I

Engineering Geology, 12 (1978) 143-161 143CD Elsevier Scientific Publishing Company, Amsterdam - Printed in The Nctherlllnds

DECISION ANALYSIS APPLIED TO ROCK TUNNELEXPLORATION

HERBERT H. EINSTEIN, DUANE A. LABRECHE, MICHAEL J. MARKOW andGREGORY B. BAECHER

Department of Ciuil Engineering, Massachusetts Institute of Technology, Cambridge,Mass. 02139 (U.S.A.)

(Received November 16, 1976; revised version accepted August 10, 1977)

ABSTRACT

Einstein, H.H., Labreche, D.A., Markow, M.J. and Baecher, G.B., 1978. Decision analysisapplied to rock tunnel exploration. In: W.R. Judd (Editor), Near Surface UndergroundOpening Design. Eng. Geo!., 12(1): 143-161.

Exploration planning is a process of decision making under uncertainty. The decisionif and where to explore depends on construction strategies and cost; the selection ofconstruction strategies del)ends on knowledge of geologic conditions which are not knownwith certainty before exploration is performed. The proposed application of decisionanalysis provides a relatively simple approach to the tunnel exploration problem. Theexisting knowledge of geology, the possible construction strategies and their costs, thereliability and the cost of considered exploration methods are used to establish if andwhere exploration is beneficial. The resulting hierarchy of locations where explorationis beneficial and the comparison oC expected values oC exploration Cor different explora­tion methods provides the basis for the selection of a particular site and method. Graphicaland simple numerical means have been created that make the proposed approach a con­venient and Cast tool in the hands of the decision maker.

1. INTRODUCTION

A major problem in geotechnical engineering and particularly intunneling is selection of methods and locations of exploration. The goalof exploration is to reduce uncertainty about geologic conditions, but inplanning for exploration, these uncertainties still exist. To place a valueon exploration one must place a value on "reduction of uncertainty".This can be done only by determining the effect of uncertainty reductionon construction cost.

Exploration planning is a classic problem of decision under uncertainty.Due to uncertainty in the environment, several actions are possible anddecisions must be based on potential cost consequences. The applicationof decision analysis to exploration is thus strongly indicated. Whilestatistical modeling and decision analysis have been applied to certainproblems in mining exploration, progress in applying such techniques to

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geotechnical engineering, and tunneling in particular, has been slow. Themajor reason for this is lack of stochastic models for geotechnical features.Most present approaches are based either on intuition or on simplifications,and many contain inconsistencies which affect decisions (Lindner, 1975).

The present paper introduces a consistent use of decision analysis inexploration for rock tunneling. A brief overview of principles of decisionanalysis is given first, followed by spec:ific application to tunnel exploration.An example concludes the paper.

2. DECISION ANALYSIS

Two topics are covered in this section: (1) principles of decisionanalysis and (2) major elements of decision theory.

2.1. Principles of decision analysis

Fig.1 presents the decision-analysis cycle consisting of deterministic,probabilistic and information phases leading to a decision (which can be adecision to collect more information). Each of the phases consists of steps,also shown in Fig.1. Most steps are described in detail in the application totunnel exploration (Section 3) and comments here are made only to clarifythe overall concept.

In the deterministic phase one enumerates courses of action and outcomesthey may lead to 1-3. Outcomes depend both on decision variables that canbe controlled by the decision maker and state variables representing theuncontrollable environment (4). Variables and outcomes are related to eachother in a model (5). Outcomes are compared with one another through an

,I,1

f~

Qath.,now information

Oet~,mlnl$tlc Phase

I, Define problem Qnd I,m," ofI"Vestloa •• on

2 Alternative courses of action

3, Outcomes of eoch ollornO'l\tC

4. Selec' decision and siote vanablos

5 Relale outcomes and variables

6. Method of compOtlnq rolaf. vovOlues of each outcome

7 Time preference

8 Dominated alternotlyes ellmlnoted

9, Sensl' lVlfy of outcome to vorlObles

Probabilistic Ptlase

I [.pre" uncortolnly In variables bymcons of probabilities

2. P'ababll,'!oc model

3, Es'abllSh ,olal,ve value 01p,obobllll~tl( outcomes

4_ Probabilistic sensitivity analySIs

Information Phase

I Value of perfect information

2 E.. aluate various informationcollection schemes

Fig.I. The decision analysis cycle (after SlaCl von Holstein, 1973).

