 1 1 1 1 1 20 27.8 2.1 90 1.8 0 1 0 0 1 1 1 0 0 21 30 3.7 88.7 6.6 1 1 1 0 1 1 1 1 1 22 40.6 2.6 66.6 3.7 1 1 1 1 1 1 1 1 1 23 11 5 115 5.7 0 0 0 0 0 0 0 1 1 24 59.82 7.31 85.8 2.78 1 1 1 1 1 1 1 1 1 25 7.5 3.7 52 1.3 0 0 0 0 0 0 0 1 0 26 11 4.9 105 4.7 0 0 0 0 0 0 0 1 1 27 57.6 5 120 5.1 1 1 1 1 1 1 1 1 1 BC brittleness coefficient criterion, SC stress coefficient criterion, EEI elastic energy index criterion Table 2.7 Confusion matrices of developed models in testing stage Model Confusion matrix Number of misclassified cases GA-ENN Predicted No Yes 4 Actual No 7 3 Yes 1 16 GEP Predicted No Yes 4 Actual No 9 1 Yes 3 14 C4.5 Predicted No Yes 5 Actual No 6 4 Yes 1 16 Russenes criterion Predicted No Yes 7 Actual No 7 3 Yes 4 13 Hoek criterion Predicted No Yes 8 Actual No 7 3 Yes 5 12 Stress coefficient criterion Predicted No Yes 9 Actual No 7 3 Yes 6 11 Brittleness coefficient criterion Predicted No Yes 9 Actual No 1 9 38

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2.5. Sensitivity Analysis In this section a sensitivity analysis is performed to evaluate the effects of input parameters on rockburst prediction models. To this end, the relevancy factor (Kamari et al. 2015) was used which is calculated by Eq. 2.19. ∑𝑛 (𝐼 −𝐼̅ )(𝑃 −𝑃̅) 𝑟 = 𝑖=1 𝑖,𝑘 𝑘 𝑖 (2.19) √∑𝑛 𝑖=1(𝐼 𝑖,𝑘−𝐼 𝑘̅ )2∑𝑛 𝑖=1(𝑃 𝑖−𝑃̅)2 where 𝐼 and 𝐼̅ are the 𝑖th and average values of the 𝑘th input parameter, respectively, 𝑃, and 𝑖,𝑘 𝑘 𝑖 𝑃̅ are the 𝑖th and average values of the predicted rockburst., respectively, and 𝑛 is the number of rockburst events. The higher 𝑟 value the more influence the input has in predicting the output value. Fig. 2.9 shows the 𝑟 values. As can be seen in this figure, the maximum tangential stress (MTS) is the most influential parameter in rockburst prediction, and uniaxial compressive strength (UCS) has the lowest impact. These results are in agreement with those obtained by others in a recent study (Li et al. 2017). 0.7 0.658 0.6 0.5 0.4 0.360 r 0.305 0.3 0.243 0.2 0.1 0.0 T UTS UCS EEI Input parameter Figure 2.9 Relevancy factor of each input parameter 2.6. Discussion A supplementary explanation regarding the proposed three models is contained in this section. As previously mentioned, this is the first attempt in the application of ENNs in earth sciences, and its results were promising. Accordingly, it is highly recommended to check the applicability of ENNs in combination with other meta-heuristic algorithms, as hybrid models, for different aims (e.g. classification, prediction, and minimization) for mining and 40

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geotechnical engineering applications. However, as a black-box method like ANN, GA-ENN neither can provide any equation nor a visual pattern for users. This may be considered as a disadvantage for this algorithm, but it is possible to overcome this issue by using this technique to find some optimum coefficients of the multiple non-linear regressions in future studies. In contrast to GA-ENN, C4.5 has a very simple modelling mechanism. Its tree structure easily can be adopted as a guide by engineers in the projects to predict the rockburst occurrence just by tracking the values of inputs within the branches of the tree. In some cases, this algorithm may provide large and complex trees according to the defined controlling parameters, which finally decrease the applicability of the developed trees. Besides, C4.5 algorithm on account of its innate PCA characteristic may remove some input parameters during the training stage to increase the accuracy of the final output. Hence, the process of C4.5 modelling requires extensive modelling experiences. The common multiple non-linear regressions need a pre- defined mathematical structure, while the GEP algorithm is able to find the latent relationship between the input and output parameters without any presupposition. This can be introduced as the most important characteristic of GEP algorithm compared with the GA-ENN and C4.5 algorithms. In addition, GEP does not have the limitations of previous methods and is more practical. In the end, it is worth mentioning that the developed models are valid just in the defined ranges of values of inputs and for the new datasets out of these ranges, the models should be adjusted again. 2.7. Summary and Conclusions This study was intended to assess rockburst hazard in deep underground openings using three renowned data mining techniques including GA-ENN, C4.5, and GEP. A database including the maximum tangential stress of the surrounding rock, the uniaxial tensile strength of rock, the uniaxial compressive strength of rock and the elastic energy index of 134 rockburst experiences in various underground projects was compiled. After a statistical analysis, the GA- ENN, C4.5, and GEP models were developed based on training datasets. In the following, the prediction performance of the models was evaluated by applying unused testing datasets. The results of the new models were compared with five conventional rockburst prediction criteria via performance indices of root mean squared error (RMSE) and percentage of the successful prediction (PSP). Finally, a sensitivity analysis was conducted to know about the influence of input parameters on rockburst using relevancy factor. The following conclusions has been drawn: 41

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134 35 133.4 9.3 2.9 Yes References Adoko AC, Gokceoglu C, Wu L, Zuo QJ (2013) Knowledge-based and data-driven fuzzy modeling for rockburst prediction. International Journal of Rock Mechanics and Mining Sciences 61:86–95 Akdag S, Karakus M, Taheri A, et al (2018) Effects of Thermal Damage on Strain Burst Mechanism for Brittle Rocks Under True-Triaxial Loading Conditions. Rock Mechanics and Rock Engineering 51(6):1–26 Armaghani DJ, Faradonbeh RS, Rezaei H, et al (2016) Settlement prediction of the rock- socketed piles through a new technique based on gene expression programming. Neural Computing and Applications 29(11):1115-1125 Aryafar A, Mikaeil R, Haghshenas SS, Haghshenas SS (2018) Application of metaheuristic algorithms to optimal clustering of sawing machine vibration. Measurement: Journal of the International Measurement Confederation 124:20–31 Berthold M, Hand D (eds) (2003) Intelligent data analysis: an introduction. Second edition. Springer Science & Business Media, 2 Breiman L, Friedman J, Stone CJ, Olshen RA (1984) Classification and Regression Trees. CRC press Bui DT, Pradhan B, Lofman O, Revhaug I (2012) Landslide Susceptibility Assessment in Vietnam Using Support Vector Machines, Decision Tree, and Naıve Bayes Models. Mathematical Problems in Engineering Coimbra R, Rodriguez-Galiano V, Olóriz F, Chica-Olmo M (2014) Regression trees for modeling geochemical data-An application to Late Jurassic carbonates (Ammonitico Rosso). Computers and Geosciences 73:198–207 Dancy C, Reidy J (2004) Statistics Without Maths for Psychology. Pearson Education Limited, New York Dong L, Li X, Peng K (2013) Prediction of rockburst classification using Random Forest. Transactions of Nonferrous Metals Society of China 23(2):472–477 45

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Statement of Authorship Title of Paper Application of self-organising map and fuzzy c-mean techniques for rockburst clustering in deep undreground projects Publication Status Published Accepted for Publication Submitted for Publication Unpublished and Unsubmitted work written in manuscript style Publication Details Shirani Faradonbeh R, Shaffiee Haghshenas S, Taheri A, Mikaeil R (2020) Application of self-organising map and fuzzy c-mean techniques for rockburst clustering in deep undreground projects. Neural Computing and Applications 32(12):8545–8559 Principal Author Name of Principal Author (Candidate) Roohollah Shirani Faradonbeh Contribution to the Paper Literature review and database preparation, statistical analysis, development of models and preparation of the manuscript Overall percentage (%) 80% Certification: This paper reports on original research I conducted during the period of my Higher Degree by Research candidature and is not subject to any obligations or contractual agreements with a third party that would constrain its inclusion in this thesis. I am the primary author of this paper. Signature Date 17 June 2021 Co-Author Contributions By signing the Statement of Authorship, each author certifies that: i. the candidate’s stated contribution to the publication is accurate (as detailed above); ii. permission is granted for the candidate in include the publication in the thesis; and iii. the sum of all co-author contributions is equal to 100% less the candidate’s stated contribution. Name of Co-Author Sina Shaffiee Haghshenas Contribution to the Paper Model Development, review of the manuscript Signature Date 21 June 2021 Name of Co-Author Abbas Taheri Contribution to the Paper Research supervision, review and revision of the manuscript Signature Date 21 June 2021 Name of Co-Author Reza Mikaeil Contribution to the Paper Review of the manuscript Signature Date 21 June 2021 52

