J. Cent. South Univ. Technol. (2008) 15: 498-502
DOI: 10.1007/s11771-008-0094-8
Over-excavation forecast of underground opening by using Bayes discriminant analysis method
GONG Feng-qiang(宫凤强)1, 2, LI Xi-bing(李夕兵)1, 2, ZHANG Wei(张 伟)1
(1. School of Resources and Safety Engineering, Central South University, Changsha 410083, China;
2. Hunan Key Laboratory of Resources Exploitation and Hazard Control for Deep Metal Mines,
Changsha 410083, China)
Abstract: A method to forecast the over-excavation of underground opening by using the Bayes discriminant analysis(BDA) theory was presented. The Bayes discriminant analysis theory was introduced. Based on an engineering example, the factors influencing the over-excavation of underground opening were taken into account to build a forecast BDA model, and the prior information about over-excavation of underground opening was also taken into consideration. Five parameters influencing the over-excavation of opening, including 2 groups of joints, 1 group of layer surface, extension and space between structure faces were selected as geometric parameters. Engineering data in an underground opening were used as the training samples. The cross-validation method was introduced to verify the stability of BDA model and the ratio of mistake-discrimination was equal to zero after the BDA model was trained. Data in an underground engineering were used to test the discriminant ability of BDA model. The results show that five forecast results are identical with the actual situation and BDA can be used in practical engineering.
Key words: underground opening; over-excavation; Bayes discriminant analysis; forecast
1 Introduction
Over-excavation in underground opening is a key problem during the excavation and often happens in large-section tunnels. An important task in all rock excavations is to get a stable contour. The most common technique to blast contours today is to use the smooth blasting. Serious over-excavation often takes place when the tunnel drills through some areas where the geological conditions are very complicated and the excavating technique is badly chosen. In fact, over-excavation will take place when the shear strength of key-block preferred planes is less than its skipping force[1-5]. At present, the over-excavation forecast of unstable block is still an important project, which directly influences project construction time and investment. Many researchers have been involved in this field and provided many methods, such as rock stability theory, smooth blasting technique, and so on[5-8]. The distribution form of the largest over-excavation value of surrounding rock of tunnels in Ⅲ, Ⅳ and Ⅴ type rock masses was verified to be a fractal distribution and the character of the sample was researched through the largest over-excavation value of sample with 360 sections of data coming from three different tunnels and based on the highway tunnel surrounding rock classification[9]. The probability distributions of over-excavating for different series of surrounding rocks were put forward through numerical and physical statistics and analyses. At the same time, the distribution laws of over-excavation at different portions on tunnel sections for different series of surrounding rocks were also obtained[10-12]. Based on discontinuity plane mapping theory and key-block theory the over-excavation of doubled arch tunnel was forecasted by using the analytical discontinuity plane characteristic parameters. At the same time, the appearance probability and the location of over-excavation block in doubled arch tunnel were forecasted[5]. Based on the theory of geological statistical model, the theory of wavelet neural network and the theory of the opening over-excavation, the prediction of block over-excavation in the openings was made[6]. A mapping method was put forward based on the mechanics mechanism of rock fracture and this technology was applied to the forecast of over-excavation in jointed rock masses openings[13]. In this work, Bayes discriminant analysis(BDA) theory was introduced and applied to forecasting the over- excavation of underground opening. The results show that this technology can be used in practical engineering.
2 Bayes discriminant analysis theory
2.1 Bayes discriminant analysis theory method
The theory of Bayes discriminant analysis can be generalized as follows[14]: suppose there are k collectivities with p member indexes: G1, G2, …, Gk (k≥2), and Gα-N (μα, Σα), the covariance matrix Σα＞0 (α=1, …, k), and the prior probability of sample X=(x1, x2, …, xp)T coming from collectivity Gα is qα, then there exists q1+q2+…+qk=1.
The square distance between sample X and collectivity Gα is defined as
(1)
The square difference of Mahalanobis distance between sample X and collectivity Gp and Gq is defined as
d2(X, Gq)-d2(X, Gp)=-2[Wq(X)-Wp(X)] (2)
where discriminant function Wp(X)=(Σ-1μp)TX-0.5μpT? Σ-1μp, Wq(X)=(Σ-1μq)TX-0.5μqTΣ-1μq.
The probability density of sample X coming from collectivity Gα is
(3)
Based on Bayes theory, the posterior probability of sample X belonging to collectivity Gα is
(4)
The square distance between sample X and collectivity Gα can also be defined as
(5)
if q1=…=qk=1/k
if Σ1, …, Σk are not equal completely
where
if Σ1=…=Σk=Σ
if q1=…=qk=1/k
(6)
if q1, …, qk are not equal completely
(7)
and the posterior probability of X belonging to collectivity Gα can be obtained as
(8)
The Bayes discriminant criterion can be written as
(9)
In fact, the expected values μα and Σα are unknown and their estimated values can be obtained from training samples. Suppose there is one sample 
= (
) coming from Gα (where nα is the number of training sample from Gα), then the unbiased estimation of μq can be defined as 
and
(10)
For the estimation of Σα, there are the following two aspects:
1) When Σ1=Σ2=…=Σk=Σ, the estimation of Σα is defined as Sp, and
(11)
where n=n1+n2+…+nk.
