SynopsisThe Mathews method of predicting open-stope stabilitywas first proposed in 1980 The initial stability graphwas based on a limited number of case studies prim-arily from deep North American steeply dipping openstopes in strong rocks of medium to good quality Sincethen new data have been added by various practitionersto modify update and validate the method and supportits use as a preliminary open-stope design tool

The original Mathews method has been extendedwith use of a significantly increased database of miningcase histories The format of the Mathews stabilitygraph has been changed to reflect the broader range ofstope geometries and rock mass conditions now cap-tured within the database The extended database nowcontains in excess of 400 case histories

Logistic regression has been performed on this largerdatabase to delineate and optimize placement of thestability zones statistically Isoprobability contourshave been generated for all stability outcomes Theadvantage of the logistic regression lies in its ability tominimize the uncertainties reflected in the methodthrough the use of maximum likelihood estimates Therisks associated with use of the Mathews method cannow be quantified and the true statistical significanceof the stability zones understood

Mathews stability graph method

The Mathews method1 is based on a stability graph relatingtwo calculated factors the Mathews stability number Nwhich represents the ability of the rock mass to stand upunder a given stress condition and the shape factor S orhydraulic radius which accounts for the geometry of the sur-face

The principal concept behind the stability graph is that thesize of an excavation surface can be related to the rock masscompetency to give an indication of stability or instabilityThe stability graph presents numerous excavation surfacesthat have a specified range of stabilities The stability numberforms the y-axis of the stability graph and is a measure of rockmass quality around the excavation several adjustmentsbeing applied to take into consideration induced stresses andexcavation orientation

Stability graphs deal with the individual surfaces of anexcavation rather than entire excavations For a typical rec-tangular excavation five stope surfaces are considered for thestability graphmdashfour sidewalls and the crown (or back) Onceplotted the stability data can be zoned and boundary lines canbe drawn on to the graph to define the stability zones

The initial stability zones and graph devised by Mathews

were based on 50 case histories1 The zones of stability weredefined from the scatter of the real mining data and thesezones were then used to predict the stability of planned exca-vations The original Mathews stability graph was dividedinto stable potentially unstable and potential caving zonesaccording to the scatter of the stability data (Fig 1) Thethree stability zones were separated by transitional zones toreflect the transition between stability classes and uncertaintyin the boundaries

The Mathews method utilizes a modified form of theNorwegian Geotechnical Institutersquos (NGI) engineering classi-fication2 the Q system to characterize rock mass qualityThe modified Q value Q cent is calculated from the results ofstructural mapping or geotechnical core logging of the rockmass according to the Q classification system but with theassumption that the joint water reduction parameter and thestress reduction factor are both equal to one (equation 1)The quality of the rock mass is defined by

(1)

where RQD is the rock quality designation index developedby Deere in 1964 and is based on a modified core recoverypercentage3 Jn is joint set number Jr is joint roughness andJa is joint alteration

The Mathews stability number is determined by adjustingthe Q cent value for induced stresses discontinuity orientationand the orientation of the excavation surface (equation 2)The stability number is defined as

N = Q cent acute A acute B acute C (2)

cent = acuteQ

J

J

JRQD

n

r

a

A27

Extending the Mathews stability graph for open-stope design

C Mawdesley R Trueman and W J Whiten

Manuscript first received by the Institution of Mining andMetallurgy on 7 February 2000 revised manuscript received on 22August 2000 Paper published in Trans Instn Min Metall (Sect AMin technol) 110 JanuaryndashApril 2001 copy The Institution of Miningand Metallurgy 2001

Fig 1 Three stability zones of original Mathews stability graphAfter Stewart and Forsyth8

where Q cent is NGI modified Q value A is rock stress factor B isjoint orientation adjustment factor and C is surface orienta-tion factor

The rock stress factor is determined from the ratio of theintact rock strength (unconfined compressive strength) to theinduced compressive stress at the centre-line of the stopesurface The induced stresses can be determined by use of atwo-dimensional stressndashdisplacement analysis package or esti-mated from published two-dimensional stress distributions Agraph relating the strength to stress ratio and rock stress fac-tor was developed by Mathews et al1 (Fig 2) In the originalMathews graph1 the joint orientation adjustment factor is ameasure of the relative difference in dip between the stopesurface and the critical joint set The surface orientation fac-tor considers the inclination of the excavation surface and itsinfluence on stability (Fig 2)

The geometry of the excavation is considered by calculat-ing the shape factor or hydraulic radius of the surface Theshape factor of an excavation surface is defined as the area ofthe stope surface divided by the length of its perimeter (inmetres)

Historical overview

The Mathews stability graph method for open-stope designwas first proposed for mining at depths below 1000 m1

A number of authors have since collected new data from avariety of mining depths and rock mass conditions to extendthe method and test its validity4ndash9 The modifications to anddevelopments of the Mathews method since its inceptionrelate largely to changes in the position and number of thestability zones represented on the stability graph with theaddition of more data and changes to the formulation of theMathews factors9

The original Mathews graph contained three distinct zonesseparated by transitions8 In the Potvin4 modified stabilitygraph the number of zones was reduced to a stable and acaved zone separated by a transition (Fig 3) The choice ofthe word lsquocavedrsquo by Potvin to represent what is essentially anunstable zone was challenged by Stewart and Forsyth8 whonoted that the term has a fixed meaning in mining termino-logy from which Potvinrsquos modified graph appears to depart8

Potvin4 collected additional case histories and modified the

A28

Fig 2 Adjustment factors for determination of Mathews stability number After Mathews et al1

Fig 3 Potvinrsquos modified Mathews stability graph based on 175case histories After Potvin4

way in which the joint orientation factor and the gravityadjustment factor were calculated This variant of the originalMathews method is referred to as Potvinrsquos modified stabilitygraph method Potvin also extended the method to considersupported excavations a development that was taken furtherby Nickson5 Nickson5 and Hadjigeorgiou and co-workers6

added further supported and unsupported case histories tothe stability database using Potvinrsquos modified method andwere the first to consider statistical zone definition

Stewart and Forsyth8 updated the Mathews stability graphin 1995 and proposed four stability zonesmdashpotentially stablepotentially unstable potential major failure and potential cav-ing zonesmdashseparated by three transitions (Fig 4) The zoneof potential caving approximated by Stewart and Forsyth wasbased on Laubscherrsquos caving stability graph10 No data wereavailable to validate the position of the caving line on theupdated Mathews graph which is meant to represent the truecaving situation as opposed to Potvinrsquos caved zone4 As partof an international study of block and panel caving theauthors have back-analysed a number of caving case historiesand delineated a caving zone on the Mathews graph thatreflects the true caving case defined as continuous cave pro-pagation At this stage the results remain confidential tosponsors of the project