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Chapter 3 Application of Self-Organizing Map and Fuzzy c- mean Techniques for Rockburst Clustering in Deep Underground Projects Abstract One of the main concerns associated with deep underground constructions is the violent expulsion of rock induced by unexpected release of strain energy from surrounding rock masses that is known as rockburst. Rockburst hazard causes substantial damages to the foundation of the structure, equipment and can be a menace to the safety of workers. This study was intended to find the latent relationship between the rockburst-related parameters based on the compiled data samples from deep underground projects using two robust clustering techniques of self- organizing map (SOM) and fuzzy c-mean (FCM). The parameters of maximum tangential stress, uniaxial compressive strength, uniaxial tensile strength, and elastic energy index were considered as input parameters. SOM model could classify data samples into four distinct classes (clusters) and the rockburst intensities were identified precisely. FCM also proved its performance in clustering task with high convergence speed and acceptable accuracy. Having a comparison, the results of SOM and FCM models were compared with ones calculated from five empirical criteria of Russenes, Hoek, tangential stress, elastic energy index, and rock brittleness coefficient. At best, the empirical criteria of Hoek and tangential stress coefficient could predict rockburst intensity with the accuracy of 56.90 %. By analyzing the SOM results as the best model, it was turned out that the maximum tangential stress around the openings has a crucial role in rockburst clustering and has the most influence on the occurrence of strong and moderate rockburst types. Hence, it was recommended as a possible solution to control these types of rockbursts by optimizing the diameter and shape of the underground openings. Keywords: Rockburst, Self-organizing map, Fuzzy c-mean, Empirical criteria 53

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3.1. Introduction Nowadays, there are many important mining and civil projects such as hard rock mines, hydropower stations, nuclear power plants, and water conveyance and transportation tunnels under construction in the deep ground condition all over the world. It is proved that by increasing of the depth, in-situ stresses would show a linear or non-linear increment accompanied by the increase of groundwater, osmotic pressure, ground temperature, and the strength of rock (Sun and Wang 2000; Jian et al. 2012). For instance, by reaching the mining depth to about 1000 m, the in-situ stresses induced by overburden, geological condition, and mining operation may lead to stress concentration and subsequently bursting and failure (Weng et al. 2017; Akdag et al. 2018). Therefore, engineering activities in the deep underground environment is challenging and difficult due to rockburst and seismic events, the inrush of water, gas, and large-scale collapses (Feng et al. 2016). Among them, rockburst accidents are known as the most critical geotechnical disaster in many countries which leads to injuries and loss of life, damage to property, delays in project activities as well as enormous economic losses (Blake and Hedley 2003; Li et al. 2007; He et al. 2017). Hence, it is important to predict and control rockburst hazards underground. The instantaneous release of large amounts of strain energy stored in overstressed rock mass cause an unexpected and violent failure which is known as rockburst phenomenon (Blake and Hedley 2003). With respect to this definition, either the presence of high-levels of in-situ stresses exceeding the rock strength or the external triggering factors, e.g. mine extraction could provide the necessary circumstances for rockburst occurrence (Yan et al. 2015). From the perspective of mining, the rockbursts can be classified into three groups (see Fig. 3.1) (Blake and Hedley 2003; Castro et al. 2012; He et al. 2015): • Strain bursts caused by the local concentration of high-stress at the edge of mining openings frequently occur during drilling for blasting or reinforcement. The consequences of strain bursts range from the ejection of small pieces of rock to the large-scale collapse of an opening as it tries to achieve a more stable shape. In civil engineering activities, the strain bursts are a common type of rockburst. • Pillar bursts caused by exceeding the stress exerted on a support pillar from its strength are frequent in the sizeable mined-out area. • Fault-slip bursts caused by the slippage along a geological plane have the mechanism like an earthquake and different magnitude and damage range. 54

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Strain burst Stress concentration behind the face 0 = No confinement 3 against the face s Pillar burst Changing driving shear stress t Geological Stress change acting feature upon a locking point Fault-slip burst Changing clamping normal stress s N Figure 3.1 Schematic representation of rockburst types and the effect of confinement (Zhou et al. 2018) Many researches have been carried out during the last decades by scholars not only on the understanding of the rockburst mechanism but also on developing reliable techniques to predict and mitigate its hazards. In terms of rockburst mechanism, many theories have been proposed to assess the stability and deformation localization of rock masses but most of them are assumptive and empirical (Shi et al. 2010; Tang et al. 2010; Jian et al. 2012; Cai 2016a). From the standpoint of prediction, the rockburst studies can be categorized into two following groups: • Strength-based criteria: These criteria such as Turchaninov criterion (Turchaninov et al. 1972), Russenes criterion (Russenes 1974), Hoek criterion (Hoek and Brown 1980), Barton criterion (Barton et al. 1974), rock brittleness coefficient criterion (Wang et al. 1998), tangential stress criterion (Wang et al. 1998) and so on are rates composed of uniaxial compressive strength, uniaxial tensile strength, maximum tangential strength, axial stress around the opening, and in-situ stresses. The defined rates show specific types of rockbursts (see Table 3.1). • Energy-based criteria: As the strain energy has a vital role in the occurrence of rockburst events, some scholars attempted to develop other criteria experimentally based on energy theory and consider both the strain energy accumulated in the rock specimen during the loading process and dissipated energy after deformation and failure. A summary of the most common energy-based criteria is listed in Table 3.1. 55

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Table 3.1 Most common strength- and energy-based criteria for the prediction of rockburst intensity Type Criterion Equation None Light Moderate Strong Strength Russenes criterion 𝜎 𝜃 <0.25 0.25 0.33−0.55 >0.55 (Russenes 1974) 𝜎 based 𝑐 −0.33 Barton et al. 𝜎 𝑐 >5 (2.5−5] − ≤2.5 (Barton et al. 1974) 𝜎 1 Hoek criterion 𝜎 𝑐 >3.5 2.0−3.5 1.7−2.0 <1.7 𝜎 (Hoek and Brown 1980) 𝜃 Tangential stress coefficient 𝜎 𝜃 ≤0.3 0.3−0.5 0.5−0.7 >0.7 𝜎 (Wang et al. 1998) 𝑐 Rock brittleness coefficient 𝜎 𝑐 >40 26.7−40 14.5−26.7 <14.5 𝜎 (Wang et al. 1998) 𝑡 Energy 𝐴 2 − >1.5 1.2−1.5 1.0−1.20 Brittleness index modified 𝐴 based 1 (BIM) (Aubertin et al. 1994) Burst energy coefficient (Li 𝑊 𝑒 ≤1 − − − 𝑊 et al. 1996) 𝑝 Elastic energy index (Wang 𝐸 𝑅 <2.0 2.0−3.5 3.5−5.0 >5.0 𝐸 et al. 1998) 𝐷 Mo criterion (Mo et al. 2014) 2(𝐸 𝑃−𝐸 𝑇) ≤1 − − − 3𝐸 𝑋 𝜎 : maximum tangential stress, 𝜎 : uniaxial compressive strength, 𝜎 : major principal stress, 𝜎: tensile strength, 𝐴 : elastic 𝜃 𝑐 1 𝑡 1 energy stored in the rock, 𝐴 : energy given by the total area below the stress-strain curve, 𝑊: the stored energy in the rock 2 𝑒 during loading before peak strength, 𝑊: the pre-peak dissipated energy during the failure process, 𝐸 : the elastic energy 𝑝 𝑃 accumulated, 𝐸 : the dissipated energy, 𝐸 : the post-peak dissipative strain energy 𝑇 𝑋 The next imperative issue concerning the rockburst study is providing solutions for its prevention and control. From this perspective, most of the studies focus on the use of microseismic monitoring systems, energy-absorbing bolts as well as some strategies to optimize the mining layout, blasting operation, and supporting system (Jha and Chouhan 1994; Frid 1997; Dou et al. 2009; Liu et al. 2013; He et al. 2014; Li et al. 2017; Zhao et al. 2017). According to the complex mechanism of rockburst and a large number of effective parameters on it, empirical criteria (especially the strength-based ones) could not show satisfactory results (Liu et al. 2013; Li et al. 2017; Zhao et al. 2017). On the other hand, developing energy-based criteria need to do an extreme experimental study which is a time-consuming and expensive process. Hereupon, the application of machine learning (ML) techniques thanks to their ability to deal with the complex non-linear problems and applying several input variables have been used widely to predict rockburst hazard in recent years. Feng and Wang (1994) for the first time used the artificial neural networks (ANNs) successfully to predict the intensity and location of rockburst. Following their success, further studies were carried out by other scholars 56

