Effect of Prior Knowledge on Site-Specific Selection of Regression Model for Characterization of Geotechnical Properties

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(1)Geotechnical Safety and Risk V T. Schweckendiek et al. (Eds.) © 2015 The authors and IOS Press. This article is published online with Open Access by IOS Press and distributed under the terms of the Creative Commons Attribution Non-Commercial License. doi:10.3233/978-1-61499-580-7-945. 945. Effect of Prior Knowledge on Site-Specific Selection of Regression Model for Characterization of Geotechnical Properties Adeyemi Emman ALADEJARE and Yu WANG Department of Architecture & Civil Engineering, Shenzhen Research Institute, City University of Hong Kong, Hong Kong. Abstract. During geotechnical site characterization, many geotechnical properties might be difficult to measure directly and have to be estimated using indirect measurement and regression models. For example, when there is no possibility of direct compression test, geotechnical engineers and practitioners may utilize regression models (i.e. equations) to estimate the uniaxial compressive strength, UCS, of rock from point load index, I() . However, there are many equations relating I() to UCS in the literature. This leads to the problem of how to select the most appropriate model for a particular rock deposit out of the numerous models available. This study presents a method that rationally compares different regression models and selects the most appropriate model for a specific site or deposit considered herein. The most appropriate model is selected using only a limited number of site-specific I() data. The selected model is then used in a Bayesian framework to integrate the prior knowledge about UCS with the limited number of site-specific I() data available for probabilistic characterization of UCS. The approach is shown to perform properly, particularly when the prior knowledge reflects information from the site. Keywords. Regression model, geotechnical properties, occurrence probability, probabilistic characterization, prior knowledge. 1. Introduction The geotechnical properties of rock play a significant role in the design and construction involving rock. UCS of rock is one of the most commonly used geotechnical properties in engineering geology projects. UCS can be determined directly (e.g., International Society of Rock Mechanics (ISRM), 2007). However, direct determination of UCS involves coring and cutting of samples which increases the difficulty of such tests. Other simpler strength indexes have been proposed to indirectly estimate UCS (e.g., Basu and Kamran, 2010). Is() has been found to be an efficient index in estimating UCS (e.g., Broch and Franklin, 1972). With varying regression equations in the literature, it is obvious that there is no unique regression model which can be used for even the same rock type. There is need for a rational method of selecting appropriate model for a specific site. Approaches in the past relied on both Is() and UCS data to generate and. evaluate regressions for specific site/deposit (e.g., Yasar et al. 2010). When UCS cannot be determined directly, there is need to evaluate the regression model using only the limited Is() data. Indeed, regression model is mostly needed when there are no UCS data at all. Selecting appropriate model, from available models, for estimating UCS for specific site based on limited Is() data available becomes a difficult task. To address the challenge, this paper presents an approach for selecting site-specific regression model based on limited Is() data available from site, and probabilistic characterization of UCS using the selected model. The proposed approach combines the limited Is() data with the prior knowledge under a Bayesian framework. The methodology involves formulation of likelihood model for selection of regression model and probabilistic characterization of UCS. A study is then performed to explore the effects of prior.

(2) 946. A.E. Aladejare and Y. Wang / Effect of Prior Knowledge on Site-Specific Selection of Regression. knowledge on selection of regression model and probabilistic characterization of UCS.. 2. Selection of Site-Specific Regression Model The approach in this paper will utilize the limited site Is() data to select the regression model that is suitable for estimating the UCS of such site. Selection of site-specific regression model is done by comparing the occurrence probability ( | ) of the candidate models given the observed data from the site (e.g., Cao and Wang, 2014a; Cao and Wang, 2014b; Wang and Aladejare, 2015), and the appropriate model is the one with the maximum probability. ( | ) can be obtained using the Bayes’ theorem, as expressed in Eq. (1). ( | ). Variability is unavoidable when dealing with natural materials like rocks, because rocks are inherently heterogeneous (e.g., Sari, 2009). Therefore, variability of UCS and the transformation uncertainty of using regression model for estimating UCS from Is() data are considered in formulating the likelihood function. UCS is taken to follow a normal distribution with a

(3) expressed in Eq. (3). = + . in which denote site-specific Is() data , is the candidate model, ( | ) is the probability of observing given a candidate model and it is frequently referred to as evidence (e.g., Wang et al. 2014; Wang and Aladejare, 2015). ( ) is the prior probability of which reflects the prior knowledge of . In the absence of any prevailing prior knowledge about

