Introduction

The determination of rock properties is indispensable in mining engineering, providing necessary input for analyses and designs of mining and geotechnical structures. Rock parameters vary from location to location within a rock deposit/site and among different rock sites. This is because rocks are formed from various geological processes and these processes affect the mineral constituents, packing, and properties of rocks. This variability (often referred to as inherent variability), which is natural and arises as a result of geological processes leading to the formation of rocks is one of the uncertainties in rock parameters (e.g., Baecher and Christian 2003; Aladejare 2016; Aladejare and Wang 2017a). During the determination of characteristic values of rock parameters, there are also measurement errors which arise as a result of imperfections. The imperfections may include but are not limited to malfunctioning of test equipment and operator and procedural errors, especially when there is a deviation from the recommendations of the International Society of Rock Mechanics (ISRM) (Ulusay and Hudson 2007) or rock testing and parameter determination. There is also transformation uncertainty, which is introduced when a rock parameter (e.g., Brazilian tensile strength BTS) for mining design and analyses are estimated from indirect measurements (e.g., point load index Is(50) values) through the use of a transformation model (e.g., regression/empirical model linking Is(50) values to BTS). Using transformation models is very prominent in mining engineering because of the difficulties in performing laboratory testing for determination of some rock parameters, especially those requiring special sample preparation (e.g., Shalabi et al. 2007; Diamantis et al. 2011; Kurtuluş et al. 2010; Sharma et al. 2011; Heidari et al. 2012; Kahraman et al. 2012; Khandelwal 2013; Kallu and Roghanchi 2015; Fakir et al. 2017).

The inherent variability which is the actual variability of a rock site is usually lumped together with the measurement errors and transformation uncertainty as the total variability of estimated rock parameters from site investigation (Phoon and Kulhawy 1999). However, using total variability instead of actual variability leads to overestimation of uncertainties in rock parameters. This might lead to design errors in which the expected performance or response of rock structures varies significantly from the actual performance or response at rock sites (Christian and Baecher, 2011; Aladejare and Wang 2017b). Hence, there is a need for rational quantification of various uncertainties in estimated rock/geotechnical parameters in order to properly characterizes the inherent variability of rock properties at a specific site (Wang et al. 2016). This will help to ensure the estimation of the actual performance or response of the rock structure at a site. Performing probabilistic characterization of BTS inherent viability remains a challenge because attention has not been given to BTS unlike uniaxial compressive strength, Young’s modulus and the deformation modulus of rocks that have been researched (e.g., Wang and Aladejare 2015; 2016a; Aladejare and Wang 2019). Furthermore, there is difficulty in separating inherent variability from measurement errors for direct measurements (e.g., BTS values from Bralizian tensile tests) and transformation uncertainty arising when indirect measurements (e.g., Is(50) values) are used to estimate BTS values. This problem is further compounded by the lack of a sufficient number of site-specific test results that can ensure meaningful statistical analysis for mining designs. It is time-consuming and costly to prepare rock samples that meet the ISRM standards for most rock strength tests. In addition, for instance, it may be difficult if not impossible to extract standard cores from weak or highly fractured, thinly bedded, and/or block-in matrix rocks (Gokceoglu 2002; Aksoy et al. 2010).

Bayesian methodology provides a logical route to formulate the quantification of inherent variability of rock parameters as an inverse analysis problem. Bayesian approaches provide the platform to quantify the inherent variability of rock parameters from data observed through direct and/or indirect measurements. Bayesian approaches have been used to determine rock and geotechnical parameters as well as solve other mining engineering problems by integrating information obtained from different sources during site investigation (e.g., Wang et al. 2010; Müller et al. 2014; Wang and Aladejare 2015, 2016b; Wang et al. 2015; Cao et al. 2016; Ng et al. 2017; Aladejare and Wang 2019; Aladejare et al. 2021; 2022a, b). These previous studies used multisource information that includes prior knowledge available about a rock parameter before a rock project or laboratory analysis, site test data from indirect measurements, and transformation models relating indirect measurements to rock parameter of interest. Aladejare and Wang (2019) likened the process to data fusion, in which information from two or more data sources are combined to provide a more accurate estimation than any of the data sources individually. However, it is not common or reported in literature that indirect testing procedures (e.g., point load tests) are combined with direct testing (e.g., Brazilian tensile tests) to improve the characterization of BTS and help to reduce the effect of limited data often obtained for BTS. Ignoring direct measurements of BTS may not lead to a full depiction of the information on BTS at a site because direct measurement of rock parameters like BTS can provide vital information during the updating of the statistics and probability distributions of BTS from multiple sources of information. The statistics and probability distributions of rock parameters are helpful in understanding the updating processes and for checking the quality of information from different sources (i.e., both direct and indirect test data). It is therefore important to rationally incorporate direct and indirect measurements into the probabilistic characterization of inherent variability of rock parameters. There is no research that has performed probabilistic characterization of BTS inherent variability at a rock site through a systematic combination of results of indirect measurements like Is(50) values for which transformation uncertainty is unavoidable when it is used to estimate BTS and direct measurements of BTS which are associated with measurement errors. Logical depiction and understanding of the effects of measurement errors and transformation uncertainty are critical to the probabilistic characterization of BTS.