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objective function usually involving cost or time criteria (6,7). Sensitivitystudies (9) make it possible to identify most important variables.

The probabilistic phase is basically a "revision" of the deterministic phaseintroducing uncertainty of the state variables. Outcomes in the probabilisticphase are in the form of probability distributions (e.g., of cost).

A decision could be made at this point, but generally one wants toestablish whether further information gathering might be beneficial.

This is analyzed in the information phase. If information gathering isexpected to be beneficial, the results of this phase will also be the optimallevel of information gathering.

2.2. Major elements of decision theory*

Probability. Decision analysis is applied because of uncertainty about thestate of nature. Although the true state of nature is of course not random,information about it is limited and any predictions we make are uncertain.Experimentation can thus not be used to directly establish the level ofuncertainty, but uncertainty can be determined by our interpretation ofexperimental data.

Probability has to be viewed as a subjective degree of belief which allowsus to reflect all available information, experience and judgement (for furtherdetails on this approach see, e.g., Baecher, 1972).

Decision criteria. To judge the relative value of alternative outcomes adecision criterion must be chosen that is consistent with the situation.In our case this is a situation of risk (outcome not known with certainty,but the probabilities of alternative outcomes can be evaluated since we usea subjectivist approach). The assessment of risk and thus the selection ofdecision criteria depends on the impacts of the decision that are deemedmost important by the decision maker. The most general measure of suchimpacts are utilities which can express the attitude of the decision makertoward deviations in cost, time, quality, etc. In our case the measure ofimpact of an outcome and thus the decision criterion, is simply the expectedcost.

Preposterior analysis. This is a key element in the information phase. Theconsequences of potential future actions (collecting new information) areassessed before the action is taken. Preposterior analysis makes use ofBayes' theorem to perform the so-called Bayes Updating.

~ Thomas Bayes formulated his theorem in a paper published in 1763(see Bayes, 1958). Benjamin and Cornell (1970) give a simple two-step

*For an introductory level discussion of decision analysis sec Raiffa (1968).

146

derivation of this theorem which can be written as:

P[BjIA] = nP[AIBilP[Bj]

~P[A IBj ] P[Bj ]j;l

Or in words:

[Posterior probability] [Likelihood of the] [priOr ~ [NOrmaliZing]of Bj given the new = new.information, x probability x factorinformation, A A, gIven Bj of Bj

The likelihood function P[A IBj ] is the probability that the particularobservation A (or "data A") would be made in exploration, given that thetrue state of nature was Bj •

In Bayesian updating the prior probabilities of the state variables P[Bj ] arecombined with the likelihood function of the new information P[A IBj ] todetermine posterior - or "updated" - probabilities. These in turn are used toperform a decision analysis by calculating probabilities of the outcomes ofeach exploration alternative occurring. The value of the new informationcan be determined as the difference between expected values of totalconstruction cost and time before and after gathering the new information.A convenient means to conduct the preposterior analysis is the decisiontree whose use will be shown in the application of the decision analysis totunnel exploration (see also Section 4).

With this introduction to decision analysis, it is now possible to presentits application to tunnel exploration.

3. APPLICATION OF DECISION ANALYSIS TO TUNNEL EXPLORATION

The decision analysis phases and steps of Fig.l are now (Table I) related tothe tunnel exploration problem. A few comments will be made regarding thosesteps.

The alternatives "exploration" or "no exploration" (step 2) can be definedfor any stage of a geotechnical study, either the very first exploration oradditional exploration at later stages. The "exploration" alternative includesmany exploration strategies which are combinations of methods, locationsand numbers of explorations. An exploration method is characterized by itscost and its reliability. The reliability is the probability that the explorationresults indicate the true conditions and is represented by the likelihood ~

function in Bayes' theorem.The value of information (step 3) is the difference between expected

construction cost without exploration and expected construction cost withthe particular exploration alternative. The goal of exploration usually is toreduce the expected construction cost. However, exploration involves somecost also. The objective of the decision analysis is thus to minimize

147

TABLE I

Decision analysis for tunnel exploration; steps in the analysis of the exploration decision

Step Phase Exploration for tunnels

1

2

3

4

5

6

1

2

3

4

1

2

DeterministicDefine the decision problem

Identify the alternatives

Outcomes

Decision and state variables

Relationships between variablesand outcomes

Value (of outcomes)