(4) models considered, each model is taken to have prior probability of 1

(5) . ( ) is a normalizing constant. ( | ) is obtained from the theorem of total probability, as expressed in Eq. (2). ( | ) = (2). where ( |, , ) is the likelihood function. It is the joint conditional probability density function, PDF of for given and a set of model parameters (e.g. mean and standard of UCS). (, | ) is the prior distribution of and of UCS.. (3). where z is a standard Gaussian random variable. Given the format of regression model for estimating UCS from Is() in Eq. (4). = Is(). ( | )( ) = = 1,2, …

(6) (1) ( ). ( |, , ) × (, | ). 2.1. Likelihood function. (4). where is the conversion factor from Is() to UCS. The general transformation model for each model is as expressed in Eq. (5). Is() = + . (5). where is the inverse of in Eq. (4) and is a Gaussian random variable with zero mean and standard deviation . Combining Eq. (3) and Eq. (5) gives the general likelihood model as expressed in Eq. (6). Is() = ( + ) + . (6). When the inherent variability is independent of the transformation uncertainty (i.e. z is independent of ), Is() is a Gaussian random variable with a mean of ( ) and standard deviation of ( ) + . The site-specific point load data (i.e., Data = { Is() , = 1,2 … ..,

(7) }) can be considered as

(8) independent realizations of the Gaussian random variable Is() . The likelihood function for the site-specific Is() is expressed in Eq. (7). (Data | , , ) =.

(9) A.E. Aladejare and Y. Wang / Effect of Prior Knowledge on Site-Specific Selection of Regression #$. %&. 1 Is() 0 ( ) '*- /0 7 8 9 2 ( ) + !2"( ) + . 1. (7). 947. Using the theorem of total probability, the PDF of UCS is determined using the updated knowledge of μ and , as expressed in Eq. (10). (UCS | ). 2.2. Prior distribution. = (UCS | , ) × (, | Data)EE (10). Wang and Aladejare (2015) suggested when the prior knowledge is relatively uninformative, the prior distribution, (, | ) in Eq. (2) can be taken as a joint uniform distribution of and , with respective minimum values of :# and :# and respective maximum values of :;< and :;< , and it is expressed in Eq. (8).. where (UCS | , ) is the conditional PDF of normally distributed UCS for a given set of mean μ and standard deviation . Combining Eqs. (9) and (10) leads to Eq. (11).. (, | ) = for ? [:# , :;< ] 1 /(:;< 0 :# )(:;< 0 :# ) and ? [:# , :;< ] others 0. (8) This type of prior distribution requires relatively limited prior knowledge (e.g. reasonable ranges of rock properties available in the literature). The approach proposed is general and can also be used for relatively informative prior knowledge like an arbitrary histogram type, normal distribution etc. Effects of different prior knowledge on site-specific selection of regression model are discussed in Section 4.. A (UCS | , )( | , )(, )EE (11). The posterior PDF (UCS | ) is transformed into a large number of UCS samples using the Bayesian equivalent approach developed by Wang and Cao (2013). The approach integrates Markov Chain Monte Carlo (MCMC) simulation with the Bayesian method to generate samples of UCS. Conventional statistics are then obtained from the samples to determine the mean and standard deviation of UCS. The PDF and cumulative distribution function, CDF, of UCS are also constructed.. 4. Effects of General and Site Specific Prior Knowledge. 3. Probabilistic Characterization of UCS A Bayesian approach is developed to characterize UCS from selected model @ , using just available I() data. @ is used to update the knowledge of μ and of UCS by expressing their joint posterior PDF based on limited site I() data and prior knowledge as in Eq. (9). (, | ) = A( |, )(, ). (UCS | ) =. (9). where K is a normalizing constant. (Data|, ) is the likelihood function of @ , and (, ) is the prior distribution of μ and given in Eqs. (7) and (8) respectively.. As an illustration, a study is performed in this section, to explore the effects of prior knowledge on the site-specific selection of regression model and probabilistic characterization of UCS. The approach is illustrated using the nineteen I() values of granite collected from Malanjkhand Copper project, India (Mishra and Basu, 2012). There data also includes twenty UCS data as shown in Table 1. The nineteen I() data points are used in the site-specific selection of appropriate model and characterization of UCS, the twenty UCS data from direct measurement is used only for validation..