The study develops a Bayesian approach that is based on a sequential updating process to perform probabilistic characterization of the inherent variability of BTS of rock at a rock site by integrating prior knowledge with site information from both direct (BTS from Brazilian tensile tests) and indirect (e.g., Is(50) values from point load tests) measurements. Aladejare and Wang (2019) developed a Bayesian sequential updating approach that systematically combines values of rock mass rating (RMR) and tunneling quality index (Q) for the probabilistic characterization of rock mass deformation modulus. Aladejare et al. (2021) developed another Bayesian sequential updating approach that systematically combines values of three different punch tests (i.e., Schmidt rebound hardness (SRH), block punch index (BPI), and point load strength (Is(50))) for the probabilistic characterization of uniaxial compressive strength (UCS) of rock. However, both cases (i.e., Aladejare and Wang (2019) and Aladejare et al. (2021)) used indirect test data without taking the direct measurements into consideration. The proposed Bayesian approach sequentially makes use of a limited number of Is(50) values from point load tests and BTS values from Brazilian tensile tests at different updating levels. The approach quantitatively accounts for measurement errors and transformation uncertainty during the probabilistic characterization of the inherent variability of BTS at a rock site. The paper starts with the probabilistic modeling of the inherent variability of BTS at a rock site, followed by the development of the Bayesian approach. The proposed approach is then illustrated and validated using real-life and simulated data. In addition, sensitivity studies are performed to explore the effects of measurement errors on the probabilistic characterization of the inherent variability of BTS at a specific rock site.

Probabilistic modeling of inherent variability of Brazilian tensile strength

For the probabilistic modeling of the inherent variability of Brazilian tensile strength in a rock formation or deposit, this study uses a normal random variable to represent the BTS. Many studies in the literature have modeled rock properties as normal random variables (e.g., Sari and Karpuz 2006; Wang and Aladejare 2015; 2016a). Therefore, let \({\mu }_{B}\) and \({\sigma }_{B}\) denote the mean and standard deviation of BTS, respectively. Then, BTS can be expressed as (Cao et al. 2016; Wang and Aladejare 2016b):

$$\mathrm{BTS}={\mu }_{B}{+\sigma }_{B}Z$$
(1)

where Z = standard normal random variable. Using Eq. (1), the inherent variability of BTS in a rock formation or deposit is quantified by \({\mu }_{B}\) and \({\sigma }_{B}\). In the next section, the Bayesian approach which is based on multi-level updating is presented. The approach updates the knowledge of \({\mu }_{B}\) and \({\sigma }_{B}\) for the probabilistic characterization of the inherent variability of BTS at a particular rock site based on prior knowledge and information from both Is(50) data and direct BTS measurements from Brazilian tensile strength tests.

Bayesian approach

Bayesian framework

Test results from different laboratory experiments might not be obtained simultaneously during laboratory investigation into the properties of rock. For example, in this study, Is(50) values are measured in the laboratory through point load tests while direct BTS measurements are obtained from Brazilian tensile tests in the laboratory. It is of practical interest to note the statistics (e.g., \({\mu }_{B}\) and \({\sigma }_{B}\)) of BTS according to indirect testing data (e.g., Is(50) values) from the point load tests and how these statistics change as BTS values from Brazilian tensile tests are taken into account. Under the proposed Bayesian framework, the updating of \({\mu }_{B}\) and \({\sigma }_{B}\) is partitioned into two levels and is sequentially performed to see the evolution of the statistics of \({\mu }_{B}\) and \({\sigma }_{B}\) at each updating level. Note that updating of \({\mu }_{B}\) and \({\sigma }_{B}\) performed at one level using both Is(50) from indirect point load tests and BTS from direct Brazilian tensile tests (e.g., Chen and Gilbert 2017; Cao et al. 2016) provides the same probabilistic characterization results as those obtained from the Bayesian approach that is based on sequential updating, but it cannot reflect the evolution of the statistics of BTS as direct BTS measurements are considered in addition to Is(50) values. In this study, Is(50) values are used at the first updating level to update the knowledge of \({\mu }_{B}\) and \({\sigma }_{B}\) for the probabilistic characterization of the inherent variability of BTS. Direct BTS measurements are then added at the second updating level to further update knowledge of \({\mu }_{B}\) and \({\sigma }_{B}\) of BTS.