ProbabilisticEncode uncertainty in state variables

Probabilistic model

Choose among distributions

Probabilistic sensitivityanalysis

InformationValue of perfect information

Best information gathering scheme

Determine if exploration is beneficial; if it is,what is the optimal exploration program

ExplorationNo exploration

Value of information; construction costs andexploration sites are intermediate results

Exploration costExploration reliabilityConstruction method costsGeology

Effect of geology and exploration onexpected construclion cost; decision tree

Expected cost

Prior probabilities of geologic states

Effect of geology and exploration onexpected construction cost; decision tree

e.g., mean of cost·time scattergrams

Critical ranges of probabilities, explorationreliability, and constructions costsestablished

Probabilistic sensitivity analysis

Exploration method and configuration(geometry along tunnel)

exploration cost plus expected construction cost, or in other words, toestablish the maximum value that one is willing to pay for exploration.

The decision variables are the exploration methods which can bedescribed by exploration cost and reliability and the construction methodswhich are described by their costs.

The state variables are the geologic conditions affecting tunnel constructionsuch as jointing, water inflow, major defects. In this paper we are usinga simplified description of geologic conditions with the three states good.fair and poor.

The establishment of relations between variables and outcomes is themajor problem that needs to be solved in the decision analysis; it will bediscussed in detail in Section 4. At this point, it may suffice to say that

148

geologic conditions, exploration and construction costs have to berelated to each other. The value of outcomes in the present case is takento be expected cost of construction, in other words a uniattribute linearutility function over monetary cost.

In the probabilistic phase, degree of belief probabilities are assignedto the geologic states. (See Section 4 and Einstein and Vick, 1974). Theprobabilistic model should make use of the deterministic relations andin addition, by introducing uncertainties in the form of subjectiveprobabilities, produce distributions of outcomes instead of single values.One means for relating geology to construction cost in a probabilisticmanner is the Tunnel Cost Model (Moavenzadeh and Markow, 1976).

A sensitivity analysis is performed to evaluate changes in the "best"decisions and changes in predicted costs that result from variation in theinput parameters. In this way, the "sensitivity" of optimal explorationstrategies to probabilities of geologic conditions and estimates ofconstruction sequences can be determined. If the optimal strategiesare insensitive to minor fluctuations in the variables, then we say thatthe decisions are "robust", and we have more confidence in them (Somedetails on the sensitivity analysis will be given in Section 4.)

In the information phase, the expected value of perfect information(EVPI) is calculated which eliminates those sections of the tunnel in whicheven perfect information (e.g., knowing the true geologic conditionsprecisely) would not be cost effective. Then, in the remaining sections,the expected value of alternative imperfect exploration schemes isevaluated.

We have now established the general procedure with which the decisionanalysis can be applied to the tunnel exploration. To make the procedurepractically useable some details of the decision analysis need to bedescribed.

4. SOl\IE DETAILS OF THE DECISION ANALYSIS

Facets of the decision analysis approach that need further discussionspecifically related to tunneling are the relationships among input variablesand outcome, the comparison among outcomes, and the sensitivityanalyses. A tool for relating variables and outcomes and for comparingdifferent outcomes is the decision tree:

4.1. Decision tree

The relationship among variables can be organized in a tree structure(Fig.2) for easier manipulation.

The expected cost of the no exploration case is computed by: (1)multiplying the cost of any of the construction strategies in a particulargeology by the originally estimated subjective probability of thatgeology; (2) summing these "expected costs" for each construction

149

EXPLORATION EXPLORATION CONSTRUCTION TUNNELMETHOD RESULTS METHOD GEOLOGY COST (5)

64051

lOO

...-E--~~ ~OO

1500

tPrior

PrObablilf les

lOO _

20'0 ~52 300

.2

~500

~OO51

.6 52 60053 600

51

t 5253

1Probablliliesof Expioralion

Results

I 51551 Posterior

ProbabilitIes515 52 543

.34 53 589

J60351

30052 60053 1000

SI52S3

Fig.2. Exploration decision tree.

strategy (e.g., 81 = 640); and (3) selecting the construction strategy withminimum expected cost (83 or 82 =620).