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using novel ML techniques (Xie and Pan 2007; Gao 2010; Shi et al. 2010; Zhou et al. 2010; Zhang et al. 2011; Li and Liu 2015). It should be mentioned that most of the used ML techniques to assess rockburst phenomenon such as ANNs have a complicated internal structure and their results are not easy to use in practice. As such, they have just focused on the prediction task. Although these studies have been considered as potential solutions to the rockburst problem, they could not solve it completely. In fact, due to the high level of uncertainty and ambiguity in relation to the rockburst phenomenon, the supervised techniques such as ANNs are not able to properly assess such problems. Unsupervised learning algorithms are other branches of machine learning algorithms which can detect the hidden patterns in the database by checking the commonalities between the unlabelled datasets. The most common types of these algorithms are clustering techniques. Due to the complicated environment of rockburst hazard, unsupervised learning algorithms can be used to categorize the datasets into several distinct clusters for better analyzing. In this regard, Xie and Pan (2007) clustered the rockburst events successfully based on grey whitenization weight function according to the grey incidence matrix. In addition, an ant colony clustering optimization model was proposed by Gao (2010) to predict rockburst classes. In another study, Chen et al. (2013) proposed a new quantitative classification method for rockburst using hierarchical clustering analysis. The results of the above studies were in good agreement (i.e. accuracy above 80%) with the practical records which show the capability of such techniques for rockburst assessment. However, there are few studies in the application of unsupervised learning algorithms for rockburst assessment, and models with the higher level of accuracy are needed. The current study focuses on the applicability of self-organizing map (SOM) and fuzzy c-mean (FCM) algorithms as two unsupervised clustering techniques in order to cluster and identify rockburst intensity simultaneously based on compiled datasets from deep underground openings. The SOM algorithm is a robust data mining tool with the ability to discover the non-linear relationships among high-dimensional data and picturing and clustering them on a low- dimensional space. Fuzzy c-mean (FCM) is also a renowned clustering technique that is similar to the k-means algorithm and using a generalized least-squares objective function creates fuzzy partitions for a set of the numerical dataset. Application of SOM and fuzzy c-mean algorithms in mining and geotechnics fields are limited to few studies (Das and Basudhar 2009; Rad et al. 2012; Mikaeil et al. 2018). In this study, the most influential parameters on the occurrence of rockburst, i.e. the maximum tangential stress, the uniaxial compressive strength, the uniaxial tensile strength, and the elastic energy index were considered as input parameters. The process 57

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of clustering of rockburst datasets using SOM and FCM algorithms was conducted based on the 58 data samples. Afterwards, for the sake of checking the applicability of empirical criteria, five strength-based of them were selected and finally, their accuracy in clustering the rockburst data samples was evaluated. 3.2. Methodology 3.2.1. Self-Organizing Map Approach In recent years, computational intelligence has been used as a powerful tool to deal with complex industrial and scientific problems (Armaghani et al. 2016; Faradonbeh et al. 2016; Khandelwal et al. 2016, 2017; Mikaeil et al. 2018). Undoubtedly, artificial neural networks (ANNs) are one of the most essential components of computational intelligence (Salemi et al. 2018; Aryafar et al. 2018). ANNs with a wide range of applications such as image processing, pattern recognition, time series prediction, control and robotic systems have a crucial role in scientific and practical areas. ANNs are efficient tools in dealing with complex systems, among which classic inferential and argumentative methods have not this ability. In recent years, ANNs have been used extensively in linear and non-linear problems in different sciences especially in earth sciences (Mohamad et al. 2016; Mahdevari et al. 2017). The self-organizing map (SOM), as an unsupervised algorithm, was proposed by Kohonen (1990) and is a specific type of ANNs which can be used efficiently in statistical and visual data analyses, especially for high-volume and non-uniform data. This method is based on some characteristics of the human brain that follows a specific classifying and mapping procedure (i.e. topographic mapping) to link the input signals to the corresponding processing area (Kohonen 1990; Yu et al. 2015). In the Kohonen model, the tasks of SOM are implemented by a number of neurons, which are placed together in a one-dimensional or two-dimensional (flat) topology and have a reciprocal behavior. Contrary to other artificial neural networks, SOM is composed of two layers, including an input layer and Kohonen layer (competitive layer) which are schematically shown in Fig. 3.2. The process of SOM training has three main phases of competition, cooperation, and adaptation. In the first phase, there is a competition among the neurons, and a neuron with the closest weight vector to the input signal vector will be selected as the winner, known as the best matching unit (BMU). Considering the input signal vector 𝑋 = [𝑥 ,𝑥 ,𝑥 ,…,𝑥 ]𝑇 and the weight vector 𝑊 = [𝑤 ,𝑤 ,𝑤 ,…,𝑤 ]𝑇, the distance between 1 2 3 𝑛 1 2 3 𝑛 these two vectors is defined mathematically as Euclidean distance and can be computed by the following equation: 58

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𝐷 = ‖𝑋−𝑊‖ = ∑𝑛 (𝑋 −𝑊)2 (3.1) 𝑖=1 𝑖 𝑖 The so-called winner neuron (BMU) has the smallest D. In cooperation phase, the neurons which are located in the immediate vicinity of the BMU are recognized and then in the adaptation phase, these neurons are adjusted using Eq. 3.2 to shape a particular pattern on a plane (this pattern belongs to a particular feature of input signal vector). 𝑊 = 𝑊 +𝜂[𝑋(𝑡)−𝑊(𝑡)] (3.2) (𝑡+1) 𝑡 Where 𝜂 is learning rate function that ranges between 0 and 1. During the process of training, the data samples of input layer obtain a certain weight equal to 𝑊 and the weight vectors of the BMU and relevant neighbours progressively will be more similar to the input data. Finally, the input data samples are attracted to the corresponding neurons on the competitive layer and the algorithm will be ceased by meeting the stopping condition (i.e. the maximum number of iteration) (Das and Basudhar 2009; Yu et al. 2015; Mikaeil et al. 2018a). More details concerning the SOM algorithm and its mathematical foundation can be found in the studies of Hagan et al. (1996) and Demuth et al. (2014). Figure 3.2 A schematic model of self-organizing map network (Malondkar et al. 2018) 3.2.2. Fuzzy C-Mean Approach Zadeh first proposed the fuzzy science as a multi-valued logic versus the classic logic under the title “Fuzzy sets theory” (Zadeh 1996). The fuzzy logic can deal with problems in which due to the lack of knowledge and understanding of humans, it is complicated to identify and understand the system. Fuzzy clustering is one of the most important applications of the fuzzy 59

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logic in various sciences. Fuzzy c-mean (FCM) is one of the clustering techniques which was first proposed by Bezdek (1981) based on the iterative optimization. In fact, FCM is the advanced version of hard c-means clustering in which unlike the classic clustering, the membership degree of data in a cluster can have a value in the range of [0,1]. The process of FCM clustering can be summarized in four steps below: Step 1: The number of classes (𝑐) is determined. This is worth mentioning that the numerical value of 𝑐 is larger than or equal to 2 and smaller than or equal to 𝑛 (the number of data samples). Then, the value of the weight parameter (𝑚′) which defines the amount of fuzziness of the clustering process must be determined. This parameter has a significant role in the optimization process. The optimization process in the FCM algorithm can continue for 𝑟 iterations, where 𝑟 = 0,1,2,…,𝑛. Step 2: The centers of clusters in each iteration are calculated. Step 3: After determining the centers of clusters, the partitioned matrix for the 𝑟𝑡ℎ iteration is updated in the form of 𝑈̃(𝑟) using Eqs. 3.3-3.8. −1 (𝑟) 2 𝜇(𝑟+1) = [∑𝑐 (𝑑 𝑖𝑘 )(𝑚′−1)] for 𝐼 = 𝜑 (3.3) 𝑖𝑘 𝑗=1 (𝑟) 𝑘 𝑑 𝑗𝑘 𝜇(𝑟+1) = 0 for all classes 𝑖 where 𝑖 ∈ 𝐼̃ (3.4) 𝑖𝑘 𝑘 (𝑟) 𝐼 = {𝑖|2 ≤ 𝐶 < 𝑛 ; 𝑑 = 0} (3.5) 𝑘 𝑖𝑘 𝐼̃ = {1,2,…,𝑐}−𝐼 (3.6) 𝑘 𝑘 ∑ 𝜇(𝑟+1) = 1 (3.7) 𝑖∈𝐼 𝑘 𝑖𝑘 where 𝑑 is the Euclidean distance between the centre of 𝑖𝑡ℎ cluster and 𝑘𝑡ℎ data and 𝜇(𝑟+1) is 𝑖𝑘 𝑖𝑘 the membership degree of 𝑘𝑡ℎ data in the 𝑖𝑡ℎ cluster for 𝑟+1 iteration. Step 4: In the final step, the accuracy of clustering must be evaluated. In this regard, the minimum acceptance precision (𝜀 ) is defined and only after satisfying the Eq. 3.8, the 𝐿 algorithm will be ceased; otherwise, the algorithm is returned to the second step and the optimization process is iterated until an appropriate level of accuracy is achieved (Bezdek 1981; Caldas et al. 2017). ‖𝑈̃(𝑟+1) − 𝑈̃(𝑟) = 𝜀 ‖ (3.8) 𝐿 60