For the two updating levels of the proposed Bayesian approach, the posterior distributions of \({\mu }_{B}\) and \({\sigma }_{B}\) are used to reflect their updated knowledge (e.g., Cao et al. 2016; Aladejare et al. 2021; 2022a) as follows:

$$P\left({\mu }_{B}, {\sigma }_{B}|{DATA}_{I}\right)={C}_{I}P\left({\mu }_{B}, {\sigma }_{B}\right)P\left({DATA}_{I}|{\mu }_{B}, {\sigma }_{B}\right)$$
(2)
$$P\left({\mu }_{B}, {\sigma }_{B}|{DATA}_{I}, {DATA}_{II}\right)={C}_{II}P\left({\mu }_{B}, {\sigma }_{B}|{DATA}_{I}\right)P\left({DATA}_{II}|{\mu }_{B}, {\sigma }_{B},{DATA}_{I}\right)$$
(3)

where \({DATA}_{I},={\{{\mathrm{Is}}_{(50)}^{*}}_{1},\) \({{\mathrm{Is}}_{(50)}^{*}}_{2},\dots,{{\mathrm{Is}}_{\left(50\right)}^{*}}_{{k}_{I}}\}\) is a set of \({\mathrm{Is}}_{(50)}^{*}\) of \({k}_{I}\) different Is(50) values measured for a rock formation or deposit or site; \({DATA}_{II}=\{{\mathrm{BTS}}_{1}^{*},{\mathrm{BTS}}_{2}^{*},\dots,{\mathrm{BTS}}_{{k}_{II}}^{*}\}\) is a set of \({\mathrm{BTS}}^{*} \mathrm{of}\) \({k}_{II}\) different \(\mathrm{BTS}\) values obtained by performing Brazilian tensile tests on rock samples recovered from a rock formation, deposi,t or site; the superscripted asterisk (*) indicates the measured value; \(P\left({\mu }_{B}, {\sigma }_{B}\right)\) is the prior distribution that reflects the information on \({\mu }_{B}\) and \({\sigma }_{B}\) in the absence of any site-specific test data and is taken as the joint uniform distribution of \({\mu }_{B}\) and \({\sigma }_{B}\) in this study, which are completely defined by their respective maximum (i.e., \({\mu }_{{B}_{max}}\) and \({\sigma }_{{B}_{max}}\)) and minimum (i.e., \({\mu }_{{B}_{min}}\) and \({\sigma }_{{B}_{min}}\)) possible values (Aladejare and Idris 2020); \(P\left({\mu }_{B}, {\sigma }_{B}|{DATA}_{I}\right)\) is the posterior distribution of \({\mu }_{B}\) and \({\sigma }_{B}\) in the first updating level (i.e., updating level I), and it is taken as the prior distribution of \({\mu }_{B}\) and \({\sigma }_{B}\) in the second updating level (i.e., updating level II); \(P\left({DATA}_{I}|{\mu }_{B}, {\sigma }_{B}\right)\) and \(P\left({DATA}_{II}|{\mu }_{B}, {\sigma }_{B},{DATA}_{I}\right)\) are likelihood functions of the two updating levels that quantify the information on \({\mu }_{B}\) and \({\sigma }_{B}\) provided by \({DATA}_{I}\) and \({DATA}_{II}\), respectively; and \({C}_{I}\) and \({C}_{II}\) are normalizing constants that are independent of \({\mu }_{B}\) and \({\sigma }_{B}\). Substituting Eq. (2) into Eq. (3) gives

$$P\left({\mu }_{B}, {\sigma }_{B}|{DATA}_{I}, {DATA}_{II}\right)={{C}_{I}C}_{II}P\left({\mu }_{B}, {\sigma }_{B}\right)P\left({DATA}_{I}|{\mu }_{B}, {\sigma }_{B}\right)P\left({DATA}_{II}|{\mu }_{B}, {\sigma }_{B},{DATA}_{I}\right)$$
(4)

Equation (4) quantifies the combined or updated information on \({\mu }_{B}\) and \({\sigma }_{B}\) from the prior knowledge, point load Is(50), data (i.e., \({DATA}_{I}\)), and BTS values (i.e., \({DATA}_{II}\)).  