The expected cost for the exploration cases are computed similarly by:(1) calculating the posterior probability of each geologic state conditionedon each possible result of exploration (e.g., if the exploration programindicates "fair" geologic conditions, the probabilities of "poor", "fair",and "good" conditions might be 0.09,0.78, and 0.13, respectively);(2) determining the expected cost (Le., probability times cost of any ofthe construction strategies in the particular geology) for each explorationresult (analogous to step 1 above: 300 x 0.09, and so on, and thesesummed); (3) selecting the minimum expected cost construction strategyfor each exploration result; (4) finally, weighing each minimum coststrategy by the probability of the corresponding exploration result andsumming (515 x 0.34 + 603 x 0.46 + 680 x 0.20 = 588). Adding the costof the particular type of exploration to this sum yields the expectedtotal cost of that exploration strategy (588 plus cost of exploration).

150

The expected value of exploration (or expected value of sampleinformation EVSI) is the difference between the expected cost of the beststrategy (step 4) and the expected cost of the "no-exploration" case(Le., 32 minus cost of exploration).

It should be kept in mind that the analysis expressing these relations hasto be performed for each location of possible exploration.

4.2. Sensitivity analysis

In the sensitivity analysis the above described relation (tree) betweenvariables and outcome is used. All the variables are varied and the EVSI foreach combination of variables is determined. The range of variation of theconstruction costs is usually estimated in a preliminary analysis (see alsoSections 5.1 and 5.2), the probabilities of geologic conditions vary betweencompletely reliable to completely ambiguous.

The exploration reliability is expressed in the form of a reliability matrix,a matrix of likelihood functions as shown in Table II. The likelihoods orreliabilities are the result of subjective assessment of the performance of anexploration method in a certain geologic condition (e.g., the methoddescribed in Table II has an 0.5 ... reliability showing "poor" conditionsif the real conditions are "poor"). In assessing the reliability one has tokeep in mind that we consider entire segments; the reliability of a singleboring may thus decrease as the segment length increases. As mentionedbefore, the reliabilities vary between completely reliable (diagonalprobabilities = 1, rest = 0) and completely ambiguous (all probabilitiesequal1/n, where n = number of geologic states).

The construction costs of a certain construction strategy in certaingeologic conditions can also be represented in matrix form (Table III).Assuming that the strategies are ideal for the geologies along the diagonal,one can form a so-called penalty matrix with O-penalties along thediagonal (upper part of Table IV). If a strategy is the best for all geologicconditions, a row of O-penalties would occur; for the numerical example

TABLE II

Reliability matrix·

Exploration resultEG E F Ep

Geologic states

G 0.6

F 0.3

p I 0.2

0.2

0.6

0.3

0.2

0.1

0.5

• The numbers in the reliability matrix are the P[ true geologic statelexploration result].For example, the probability of poor geology at a particular site given that the explorationshowed poor geology is (circled): P[PIEpl " 0.5.

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TABLE III

Construction cost matrix

GeologyG F P G F P

81 CIG ClI: ClP 81 200 500 1500

Construction strategy 82 C2G C2F C2P e.g. 82 300 600 1000

83 C3G C3F C3P 83 500 600 800

in Table IV, strategy 81 is the best for "good" and "fair" geology,stratef,'Y 83 for "poor" geology.

The contour diagram of Fig.3 is a convenient graphical means forcarrying out the calculations of the decision tree when there are onlythree geologic states. The prior probabilities of each state (e.g., for "good","fair", "poor") are plotted along the three axes, and the expected valueof exploration is plotted as contours over the triangular grid. One such plotdescribes thus the expected value of exploration for a specified penaltymatrix and reliability matrix.

The value of perfect information (as introduced in Section 4.1) for agiven set of prior probabilities equals the minimum expected penalty costover possible construction strategies minus the penalty associated withperfect knowledge of the geologic state, which is zero. Calling this the"expected value of perfect information" (EVPI),

where PG,F,p = prior probabilities (of good (G), fair (F), poor (P)conditions); PSG = penalty cost of strategy S, in good (G) geologicconditions.

TABLE IV

Penalty matrix

G F P G F P

81 ClG-ClG CIF-C 2F CIP-C 3P 0 Pu, PIP

82 C2G-CIG C2~·-C2F C2P-C 3P ~ P2G 0 P2P

83 CaG-ClG C3F-C 2F C3P-C 3P PaG P3~' 0

e.g., from Table III: G F P

81 0 0 700

82 100 100 200

83 300 100 0

152

Good10

10 2 (90),.~

20" (80)

\30"\ (70)

40\ (60l

50 50 \40 -t

~

30 \~~'"

20 •

/10

10 10Fo" 10 20 30 40 50 (GO) (70) (80l (90l Poet

(901 (80) (70) (60) 50 40 30 20 10

Penalty Malti.,

G F P

51

52

53

P'G , 0 p,. '100 p'p '00

Pz. '100 PZF' : 0 P2 • '\00

P)Q '100 p). '100 p]P : 0

Fig.3. Contour diagram and corresponding penalty matrix (for perfect information, i.e.,100% reliability of exploration).