2) When Σ1≠Σ2≠…≠Σk≠Σ, the estimation of Σα is defined as Sα, and
(12)
The prior probability can be obtained by the following two methods.
1) Suppose q1=q2=…=qk=1/k.
2) q1, q2, …, qk are allocated by the proportion of training samples of collectivity Gα to all samples, i.e.
(13)
2.2 Cross-validation method
Considering training samples, the cross-validation method was introduced to estimate the reliability of discriminant criterion[15]. For example, suppose there are two collectivities G1 and G2, and the numbers of their training samples are n1 and n2, respectively. The principle of cross-validation method is to eliminate one sample from all samples firstly, and then the other surplus samples are used to build a discriminant criterion. This criterion is used to discriminate the sample eliminated. The processes of cross-validation method can be expressed as follows.
1) Eliminate one sample from the training samples of collectivities G1 and build one discriminant criterion with n1+n2-1 samples;
2) Discriminate the eliminated sample with discriminant criterion built in process 1);
3) Repeat the processes 1) and 2), and define the mis-discrimination number as 
when all the training samples of collectivity G1 are eliminated;
4) Repeat processes 1), 2) and 3) for collectivity G2 and define the mis-discrimination number as 
when all training samples of collectivity G2 are eliminated;
5) The ratio of mis-discrimination (η) can be calculated by
(14)
3 Over-excavation forecast model for under- ground opening
3.1 Indexes of tunnel over-excavation
There are many factors influencing the over- excavation of tunnel and the natural factors play an important role in it. Some natural factors, including the geological condition of surrounding rock mass, the cutting of structural face, the developing condition of joint, will influence the over-excavation of tunnel directly. Therefore, the geometrical characters of surrounding rock mass must be investigated carefully before forecasting the over-excavation of underground opening. Based on these investigations, the scope of over-excavation volume can be determined and the over-excavation condition can be forecasted reasonably. Fig.1 shows an over-excavation sketch of underground opening.
Because the engineering geological condition is very complex in different zones, the corresponding geometric parameters should be selected according to thein-site engineering condition before investigating surrounding rock mass. The engineering data in Ref.[6] were introduced to show how the BDA method was applied in practice. Five parameters, including 2 groups of joint, 1 group of layer surface, extension and space between structural faces, were selected as geometric parameters that influence the over-excavation of opening. Therefore, 8 specific parameters listed in Table 1 were regarded as input parameters in BDA model.
Fig.1 Over-excavation sketch of underground opening
Table 1 Training samples of Bayes discriminant analysis method
The over-excavation volume of block was divided into 4 types, i.e. 4 collectivities: 1) V＞2.4 m3, the model output is G1; 2) V=2.4-2.1 m3, the model output is G2; 3) V=2.1-1.8 m3, the model output is G3; 4) V＜1.8 m3, the model output is G4. The volume classification of over- excavation block can be modified in different engineering examples, and the classification standards in Ref.[6] were introduced in this work.
3.2 Engineering example data
It is a hydraulic power station engineering[6]. According to the design plan, the water-storage altitude of up-reservoir is 470.00 m and the water-storage altitude of under-reservoir is 79.80 m, and total length is 2 489.24- 3 103.67 m. There are two water diversion tunnels and two trail water tunnels. The water volume is 5.0×107 m3, and the installed power is 800 MW.
3.3 Application of BDA model
Referring to actual information of different structural faces of this engineering, 21 samples were selected as the training samples of BDA model (listed in Table 1). The prior probability was allocated by the proportion of training samples, and then q1=6/21, q2=2/21, q3=9/21 and q4=4/21. Suppose Σ1=Σ2=Σ3=Σ4, discriminant function Wα(X) (α=1, 2, 3, 4) can be obtained as
W1(X)=-357.021-1.388X1+2.201X 2+3.309X3+4.716X4-
0.219X5+5.595X6+0.414X7-3.215X8;
W2(X)=-506.234+0.039X1+6.568X2+5.734X3+
16.766X4-1.893X5+1.547X6+1.588X7-9.432X8;
W3(X)=-484.630+0.845X1+7.744X2+5.375X3+
19.553X4-2.468X5-0.237X6+1.960X7-10.874X8;
W4(X)=-713.889+1.795X1+10.144X2+7.452X3+
29.410X4-3.852X5-2.658X6+2.520X7-15.503X8.
Trained with training samples, the outputs of BDA model are the same as expected outputs and the ratio of mis-discrimination equals zero with cross-validation method. The results show that BDA model has stable discriminant ability.