Significant developments and modifications have beenmade to the stability graph method extending its use intocable-supported excavations4ndash7 and more recently to takeaccount of fault-driven instability11 Potvinrsquos modifiedapproach as applied to unsupported stopes was comparedwith the original Mathews method for 180 case histories fromMt Charlotte9 The variability of the results the apparentpredictive accuracy and the accuracy of the defined stabilityzones (in terms of misclassification) of the two methods wereinvestigated From these case studies Trueman et al9 con-cluded that Potvinrsquos modifications to the method ofcalculating the stability number resulted in no appreciabledifference in the predictive capability of the technique forunsupported excavationsmdasha conclusion that was in generalagreement with that of Stewart and Forsyth8 The originalmethod of determining the Mathews stability number and the

adjustment factors as proposed by Mathews et al1 hastherefore been followed here

Historically the stability zones on the Mathews graph havebeen defined by eye14 As new data were added the natureand placement of the stability zones were modified accord-ingly Nickson5 was the first to attempt to determine theposition of the boundaries statistically He applied a discrimi-nant analysis to the three-dimensional multivariate stabilitydatabase and utilized Mahalanobisrsquo distance to separate thedata into two groups Nickson derived a linear separationbetween stable and caved unsupported histories using a loga-rithmic transformation Again it should be noted that theterm lsquocavedrsquo does not represent true caving No unstablecases were considered in the analysis and separation linesbetween stable and unstable or unstable and caving zoneswere not determined Nickson compared his statisticallydetermined stablendashcaved boundary with Potvinrsquos proposedtransition zone Nickson recommended on the basis of hisresults that Potvinrsquos transition zone be used for the design ofunsupported stope surfaces Hadjigeorgiou and co-workers6

collected further stability data repeated the discriminantanalysis and obtained similar results Again no changes toPotvinrsquos transition zone were proposed The transition casehistories were not included in either statistical analysis andthe calculated boundary was only between the stable andcaved case histories

The position of the stability zones on the stability graph iscritical to the reliability of the method and a technique wassought to optimize their positioning Mathematical calcula-tion of the width and position of the stability zones is animportant step towards improving the reliability of theMathews method Removal of the subjectivity involved inzone definition is important in maximizing the value of themethod as a design tool

Extension of the Mathews method

Back-analyses of 180 open-stope surfaces at Mt Charlotte bythe Mathews method confirmed the validity of using the tech-nique as a predictive tool9 These data when combined withother Australian case histories in 1998 resulted in a com-bined stability database for more than 400 stope surfacesThe additional case histories included much larger stopesthan had previously been documented for the method andthey extend the Mathews graph to a hydraulic radius of 55 mcompared with the previous maximum hydraulic radius of23 m8 The expanded database covers a broader spectrum ofrock mass and in-situ stress conditions with stability numbersin the range from 0005 to 700 The background to the MtCharlotte data has already been given by Trueman et al9 thepurpose of the present contribution is to communicate a newapproach for the statistical treatment of stability data and forthe calculation of stability zones and isoprobability contourson stability graphs

A logistic regression analysis12 was carried out on theextended database to delineate the zones of stability statisti-cally and to determine isoprobability contours for stableminor failure and major failure scenarios All the case histo-ries adhere to the original Mathews method of determiningthe adjustment factors By the use of logistic regression theuncertainties in the application of the Mathews method canbe quantified over a larger range of stope geometries and rockmass conditions than was previously possible The extendedMathews stability graph (Fig 5) contains the new data andstatistically determined stability zones A logndashlog graph hasbeen used rather than the traditional logndashlinear plot of theMathews method as it was found to give a clearer picture ofthe zoning The logndashlog plot has linear stability boundaries

A29

Fig 4 Mathews stability graph as modified by Stewart andForsyth8

and the shape factor range of the case histories is separatedbetter on a log shape factor x-axis The new logndashlog stabilitygraph enables individual cases to be clearly distinguished aswell as presents the increased range of shape factors capturedwithin the stability database

Logistic regression analysisLogistic regression12 provides a method of calculating theprobability that a particular stope of a given dimension andgeometry will be stable It also shows how the probability ofstability is altered by a change in rock mass quality or inducedstress The regression analysis allows statistical definition ofthe different zones of stability by optimizing the number ofdata points that report to the correct zones and minimizingmisclassification

Logistic regression analyses are typically used to analysetrue or false values of the dependent variable but can easilybe extended to include estimates of the proportion truefalseA logistic regression analysis was undertaken because prob-lems with ordered outcomes are not easily modelled bytraditional regression techniques and ordinary linear regres-sion does not consider the interval nature of the dependentvariable13

The stability data were specified in terms of three variablesshape factor (or hydraulic radius) Mathews stability numberand stability (whether it was stable a failure or a majorfailure) Logistic regression of these values fits a maximumlikelihood model calculates a separation line of best fitand produces a predicted stability value The maximumlikelihood method is used to maximize the value of the log-likelihood function by estimating the value of the parametersDetermination of the maximum likelihood estimates for thelogistic regression model is an iterative process12 The stabil-ity data do not separate perfectly into each stability zone and

hence the actual values of the dependent variable (stability)differ from the logit values The logit probability values areused to optimize the boundaries of the three stability zones onthe stability graph The predicted stability values can be com-pared with the original values and the nature of the residualscan be analysed to delineate the stability zones and to con-sider misclassification

One of the benefits of logistic regression is that it providesthe predicted probabilities of event occurrence based on thelogit model By using the predicted probabilities of eventoccurrence calculated for each stability class isoprobabilitycontours of risk can be generated for the stability dataIsoprobability contours are an important component of theMathews stability graph method as they represent the esti-mated probability of event occurrence on the stability graph

Logit modelSeveral models can be applied to the data when logisticregression is undertaken For the stability data a modifiedbinary logit model was used in which the logit function is atype of probability distribution13 For the interpretation ofprobabilities the logit model is more appropriate than the tra-ditional linear model given by ordinary least squares121314

Logit analysis produces probabilities after a nonlinear trans-formation for a categorical response for the stability data twomain discrete categories can be identified (stable and majorfailure)14 and in-between probability values can be inter-preted as the failure zone The logit probability value is thenatural logarithm of the odds where the odds indicate therelative probability of falling into one or two categories onsome variable of interest14 The logit model of the stabilitydata was produced in MATLAB (version 51) using a routinedeveloped by Holtsberg15 The regression procedure enabledthe optimization of the stability zone boundaries and the gen-