Likelihood function for indirect measurements

Point load test measurements, \({\mathrm{Is}}_{(50)}^{*}\), can be related to \({\mathrm{BTS}}^{*}\) values from Brazilian tensile strength tests by regression model from Sulukcu and Ulusay (2001) as given in Eq. (5).

$$\mathrm{BTS}={2.30\mathrm{I}s}_{(50)}$$
(5)

Equation (5) above can be transformed to incorporate the transformation uncertainty which is introduced to the estimation of parameters when regression models are used for parameter estimations. The transformation model of Eq. (5) is given as

$${\mathrm{Is}}_{(50)}^{*}=a{\mathrm{BTS}}^{*}+{\varepsilon }_{T}$$
(6)

where \(a=0.4348\); and \({\varepsilon }_{T}\) is a normal random variable that has a mean of zero and a standard deviation of \({\sigma }_{{\varepsilon }_{T}}=\) 2.2700 and it represents the transformation uncertainty associated with Eq. (6).

In Eq. (6), \({\mathrm{BTS}}^{*}\) represents the measured value, rather than the actual value of BTS that is modeled by Eq. (1) of Brazilian tensile strength obtained from Brazilian tensile strength tests because regression models are usually developed by fitting the empirical relationship between rock parameters based on their respective measurements. Therefore, \({\mathrm{BTS}}^{*}\) is associated with measurement errors which arise during Brazilian tensile tests and can be expressed as a summation of the measurement error \({\varepsilon }_{M}\) and the actual value (i.e., BTS) of Brazilian tensile strength, as follows:

$${\mathrm{BTS}}^{*}=\mathrm{BTS}+{\varepsilon }_{M}=\left({\mu }_{B}{+\sigma }_{B}Z\right)+{\varepsilon }_{M}$$
(7)

where \({\varepsilon }_{M}\) is a normal random variable. The mean and standard deviation of \({\varepsilon }_{M}\) denoted by \({\mu }_{{\varepsilon }_{M}}\) and \({\sigma }_{{\varepsilon }_{M}}\), respectively, are taken as 1 and 0.1 MPa. These adopted statistics for \({\varepsilon }_{M}\) are consistent with the typical values reported in the literature for most geotechnical parameters (e.g., Phoon and Kulhawy 1999; Shen et al. 2018). The measurement errors are incorporated in this study because of its effects on the characterization of inherent variability of rock/geotechnical parameters.

Combining Eqs. (6) and (7) gives

$${\mathrm{Is}}_{(50)}^{*}=a{\mu }_{B}+{a\sigma }_{B}Z+a{\varepsilon }_{M}+{\varepsilon }_{T}$$
(8)

Equation (8) presents a likelihood model that relates \({\mu }_{B}\) and \({\sigma }_{B}\) to site-specific \({\mathrm{Is}}_{(50)}^{*}\) data. The likelihood model explicitly considers the measurement errors and transformation uncertainty for inferring the inherent variability of BTS based on \({\mathrm{Is}}_{(50)}^{*}\) data. Since the inherent variability, measurement errors, and transformation uncertainty are from different sources, they are assumed to be mutually independent. Therefore, \({\mathrm{Is}}_{(50)}^{*}\) can be taken as a normal random variable with a mean of \(\mathrm{a}{\upmu }_{B}+\mathrm{a}{\upmu }_{{\varepsilon }_{M}}\) and a standard deviation of \(\sqrt{{{a}^{2}\sigma }_{B}^{2}+{{a}^{2}\sigma }_{{\varepsilon }_{M}}^{2}+{\sigma }_{{\varepsilon }_{T}}^{2}}\). The likelihood function for indirect measurement at updating level I \(P\left({DATA}_{I}|{\mu }_{B}, {\sigma }_{B}\right)\) is thus given as

$$P\left({DATA}_{I}|{\mu }_{B}, {\sigma }_{B}\right)=\prod_{j=1}^{{k}_{I}}\frac{1}{\sqrt{2\pi }\sqrt{{{a}^{2}\sigma }_{B}^{2}+{{a}^{2}\sigma }_{{\varepsilon }_{M}}^{2}+{\sigma }_{{\varepsilon }_{T}}^{2}}}\times \mathrm{exp}\left\{-\frac{1}{2}{\left[\frac{{{\mathrm{Is}}_{(50)}^{*}}_{j}-(a{\mu }_{B}+ {a\mu }_{{\varepsilon }_{M}})}{\sqrt{{{a}^{2}\sigma }_{B}^{2}+{{a}^{2}\sigma }_{{\varepsilon }_{M}}^{2}+{\sigma }_{{\varepsilon }_{T}}^{2}}}\right]}^{2}\right\}$$
(9)