The data of adit 4 of this engineering were used to test the actual discriminant ability of BDA. It can be seen from Table 2 that the forecast results are equal to G1 and over-excavation volume is large. In actual investigation[6], the big cutting block is generated by the large dip angle of adit 4 with two joints, which shows that the forecasting results are concordant with actual ones.
4 Conclusions
1) A method is presented to forecast the overexcavation of underground opening by using the Bayes discriminant analysis(BDA) model. The factors influencing the over-excavation of underground opening and prior information about over-excavation are taken into account to build a forecast BDA model and cross-validation method is introduced to verify the stability of BDA model.
Table 2 Forecast results of Bayes discriminant analysis method
2) The BDA theory is applied to the over-excavation forecasting of underground opening and some problems should be researched in the future. For example, how to select the influencing factors of over-excavation and the discriminant factors. In the future work, some representative samples should be obtained and applied in the BDA model. The forecasting results should be more concordant with actual condition.
References
[1] CHAKRABORTY A K, RAINA A K, RAMULU M, CHOUDHUR Y P B, HALDAR A, SAHOO P, BANDOPADHYAY C. Development of rational models for tunnel blast prediction based on a parametric study [J]. Geotechnical and Geological Engineering, 2004, 22(4): 477-496.
[2] MAERZ N H, IBARRA J A, FRANKLIN J A. Overbreak and underbreak in underground openings (part 1): Measurement using the light section method and digital image processing [J]. Geotechnical and Geological Engineering, 1996, 14(4): 307-323.
[3] FRANKLIN J A, MAERZ N H, IBARRA J A. Overbreak and underbreak in underground openings (part 2): Causes and implications [J]. Geotechnical and Geological Engineering, 1996, 14(4): 325-340.
[4] WU J M, MAHTAB A. Overbreak forecast of underground opening in jointed rock mass [J]. Journal of Engineering Geology, 1999, 7(1): 3-8.(in Chinese)
[5] WEI Ji-hong, SUN Shao-rui, PENG Jian-zhong, CHEN Xian-chun, WU Ji-min, CHEN Yong-hui. Study on forecasting and evaluation of overbreak in double-arch tunnels of highway [J]. Chinese Journal of Rock Mechanical and Engineering, 2006, 25(S1): 3324-3329. (in Chinese)
[6] SUN Shao-rui, WU Ji-min, WEI Ji-hong. The predication of overbreak block in the openings based on wavelet neural network of geology statistical model [J]. Chinese Journal of Rock Mechanical and Engineering, 2003, 22(8): 1344-1349. (in Chinese)
[7] MAHTAB M A, ROSSLER K, KALAMARAS G S, GRASSO P. Assessment of geological overbreak for tunnel design and contractual claims [J]. International Journal of Rock Mechanics and Mining Sciences, 1997, 34(3/4): 185.e1-185.e13.
[8] SINGH S P, XAVIER P. Causes, impact and control of overbreak in underground excavations [J]. Tunnelling and Underground Space Technology, 2005, 20(1): 63-71.
[9] SU Yong-hua, SUN Xiao-ming, ZHAO Ming-hua. Fractal character of over-excavation of tunnel surrounding rock[J]. Journal of China University of Mining and Technology, 2006, 35(1): 89-93. (in Chinese)
[10] WANG Ming-nian, GUAN Bao-shu. Statistic characteristic research of over-excavation and under-excavation in surrounding rock of high road tunnel and its effect on reliability degree [J]. Chinese Journal of Geotechnical Engineering, 1997, 19(1): 83-88. (in Chinese)
[11] SHE Jian, ZHONG Xin-zhi. Statistic characteristic research of over-excavation and under-excavation highroad tunnel [J]. Journal of Chongqing Traffic College, 2000, 19(2): 15-20. (in Chinese)
[12] ZHAO Hui-bing. Statistic characteristic research of over-excavation and under-excavation in tunnel constructed by blasting method [J]. Journal of Railroad Engineering, 1999, 8(3): 106-110. (in Chinese)
[13] WU Ji-min, SUN Shao-rui, WEI Ji-hong. Forecast of overbreak in opening based on discontinuity surface mapping of mechanics mechanism [J]. Rock and Soil Mechanics, 2004, 25(8): 1220-1224. (in Chinese)
[14] WANG Xue-min. Applied multivariate statistic [M]. Shanghai: Shanghai University of Finance and Economics Press, 1999. (in Chinese)
[15] FAN Jin-cheng, MEI Chang-lin. Data analysis [M]. Beijing: Science Press, 2002. (in Chinese)
Foundation item: Project(50490274) supported by the National Natural Science Foundation of China
Received date: 2007-11-26; Accepted date: 2008-01-30
Corresponding author: GONG Feng-qiang, PhD; Tel: +86-731-8879612; E-mail: fengqiangg@126.com
(Edited by CHEN Wei-ping)