A30

Fig 5 Extended Mathewsrsquo stability graph based on logistic regression

eration of isoprobability contours by use of the predicted logitprobability values determined from the logit model

A three-level logit model was used in the investigation ofthe Mathews stability data to reflect the three stability classesLogit models are typically applied to yesno outcomes (givinga binary model) but for the present analysis a third inter-mediate category was used The three-level logit modelproduced two separation lines between the three stabilityclasses The fact that the two lines were parallel is a functionof the three-level model The use of a three-level logit modelto investigate the stability data was verified by running dualbinary logit models to determine the stablendashfailure and thefailurendashmajor failure boundaries separately These boundarieswere found to be very close to parallel

For a binary logit model the dependent variable is assigneda value of either zero or one With three separate stability cat-egories an intermediate value needed to be determinedA value of 05 was used for the failure region which was con-sidered intermediate between stable and major failure Thelogit model was run for different values of the dependent vari-able categories and the distribution of the logit values soobtained was examined closely to consider overlap and spac-ing between the stability classes The values assigned to eachstability class as a result are stable 06ndash1 failure 04ndash06and major failure 0ndash04

Initial values of 1 05 and 0 were used to define the threestability categories and for each point a logit probability valuewas calculated The boundaries between the stability classescan then be determined at an appropriate probability forgiven site conditions or design requirements Alternativelygiven the equal probability lines actual counts of the numberof points in each class can be made giving a distribution ofpoints versus the probability value A graph of these results isa useful way of determining the stability zone boundaries

The inclination position and probabilities of the twoboundaries determined from the two separate binary logitmodels were compared with the two boundaries determinedfrom the single three-level logit model The observed differ-ences in the position and inclination of the boundaries wereminor and the three-level model was adopted

The general form of the logit function can be seen inequation 3 The logit model is similar to the traditional linearregression model or the general linear model for ANOVAexcept that the response is the log odds rather than the metricdependent variable14 In logit modelling the conditional log

odds of the dependent variable are expressed as a linear func-tion of a set of explanatory variables (equations 3 and 4)14 Inthe logit model of the stability data the probability of stabilityis expressed as a linear function of the shape factor theMathews stability number and a constant (equation 5)

To fit the logit model the unknown parameters a b 1 b 2 hellipb k (in equation 3) are estimated by the maximum likelihoodmethod With the use of the estimated values for a b 1 and b 2(equation 5) the estimated probability or predicted risk canbe determined (equation 4) The general form of the logitfunction is

z = a + b 1X1 + b 2X2 + b 3X3 + hellip + b kXk= a + aring b kXk (3)

(4)

where z is predicted log odds value a is a constant b 12 arenumerical coefficients and ƒ(z) is predicted logit probabilityvalue For the case of open-stope stability equation 3becomes

z = a + b 1 lnS + b 2 lnN (5)

The maximum likelihood statistic is used to define the incli-nation of the stability zone boundaries By examination of thecumulative and probability density functions of the logit valuesfor each class the position of the lines delineating each of thestability zones can be optimized according to the needs of par-ticular applications The criteria used to place the zones can bespecified and the resulting probabilities for each stability classwill then be known By using the cumulative density functionsfor each class as determined in Fig 7 the position of the sta-bility zones can be optimized for certain criteriamdashfor exampleminimized misclassification (either numerically or propor-tionally) or similarly a more conservative limit of zeromisclassification This method is a significant improvement onearlier methods of delineating stability zones in that the prob-able stability outcomes for a boundary at any position of thestability graph can be quantified statistically The probabilitydensity functions provide data on the degree of misclassifica-tion of the stability data which is an important consideration

f ze z

( ) =+( )-

1

1

A31

Fig 6 Cumulative frequency graph of logit values for stable failure and major failure points

when evaluating the confidence of a stability estimate and thepotential for misclassification

Delineation of stability zonesA cumulative distribution curve can be plotted for the logitprobability values of the data for each stability class (Fig 6)The inverse cumulative distribution curve of each class is alsoplotted The point of intersection of the stable line and theinverse failure line is termed the crossover point Thecrossover points on the cumulative distribution graph repre-sent the logit probability value that will define the separationline that will have the same proportion of misclassified pointson either side of the line The cumulative frequency graph inFig 6 shows that for a logit probability of 0865 there is a20 mismatch of stable and failure cases This means that20 of the stable points lie below the separation line definedby a logit value of 0865 (in either the failure or the majorfailure zone) and 20 of the failures lie above the line in thestable zone

The true significance of the stablendashfailure line can be deter-mined from the probability density functions for each stabilityclass shown in Fig 7 On the stablendashfailure boundary with alogit value of 0865 there is 48 probability of both stableconditions and failure and a 4 probability of a major failureThis can be directly converted into misclassification propor-tions ie if a designed surface classed as stable plottedexactly on the line there is a 52 probability that it would be

a failure or major failure ie would have been misclassifiedThe positions of the separation lines on the stability graph

were determined from the logit value at the crossover pointson the cumulative frequency plots Data of the type usedhere originating from multiple sources typically include thepossibility of outliers The location of outliers needs to beconsidered because the optimized location of the boundaryobtained from the cumulative plot does not specifically con-sider the distribution of the outliers within the other stabilityzones If a slight shift in the stable boundary would remove asignificant number of failure points from the stable zone for acorrespondingly lower number of stable points moving intothe failure zone a better boundary could exist than the opti-mized one indicated by the cumulative crossover logit valueThe accuracy of the model can be assessed through signifi-cance testing goodness-of-fit measures and by considering

the magnitude of the residuals Measures that have been pro-posed for the assessment of predictive efficiency in logisticregression have been detailed in the work of DeMaris14

Ultimately the decision on the criteria for placing the separa-tion lines will depend on the degree of confidence in theaccuracy of the data and the cost of failure for a givenscenario

The consequences of failure and major failure are highlysite-specific and the level of certainty required in a stableexcavation must be determined by considering the cost of anunplanned failure or major failure The consequences ofmisclassification and the probabilities of a given stabilityoutcome determined from the isoprobability contours willdepend on the data available First sufficient failure andmajor failure case studies must be available to define the sta-bility zone boundaries accurately on the stability graph andsecond adequate site-specific cases are required for accurateprediction of the consequences The lower number of failureand major failure case histories compared with stable caseswithin the stability database causes a range of confidence inthe accuracy of predicted stability zones Fewer case historiesreduce confidence in the results Mining (with the exceptionof caving) is focused principally on the production of stableexcavations The stable zone on the stability graph is definedby the greatest number of case histories and accordingly ourconfidence is highest for designs that utilize this region of thestability graph