Likelihood function for direct measurements

Updating level II uses the direct BTS measurements (i.e., \({DATA}_{II}\) = {\({\mathrm{BTS}}_{1}^{*}\), \({\mathrm{BTS}}_{2}^{*}\), …., \({\mathrm{BTS}}_{{k}_{II}}^{*}\)}) from Brazilian tensile tests as input in the Bayesian framework. The likelihood function \(P\left({DATA}_{II}|{\mu }_{B}, {\sigma }_{B},{DATA}_{I}\right)\) of updating level II is used to quantify the information provided by \({DATA}_{II}\). From Eq. (7), \({\mathrm{BTS}}^{*}\) is a normal random variable with a mean of \({\mu }_{B}+ {\mu }_{{\varepsilon }_{M}}\) and a standard deviation of\(\sqrt{{\sigma }_{B}^{2}+{\sigma }_{{\varepsilon }_{M}}^{2}}\). At a given rock formation or site, the statistics of BTS (i.e., \({\mu }_{B}\) and \({\sigma }_{B}\)) is used to determine the probability density function (PDF) of \({\mathrm{BTS}}^{*}\). The BTS is independent of Is(50) data (i.e., \({DATA}_{I}\)) for a given set of \({\mu }_{B}\) and \({\sigma }_{B}\). Therefore, the \(P\left({DATA}_{II}|{\mu }_{B}, {\sigma }_{B},{DATA}_{I}\right)\) can be simplified as \(P\left({DATA}_{II}|{\mu }_{B}, {\sigma }_{B}\right)\) in Bayesian framework and expressed as:

$$P\left({DATA}_{II}|{\mu }_{B}, {\sigma }_{B}\right)=\prod_{j=1}^{{k}_{II}}\frac{1}{\sqrt{2\pi }\sqrt{{\sigma }_{B}^{2}+{\sigma }_{{\varepsilon }_{M}}^{2}}}\times \mathrm{exp}\left\{-\frac{1}{2}{\left[\frac{{\mathrm{BTS}}_{j}^{*}-({\mu }_{B}+ {\mu }_{{\varepsilon }_{M}})}{\sqrt{{\sigma }_{B}^{2}+{\sigma }_{{\varepsilon }_{M}}^{2}}}\right]}^{2}\right\}$$
(10)

The updated information in terms of the posterior distribution of \({\mu }_{B}\) and \({\sigma }_{B}\) is obtained based on prior information about BTS, point load test data, and Brazilian tensile test data using Eqs. (4), (9), and (10). Thus, the posterior distribution quantifies the integrated information from the three sources. Using the Eqs. for a specific rock site, equivalent samples of BTS can be generated for probabilistic characterization of the inherent variability of BTS.

Sample simulation from the Bayesian approach

The PDFs of BTS in the updating levels I and II based on the updated information on \({\mu }_{B}\) and \({\sigma }_{B}\) from Eqs. (2) and (3) can be expressed as Eqs. (11) and (12), respectively.

$$P\left(\mathrm{BTS}|{DATA}_{I}\right)=\iint P\left(\mathrm{BTS}|{\mu }_{B}, {\sigma }_{B}\right)P\left({\mu }_{B}, {\sigma }_{B}|{DATA}_{I}\right){d}_{{\mu }_{B}}{d}_{{\sigma }_{B}}$$
(11)
$$P\left(\mathrm{BTS}|{DATA}_{II},{DATA}_{I}\right)=\iint P\left(\mathrm{BTS}|{\mu }_{B}, {\sigma }_{B}\right)P\left({\mu }_{B}, {\sigma }_{B}|{DATA}_{II},{DATA}_{I}\right){d}_{{\mu }_{B}}{d}_{{\sigma }_{B}}$$
(12)

where \(P\left(\mathrm{BTS}|{\mu }_{B}, {\sigma }_{B}\right)\) is a normal PDF for a given set of \({\mu }_{B}\) and \({\sigma }_{B}\) to reflect the normal distribution assumed for BTS; \(P\left({\mu }_{B}, {\sigma }_{B}|{DATA}_{I}\right)\) and \(P\left({\mu }_{B}, {\sigma }_{B}|{DATA}_{II},{DATA}_{I}\right)\) are the posterior distribution at updating levels 1 and II, respectively. A large number of BTS samples is generated from Eqs. (11) and (12) to numerically quantify the updated PDFs of BTS in updating levels I and II, respectively, using Markov chain Monte Carlo (MCMC) simulation. This is consistent with the methodology in many previous studies for solving complicated PDFs such as those presented in Eqs. (11) and (12) (e.g., Beck and Au 2002; Zhang et al. 2012; Wang and Cao 2013; Juang et al. 2013; Peng et al. 2014; Wang and Aladejare 2016a). Based on the simulated samples of BTS in each updating level, conventional statistical analyses are performed on the samples to evaluate the respective statistics and PDFs of BTS in each updating level. The statistical estimates of BTS summarize the updated knowledge on their respective inherent variability at the site, where Is(50) data and direct BTS measurements from Brazilian tensile tests are obtained. The proposed Bayesian approach provides insights into the evolution of the statistics and probability distribution of BTS as direct and indirect measurements are used. Understanding such evolution is helpful in assessing the quality of information from point load tests and Brazilian tensile tests. It also helps to discover the consistency or otherwise of the information from both tests.