(10) 948. A.E. Aladejare and Y. Wang / Effect of Prior Knowledge on Site-Specific Selection of Regression. To perform site-specific selection of regression model and probabilistic characterization of UCS, some models in the literature are utilized. For example, the model developed by Broch and Franklin (1972) is given in Eq. (12) and adopted as model 1, & . UCS = 23.7 Is(). (12). The least square regression analysis of Is() and UCS data points reported by Ghosh and Srivastava (1991) gave a conversion factor of 13.85. The model is given in Eq. (13) and taken as model 2, . UCS = 13.85 Is(). (13). Chau and Wong (1996) developed a regression model given in Eq. (14). UCS = 16.31 Is(). (14). The regression model is named model 3, H , in this study. The constant factor of 16.31 is obtained through the re-analysis of their 21 data points. Tugrul and Zarif (1999) developed a regression given in Eq. (15) It is adopted as model 4, J . UCS = 15.25 Is(). (15). Table 2 summarizes the parameters of each models used in this study. is calculated using the original data reported by the authors. Table 3 summaries the prior information of UCS used in this study.. Table 1. Laboratory test results of granite from Malanjkhand Copper project, India (from Mishra and Basu (2012)) Specimen No G1 G2 G3 G4 G5 G6 G7 G8 G9 G10 G11 G12 G13 G14 G15 G16 G17 G18 G19 G20. KL(MN) (MPa) 8.35 10.85 10.02 9.92 11.73 14.13 10.63 6.93 8.49 7.87 8.41 7.85 5.99 Invalid 7.29 11.36 9.23 6.92 9.72 5.66. UCS (MPa) 139.04 177.37 167.17 176.75 160.82 198.15 148.34 117.95 134.76 124.89 138.22 130.06 122.74 201.73 153.55 182.33 150.42 127.47 158.69 91.48. Table 2. Parameters of the models compared Parameter OP OQ OR 0.042 0.072 0.061 . 1.500. 1.424. 2.073. OT 0.066 0.292. Table 3. Sets of prior knowledge of UCS used in this study Parameter General Prior Site-specific Knowledge Prior Knowledge 3.65 129 :# (MPa) 355 337 :;< (MPa) 0 0 :# (MPa) 59 35 :;< (MPa). Kulhawy (1975) reported the ranges of UCS values for igneous rock as from 3.65 MPa to 355 MPa. The six-sigma rule is used to estimate the maximum standard deviation which gives 59 MPa. The minimum standard deviation of UCS is taken as 0 MPa because of its non-negative physical meaning. This set of ranges (i.e. μ [3.65 MPa, 355 59 MPa)] is taken as general prior knowledge. This general range is reduced using site I() information. The range of I() compiled for granite in the literature is from 0.58 MPa to 14.85 MPa (e.g., Momeni et al., 2015, Tugrul and Zarif, 1999). This range is combined with the range of site I() , to estimate the likely range of UCS at this site by linear interpolation, since the relation between UCS and I() is linear (see Eqs. (12-15)). This.