Size bias is an issue for the stable failure and major failureclasses within the stability database Mining is focused onstability so the imbalance in subset sizes mirrors the realdistribution of stability cases within operational mines TheMathews stability graph method is moreover a non-rigoroustechnique The quality of the end-productmdashin this case a sta-bility estimate (including probabilities)mdashcan only be as goodas the quality of the underlying data on which the methodwas formulated One suggestion has been to apply a secondoptimization to consider size bias effects on the definition ofthe stability zones This could be undertaken on a larger data-base and investigated by considering a random subsamplefrom the stability database

The logit model allows the direct effect of moving thestability zone boundaries to be quantified in terms of risk orprobability of a specified stability outcome Once a costndashrisk

A32

Fig 7 Probability density functions for stability data determined from logit probability values

relationship has been developed for a site an appropriate levelof risk can be determined and the stability zone boundariesand isoprobability contours on the Mathews graph can betailored to the specific site A costndashrisk analysis is now possi-ble and the optimal design limits can be chosen for individualoperations

Isoprobability contoursAlthough the stability zones can be defined statistically anumber of case histories report to the wrong zones This is to

be expected given the inherent variability of rock massesdata that can be somewhat subjective and the fact that thedesign technique is non-rigorous Diederichs and Kaiser16

proposed the drawing of isoprobability contours to accountfor the uncertainties inherent in the design limitsIsoprobability contours allow the probabilities of stabilityfailure and major failure for a design surface to be obtaineddirectly off the stability graph (an example is given in theAppendix)

The probability density functions have been determined

A33

Fig 8 Isoprobability contours for stable excavation based on logistic regression

Fig 9 Isoprobability contours for failure based on logistic regression

for each of the stability classes (Fig 7) from the predictedlogit values From the probability density functions the iso-probability contours were produced for the three separatelevels of stabilitymdashstable (Fig 8) failure (Fig 9) and majorfailure (Fig 10)

Accuracy and use of extended Mathewsmethod

The Mathews stability graph encompasses a broad range ofopen-stoping experience and is essentially a self-validatingmodel Nevertheless the validity and accuracy of theMathews method depend on the quality and quantity of thestability data that it contains The value of the documentedstability data has been effectively increased by undertaking aleast-squares regression to treat the uncertainties and subjec-tivity of the data

The application of statistical methods in delineating stabil-ity zones and generating isoprobability contours has removedthe subjectivity from the delineation of the zones The appli-cation of regression techniques to a sufficiently large stabilitydatabase is currently the best option for minimizing the influ-ence of subjective data

Major advantages of statistical determination of the zoneson the stability graph are optimization of the design curvesand the ability to evaluate the accuracy of the model Theprecise nature of the risks associated with the position of thestability zone boundaries can be quantified This removes theneed for intermediate transition zones to be defined on thestability graph as general indicators of higher variability inthe stability outcomes Through the logit model the exactmeaning of the stability zone boundaries in terms of theprobability of a given stability outcome is precisely known

Continual improvement of the accuracy of the model forall subsequent modifications is critical for optimization of thefuture use of the Mathews method in stope design Throughstatistical techniques the value of the stability data can bemaximized despite the inherent subjectivity From the resultsobjective interpretations can be made to improve what isacknowledged to be an empirical non-rigorous yet increas-ingly powerful design tool

An example of how to use the Mathews stability graphmethod to assess the stability of an open stope is outlined inthe Appendix

Conclusions

The database for the Mathews method has been extended tomore than 400 case histories such that it now includes open-stoping experience for a broad range of surface geometriesand rock mass conditions The extended Mathews graph is alogndashlog plot with linear stable failure and major failure zonesrather than the traditional logndashlinear format with multiplecurvilinear stability boundaries The extension of the data-base has provided a wide-ranging data-set to which statisticalmethods can be applied

Logistic regression was used to optimize the placement ofstability zones and produce isoprobability contours for theprediction of stope surfaces that are stable or exhibit minor ormajor failures The advantage of logistic regression lies in itsability to minimize the uncertainties reflected in the methodthrough the use of maximum likelihood estimates

The application of statistics does not change the subject-ivity degree of reliability or lack of rigour inherent in theMathews method Statistics can be used to take into accountsome of the inherent variability within the data but the use ofstatistical regression must not be mistaken as adding a greaterlevel of rigour to the Mathews stability graph methodLogistic regression should be viewed as an objective means tocalculate zone boundaries and isoprobability contours for theavailable stability data Through logistic regression the risksassociated with using the technique can be quantified and thestatistical significance of the stability zones becomes known

The authors caution that both the Mathews stability graphmethod and subsequent statistical treatment rely on data thatare not precise and care must be taken not to promote over-confidence in a result without recognizing the nature of thedata underlying the method The preferred approach toincreasing the accuracy of the method is to increase the sizeand quality of the stability database and the authors are pur-suing this

The strength of the extended Mathews method lies in the

A34

Fig 10 Isoprobability contours for major failure based on logistic regression

improved stability graph with statistically determined stabilityzone boundaries and isoprobability contours The uncertain-ties inherent in the design technique can now be quantifiedover a larger range of stope surface geometries and rock massconditions than was previously the case This makes theextended Mathews method a powerful tool for risk assess-ment and optimization of open-stope design

Acknowledgement

The work reported here is a component of research fundedby the Australian Research Council The authors thankProfessor E T Brown for his review of the work

References1 Mathews K E et al Prediction of stable excavation spans formining at depths below 1000 m in hard rock Report to CanadaCentre for Mining and Energy Technology (CANMET)Department of Energy and Resources 19802 Barton N Lien R and Lunde J Engineering classification ofrock masses for design of tunnel support Rock Mechanics 6 1974189ndash2363 Bieniawski Z T Engineering rock mass classifications a completemanual for engineers and geologists in mining civil and petroleum engi-neering (New York Wiley 1989) 251 p4 Potvin Y Empirical open stope design in Canada PhD thesisUniversity of British Columbia 19885 Nickson S D Cable support guidelines for underground hardrock mine operations MAppSc thesis University of BritishColumbia 19926 Hadjigeorgiou J Leclair J and Potvin Y An update of the sta-bility graph method for open stope design Paper presented at CIMRock Mechanics and Strata Control session Halifax Nova Scotia14ndash18 May 19957 Potvin Y Hudyma M R and Miller H D S Design guidelinesfor open stope support CIM Bull 82 no 926 1989 53ndash628 Stewart S B V and Forsyth W W The Mathews method foropen stope design CIM Bull 88 no 992 1995 45ndash539 Trueman R et al Experience in Australia with the Mathewsmethod for open stope design CIM Bull 93 no 1036 2000 162ndash710 Laubscher D H A geomechanics classification system for therating of rock mass in mine design Jl S Afr Inst Min Metall 901990 257ndash7311 Suorineni F T Tannant D D and Kaiser P K Fault factorfor the stability graph method of open-stope design Trans Instn MinMetall (Sect A Min industry) 108 1999 A92ndash10612 Menard S Applied logistic regression analysis (Beverley HillsSage 1995) 98 p Sage University Paper Series on QuantitativeApplications in the Social Sciences no 07-10613 Liao T L Interpreting probability models logit probit andother generalized linear models (Beverley Hills Sage 1984) 88 pSage University Paper Series on Quantitative Applications in the SocialSciences no 07-10114 DeMaris A Logit modelingndashpractical applications (BeverleyHills Sage 1992) 87 p Sage University Paper Series on QuantitativeApplications in the Social Sciences no 07-08615 Holtsberg A Stixboxmdasha statistics toolbox for Matlab andOctave Logitfitm Loddsm and Loddsinvm Available athttpwwwmathslthsematstatstixboxcontentshtml 1999 (cited10200)16 Diederichs M S and Kaiser P K Rock instability and riskanalyses in open stope mine design Can Geotech J 33 1996431ndash9