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3.3. Results and Discussion 3.3.1. Rockburst Data In this study, a total of 58 rockburst events were compiled from the literature belong to various underground openings all around the world (Jian et al. 2012; Dong et al. 2013; Adoko et al. 2013). Due to difficulties in recording the rockburst-related parameters and the incompleteness of the data, it was tried to consider the most important parameters for further analyses. Recently, Zhou et al. (2018) have provided a state-of-the-art literature review about the application of different uncertainty theory, unsupervised learning and supervised learning algorithms in rockburst studies. In their study, maximum tangential stress (MTS) around the underground openings, uniaxial compressive strength (UCS) of rock, uniaxial tensile strength (UTS) of rock, and elastic energy index (EEI) were identified as the most common parameters for rockburst assessment. Maximum tangential stress around the excavation is a key factor that is affected by the rock stress, groundwater, shape, and diameter of excavation (Palmstrom 1995). Since it would not be possible to measure these four factors in association with rockburst occurrence, maximum tangential stress can be considered as a good representative of those factors. This parameter usually is calculated based on numerical analysis or the information obtained from in-situ stress tests (e.g. hollow inclusion strain gauge method) and the following equation (Zhao et al. 2017): 1 𝑎2 1 3𝑎4 𝜎 = (𝜎 +𝜎 )(1+ )− (𝜎 −𝜎 )(1+ )𝑐𝑜𝑠2𝜃 (3.9) 𝜃 2 𝐻 𝑉 𝑟2 2 𝐻 𝑉 𝑟4 where 𝜎 , 𝜎 , and 𝜎 denote the tangential stress, the major horizontal principal stress, and 𝜃 𝐻 𝑣 vertical stress, respectively. The parameters of 𝑟 and 𝑎 denote the tunnel’s radius and the distance between the point of rockburst occurrence to the center of the tunnel, and 𝜃 represents the angle between the virtual line connecting the point of rockburst occurrence and the center of the tunnel and horizontal axis. The strength parameters i.e. the uniaxial compressive strength and the uniaxial tensile strength also are indicators which show the capability of rocks to store elastic strain energy before failure as well as their brittleness and indirectly, could describe the effect of joints and block size of rock mass (Liu et al. 2013). These two parameters can be easily measured using the related laboratory tests based on the collected rock samples from the case studies. As mentioned before, several energy-based indices have been proposed and most of them are correlated with each other and similarly related to rockburst occurrence. Among them, elastic energy index (EEI) is the most common energy criterion to assess rockburst. EEI is the ratio of stored energy to that dissipated during a single loading-unloading cycle under 61

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uniaxial compression (Kidybiński 1981). This parameter also can be measured directly using the double-hole method or indirectly using the rebound method. Therefore, in the current study, four parameters of maximum tangential stress, uniaxial compressive strength, tensile strength, and the elastic energy index were adopted as input parameters for modelling. The goal parameter is the rockburst intensity. Rockburst is a qualitative parameter that in such studies rarely is introduced as a binary problem (i.e. “1” for rockburst occurrence, “0” otherwise) (Li et al. 2017; Shirani Faradonbeh and Taheri 2019) and mostly is measured and assessed based on four classes of intensities which their description are given in Table 3.2. Table 3.2 provides an empirical classification of characteristic behaviour of underground openings subjected to various rockburst intensities that can be used as a standard for rockburst measuring and further predictions. The statistical features of all collected rockburst datasets and abbreviation of parameters are listed in Table 3.3. Fig. 3.3 shows the rockburst classes in regard to each input parameter. In an ideal manner, each input parameter value should belong to only one class in order to have an easy clustering process. According to Fig. 3.3, it is apparent that some parameters values belong to more than one class which shows that these values do not have distinct boundaries between four classes of rockburst. So, it is not practicable to cluster the rockburst events precisely just by considering one input parameter. It may be possible to cluster the datasets by a combination of several parameters. In the following section, it is tried to cluster the datasets into several distinct groups using SOM and fuzzy c-mean techniques. Table 3.2 Empirical classification of rockburst based on its intensity (Jian et al. 2012; Liu et al. 2013) Rockburst Descriptive characteristic behaviours of the tunnels intensity None No sound of rock burst and absence of rock burst activities May cause loosening of a few fragments. The surrounding rock will be deformed, cracked or Light rib-spalled. There would be a weak sound, but no ejection phenomenon Spalling and falls of thin rock fragments. The surrounding rock will be deformed and fractured; Moderate there may be a considerable number of rock chip ejections and loose and sudden destructions, accompanied by crisp crackling and often presented in the local cavern of surrounding rock Loosening and falls, often as a violent detachment of fragments and platy blocks. The surrounding rock will be bursting severely and suddenly thrown out or ejected into the tunnel, Strong accompanied by strong bursts and roaring sound, and will expand rapidly to the deep surrounding rock 62

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Table 3.3 Descriptive statistics of collected rockburst dataset Input parameter Statistical feature 𝜎 𝜎 𝜎 𝑊 𝜃 𝑐 𝑡 𝑒𝑡 Abbreviation T UCS UTS EEI Unit MPa MPa MPa Dimensionless Minimum 2.6 20 1.3 1.1 Maximum 167.2 263 22.6 9 Mean 49.752 114.592 6.039 4.553 Variance (n) 1184.511 2673.039 18.545 4.332 Standard deviation (n) 34.417 51.701 4.306 2.081 Output parameter (rockburst intensity) None Light Moderate Strong Abbreviation N L M S Number of samples 22 4 19 13 4 4 3 3 l e l e b b a a l s s 2 l s s 2 a a l c l c t s t s r u 1 r u 1 b b k k c c o o R R 0 0 0 30 60 90 120 150 180 0 50 100 150 200 250 T UCS 4 4 l l e e b3 b3 a a l s l s s s a a l l c2 c2 t t s s r r u u b b k c1 k c1 o o R R 0 0 0 5 10 15 20 25 30 0 2 4 6 8 10 UTS W et Figure 3.3 Rockburst class regarding each input parameter (1: None, 2: Light, 3: Moderate, 4: Strong) 63

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3.3.2. Implementation of SOM Technique For SOM modeling, 58 datasets with the main parameters of T, UCS, UTS, EEI, and the corresponding rockburst intensities were used, and the process of modeling was conducted in MATLAB software environment. First, all the 58 datasets related to the four mentioned parameters were normalized between 0 and 1 and considered as input data. Then, the controlling parameters were determined. These parameters have a significant role in the acceleration and improvement of convergence of algorithm in reaching the optimum response. In this study, in accordance with trial and error procedure and other scholars’ suggestions (Chen and Kuo 2017; Mikaeil et al. 2018a, b), the optimum values of 100, 4, and 90 were obtained for controlling parameters of maximum iteration (epochs), Initneighbor (initial neighborhood size), and cover steps (the number of training steps for initial covering of the input space), respectively. Afterwards, the number of neurons (classes) in the competitive layer was defined as 4 (i.e. none, light, moderate, and strong). Eventually, by adjusting the required parameters, the algorithm was implemented for 100 iterations, and the results were obtained. By stopping the algorithm, 58 datasets were absorbed by 4 neurons (classes) on a two-dimensional lattice structure, and the classification process was completed. Fig. 3.4, as the hits plot, shows the number of data samples absorbed by each neuron. In Fig. 3.4, the axes show the Euclidean distance between classes. According to this figure, the four obtained classes have distinct boundaries. Besides, it can be seen obviously that the third neuron (class) is the most successful neuron in absorbing input data (by absorbing 22 data samples). In addition, the fourth, second and first neurons absorbed 19, 13, and 4 data samples, respectively. After assessing the contents of each class, it was found that all rockburst events in the first class (4 cases) belong to the light type and similarly, rockburst events in the second class (13 cases) belong to the strong type, rockburst events in the third class (22 cases) belong to the none type, and rockburst events in the fourth class (19 cases) belong to the moderate type. These four classes were labelled and shown in Fig. 3.4. This figure shows that SOM algorithm could classify the data samples into four classes in such a way that its results have the absolute consistency with the measured rockburst intensities by the operators in the field. 64

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Figure 3.4 Hits plot for SOM model In pursuance of more transparency, weighted distances between neighboring neurons were measured and displayed in Fig. 3.5. The axes in Fig. 3.5 show the weighted distances between neurons. The darker colors show that neurons (classes) are closer to each other and vice versa. For example, the distance between the first class (light) and the second one (strong) is less than the distance between the second class (strong) and the third one (none). As such, the distance between first class (light) and the third one (none) is less than the distance between the third class (none) and the fourth one (moderate). From another point of view, the distances between classes are in agreement with the definitions given in Table 3.2 for rockburst intensities. According to Fig. 3.5, the second and third classes have the maximum distance which can be referred to the rockburst characteristics explained in Table 3.2 for None and Strong types. To evaluate the relative importance of the input parameters for rockburst clustering using SOM, the weights of parameters corresponding to each class are shown graphically in Fig. 3.6. The darkness of the colors shows the high influence of the parameter on that class. By this figure, maximum tangential stress (T) has a high influence on the strong (the second class) and moderate (the fourth class) rockburst events, respectively. On the other hand, the parameters of UCS, UTS, and EEI similarly have a high influence on the moderate rockburst events (the fourth class). 65