Illustration of the proposed Bayesian approach using real rock site

For illustration, the proposed approach was applied to probabilistically characterize the inherent variability of Brazilian tensile strength at a real rock site in India (Mishra and Basu 2012). Table 1 shows the values of \({\mathrm{Is}}_{(50)}^{*}\) and \({\mathrm{BTS}}^{*}\) obtained for the site.

Table 1 Laboratory test results of granite from Malanjkhand Copper Project in the state of Madhya Pradesh, India (from Mishra and Basu 2012)
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For the purpose of illustration, 10 data of \({\mathrm{BTS}}^{*}\) were randomly drawn and combined with the 19 \({\mathrm{Is}}_{(50)}^{*}\) as input in the proposed approach from the data in Table 1. Note that it is often the case that for most mining engineering projects, there is a possibility to perform more point load measurements compared to Brazillian tensile strength tests. In addition, a joint uniform distribution with \({\mu }_{{B}_{min}}=4.39\) MPa; \({\mu }_{{B}_{max}}=\) 26.05 MPa; \({\sigma }_{{B}_{min}}=\) 0.91 MPa; and \({\sigma }_{{B}_{max}}=\) 22.87 MPa was used as the prior distribution in this example, which is consistent with the typical range of BTS of igneous rocks (Aladejare and Wang 2017a). Using the prior knowledge, the 10 randomly drawn data of \({\mathrm{BTS}}^{*}\) and the 19 \({\mathrm{Is}}_{(50)}^{*}\) data from Table 1, a total of 20,000 equivalent samples of BTS were generated in updating levels I and II, respectively, to probabilistically characterize the inherent variability of Brazilian tensile strength at the rock site. Then, the equivalent samples of BTS are compared with the 20 \({\mathrm{BTS}}^{*}\) values from the site. To avoid confusion in this study, only the samples generated in updating level II are shown in this example.

Characterization of the inherent variability of Brazilian tensile strength

Figure 1 shows the 20,000 samples of BTS generated by MCMC simulation in updating level II, based on which a histogram of BTS was constructed. The histogram is presented in Fig. 2, and it includes the 95% confidence interval (i.e., [10 MPa, 22 MPa]) ranging from 2.5 to 97.5% quantiles of BTS and estimates of the mean and standard deviation of BTS samples (i.e., mean = 15.24 MPa and standard deviation = 2.74 MPa). Figure 3 shows the PDF of BTS estimated from the histogram by a solid line, and the PDF of BTS calculated from direct numerical integration by Eq. (12), which is shown by open squares. The two PDFs estimated from the simulated samples of BTS and the direct numerical integration agree well with each other. This validates the Bayesian approach and indicates that the simulated samples in Fig. 1 can sufficiently depict the PDF of BTS after the multi-level updating. By integrating the prior information, \({\mathrm{Is}}_{(50)}^{*}\) data and \({\mathrm{BTS}}^{*}\) data in the proposed approach, the statistics and PDF of BTS satisfactorily characterize the inherent variability of BTS of the granite at the rock site. In addition, Fig. 4 shows the estimated mean and standard deviation of BTS in the two updating levels. It is shown that incorporating \({\mathrm{BTS}}^{*}\) values in updating level II, the mean and standard deviation values of BTS improve and become more consistent with the site BTS. This indicates that integrating the information provided by \({\mathrm{Is}}_{(50)}^{*}\) data and \({\mathrm{BTS}}^{*}\) data, the simulated samples have reduced uncertainty. In addition, the proposed approach is useful in assessing the type of information available during rock parameter characterization and how incorporating information from multiple sources can impact site characterization in terms of reduction in uncertainty.