Appendix

Using the Mathews stability graph method

The use of the stability graph for stability assessment in open-stope design is outlined here in an example from CSA minein Cobar taken from the work of Stewart and Forsyth8 Theexample has been extended to illustrate the use of isopro-bability contours in quantifying the uncertainty of stabilitywithin the stope design process

An open stope is planned at a depth of 1000 m in an ore-body that is 25 m wide and dips at 80deg Because ofoperational constraints the preferred stope length is 30 m and

the stope height is 75 m The geometry of the planned openstope is shown in Fig 1 below

The average unconfined compressive strength of the intactrock is 120 MPa The principal joint set is flat-dipping andclosely spaced (average spacing is 10 cm) The joint surfacesare unaltered but display surface staining Rock mass qualitydata have been collected and are summarized in AppendixTable 1 to the requirements of the NGI Q classificationsystem2

The first consideration when looking at the use of theMathews stability graph in excavation design is to considerhow current conditions compare with the data on which thetechnique was based The non-rigorous and empirical natureof the Mathews method must be considered and the strengthand predictive capability of the method relies accordingly onthe extent of the stability database the method should not beused to predict stability outside the range of experience fromwhich it was developed

The first step in use of the Mathews method at the designstage is to determine the S and N values for each surface ofthe excavation In this example four surfaces are examinedmdashthe footwall hanging-wall crown (or roof) and end-walls ofthe stope As the principal joint set is flat-lying the two end-walls of the stope are identical cases however if this were notso the stability of each end-wall would need to be investi-gated separately

A35

Fig 1 Diagram of stope layout After Stewart and Forsyth8

Table 1 Rock mass quality data used to determine Q cent value

Item Description Value

Rock quality Good RQD = 85Joint sets One joint set and random Jn = 3Joint roughness Rough or irregular undulating Jr = 3Joint alteration Unaltered with surface staining Ja = 1

where Q cent is NGI modified Q value A is rock stress factor B isjoint orientation adjustment factor and C is surface orienta-tion factor

The rock stress factor is determined from the ratio of theintact rock strength (unconfined compressive strength) to theinduced compressive stress at the centre-line of the stopesurface The induced stresses can be determined by use of atwo-dimensional stressndashdisplacement analysis package or esti-mated from published two-dimensional stress distributions Agraph relating the strength to stress ratio and rock stress fac-tor was developed by Mathews et al1 (Fig 2) In the originalMathews graph1 the joint orientation adjustment factor is ameasure of the relative difference in dip between the stopesurface and the critical joint set The surface orientation fac-tor considers the inclination of the excavation surface and itsinfluence on stability (Fig 2)

The geometry of the excavation is considered by calculat-ing the shape factor or hydraulic radius of the surface Theshape factor of an excavation surface is defined as the area ofthe stope surface divided by the length of its perimeter (inmetres)

Historical overview

The Mathews stability graph method for open-stope designwas first proposed for mining at depths below 1000 m1

A number of authors have since collected new data from avariety of mining depths and rock mass conditions to extendthe method and test its validity4ndash9 The modifications to anddevelopments of the Mathews method since its inceptionrelate largely to changes in the position and number of thestability zones represented on the stability graph with theaddition of more data and changes to the formulation of theMathews factors9

The original Mathews graph contained three distinct zonesseparated by transitions8 In the Potvin4 modified stabilitygraph the number of zones was reduced to a stable and acaved zone separated by a transition (Fig 3) The choice ofthe word lsquocavedrsquo by Potvin to represent what is essentially anunstable zone was challenged by Stewart and Forsyth8 whonoted that the term has a fixed meaning in mining termino-logy from which Potvinrsquos modified graph appears to depart8

Potvin4 collected additional case histories and modified the

A28

Fig 2 Adjustment factors for determination of Mathews stability number After Mathews et al1

Fig 3 Potvinrsquos modified Mathews stability graph based on 175case histories After Potvin4

way in which the joint orientation factor and the gravityadjustment factor were calculated This variant of the originalMathews method is referred to as Potvinrsquos modified stabilitygraph method Potvin also extended the method to considersupported excavations a development that was taken furtherby Nickson5 Nickson5 and Hadjigeorgiou and co-workers6

added further supported and unsupported case histories tothe stability database using Potvinrsquos modified method andwere the first to consider statistical zone definition

Stewart and Forsyth8 updated the Mathews stability graphin 1995 and proposed four stability zonesmdashpotentially stablepotentially unstable potential major failure and potential cav-ing zonesmdashseparated by three transitions (Fig 4) The zoneof potential caving approximated by Stewart and Forsyth wasbased on Laubscherrsquos caving stability graph10 No data wereavailable to validate the position of the caving line on theupdated Mathews graph which is meant to represent the truecaving situation as opposed to Potvinrsquos caved zone4 As partof an international study of block and panel caving theauthors have back-analysed a number of caving case historiesand delineated a caving zone on the Mathews graph thatreflects the true caving case defined as continuous cave pro-pagation At this stage the results remain confidential tosponsors of the project