Fig.3 is the contour diagram for the penalty matrix shown in the lowerpart of this figure.

The expected value of sample information (EVSl) is less than the EVPIby some factor which depends on the reliability of the explorationtechnique. For 100% reliable technique, this factor is 1.0, and diminishesto zero for a uniform reliability matrix (Le., one for which all R jj = lIn,where n is the number of geologic states). The EVS/ can be calculated by:

(1) Determining the updated probabilities of the geological stateconditioned on each possible exploration outcome (Bayes' theorem).

(2) Determining the minimum expected penalty for each updatedprobability distribution.

(3) Averaging over the probability of exploration yielding each possibleresult (Le., taking the weighted sum).

This procedure can be performed with a simple computer program andthe contour plot can be correspondingly restructured. FigA shows thecontour plot for the same penalty matrix as in Fig.3, but for a 90%reliability of exploration as expressed by the reliability matrix in FigA.(It should be noted that 90% reliability refers to the diagonal, Le., anexploration result "good" has 90% probability of representing the true

153

Good10

ConlOUt 40 is Ih. Conlour Corresponding 10 Explcwation Cosl.5eQmtnls which Piol in th. U'lshod.d Zone are therefor.Polenliolly Voluobl. Explcrotion SitlS.

R.I=ility MOlflx:

Ep

G

F

P

09 01 0

0 0.9 01

0 01 09

Fig.4. Contour diagram for reduced reliability of exploration and with exploration costcontour (same penalty matrix as for Fig.3, but 90% reliability as expressed by reliabilitymatrix).

geologic condition "good". Thus reliability matrices with differentnon-diagonal members can also represent 90% reliability of exploration.)

One of the major uses of the sensitivity analyses and the contour plotsis in locating the sites where exploration is beneficial: the explorationvalue contour corresponding to the exploration cost is drawn as in thecontour diagram in FigA. The geologic probabilities of the particulartunnel segment can then be plotted. If they fall inside the explorationcost contour, exploration is beneficial.

5. DESCRIPTION 01<' THE DECISION ANALYSIS PROCEDURE FOR TUNNELEXPLORATION

Up to this point, the theoretical development of the exploration decisionproblem has been described as well as important tools which will be usedin the analysis. Based on this step by step formulation of the problem,a procedure for analysis was developed and will now be described.

154

5.1. The problem of cyclic interdependence of construction and explorationand its solution

One of the advantages of our approach is the segmentation of the tunnelgeology. Geologic conditions are assumed to be constant over a certainlength of tunnel; i.e., the geologic variables and their probabilities ofoccurrence are constant (which of the geologic conditions actually willoccur is however uncertain as expressed by the probability). Thesegmentation corresponds well to construction procedures that are modifiedin discrete steps; it has the additional advantage that segments in which theexploration would be beneficial can be identified and ranked. The discrete(segmented) character of the construction and exploration procedures has,however, some disadvantages that need to be overcome.

The construction cost within any segment cannot be assumed to besimply the cost of driving the tunnel through the type of rock which existsin that segment. In general, the construction cost is dependent not onlyon the geologic conditions in a given segment, but also on the conditionsin previous, and possibly later, sections of the tunnel. (e.g., the switchingfrom full face excavation to heading and benching and back to fullexcavation in crossing a shear zone may result in a greater cost than ifheading and benching had been used throughout. This is due to costsrelated to switching which may more than offset the greater cost ofheading and benching). Also, since the improvement of knowledge ofgeologic conditions by exploration will influence one's choice ofconstruction method, it is possible that exploration in one segment canaffect the entire tunnel. Therefore, it is not sufficient to analyze the valueof exploration on a simple segment by segment basis; it must be done on anentire tunnel basis.