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FCM and examining different combinations of control parameters, the values of 100, 0.00001, and 2 were obtained for the maximum iteration, 𝜀 , and 𝑚′, respectively. Then, the algorithm 𝐿 was implemented based on the determined values and the variations of cost function was recorded that is shown in Fig. 3.7. According to this figure, up to iteration No. 30, the cost value gradually reduces and then becomes constant till iteration No. 33. In this iteration, the cost value and the precision level are equal to 1.8014 and 0.0001, respectively in which the precision level is larger than defined 𝜀 = 0.00001. So, because the Eq. 3.8 is not satisfied yet, 𝐿 FCM algorithm continues and in iteration No. 36 by reaching to the cost value of 1.8013 and 𝜀 = 0, the algorithm is stopped. It means that FCM was able to classify 58 data samples 𝐿 into four classes (clusters). 3.5 3.0 2.5 Cost= 1.8014, eL=0.001 t s2.0 o c t s e1.5 B Cost= 1.8013, eL=0 1.0 0.5 0.0 0 5 10 15 20 25 30 35 40 Iteration Figure 3.7 Variations of cost value during FCM modelling Table 3.4 presents the membership degrees of each data sample for each class created by FCM. FCM is based on the minimization of the objective function and in its algorithm, the membership degree has an inverse relationship with the Euclidean distance. So, the sample with higher membership degree (or lower Euclidean distance) value in a class will belong to that class. By comparing the values listed in the rows in Table 3.4, it was turned out that the first class, the second class, the third class, and the fourth class have 5, 9, 26, and 18 data samples, respectively. For instance, membership degrees of sample No. 31 is 0.033, 0.087, 0.648, and 0.231 for the first, second, third and fourth classes, respectively, which based on the above explanation, this sample belongs to the third class. In other words, Table 3.4 gives information like the hits plot of SOM model. Then, by counting the majority of rockburst types in each class, the classes were nominated as Table 3.5, i.e. the first class is known as “light”, 67

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the SOM and FCM techniques along with the ones obtained from empirical criteria are given in Table 3.6. To have a quantitative insight regarding the performance of the developed models, five performance metrics i.e. accuracy rate (Grinand et al. 2008), Cohen’s Kappa coefficient (Kappa) (Cohen 1960), precision, recall, and F1 score (Zhou et al. 2016) were calculated for different models based on the confusion matrices obtained from Table 3.6 (see Table 3.7) for each model. Accuracy rate is a primary criterion for evaluating the model, which is defined as the ratio of truly classified samples to the total number of samples. Ideally, this value equals 100%. The Kappa coefficient is a more robust index than accuracy rate that measures the proportion of precisely classified cases after removing the probability of chance agreement. Hence, Kappa is always somewhat lower than the accuracy rate, and according to the scale proposed by Landish and Koch (Landis and Koch 1977), a Kappa higher than 0.4 shows a good agreement. Precision is another metric that measures the accuracy of the model when it predicts a specific class. The ratio of correctly classified cases of a class by the model is defined as the recall. The F1 score is the harmonic mean of precision and recall metrics that its best value is 1. For all five metrics, a higher value shows the better performance. Fig. 3.8 compares the models in terms of different performance indices. As can be seen from this figure, the SOM model could classify the rockburst events exactly with 100% value for all performance indices that show the high potential of this algorithm for dealing with such a complex geotechnical problem. In other words, SOM succeeded to find the latent relationship between the input parameters and the corresponding output and placed all data samples in their proper clusters. In this study, FCM classified the data samples during 36 iterations with a satisfactory precision level and proved its capability in dealing with geotechnical problems. However, in some cases, FCM was not able to place some data samples in proper clusters and finally showed a lower accuracy than the SOM model. For example, FCM placed the samples No. 3 and 4 in the third class (moderate), while in the field they have been measured as none (N) and moderate (M) rockburst types, respectively. In another case, both samples of 25 and 26 have been measured as the strong rockbursts in the field, while FCM put them in different classes of light and moderate, respectively. On the other hand, among the five conventional rockburst criteria, Hoek criterion showed slightly better performance than others, while rock brittleness coefficient identified as the worst model for clustering. Besides, the obtained Kappa values for EEI (33.3%) and rock brittleness coefficient (1.8%) are lower than 0.4 (40%), and according to Landish and Koch (Landis and Koch 1977), these models show a poor agreement and arbitrary classification, respectively. Hence, these models could not be used reliably to 70

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classify and predict rockburst intensity. It should be noted that the empirical methods have been developed based on specific case studies and some engineering judgments and consider few input parameters, while the datasets compiled in this study have a broad range of rock properties and locations. As mentioned in the introduction section, few studies have been done in relation to the application of unsupervised learning algorithms for assessing rockburst hazard. Among them, Xie and Pan (2007) and Gao (2010) could classify the rockburst events with grey whitenization weigh function cluster approach and ant colony clustering algorithm with the accuracy values of 80% and 83.3%, respectively. They used the maximum tangential stress, uniaxial compressive strength, uniaxial tensile strength, and elastic energy index as input parameters in their studies like the current study. Therefore, it can be concluded that the results obtained from SOM algorithm are more reliable and this method could be considered as a high-performance clustering system in geoscience, especially in assessing rockburst hazard. It is worth mentioning that the results of this study can provide feasible measures to prevent rockburst hazards. Since each of input parameters plays different roles, some indications can be extracted. As mentioned in section 3.3.2, the maximum tangential stress (T) has a significant impact on the occurrence of strong and moderate rockbursts, respectively, whereas other input parameters mostly affect moderate rockbursts. Large values of 𝑇 could led to more intense rockbursts in underground openings. As discussed by Palmstrom (1995) and Shirani Faradonbeh and Taheri (2019), the tangential stress around the openings is the representative of four components of rock stress, groundwater, the shape of the structure, and diameter. Therefore, it is very important to control these four parameters. With respect to difficulties in controlling the rock stress and groundwater pressure, it is easier to control maximum tangential stress indirectly by optimizing the shape and diameter of underground openings in practical projects. It can be a primary measure to control rockburst. Table 3.6 Results of clustered data samples using different models No. Measured Russenes Hoek Tangential Brittleness EEI FCM SOM 1 M M L L M M M M 2 S S S S M S S S 3 N N N N M S M N 4 M M L L N S M M 5 M M M M N S M M 6 M M L L N S M M 7 S S S M S M S S 8 L M L L S S L L 9 L M L L S M S L 10 N N N N L L N N 71

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11 N N N N L L N N 12 N N N N L L N N 13 M M L L M M M M 14 M M L L M S M M 15 S S S S S S S S 16 N N N N S N N N 17 N L N N M M M N 18 N N N N M M N N 19 S M M M S S M S 20 M S S S M M S M 21 S S S S S S S S 22 N S S S S N N N 23 N N N N M M N N 24 L M L L M S L L 25 S L N N M S L S 26 S L L L M S M S 27 N N N N M M M N 28 M S S S M M S M 29 S S S S M M S S 30 M M M M N S M M 31 M S M M L M M M 32 M M L L N M M M 33 N L L L N N N N 34 M M L L N S M M 35 M M L L N M M M 36 M S M M L M M M 37 M S S M M M M M 38 N M L L N N N N 39 S S S M M S M S 40 S M L L M S M S 41 N N N N S N N N 42 N N N N L L N N 43 N N N N M M M N 44 N S S S S N N N 45 N N N N M M N N 46 L M L L M S L L 47 S L N N M S L S 48 S L L L M S M S 49 M S S M M S M M 50 N N N N S N N N 51 M M L L M S M M 52 N N N N S N N N 53 M S S M M S M M 54 N L N N S N N N 55 S S S M S M S S 56 N M L L S N N N 57 M S M M M S M M 58 N N N N S N N N N none, L light, M moderate, S strong 72

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Table 3.7 Confusion matrix for different models No. Model Confusion matrix No. Model Confusion matrix 1 Russenes Predicted 5 EEI Predicted N L M S N L M S Actual N 15 3 2 2 Actual N 11 4 6 1 L 0 0 4 0 L 0 0 1 3 M 0 0 11 8 M 0 0 9 10 S 0 4 2 7 S 0 0 3 10 2 Hoek Predicted 6 FCM Predicted N L M S N L M S Actual N 17 3 0 2 Actual N 18 0 4 0 L 0 4 0 0 L 0 3 0 1 M 0 9 5 5 M 0 0 17 2 S 2 3 1 7 S 0 2 5 6 3 Tangential Predicted 7 SOM Predicted N L M S N L M S Actual N 17 3 0 2 Actual N 22 0 0 0 L 0 4 0 0 L 0 4 0 0 M 0 9 8 2 M 0 0 19 0 S 2 3 4 4 S 0 0 0 13 4 Brittleness Predicted N L M S Actual N 2 4 7 9 L 0 0 2 2 M 7 2 10 0 S 0 0 8 5 100 90 80 ) % 70 ( x 60 e d 50 n i e 40 c n a 30 m r 20 o f r 10 e P 0 Tangenti Brittlene Russenes Hoek EEI FCM SOM al stress ss Accuracy (%) 56.9 56.9 56.9 29.3 51.7 75.8 100 Precision (%) 50 61 57 23 47 73 100 Recall (%) 45 64 62 25 43 73 100 F1 score (%) 47 62 59 24 45 73 100 Kappa (%) 40.2 43.7 42.9 1.8 33.3 65.3 100 Figure 3.8 Comparison of the proposed models’ performance for rockburst clustering based on five indices 73