Significant developments and modifications have beenmade to the stability graph method extending its use intocable-supported excavations4ndash7 and more recently to takeaccount of fault-driven instability11 Potvinrsquos modifiedapproach as applied to unsupported stopes was comparedwith the original Mathews method for 180 case histories fromMt Charlotte9 The variability of the results the apparentpredictive accuracy and the accuracy of the defined stabilityzones (in terms of misclassification) of the two methods wereinvestigated From these case studies Trueman et al9 con-cluded that Potvinrsquos modifications to the method ofcalculating the stability number resulted in no appreciabledifference in the predictive capability of the technique forunsupported excavationsmdasha conclusion that was in generalagreement with that of Stewart and Forsyth8 The originalmethod of determining the Mathews stability number and the

adjustment factors as proposed by Mathews et al1 hastherefore been followed here

Historically the stability zones on the Mathews graph havebeen defined by eye14 As new data were added the natureand placement of the stability zones were modified accord-ingly Nickson5 was the first to attempt to determine theposition of the boundaries statistically He applied a discrimi-nant analysis to the three-dimensional multivariate stabilitydatabase and utilized Mahalanobisrsquo distance to separate thedata into two groups Nickson derived a linear separationbetween stable and caved unsupported histories using a loga-rithmic transformation Again it should be noted that theterm lsquocavedrsquo does not represent true caving No unstablecases were considered in the analysis and separation linesbetween stable and unstable or unstable and caving zoneswere not determined Nickson compared his statisticallydetermined stablendashcaved boundary with Potvinrsquos proposedtransition zone Nickson recommended on the basis of hisresults that Potvinrsquos transition zone be used for the design ofunsupported stope surfaces Hadjigeorgiou and co-workers6

collected further stability data repeated the discriminantanalysis and obtained similar results Again no changes toPotvinrsquos transition zone were proposed The transition casehistories were not included in either statistical analysis andthe calculated boundary was only between the stable andcaved case histories

The position of the stability zones on the stability graph iscritical to the reliability of the method and a technique wassought to optimize their positioning Mathematical calcula-tion of the width and position of the stability zones is animportant step towards improving the reliability of theMathews method Removal of the subjectivity involved inzone definition is important in maximizing the value of themethod as a design tool

Extension of the Mathews method

Back-analyses of 180 open-stope surfaces at Mt Charlotte bythe Mathews method confirmed the validity of using the tech-nique as a predictive tool9 These data when combined withother Australian case histories in 1998 resulted in a com-bined stability database for more than 400 stope surfacesThe additional case histories included much larger stopesthan had previously been documented for the method andthey extend the Mathews graph to a hydraulic radius of 55 mcompared with the previous maximum hydraulic radius of23 m8 The expanded database covers a broader spectrum ofrock mass and in-situ stress conditions with stability numbersin the range from 0005 to 700 The background to the MtCharlotte data has already been given by Trueman et al9 thepurpose of the present contribution is to communicate a newapproach for the statistical treatment of stability data and forthe calculation of stability zones and isoprobability contourson stability graphs

A logistic regression analysis12 was carried out on theextended database to delineate the zones of stability statisti-cally and to determine isoprobability contours for stableminor failure and major failure scenarios All the case histo-ries adhere to the original Mathews method of determiningthe adjustment factors By the use of logistic regression theuncertainties in the application of the Mathews method canbe quantified over a larger range of stope geometries and rockmass conditions than was previously possible The extendedMathews stability graph (Fig 5) contains the new data andstatistically determined stability zones A logndashlog graph hasbeen used rather than the traditional logndashlinear plot of theMathews method as it was found to give a clearer picture ofthe zoning The logndashlog plot has linear stability boundaries

A29

Fig 4 Mathews stability graph as modified by Stewart andForsyth8

and the shape factor range of the case histories is separatedbetter on a log shape factor x-axis The new logndashlog stabilitygraph enables individual cases to be clearly distinguished aswell as presents the increased range of shape factors capturedwithin the stability database

Logistic regression analysisLogistic regression12 provides a method of calculating theprobability that a particular stope of a given dimension andgeometry will be stable It also shows how the probability ofstability is altered by a change in rock mass quality or inducedstress The regression analysis allows statistical definition ofthe different zones of stability by optimizing the number ofdata points that report to the correct zones and minimizingmisclassification

Logistic regression analyses are typically used to analysetrue or false values of the dependent variable but can easilybe extended to include estimates of the proportion truefalseA logistic regression analysis was undertaken because prob-lems with ordered outcomes are not easily modelled bytraditional regression techniques and ordinary linear regres-sion does not consider the interval nature of the dependentvariable13

The stability data were specified in terms of three variablesshape factor (or hydraulic radius) Mathews stability numberand stability (whether it was stable a failure or a majorfailure) Logistic regression of these values fits a maximumlikelihood model calculates a separation line of best fitand produces a predicted stability value The maximumlikelihood method is used to maximize the value of the log-likelihood function by estimating the value of the parametersDetermination of the maximum likelihood estimates for thelogistic regression model is an iterative process12 The stabil-ity data do not separate perfectly into each stability zone and

hence the actual values of the dependent variable (stability)differ from the logit values The logit probability values areused to optimize the boundaries of the three stability zones onthe stability graph The predicted stability values can be com-pared with the original values and the nature of the residualscan be analysed to delineate the stability zones and to con-sider misclassification

One of the benefits of logistic regression is that it providesthe predicted probabilities of event occurrence based on thelogit model By using the predicted probabilities of eventoccurrence calculated for each stability class isoprobabilitycontours of risk can be generated for the stability dataIsoprobability contours are an important component of theMathews stability graph method as they represent the esti-mated probability of event occurrence on the stability graph

Logit modelSeveral models can be applied to the data when logisticregression is undertaken For the stability data a modifiedbinary logit model was used in which the logit function is atype of probability distribution13 For the interpretation ofprobabilities the logit model is more appropriate than the tra-ditional linear model given by ordinary least squares121314

Logit analysis produces probabilities after a nonlinear trans-formation for a categorical response for the stability data twomain discrete categories can be identified (stable and majorfailure)14 and in-between probability values can be inter-preted as the failure zone The logit probability value is thenatural logarithm of the odds where the odds indicate therelative probability of falling into one or two categories onsome variable of interest14 The logit model of the stabilitydata was produced in MATLAB (version 51) using a routinedeveloped by Holtsberg15 The regression procedure enabledthe optimization of the stability zone boundaries and the gen-

A30

Fig 5 Extended Mathewsrsquo stability graph based on logistic regression

eration of isoprobability contours by use of the predicted logitprobability values determined from the logit model

A three-level logit model was used in the investigation ofthe Mathews stability data to reflect the three stability classesLogit models are typically applied to yesno outcomes (givinga binary model) but for the present analysis a third inter-mediate category was used The three-level logit modelproduced two separation lines between the three stabilityclasses The fact that the two lines were parallel is a functionof the three-level model The use of a three-level logit modelto investigate the stability data was verified by running dualbinary logit models to determine the stablendashfailure and thefailurendashmajor failure boundaries separately These boundarieswere found to be very close to parallel