The decision if and where exploration is necessary depends on theplanned construction strategy and the projected construction cost. Theprojected construction cost depends on the knowledge of geologicconditions which means that construction cost in tum depends onexploration. The exploration decision problem is thus cyclically intertwinedwith the tunnel construction cost and construction strategy. In order tobreak into the cycle, an approximation had to be made:

Each segment is analyzed individually, but is always considered as a partof an entire tunnel profile: within the segment being analyzed, thegeologic state successively takes on each possible value, while the remainderof the tunnel takes on the most likely value. E.g., in analyzing segment 3 inFig.5, the profiles of interest are (symbolically): (1) CBACA; (2) CBBCA;(3) CBCCA. The different geologies within the analyzed segment producean effect on total construction cost from which in tum the value ofexploration in that segment can be determined (with the tree or the contourplot as explained in Section 4).

3 4 5

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Prior Geoloole Probabilities of Condition;

A 0 .2 .6 0 7

B .3 .7 3 .1 2

C .7 I ,I 9 I

Fig.5. Segmented tunnel with prior probabilities of geologic conditions A. n, C.

5.2. Construction cost

After establishing the possible profiles by in tum applying the proceduredescribed above to each segment, the cost of construction for each profilewill be estimated. This can be done easily if records of past performance fortypical geologic conditions and construction strategies are available, or byusing the Tunnel Cost Model (see Moavenzadeh et a1., 1976) or any othercost estimation tool. Next a construction cost matrix is created as shownin Table III, but now relating construction cost to entire geologic profilesinstead of a single geological state. The construction cost matrix is thentransformed into a penalty matrix (analogous to Table IV).

5.3. Exploration value

Once the penalty matrix is formed, the contour diagram is drawn asdescribed in Section 4.2. The exploration-cost contour is also drawn todefine the explore and no-explore regions. By plotting the priorprobabilities for the segment being analyzed, a decision can be made withrespect to exploration or no-exploration, and for those segments in whichexploration is beneficial, an expected value of exploration can bedetermined. For example, if a segment had construction and explorationcosts as characterized by the plot in Figo4, and prior probabilities of0.3-004-0.3 (good-fair-poor), it would have an expected value ofexploration of 50. This value can be used to establish a hierarchy forexploration in individual segments.

This procedure is usually satisfactory since once the decision is made toexplore in a particular segment, the exploration will be performed and theresults used to update our knowledge on geologic states. Only then will adecision be made where and how to explore further. There are instances,however, where the decision will involve a combination of segments that areto be explored and even differerit exploration methods in different segments.The analysis described in this paper does not deal with these so-calledexploration strategies.

156

5.4 Other aspects of the decision analysis procedure for tunnel exploration

In this section, we shall shortly discuss aspects of the decision analysisprocedure that have not been commented upon either because they areneglected in the simplified approach presented here or because the clarityof the description would have been reduced.

The variables that we consider (exploration cost, exploration reliability,construction cost and geologic conditions) are only four of the possible sixvariables. The other two are length and location of the considered segments.The possible importance of these two variables has been indirectly mentionedin the discussion on cyclic interdependence of construction and exploration.The effect of these variables is not fully understood, but seems to depend onthe cost differences between different methods and on the cost and numberof construction method changes.

As mentioned before, the analysis of exploration strategies - combinationsof exploration in different segments and with different methods has not beentreated here because of the frequent step-by-step character of explorationplanning which makes consideration of exploration strategies unnecessary.

A last clarifying point concerns the number of geologic states which we,in this paper, have limited to three. This is the maximum that can be handledgraphically. By using numerical methods, any number of geologic states thatare considered important can be handled. It should be noted, however, thatin the exploration planning stage, simplifications and thus three or even onlytwo geologic states will usually be satisfactory.

5.5 Summary of the decision analysis procedure

The decision analysis procedure can thus be summarized as follows:(1) Determination of possible geologic states, segmentation and assignment

of (subjective) prior probabilities to the geologic states.(2) Selection of possible construction strategies and of their costs

(including transition costs) in the various geologic states.(3) Determination of construction costs f'Jr all possible geologic profiles

based on (1) and (2) above.(4) Analysis for each segment consisting of:(a) Variation of geologic states in the considered segment, with the

remainder of the geologic profiles having the most likely geologic states.This results in a set of geologic profiles.

(b) Determination of construction cost and penalty matrices for the set ofgeologic profiles determined in (a).

(c) Determination of the values of exploration by drawing the contourplots as described in section 4.2.

(d) Drawing of the exploration cost contour and plotting of priorprobabilities. This yields an exploration value, which indicates if explorationis beneficial and provides a means for ranking different segments with regardto benefit of exploration.