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3.5. Summary and Conclusions Many empirical equations have been proposed by researchers to predict rockburst intensities in recent years. However, according to the literature, they are not sufficient and reliable. The maximum tangential stress, uniaxial compressive strength, uniaxial tensile strength, and elastic energy index are the most common input parameters which are used to predict rockburst intensity. In this study by considering these four parameters, it was attempted to apply two novel clustering techniques namely self-organizing map (SOM) and fuzzy c-mean (FCM) to 58 rockburst data samples that are collected from several underground projects to classify and determine rockburst intensity. In addition, the capability of five common empirical criteria was assessed. Five performance metrics including accuracy rate, precision, recall, F1 score, and Kappa were used to assess the performance of the proposed models. The SOM algorithm with its especial mechanism classified all data into 4 distinct clusters and predicted rockburst intensity with the accuracy rate, precision, recall, f1 score, and Kappa values equal to 100 %. In addition, SOM indicated that the distances between classes are consistent with the intensities that are described by engineers. The evaluation of the weights of input parameters in each created class by SOM showed the high influence of maximum tangential stress (T) of surrounding rock mass on the clustering process, especially on the occurrence of strong and moderate rockburst events. Therefore, to tackle the rockburst problem, it is recommendable to optimize the shape and diameter of the underground openings. Despite the high and acceptable accuracy rate of FCM model (75.86 %), this method was not able to classify some data samples in appropriate clusters. Nevertheless, FCM outperformed the five empirical criteria that were studied in this research. Among the empirical criteria, Hoek criterion and tangential stress coefficient showed better performance in clustering the rockburst datasets, while rock brittleness coefficient criterion showed the lowest performance. Finally, it can be concluded that the SOM and FCM algorithms are strong enough to discover the latent relationships between the independent parameters and the corresponding dependent one. Specifically, in geoscience, we deal with high-complex and non-linear problems which there is no definite solution for them, and these kinds of algorithms can help engineers to have an insight into the hazards. 74

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Statement of Authorship Title of Paper The propensity of the over-stressed rock masses to different failure mechanisms based on a hybrid probabilistic approach Publication Status Published Accepted for Publication Submitted for Publication Unpublished and Unsubmitted work written in manuscript style Publication Details Shirani Faradonbeh R, Taheri A, Karakus M (2021) The propensity of the over- stressed rock masses to different failure mechanisms based on a hybrid probabilistic approach. Tunnelling and Underground Space Technology x(x):x–x. Note: Under review [the revised format submitted on 15 June 2021] Principal Author Name of Principal Author (Candidate) Roohollah Shirani Faradonbeh Contribution to the Paper Literature review and database preparation, statistical analysis, development of models and preparation of the manuscript Overall percentage (%) 80% Certification: This paper reports on original research I conducted during the period of my Higher Degree by Research candidature and is not subject to any obligations or contractual agreements with a third party that would constrain its inclusion in this thesis. I am the primary author of this paper. Signature Date 17 June 2021 Co-Author Contributions By signing the Statement of Authorship, each author certifies that: i. the candidate’s stated contribution to the publication is accurate (as detailed above); ii. permission is granted for the candidate in include the publication in the thesis; and iii. the sum of all co-author contributions is equal to 100% less the candidate’s stated contribution. Name of Co-Author Abbas Taheri Contribution to the Paper Research supervision, review and revision of the manuscript Signature Date 21 June 2021 Name of Co-Author Murat Karakus Contribution to the Paper Review and revision of the manuscript Signature Date 21 June 2021 81

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Chapter 4 The Propensity of the Over-Stressed Rock Masses to Different Failure Mechanisms Based on a Hybrid Probabilistic Approach Abstract The simultaneous impact of excavation-induced stress concentration and mining disturbances on deep underground mines/tunnels can result in severe and catastrophic failure like strain bursting. In this regard, the proper measurement of proneness to different rock failure mechanisms has great importance in terms of safety and economics. This study proposes a practical hybrid gene expression programming-based logistic regression (GEP-LR) model, as a multi-class classifier, to detect the failure mechanism (i.e. squeezing, slabbing and strain burst) in hard rock based on four intact rock properties. Three non-linear binary models are developed to predict the occurrence/non-occurrence of each failure mechanism. The logistic regression technique is linked to the developed GEP models to measure the occurrence probability of each failure mechanism. Finally, the failure mechanism that has the maximum probability of occurrence is selected as the predicted output. The performance analysis of the developed model shows that it is efficiently capable of detecting failure mechanisms with high accuracy. The failure mechanism detection models are presented in MATLAB codes to be easily used in practice by engineers/researchers as an initial guide for failure/stability analysis of underground openings. Finally, the validity of the proposed model is further evaluated by new datasets compiled from different studies. Keywords: Failure mechanism; Strain burst; Slabbing; Squeezing; Gene expression programming; Logistic regression 82

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4.1. Introduction The mechanical rock properties and their corresponding deformation failure mechanisms are dramatically different in deep underground than those in shallow conditions. This is due to the high geo-stress, ground-water pressure and high-temperature environment, which affect the rock mass for a long time. In this regard, many studies have been undertaken to investigate the parameters that influence the stability of underground structures using theoretical analyses, experimental and numerical simulations (Hoek and Brown 1980; Barla et al. 2011; Saadat and Taheri 2020; Li et al. 2020; Shirani Faradonbeh et al. 2021). Rock fracturing around deep excavations is mostly governed by the rock type, rock mass jointing degree and its orientation relative to the excavation free faces, the geometry of the excavation, in-situ stress magnitude and its orientation relative to the excavation direction (Wagner 2019). In deep mining and geotechnical projects, the highly uncertain governing factors are coupled to the stress distribution around the excavations, making the failure mechanism prediction one of the most challenging issues in terms of safety, the economic viability of the projects etc. The dominant failure mechanism in deep mining/tunnelling projects is strain burst or slabbing rather than shearing or squeezing (Fairhurst and Cook 1966). Palmstrom and Stille (2007) give a summary of different failure mechanisms and their characteristics in the underground. Also, a brief description of the common failure mechanisms in underground projects is presented below. One of the common failure mechanisms is squeezing, a non-violent rock behaviour/failure mechanism, which is characterised as a large time-dependent deformation associated with creep induced by over-stressing of massive rocks (Kabwe and Karakus 2020; Kabwe et al. 2020). These massive rocks usually have a high percentage of micaceous or clay minerals. Squeezing creates a plastic zone around the underground openings, which will result in cross- sectional area reduction during an aseismic process. The potential of rocks to squeezing is influenced by different parameters such as the geological conditions, rock mass mechanical properties, in-situ stresses, groundwater pressure, the geometry of the opening and the supporting system (Aydan et al. 1993; Barla 1995). Fig. 4.1a shows an example of a highly deformed cross-section of the Saint Martin access adit (Lyon–Turin base tunnel) induced by squeezing. According to Ortlepp (1997), slabbing refers to the formation of the densely spaced stress-induced slabs (onion-skin-like fractures) on the boundary of an underground opening (i.e. roof and sidewalls). The spacing of these slabs depends on the rock heterogeneity, rock strength, as well as in-situ stresses (Li et al. 2011). This failure mechanism is more common in moderate to hard over-stressed massive rocks and initiates in excavated regions having high 83

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maximum tangential stresses by creating a local V-shaped notch on the opening boundary (Ortlepp 2001). Fig. 4.1b displays the slabbing failure in the roof of a mine drift excavated in quartzite at 1000 m depth. Strain burst is a term for the much more violent fracturing of rocks than slabbing accompanied by the high seismicity, rock chips ejection and sudden release of strain energy that can pose a serious threat to workers, equipment and project life (Fig. 4.1c). The coupled static-dynamic loading conditions induced by stress redistribution after excavations and the dynamic disturbances generated by drilling and blasting, roof collapse, fault-slip, etc. provide a high- stress zone around the openings, which in turn triggers the strain bursting proneness effectively (Akdag et al. 2018; Shirani Faradonbeh et al. 2019; Shirani Faradonbeh et al. 2020; Wang et al. 2020). Many factors affect the bursting proneness of rocks, and owing to its vague mechanism, strain burst is known as a high-complex non-linear problem and difficult to predict (He et al. 2015; Shirani Faradonbeh and Taheri 2019). Among these influential factors, the intact rock properties have a critical role in the occurrence of this phenomenon in the deep underground. The uniaxial compressive strength (𝜎 ) and tensile strength (𝜎 ) are among the 𝑐 𝑡 most prominent intact rock properties which can be used for assessing the rock capacity to store elastic strain energy (Munoz et al. 2016; Munoz and Taheri 2017; Shirani Faradonbeh et al. 2020). These parameters also represent the tensile and shear failure characteristics of rocks (Liu et al. 2013; Shirani Faradonbeh and Taheri 2019). The 𝜎 and 𝜎 have been used frequently 𝑐 𝑡 in many strain burst studies as the rock brittleness index (i.e. 𝐵 = 𝜎 /𝜎 ) (Cai 2016) or potential 𝑐 𝑡 energy of elastic strain (i.e. 𝑃𝐸𝑆 = 𝜎2/2𝐸 , where 𝐸 is the unloading modulus) (Wsang and 𝑐 𝑢 𝑢 Park 2001) to evaluate the probability of strain burst occurrence and its intensity. Lee et al. (2004) investigated the interrelationship of rock strength parameters (i.e. 𝜎 and 𝜎 ) and strain 𝑐 𝑡 burst index (PES) mathematically by conducting the experimental tests on the obtained specimens from a waterway tunnel in Korea, and they proposed a strain burst chart as shown in Fig. 4.2a. In this chart, the bursting intensity is predicted based on the defined four classes of very low (VL), low (L), medium (M), and very high (VH). These classes follow the standard classification assigned for strain burst intensity which is based on visual inspection of the failure, rock ejection, sound and seismicity (Liu et al. 2013; Shirani Faradonbeh et al. 2019). In another study, by plotting the 𝜎 values against the brittleness index (𝐵 = 𝜎 /𝜎 ) values, 𝑐 𝑐 𝑡 Diederichs (2007) proposed a chart (see Fig. 4.2b) to predict the strain burst risk level. In that study, the low value of 𝐵 shows the dominance of extension cracking (spalling potential) in the damage process, while the rocks with high 𝜎 can accumulate more strain energy and 𝑐 84