Fig. 1
figure 1

Scatter plot of Brazilian tensile strength BTS

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Fig. 2
figure 2

Histogram of Brazilian tensile strength BTS

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Fig. 3
figure 3

Probability density function of Brazilian tensile strength BTS

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Fig. 4
figure 4

Statistics of Brazilian tensile strength in updating levels I and II

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Effect of assumed measurement error for test data

In this study, the measurement errors \({\varepsilon }_{M}\) in direct BTS measurements from Brazilian tensile tests are modeled as normal random variables with a mean of \({\mu }_{{\varepsilon }_{M}}=1\) MPa and \({\sigma }_{{\varepsilon }_{M}}=\) 0.1 MPa. However, correct characterization of measurement errors for specific rock parameters is difficult because of the possibility of the introduction of different sources of errors during laboratory testing of rock samples. In addition, it is also difficult to delineate the measurement errors from inherent variability because both uncertainties are usually combined into the scatterness of the rock test data. It is difficult to separate these uncertainties when there is a limited number of laboratory test results, which is often the case during laboratory investigations for the characterization of rock parameters. Assumptions can be made for measurement errors based on engineering judgements, previous experience, or typical values available in the literature. The magnitude of measurement errors assumed represents the extent of belief in the laboratory procedure and testing for characterization of rock parameters, with larger measurement errors indicating a lower degree of belief in laboratory procedures and vice versa. To explore the effects of the assumed \({\varepsilon }_{M}\) on the probabilistic characterization of the inherent variability of rock parameters, this subsection performs a sensitivity study by assuming four more different values (i.e., 0 MPa, 0.05 MPa, 0.2 MPa, and 0.3 MPa) of \({\sigma }_{{\varepsilon }_{M}}\). The four different values in addition to \({\sigma }_{{\varepsilon }_{M}}=\) 0.1 MPa used in the previous section makes a total of five different values of \({\sigma }_{{\varepsilon }_{M}}\). For each assumed \({\sigma }_{{\varepsilon }_{M}}\), the proposed Bayesian approach is used to characterize the inherent variability of BTS and estimate the mean and standard deviation of BTS. Figure 5 shows the variation of the estimated mean and standard deviation of BTS against the assumed \({\sigma }_{{\varepsilon }_{M}}\) by open circles and squares, respectively.

Fig. 5
figure 5

Effects of assumed measurement errors on estimated inherent variability of BTS

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It can be noted that the estimated mean of BTS remains almost unchanged as the value of \({\sigma }_{{\varepsilon }_{M}}\) increases. This is likely due to the fact that the mean of measurement error remains constant during the characterizations. However, as the assumed \({\sigma }_{{\varepsilon }_{M}}\) increases, the estimated standard deviation of BTS decreases considerably, which means that the estimated inherent variability of BTS decreases in this example because \({\mu }_{{\varepsilon }_{M}}\) remains almost constant. For a given set of site-specific test data (e.g., direct BTS measurements from Brazilian tensile tests), the observed data scatterness is due to inherent variability and measurement errors. When the assumed \({\sigma }_{{\varepsilon }_{M}}\) is high, more parts of the scatterness of the observed BTS data are attributed to measurement errors during the interpretation of the test data. As can be noted in Fig. 5, the proposed Bayesian approach reflects the degrees of belief in laboratory procedures in a logical way. The implication of the results in Fig. 5 is that when test data are considered perfectly accurate (i.e., \({\sigma }_{{\varepsilon }_{M}}=\) 0 MPa), the scatterness in measured data during direct measurements is totally attributed to the inherent variability, leading to an overestimation of inherent variability in the BTS. In reality, an increase in \({\sigma }_{{\varepsilon }_{M}}\) (which indicates a decrease in the degree of belief in the laboratory procedure and testing) means that the magnitude of measurement errors in the scatterness of the measured data is greater, and the estimated inherent variability of BTS decreases as shown in this example. The proposed Bayesian approach is useful in understanding the effect of the degree of belief, which reflects the assumed measurement errors in the laboratory procedure and testing for the characterization of rock parameters. This will be helpful for mining engineers and practitioners because of the difficulty in quantifying the measurement errors involved during the laboratory determination of rock parameters for a specific site.