I ,

157

(5) Evaluation of various exploration methods and establishment ofhierarchies for these exploration methods.

This procedure is now illustrated in an example.

6. APPLICATION OF DECISION ANALYSIS

In order to illustrate the procedure which has been described in Section 5,a 10,OOO-ft. example tunnel, whose characteristics are presented in Fig.6, willbe analyzed (Sections 6.1-6.5) to determine the location of potentiallyvaluable exploration sites.

6.1. Geologic states, segmentation and prior probabilities

Based on preliminary exploration data, the two geologic conditions"good" and "poor" have been determined. Also, the preliminary explorationled to a "segmentation" of the tunnel, and to prior probabili ties indicatingwhich of the two geologic conditions might exist in each segment.

6.2. Selection of construction strategies

Three construction strategies were selected for evaluation:(a) Tunnel boring machine (TBM) throughout the tunnel.(b) Drill and blast (D-B) throughout the tunnel.(c) A combination of TBM and D-B in which D-B is used in poor rock and

TBM in good rock with the following qualifications: (1) if D-B is used insegments 1 and 2, the tunnel should be completed with D-B; (2) a switchfrom D-B to TBM should be made only between segments 1 and 2, i.e., ifpoor rock occurs in segment 1 and good rock in segment 2.

Costs are based on the assumption that the original construction planscalled for TBM. Any switch of method will involve a transition cost.

The estimated construction costs for TBM and D-B as well as the methodchange or transition costs are given in Table V.

6.3. Profile construction costs

The total construction cost for each of the possible profiles is determinedusing the costs listed below. The geology of each profile, which is represented

0 :3 4

SEGMENT LENGTH 4000 1000 2000 3000(tt)

P"or Probob.1o lies

rl .8

~6

I

:3

I.2 4 7

GEOLOGY

Poor

Fig.5. Example tu nnel for exploration evaluation.

158

TABLE V

Estimated unit construction costs and transition costs

Method Estimated unit construction costs ($/ft.)Geology

J

TBMD-B

TBM to D·Bnear portal (segment 1)in other segments2nd method change

D·B to TBM

good

500800

Transition costs ($)

720,000750,000200,000

20,000

poor

30001900

by a certain percentage (probability) of good and poor conditions, and itsassociated cost are plotted in Fig.7. From Fig.7, it can be seen that eachstrategy has a range of geologic conditions (profiles) over which it is the bestconstruction method.

6.4. Analysis Of each segment

Each segment is analyzed in decision tree fashion. In this example, we usethe contour diagram method. The calculations will be shown for segment 1only.

(a) Geologic profiles obtained by varying the geology in segment 1 andkeeping the most likely geologies in the other segments: (1) GPGP; (2) PPGP.

(b) Construction cost and penalty matrices for each set of profiles (TablesVI and VII).

(c) Determination of the expected value of exploration. The reliabilitymatrix for the exploration method is shown in Table VIII.

With the prior probabilities from Fig.6 and the construction penaltieslisted in Table VII, it is now possible to compute the expected penalties forthe no exploration case:

Comb 2:D-B:

(0.8)(0) + (0.2)(0.72) = 0.144(0.8)(0.45) + (0.2)(0) = 0.36

Comb 2 has the minimum expected penalty and will thus be the selectedstrategy.

For the determination of the expected penalties of the exploration case,one computes first the posterior probabilities using the prior probabilitiesand the exploration reliability listed in Table VIlI:

P'[GIG] =0.96P'[GIP] =0.48

P'[PIG] = 0.04P'[PIP] = 0.52

159

30r--TBM

: -- O-B

I· _.- COMB 2

~ 25 .

= Ioo

"o-u:>

I t I

·---=7"""5.......• .~25'- 5-5I t

25-75 0-1Poor

% Tun-no I Good· POOl

Fig.7. Construction costs for different geologic proriles and different constructionstrategies.