(11) A.E. Aladejare and Y. Wang / Effect of Prior Knowledge on Site-Specific Selection of Regression. 4.1. Effect on selection of regression model Table 4 shows the results of the calculated evidence and occurrence probability for the four models considered using the different prior knowledge. For general prior knowledge, &, has the highest evidence and occurrence probability. For site-specific prior knowledge, H has the highest evidence and occurrence probability. To better evaluate which of the prior knowledge performed better, a plot is made in Figure 1 comprising regression lines of the models and the site-specific data used in this study. It can be observed that the site data plots more closely to the regression line of H than & . The result from using site-specific prior knowledge is more reasonable and confident than the result from using general prior knowledge. This is logical since site-specific prior knowledge reflects the characteristics from the site.. UCS (MPa). 260 240 220 200 180 160 140 120 100 80 60 40 20 0. Site data. M1. M3 M2. M4. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Is(50) (MPa) Figure 1. Plot of site data with regression lines of the models. 4.2. Effect on probabilistic characterization of UCS The models selected in each case together with their respective prior knowledge are used in probabilistic characterization of UCS. For each case, 30000 samples of UCS are generated and the results of their statistics are summarised in Table 5. Comparing with the site UCS data, both mean and standard deviation of samples from H using site-specific prior knowledge has smaller relative difference than samples generated from & using general prior knowledge. This shows that samples from H using site-specific prior knowledge is more consistent with that from the direct laboratory tests than & using general prior knowledge. 1 0,8 0,6 Site data. CDF. gives a UCS range from 129 MPa to 337 MPa. Using six-sigma rule, the maximum standard deviation of UCS is 35.0 MPa and the minimum standard deviation of UCS is taken as 0 MPa. This set of ranges (i.e. μ [129 MPa, 337 MPa) 0 MPa, 35 MPa)] is taken as site-specific prior knowledge. The nineteen I() data points are then used in the model selection approach and probabilistic characterization of UCS, using the different prior knowledge and the results are discussed in the next subsections.. 949. 0,4 General Prior Site-specific Prior. 0,2 0 0. 50. 100 150 200 250 300 350 400 UCS (MPa). Figure 2. CDFs of UCS samples. Figure 2 is a plot of the UCS CDFs estimated from the cumulative frequency diagrams of the 30,000 samples using & with general prior knowledge, H with site-specific prior knowledge and the 20 direct measurement of UCS recorded by Mishra and Basu (2012). The CDF of the samples generated from H using site-specific prior knowledge, plotted by dashed lines is closer to the CDF of direct compression test indicated by open circles than CDF of samples generated from & using general prior knowledge plotted by solid line. The information contained in the samples generated using site-specific prior knowledge is more consistent with the direct measurement by compression tests than samples generated using general prior knowledge..

(12) 950. A.E. Aladejare and Y. Wang / Effect of Prior Knowledge on Site-Specific Selection of Regression. Table 4. Sets of prior knowledge of UCS used in this study Regression model. General prior knowledge Evidence Occurrence ( |X probability (X | ) 6.55E-20 0.434 2.27E-20 0.151 4.26E-20 0.281 2.02E-20 0.134. X M1 M2 M3 M4. Site-specific prior knowledge Occurrence prob Evidence ( |X ability (X | ) 8.27E-20 0.342 1.74E-20 0.072 1.13E-19 0.468 2.84E-20 0.118. Table 5. Summary of the statistics of UCS generated using General and Site-specific prior knowledge Prior. General prior knowledge. Site-specific prior knowledge. Approaches. Uniaxial compressive test. Bayesian equivalent sample. Difference (MPa). Relative difference (%). Bayesian equivalent sample. Difference (MPa). Relative difference (%). Mean (MPa). 150.1. 216.2. 66.1. 44.0. 149.0. 1.1. 0.7. Standard deviation (MPa). 28.3. 39.6. 11.3. 40.0. 18.7. 9.6. 34.0. 5. Summary and Conclusions References The proposed approach integrates the information from regression models and sitespecific I() data to systematically select the appropriate model for specific site. The selected model is then used in a Bayesian framework for probabilistic characterization of UCS using MCMC simulation. This eliminates the difficulty in generating meaningful statistics from the usually limited number of rock property data obtained during site investigation. A sensitivity study is then performed to evaluate the effect of prior knowledge on selection of regression model and probabilistic characterization of UCS. It was shown that the proposed approach becomes more confident when the prior knowledge reflects the characteristics from site.. Acknowledgment The work described in this paper was supported by a grant from the National Natural Science Foundation of China (Project No. 51208446). The financial support is gratefully acknowledged.. Basu, A., Kamran, M. (2010). Point load test on schistose rocks and its applicability in predicting uniaxial compressive strength. Int J Rock Mech Min Sci.; 47: 823–828. Broch, E., Franklin, J.A. (1972). The point-load strength test. Int J Rock Mech Min Sci.; 28: 669-697. Cao, Z. and Wang, Y. (2014a). Bayesian model comparison and characterization of undrained shear strength. J. Geotech. Geoenviron. Eng., 140(6), 04014018. Cao, Z. and Wang, Y. (2014b). Bayesian model comparison and selection of spatial correlation functions for soil parameters. Structural Safety 49: 10–17. Chau, K.T., Wong, R.H.C. (1996). Uniaxial compressive strength and point load strength of rocks. Int. J. Rock Mech. Min. Sci.; 33: 183–188. Ghosh, D.K., Srivastava, M. (1991). Point-load strength: an index for classification of rock material. Bull Eng Geol Env; 44: 27–33. ISRM, (2007). The complete ISRM suggested methods for rock characterization, testing and monitoring: In: R Ulusay, JA Hudson, editors; 628 pp..

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