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consequently have a higher potential to bursting. In addition, the strength parameters have been used extensively to assess this hazard by different researchers using supervised and unsupervised data-mining algorithms (Pu et al. 2019). On the other hand, the modulus of rigidity is an important parameter to study the stress distribution in the rock mass. Under mining-induced disturbances, some rocks tend to react elastically, while others may show plastic deformation. However, in hard rocks, the elastic characteristics are more dominant. Therefore, they can store a great amount of elastic strain energy. This energy can be released as an excess energy with seismicity in a violent manner (Singh 1987; Shirani Faradonbeh and Taheri 2019; Shirani Faradonbeh et al. 2019; Akdag et al. 2019). Singh (1987) evaluated the relationship between the burst proneness index (𝜂 = 𝐸 /𝐸 , where 𝐸 and 𝐸 are the retained 𝑅 𝐷 𝑅 𝐷 energy and the dissipated energy during a loading-unloading cycle) and elastic modulus experimentally, and reported that the 𝜂 increases with the increase of elastic modulus. Hence, the elastic deformation parameters such as elastic modulus and Poisson’s ratio can be considered as prominent indicators for strain burst proneness measurement. As mentioned earlier, the failure mechanisms are highly dependent on intrinsic rock properties, because in deep underground conditions, the rock masses have less discontinuities, and the existing ones cannot freely slide on each other to create structurally controlled failures (i.e. the failure is stress-driven). This is while in the shallow ground (low in-situ stress conditions), the failure process is controlled by the persistence and distribution of natural fractures (discontinuities), i.e. the failure is structure-driven (Kaiser et al. 2000). Therefore, discontinuities do not have a dominant role in the stability of structures. Besides, it is quite easy and convenient to determine intact rock properties such as uniaxial compressive strength (𝜎 ), tensile strength (𝜎 ), elastic modulus (𝐸) and Poisson’s ratio (𝜈) compared with other 𝑐 𝑡 parameters such as in-situ stresses, maximum tangential stress around the openings, etc. The proper measurement of failure mechanisms at the initial stages of the project can aid engineers to optimise the project layout and provide an adequate supporting system to prevent the occurrence of irreparable damages like fatalities, destruction of supporting systems and equipment, as well as the negative impact of such failure types on the economic viability of the project. However, to the best of our knowledge, there is no practical and easy-to-use model to distinguish the failure mechanisms, especially the strain burst and slabbing, and measure the propensity of competent over-stressed rock masses to different failure mechanisms. Due to the non-linearity nature of the failure mechanisms and the complex relationship between the failure mechanisms and their corresponding influential factors, the common linear and non-linear 85

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mathematical models cannot be implemented to unveil the latent relationships between parameters. Hence, soft computing techniques can be assumed as alternative approaches to tackle this difficulty. These techniques learn from the experiences and recognise the patterns in the database automatically (Mitchell 1997). From this perspective, soft computing techniques have been used extensively in mining and geotechnical engineering (Shirani Faradonbeh et al. 2017; Zhou et al. 2018; Haghshenas et al. 2019). In this study, the gene expression programming-based logistic regression (GEP-LR) technique is proposed as a new and practical probabilistic model to measure the propensity of the competent over-stressed rock masses to different failure mechanisms including squeezing, slabbing and strain burst. The intact rock properties (i.e. 𝜎 , 𝜎 , 𝐸 and 𝜈) which can be measured easily by the common 𝑐 𝑡 laboratory tests are used as indicators for modelling. The methodology and the obtained results are discussed in detail. a a Squeezing b Squeezing zdiorencteion Squeezing direction Slabbing zone c Strain burst zone Figure 4.1 Different failure mechanisms in underground excavations: (a) squeezing (modified from Barla et al. 2010), (b) high-stress slabbing (modified from Li et al. 2011) and (c) strain burst (modified from Yan et al. 2012) 86

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underground hard rock mines (mostly in Australia) with the known failure mechanism (Lee et al. 2018). Each dataset corresponds to a specific failure mechanism (i.e. strain burst, slabbing and squeezing) defined based on the in-situ observations of the fracturing process. The definition of these failure mechanisms is as those explained in the previous section. It should be mentioned that this database only covers the intact rock properties for the competent and over-stressed rock masses and does not consider the blocky over-stressed rock masses or the competent rock masses which have not yet been over-stressed (Lee et al. 2018). According to the rock mass classification system developed by Barton et al. (1974) (i.e. the Q-system), the competent rock masses are characterised by 𝑄 > 60. The results of a minimum of five reliable tests are used for each case study to measure the intact rock properties (Sainsbury and Kurucuk 2019). The 𝜎 values in Table 4.1 have been normalised using Eq. 4.1 owing to the size-scale 𝑐 dependency of rocks (Lee et al. 2018). 𝜎 𝜎 = 𝑑 (4.1) 𝑐 (50/𝑑)0.18 where 𝜎 is the normalised uniaxial compressive strength and 𝜎 and 𝑑 are the measured 𝑐 𝑑 uniaxial compressive strength and the diameter of the tested specimen, respectively. The 𝜎 , on the other hand, has been measured using the common Brazilian test method on the 𝑡 specimens having 50 mm diameter. The elastic deformation parameters of 𝐸 and 𝜈 also have been standardised in this database to the mid-third values by considering a minimum of five reliable test results. The box-plot is a common technique to evaluate the distribution of datasets in their range of values using some statistical indices such as minimum value, first quartile (𝑄 ), second quartile/median (𝑄 ), third quartile (𝑄 ) and the maximum value. Fig. 4.3 1 2 3 demonstrates the box-plots for the intact rock properties. As can be seen in this figure, the parameters have a wide range of values, and the datasets for all parameters follow an almost normal distribution. This makes mathematical modelling more feasible and easier. 88

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4.3. Methodology and Results As mentioned earlier, soft computing algorithms, e.g. artificial neural network (ANN) and support vector machine (SVM), have shown promising results in dealing with non-linear problems in different mining and geotechnical projects. However, these techniques suffer from several limitations, such as the necessity for defining the structure of models in advance, getting trapped in the local minimum and the inability to generate the practical prediction equations (Alavi et al. 2016). Therefore, the common soft computing techniques cannot provide practical models for assessing different failure mechanisms in deep underground openings. In this regard, a new hybrid gene expression programming-based logistic regression (GEP-LR) model is proposed in this section to measure the probability of occurrence of the different failure mechanisms in underground hard rock mines as a function of intact rock properties. According to Table 4.1, the parameters of 𝜎 , 𝜎 , 𝐸 and 𝜈 are defined as quantitative input/independent 𝑐 𝑡 parameters, while the failure mechanism as the output/dependent parameter is qualitative, having three types of failure. The dependent parameter does not need to have a normal distribution regarding the independent parameters. For simplicity, the dependent parameter is labelled as “1” in the case of squeezing failure, “2” in the case of slabbing failure, and “3” in the case of strain burst failure (see Table 4.1). The failure mechanisms concerning each independent parameter can be seen in Fig. 4.4. Ideally, to have a simple classification process, every datapoint should belong to a specific failure mechanism. As can be observed in Fig. 4.4, the parameters have some values belonging to more than one class, which means that it is impossible to predict the failure mechanism merely using one of the independent parameters. However, a combination of independent parameters along with a robust multi-class classification technique can be useful for the correct classification of failure mechanisms. The following sections present a description of the GEP algorithm as a robust classifier and the hybridisation process of GEP with logistic regression (LR) to predict the occurrence probability of each failure mechanism. 91

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3 3 le b le b a l m a l m s in s in a h 2 a h 2 c c e e m m e e r r u u lia lia F 1 F 1 0 100 200 300 400 0 10 20 30 s (MPa) s (MPa) c t 3 3 le le b b a a l m l m s s in in a h 2 a h 2 c c e e m m e e r u r u lia F 1 lia F 1 0 40 80 120 160 0.0 0.1 0.2 0.3 0.4 0.5 E (GPa) n Figure 4.4 Failure mechanism with respect to each independent parameter 4.3.1. GEP-Based Binary Models As a population-based algorithm, the gene expression programming (GEP) proposed by Ferreira (2002) is a modified and improved version of the basic genetic algorithm (GA) and genetic programming (GP). GEP algorithm opens the black-box nature of the prior soft computing algorithms (e.g. ANN) by providing mathematical equations representing the latent non-linear relationship between the parameters. Due to this significant capability of the GEP algorithm, it has been used recently by different researchers to appraise various mining and geotechnical problems (Armaghani et al. 2016; Jahed Armaghani et al. 2017; Khandelwal et al. 2017; Salimi et al. 2016). In the GEP algorithm, as shown in Fig. 4.5a, the solutions are in the form of linear fixed-length coded strings/chromosomes (single-gene or multiple-gene chromosomes) consisting of two main parts of head and tail in which the genetic operators are applied on these areas to improve the quality of solutions. The head of a chromosome contains symbols representing both terminals (input parameters and constant values) and mathematical functions (e.g. +, -, × and /) and always starts with a function, whereas the tail is composed of only terminals. The head length/size (h) that affects the complexity of the solutions usually is 92