The computation of the posterior probabilities also yields the probabilitiesof exploration (the probability that the exploration result is "good" or"poor"):

P[G) = 0.67 P[P) = 0.33

The expected penalties are then computed for the possible explorationresults:

Exploration "good"Comb 2: (0.96)(0) + (0.04)(0.72) = 0.029D-H: (0.96)(0.45) + (0.04)(0) = 0.431

Exploration "poor"4 Comb 2: (0.48)(0) + (0.52)(0.72) = 0.374

D-H: (0.48)(0.45) + (0.52)(0) =0.216

Combination 2 yields the minimum expected penalty if the exploration resultis good, D-H if the exploration result is poor. The minimum expectedpenalties are now combined by taking their weighted sum (weighted by theprobability of the exploration result): 0.67 x 0.029 + 0.33 x 0.216 = 0.090.The expected value of sample information is the difference between the

160

TABLg VI

Construction cost matrix(millions of $)

J

Geologic profile

Strategy 1 2

TBI\I 15.0 25.0

D·B 12.4 16.8

COMB 2 11.95 17.52

TABLE VII

Penalty matrix(millions of $)

Geologic profile

TABL!'~ VIII

Reliability matrix

Exploration result

G PStrategy

COMB 2

I).B

1

o0.45

2

0.72

o

Geologicstate

G

P

0.8

0.15

0.2

0.85

expected costs (penalties) of the no exploration and the exploration case,the latter including the cost of performing the exploration:

EVS/ =$144,000 - ($90,000 + cost of exploration)= 54,000 - cost of exploration

Exploration is thus beneficial in segment 1 if the cost of exploration is lessthan $54,000.

6.5. All segments will have to be evaluated in the same manner to establish ahierarchy

7. CONCLUSIONS

Exploration for tunneling should lead to a reduction of construction costs.Since exploration itself involves a cost, the goal of exploration planning is tominimize the total cost of construction plus exploration. However, thedecision if and where to explore depends on construction strategies and costwhose selection in turn depends on our knowledge of the geologic conditions. ~

The exploration planning problem is thus a problem of decision underuncertainty with a cyclic nature as a complicating feature.

The proposed application of decision analysis provides a relatively simpleapproach to the tunnel exploration problem. The existing knowledge ofgeology, the possible construction strategies and their costs, the reliability

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and the cost of the considered exploration methods are used to establish ifand where exploration is beneficial. The resulting hierarchy of locationswhere exploration is beneficial and the comparison of expected values ofexploration for different exploration methods provides the basis for theselection of a particular site and method. Graphical and simple numericalmeans have been created that make the proposed approach a convenient andfast tool in the hands of the decision maker. The approach has beenpurposely kept simple and is based on several approximations, a fact whichhas to be considered if the user wants to develop it further.

EPILOGUE

The application of decision analysis to geotechnical exploration andparticularly to tunnel exploration is new and untested except for a fewexample cases. It is just the purpose of this paper and particularly of thesimplified approach presented in it to induce the practitioner to use decisionanalysis in tunnel exploration.

The soundness and applicability of decision analysis have been provenextensively in other areas where decisions under uncertainty have to betaken, notably in business administration. Since the managers usually decideon the expenditures for exploration and since they base their decisions to alarge extent on decision analysis, it seems opportune that engineers andgeologists start talking in the same language.

ACKNOWLEDGEMENTS

The research on which this paper is based has been conducted inconnection with the development of the Tunnel Cost Model (TCM)sponsored by NSF-RANN. Prof. F. Moavenzadeh is the principalinvestigator of the TCM project.

REFERENCES

Baecher, G.B., 1972. Site Exploration: A Probabilistic Approach. Ph.D. Thesis,Massachusetts Institute of TechnoloftY, Cambridge, Mass.• 515 pp.

Bayes, T. 1958. Essay toward solving a problem in the doctrine of Chaines. Biometrika,45: 293-315 (reproduction).

Benjamin. J.R. and Cornell, C.A.• 1970. Probability. Statistics and Decision for CivilEngineers. McGraw-Hili, New York, N.Y., 684 pp.

Einstein, H.H. and Vick, S.G., 1974. Geoloftical model for a tunnel cost model. Proe.Rapid Excavation and Tunneling Conf., 2nd,lI: 1701-1720.

Lindner, E.N., 1975. Exploration: Its Evaluation in Hard Rock Tunneling. j\IS Thesis,Massachusetts Institute of Technology, Cambridge, Mass., 210 pp.

Moavenzadeh, F. and Markow, M.J., 1976. Simulation model for tunnel constructioncosts. J. Contr. Div., Am. Soc. Civ. Eng., COl: 51-66.

Raiffa, H., 1968. Decision Analysis. Addison-Wesley. Reading, Mass., 309 pp.StaiB von Holstein, C.S., 1973. In: Readings in Decision Analysis. Stanford Res. Inst.,

Stanford. Calif., pp.97-216.