For a binary logit model the dependent variable is assigneda value of either zero or one With three separate stability cat-egories an intermediate value needed to be determinedA value of 05 was used for the failure region which was con-sidered intermediate between stable and major failure Thelogit model was run for different values of the dependent vari-able categories and the distribution of the logit values soobtained was examined closely to consider overlap and spac-ing between the stability classes The values assigned to eachstability class as a result are stable 06ndash1 failure 04ndash06and major failure 0ndash04

Initial values of 1 05 and 0 were used to define the threestability categories and for each point a logit probability valuewas calculated The boundaries between the stability classescan then be determined at an appropriate probability forgiven site conditions or design requirements Alternativelygiven the equal probability lines actual counts of the numberof points in each class can be made giving a distribution ofpoints versus the probability value A graph of these results isa useful way of determining the stability zone boundaries

The inclination position and probabilities of the twoboundaries determined from the two separate binary logitmodels were compared with the two boundaries determinedfrom the single three-level logit model The observed differ-ences in the position and inclination of the boundaries wereminor and the three-level model was adopted

The general form of the logit function can be seen inequation 3 The logit model is similar to the traditional linearregression model or the general linear model for ANOVAexcept that the response is the log odds rather than the metricdependent variable14 In logit modelling the conditional log

odds of the dependent variable are expressed as a linear func-tion of a set of explanatory variables (equations 3 and 4)14 Inthe logit model of the stability data the probability of stabilityis expressed as a linear function of the shape factor theMathews stability number and a constant (equation 5)

To fit the logit model the unknown parameters a b 1 b 2 hellipb k (in equation 3) are estimated by the maximum likelihoodmethod With the use of the estimated values for a b 1 and b 2(equation 5) the estimated probability or predicted risk canbe determined (equation 4) The general form of the logitfunction is

z = a + b 1X1 + b 2X2 + b 3X3 + hellip + b kXk= a + aring b kXk (3)

(4)

where z is predicted log odds value a is a constant b 12 arenumerical coefficients and ƒ(z) is predicted logit probabilityvalue For the case of open-stope stability equation 3becomes

z = a + b 1 lnS + b 2 lnN (5)

The maximum likelihood statistic is used to define the incli-nation of the stability zone boundaries By examination of thecumulative and probability density functions of the logit valuesfor each class the position of the lines delineating each of thestability zones can be optimized according to the needs of par-ticular applications The criteria used to place the zones can bespecified and the resulting probabilities for each stability classwill then be known By using the cumulative density functionsfor each class as determined in Fig 7 the position of the sta-bility zones can be optimized for certain criteriamdashfor exampleminimized misclassification (either numerically or propor-tionally) or similarly a more conservative limit of zeromisclassification This method is a significant improvement onearlier methods of delineating stability zones in that the prob-able stability outcomes for a boundary at any position of thestability graph can be quantified statistically The probabilitydensity functions provide data on the degree of misclassifica-tion of the stability data which is an important consideration

f ze z

( ) =+( )-

1

1

A31

Fig 6 Cumulative frequency graph of logit values for stable failure and major failure points

when evaluating the confidence of a stability estimate and thepotential for misclassification

Delineation of stability zonesA cumulative distribution curve can be plotted for the logitprobability values of the data for each stability class (Fig 6)The inverse cumulative distribution curve of each class is alsoplotted The point of intersection of the stable line and theinverse failure line is termed the crossover point Thecrossover points on the cumulative distribution graph repre-sent the logit probability value that will define the separationline that will have the same proportion of misclassified pointson either side of the line The cumulative frequency graph inFig 6 shows that for a logit probability of 0865 there is a20 mismatch of stable and failure cases This means that20 of the stable points lie below the separation line definedby a logit value of 0865 (in either the failure or the majorfailure zone) and 20 of the failures lie above the line in thestable zone

The true significance of the stablendashfailure line can be deter-mined from the probability density functions for each stabilityclass shown in Fig 7 On the stablendashfailure boundary with alogit value of 0865 there is 48 probability of both stableconditions and failure and a 4 probability of a major failureThis can be directly converted into misclassification propor-tions ie if a designed surface classed as stable plottedexactly on the line there is a 52 probability that it would be

a failure or major failure ie would have been misclassifiedThe positions of the separation lines on the stability graph

were determined from the logit value at the crossover pointson the cumulative frequency plots Data of the type usedhere originating from multiple sources typically include thepossibility of outliers The location of outliers needs to beconsidered because the optimized location of the boundaryobtained from the cumulative plot does not specifically con-sider the distribution of the outliers within the other stabilityzones If a slight shift in the stable boundary would remove asignificant number of failure points from the stable zone for acorrespondingly lower number of stable points moving intothe failure zone a better boundary could exist than the opti-mized one indicated by the cumulative crossover logit valueThe accuracy of the model can be assessed through signifi-cance testing goodness-of-fit measures and by considering

the magnitude of the residuals Measures that have been pro-posed for the assessment of predictive efficiency in logisticregression have been detailed in the work of DeMaris14

Ultimately the decision on the criteria for placing the separa-tion lines will depend on the degree of confidence in theaccuracy of the data and the cost of failure for a givenscenario

The consequences of failure and major failure are highlysite-specific and the level of certainty required in a stableexcavation must be determined by considering the cost of anunplanned failure or major failure The consequences ofmisclassification and the probabilities of a given stabilityoutcome determined from the isoprobability contours willdepend on the data available First sufficient failure andmajor failure case studies must be available to define the sta-bility zone boundaries accurately on the stability graph andsecond adequate site-specific cases are required for accurateprediction of the consequences The lower number of failureand major failure case histories compared with stable caseswithin the stability database causes a range of confidence inthe accuracy of predicted stability zones Fewer case historiesreduce confidence in the results Mining (with the exceptionof caving) is focused principally on the production of stableexcavations The stable zone on the stability graph is definedby the greatest number of case histories and accordingly ourconfidence is highest for designs that utilize this region of thestability graph

Size bias is an issue for the stable failure and major failureclasses within the stability database Mining is focused onstability so the imbalance in subset sizes mirrors the realdistribution of stability cases within operational mines TheMathews stability graph method is moreover a non-rigoroustechnique The quality of the end-productmdashin this case a sta-bility estimate (including probabilities)mdashcan only be as goodas the quality of the underlying data on which the methodwas formulated One suggestion has been to apply a secondoptimization to consider size bias effects on the definition ofthe stability zones This could be undertaken on a larger data-base and investigated by considering a random subsamplefrom the stability database

The logit model allows the direct effect of moving thestability zone boundaries to be quantified in terms of risk orprobability of a specified stability outcome Once a costndashrisk