Illustration of the proposed Bayesian approach using simulated data

Simulated rock site

In mining engineering, rock parameters are unavoidably associated with inherent variability. This variability is not known with certainty and is only inferred from prior knowledge like engineering judgement and measured values of rock parameters from direct and indirect laboratory tests. Therefore, a sensitivity study is performed in this section to further validate the proposed Bayesian approach. To perform the sensitivity study, \({\mathrm{Is}}_{(50)}^{*}\) and \({\mathrm{BTS}}^{*}\) values simulated from a simulated rock site with known inherent variability of BTS are used. In the simulated rock site, the inherent variability of BTS is modeled by a normal random variable with mean \({\mu }_{B}=\) 12 MPa and \({\sigma }_{B}=\) 3 MPa. \({\mathrm{BTS}}^{*}\) and \({\mathrm{Is}}_{(50)}^{*}\) data are simulated using Eqs. (7) and (8), respectively, while the measurement error is modeled by a normal random variable with \({\mu }_{{\varepsilon }_{M}}=1\) MPa and \({\sigma }_{{\varepsilon }_{M}}=\) 0.1 MPa. Ten sets of {\({\mathrm{Is}}_{(50)}^{*}\), \({\mathrm{BTS}}^{*}\)} values are simulated for different numbers of input data. In this study, four different numbers of input data for both indirect and direct measurements are considered (i.e., n = 5, 10, 20, and 30). The idea of using different numbers of input data is to explore the effect of the number of input data on the output of the proposed Bayesian approach. For a given n value, the 10 sets of {\({\mathrm{Is}}_{(50)}^{*}\), \({\mathrm{BTS}}^{*}\)} values simulated were used together with the prior knowledge in the previous section to simulate 20,000 samples of BTS using the proposed Bayesian approach. Using the 20,000 samples for each case, the estimated mean and standard deviation of BTS were obtained. The procedures were performed 30 times for each n value to obtain 30 values of estimated mean and standard deviation of BTS, and the calculations of 95% confidence intervals using conventional statistical analyses.

Results of sensitivity analysis

The results of the sensitivity analysis performed are plotted in Figs. 6 and 7. In both figures, open circles are used to plot the estimated mean and standard deviation of BTS versus the n value of indirect and direct laboratory measurements used in the sensitivity study. The figures also include the 95% confidence intervals of the estimated mean and standard deviation of BTS represented by error bars. For comparison, the figures also include the true values of \({\mu }_{B}\) and \({\sigma }_{B}\) which were used to simulate the input data, and represented by dashed lines. As the n value increases from 5 to 30, the shaded part in the middle of the error bars gradually approaches the dashed lines and then plots directly on the dashed lines. This indicates that the estimated mean and standard deviation of BTS are identical to their respective true values that were used to generate indirect and direct input data. At small n values, the values of the estimated mean and standard deviation are scattered to some extent. The scatterness is largely due to statistical uncertainties in the data, which arise as a result of relatively sparse data at such n values. However, such scatterness gradually disappears as the n value increases, which indicates that the scatterness of the estimated mean and standard deviation of BTS reduces as the number of input data in the proposed approach increases. The estimated inherent variability of BTS obtained from the proposed Bayesian approach becomes more accurate as the number of input data from laboratory investigations used in the approach increases. It can be concluded that the proposed Bayesian approach provides reasonable estimates of the mean and standard deviation of BTS from the integration of both prior knowledge and information from indirect and direct measurements from laboratory tests.

Fig. 6
figure 6

Mean values of BTS estimated using different numbers of simulated test data

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Fig. 7
figure 7

Standard deviation values of BTS estimated using different numbers of simulated test data

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Summary and conclusions

This study proposed a Bayesian approach that integrates engineering experience (i.e., prior knowledge) and measurements from point load tests (i.e., indirect test source in this study) and Brazilian tensile tests (i.e., direct test source in this study) in a logical manner for the probabilistic characterization of the inherent variability of BTS at a rock site. The proposed approach was illustrated and validated using data from a real-life rock site and a simulated rock site. In addition, a sensitivity study was performed to explore the effects of assumed measurement errors on the interpretation of test data during laboratory investigation of rock samples for characterization of rock parameters. The magnitude of measurement errors depicts the degrees of belief in the testing procedures, with increased measurement errors indicating somewhat lack of confidence in the laboratory testing procedures that produced test measurements. The results showed that the proposed Bayesian approach provides a systematic route to incorporate measurement errors into the characterization of site-specific inherent variability of rock parameters in a rational manner. With limited information from indirect and direct test measurements together with prior knowledge, the approach satisfactorily characterized the BTS in both sites. The results of the sensitivity analysis showed that the assumed measurement errors have effects on the inherent variability estimated for a given set of data from laboratory testing. As the number of input data increases, the proposed Bayesian approach reduces the statistical uncertainties and the effects of measurement errors contained in the input data. The insight from the proposed approach will be helpful during the characterization of rock parameters because it makes it logically possible to quantify the inherent variability of rock parameters without lum** it with measurement errors. With the limited number of data often obtained from laboratory testing and investigations into rock parameters, it is impractical to quantify the measurement errors and their effects on the characterization of rock parameters. This further reinforces the usefulness of the proposed approach because such effects are systematically quantified and incorporated into the interpretation of test results for the characterization of rock parameters through the Bayesian approach.