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determined by the user through a trial-and-error procedure. However, the length of the tail (t) is a function of head size and the maximum argument number (𝑛 ) and can be determined 𝑚𝑎𝑥 using the following equation: 𝑡 = ℎ(𝑛 −1)+1 (4.2) 𝑚𝑎𝑥 Fig. 4.5 schematically displays the foundation of the GEP algorithm. However, the detailed mechanism of GEP can be found in Ferreira (2002). According to Fig. 4.5, the main steps of the GEP modelling procedure can be summarised as follows: • A population of potential solutions/models initially are generated in the form of linear chromosomes using a random combination of terminals and mathematical functions following the Karva language (a language invented for reading and expressing the information encoded in the chromosomes) (Fig. 4.5a). • These coded solutions then are automatically parsed into visual tree structures known as expression trees (ETs) (Fig. 4.5b). To do so, for each gene, the first function of the head is selected as the root node, and according to its argument number, some empty sub-nodes are generated. The terminals and functions in the chromosome are then placed in the sub-nodes from top to down and left to right in each line. This process continues until a line containing terminals is formed. As the terminals have no argument, no further sub-nodes are generated. Then, the created sub-ETs for different genes are linked together using a linking function (e.g. “/” in Fig. 4.5b) to form a single large ET. The ETs ease and speed up the process of function finding and mathematical interpretation of coded chromosomes. Thereafter, the mathematical formulation of solutions is extracted for further assessment (Fig. 4.5c). • The fitness of solutions is evaluated using a fitness function defined by the user (Fig 4.5d), and if the termination criterion (i.e. the maximum number of iteration or a prescribed fitness value) did not meet, the best solutions are selected using the fitness proportionate selection technique (Ferreira 2002) to reproduce with modification (Fig. 4.5e) based on the defined ratios for genetic operators (i.e. mutation, inversion, transposition, and reproduction). As seen in Fig. 4.5f, these operators try to improve the fitness of solutions by changing an element through a gene length (i.e. mutation), inverting a fragment in the head of a gene (i.e. inversion), copying a fragment to the head of a gene (transposition), and exchanging a fragment between two chromosomes 93

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(i.e. recombination). Afterwards, improved solutions are transferred to the next generation (Fig. 4.5g). • The above process continues until the termination criterion is met. In this study, firstly, three separate GEP-based binary models are developed to predict the occurrence (i.e. “1”) or non-occurrence (i.e. “0”) of each class of failure mechanism based on the procedure explained above. GeneXproTools 5.0 computer program is used to develop the GEP models. The intact rock properties of 𝜎 , 𝜎 , 𝐸 and 𝜈 are defined as the terminals/input 𝑐 𝑡 parameters. Furthermore, the computer program is allowed to select up to ten constant values randomly in the range of [-10,10], should the performance of the solutions is improved. Finally, the following comprehensive range of mathematical functions is selected to provide a broader search space for the algorithm, and consequently, generate solutions with higher fitness values: Function set = {+,−,×,/,𝐸𝑥𝑝,𝐿𝑛,^2,^3,𝑆𝑞𝑟𝑡,3𝑅𝑡,𝑆𝑖𝑛,𝐶𝑜𝑠,𝑇𝑎𝑛,𝐴𝑡𝑎𝑛} (4.3) where 𝑆𝑞𝑟𝑡,3𝑅𝑡 and 𝐴𝑡𝑎𝑛 respectively represent square root, cube root and arctangent. As shown in Fig. 4.5d, the correlation coefficient (𝑟) is defined as the fitness function to evaluate the performance of the generated solutions. For the classification task, the learning algorithm of the GEP converts the returned value by the evolved model into “1” or “0” using a rounding threshold. If the evolved model's returned value is equal to or greater than the rounding threshold, then the record is classified as “1”, “0” otherwise. The correlation coefficient 𝑟 of the solution/model 𝑖 is calculated as follows: 𝑖 𝐶𝑜𝑣(𝑇,𝑃) 𝑟 = (4.4) 𝑖 𝜎𝑡.𝜎𝑝 where 𝐶𝑜𝑣(𝑇,𝑃) is the covariance of the target and model outputs; and 𝜎 and 𝜎 are the 𝑡 𝑝 corresponding standard deviations. As it stands, 𝑟 cannot be used directly as a fitness function since, for the fitness proportionate 𝑖 selection technique, the value of fitness must increase with efficiency. Therefore, the following equation is employed to determine the fitness 𝑓 of a solution 𝑖: 𝑖 𝑓 = 1000×𝑟 ×𝑟 (4.5) 𝑖 𝑖 𝑖 where 𝑓 ranges from 0 to 1000, with 1000 corresponding to the ideal. 𝑖 Taking into account the previously suggested values (Alavi et al. 2016; Ferreira 2002; Hoseinian et al. 2017) for other GEP parameters, including the population size, the number of 94

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genes for each chromosome, head size, linking function and the genetic operators, several preliminary runs are also performed to find the optimum solution with highest fitness value for each failure mechanism class. The obtained optimum values for the GEP parameters are listed in Table 4.2. By applying these settings to the software, and running the algorithm for 3000 generations/iterations (i.e. the termination criterion), the following optimum GEP-based binary models are achieved: 1 𝑒√𝜐 Squeezing = 𝑌 = 𝐿𝑛(𝐸)×( )×( ) ; 1 𝜎 𝑐2 𝐸3+𝜎𝑡 1 𝑌 ≥ 2.2029×10−9 Failure status = { 1 (4.6) 0 𝑌 < 2.2029×10−9 1 Slabbing = 𝑌 = [𝑡𝑎𝑛(𝐸)+𝜎 −((𝜎𝑡−4.1812 )×(𝜐−3.1256))]× 2 𝑡 2 3 1 √ tan (𝜎6 − 3 𝜐−𝜎 −𝐸)×[tan(𝜎 −0.6500𝐸 +𝜐3)−𝜐]; 𝑐 𝑐 𝑡 1 𝑌 ≥ 8.1788 Failure status = { 1 (4.7) 0 𝑌 < 8.1788 1 9.572 Strain burst = 𝑌 = 3 𝐸 +tan (−7.4736𝜐(𝜎 +𝜎 ))+ 𝜎𝑡 +sin(𝜐−𝐸)+sin(0.2490𝜎𝑐) +𝜎1/9 ; 3 𝑐 𝑡 𝑐 2 1 𝑌 ≥ 6.2353 Failure status = { 1 (4.8) 0 𝑌 < 6.2353 1 By calculating the 𝑌-values using input parameters and feeding them to the developed binary classifiers, i.e. Eqs. 4.6 to 4.8, the occurrence/non-occurrence of each failure mechanism can be predicted. However, a multi-class classifier is still needed to determine the most probable failure mechanism based on the given intact rock properties. Indeed, the GEP algorithm has been basically designed for binary classification and cannot be implemented directly for the multi-class classification tasks like failure mechanism detection, which has three classes of squeezing, slabbing and strain burst. This can be defined as a limitation of this algorithm. However, in the next section, an efficient strategy is employed to adapt the GEP algorithm for the multi-class classification task. 95

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(a) Head Tail Head Tail Create initial population - b × b a b b a b × √ b + a b b b a Gene 1 Gene 2 Linking function Root node (b) Express solutions as ETs / Root node - × b × √ b + (c) 𝑏−(𝑏×𝑎) b a Execute each program a b √𝑎+𝑏×𝑏 Gene 1 (d) Evaluate fitness Correlation coefficient (r) Gene 2 1) Mutation - b × b a b b a b Before Yes - b × b × b b a b After TTeerrmmiinnaattee?? 2) Inversion No - b × b a b b a b Before - × b b a b b a b After Select the best solutions (e) 3) Transposition - b × b a b b a b Before (f) - b a b a b b a b After Apply genetic operators 4) Recombination - b × b a b b a b Before × √ b + a b b b a Create next generation (g) - b b + a b b b a After × √ × b a b b a b (h)Logistic regression (LR) GEP score = X Calculate probabilities (ps) 1 𝑝= Print p max 1+𝑒−(𝛼+𝛽𝑋) Figure 4.5 The multi-class classification procedure used in this study Table 4.1 The settings for GEP-based models Parameter Setting Squeezing Slabbing Strain burst failure failure failure General Population size 100 100 85 Number of genes 3 3 3 Head size 8 9 9 Linking function Multiplication (×) Multiplication (×) Addition (+) Fitness function Correlation Correlation Correlation coefficient coefficient coefficient Genetic Mutation rate 0.00138 0.00138 0.00138 operators Inversion rate 0.00546 0.00546 0.00546 Transposition rate 0.00546 0.00546 0.00546 Recombination rate 0.00277 0.00277 0.00277 96