A32

Fig 7 Probability density functions for stability data determined from logit probability values

relationship has been developed for a site an appropriate levelof risk can be determined and the stability zone boundariesand isoprobability contours on the Mathews graph can betailored to the specific site A costndashrisk analysis is now possi-ble and the optimal design limits can be chosen for individualoperations

Isoprobability contoursAlthough the stability zones can be defined statistically anumber of case histories report to the wrong zones This is to

be expected given the inherent variability of rock massesdata that can be somewhat subjective and the fact that thedesign technique is non-rigorous Diederichs and Kaiser16

proposed the drawing of isoprobability contours to accountfor the uncertainties inherent in the design limitsIsoprobability contours allow the probabilities of stabilityfailure and major failure for a design surface to be obtaineddirectly off the stability graph (an example is given in theAppendix)

The probability density functions have been determined

A33

Fig 8 Isoprobability contours for stable excavation based on logistic regression

Fig 9 Isoprobability contours for failure based on logistic regression

for each of the stability classes (Fig 7) from the predictedlogit values From the probability density functions the iso-probability contours were produced for the three separatelevels of stabilitymdashstable (Fig 8) failure (Fig 9) and majorfailure (Fig 10)

Accuracy and use of extended Mathewsmethod

The Mathews stability graph encompasses a broad range ofopen-stoping experience and is essentially a self-validatingmodel Nevertheless the validity and accuracy of theMathews method depend on the quality and quantity of thestability data that it contains The value of the documentedstability data has been effectively increased by undertaking aleast-squares regression to treat the uncertainties and subjec-tivity of the data

The application of statistical methods in delineating stabil-ity zones and generating isoprobability contours has removedthe subjectivity from the delineation of the zones The appli-cation of regression techniques to a sufficiently large stabilitydatabase is currently the best option for minimizing the influ-ence of subjective data

Major advantages of statistical determination of the zoneson the stability graph are optimization of the design curvesand the ability to evaluate the accuracy of the model Theprecise nature of the risks associated with the position of thestability zone boundaries can be quantified This removes theneed for intermediate transition zones to be defined on thestability graph as general indicators of higher variability inthe stability outcomes Through the logit model the exactmeaning of the stability zone boundaries in terms of theprobability of a given stability outcome is precisely known

Continual improvement of the accuracy of the model forall subsequent modifications is critical for optimization of thefuture use of the Mathews method in stope design Throughstatistical techniques the value of the stability data can bemaximized despite the inherent subjectivity From the resultsobjective interpretations can be made to improve what isacknowledged to be an empirical non-rigorous yet increas-ingly powerful design tool

An example of how to use the Mathews stability graphmethod to assess the stability of an open stope is outlined inthe Appendix

Conclusions

The database for the Mathews method has been extended tomore than 400 case histories such that it now includes open-stoping experience for a broad range of surface geometriesand rock mass conditions The extended Mathews graph is alogndashlog plot with linear stable failure and major failure zonesrather than the traditional logndashlinear format with multiplecurvilinear stability boundaries The extension of the data-base has provided a wide-ranging data-set to which statisticalmethods can be applied

Logistic regression was used to optimize the placement ofstability zones and produce isoprobability contours for theprediction of stope surfaces that are stable or exhibit minor ormajor failures The advantage of logistic regression lies in itsability to minimize the uncertainties reflected in the methodthrough the use of maximum likelihood estimates

The application of statistics does not change the subject-ivity degree of reliability or lack of rigour inherent in theMathews method Statistics can be used to take into accountsome of the inherent variability within the data but the use ofstatistical regression must not be mistaken as adding a greaterlevel of rigour to the Mathews stability graph methodLogistic regression should be viewed as an objective means tocalculate zone boundaries and isoprobability contours for theavailable stability data Through logistic regression the risksassociated with using the technique can be quantified and thestatistical significance of the stability zones becomes known

The authors caution that both the Mathews stability graphmethod and subsequent statistical treatment rely on data thatare not precise and care must be taken not to promote over-confidence in a result without recognizing the nature of thedata underlying the method The preferred approach toincreasing the accuracy of the method is to increase the sizeand quality of the stability database and the authors are pur-suing this

The strength of the extended Mathews method lies in the

A34

Fig 10 Isoprobability contours for major failure based on logistic regression

improved stability graph with statistically determined stabilityzone boundaries and isoprobability contours The uncertain-ties inherent in the design technique can now be quantifiedover a larger range of stope surface geometries and rock massconditions than was previously the case This makes theextended Mathews method a powerful tool for risk assess-ment and optimization of open-stope design

Acknowledgement

The work reported here is a component of research fundedby the Australian Research Council The authors thankProfessor E T Brown for his review of the work

References1 Mathews K E et al Prediction of stable excavation spans formining at depths below 1000 m in hard rock Report to CanadaCentre for Mining and Energy Technology (CANMET)Department of Energy and Resources 19802 Barton N Lien R and Lunde J Engineering classification ofrock masses for design of tunnel support Rock Mechanics 6 1974189ndash2363 Bieniawski Z T Engineering rock mass classifications a completemanual for engineers and geologists in mining civil and petroleum engi-neering (New York Wiley 1989) 251 p4 Potvin Y Empirical open stope design in Canada PhD thesisUniversity of British Columbia 19885 Nickson S D Cable support guidelines for underground hardrock mine operations MAppSc thesis University of BritishColumbia 19926 Hadjigeorgiou J Leclair J and Potvin Y An update of the sta-bility graph method for open stope design Paper presented at CIMRock Mechanics and Strata Control session Halifax Nova Scotia14ndash18 May 19957 Potvin Y Hudyma M R and Miller H D S Design guidelinesfor open stope support CIM Bull 82 no 926 1989 53ndash628 Stewart S B V and Forsyth W W The Mathews method foropen stope design CIM Bull 88 no 992 1995 45ndash539 Trueman R et al Experience in Australia with the Mathewsmethod for open stope design CIM Bull 93 no 1036 2000 162ndash710 Laubscher D H A geomechanics classification system for therating of rock mass in mine design Jl S Afr Inst Min Metall 901990 257ndash7311 Suorineni F T Tannant D D and Kaiser P K Fault factorfor the stability graph method of open-stope design Trans Instn MinMetall (Sect A Min industry) 108 1999 A92ndash10612 Menard S Applied logistic regression analysis (Beverley HillsSage 1995) 98 p Sage University Paper Series on QuantitativeApplications in the Social Sciences no 07-10613 Liao T L Interpreting probability models logit probit andother generalized linear models (Beverley Hills Sage 1984) 88 pSage University Paper Series on Quantitative Applications in the SocialSciences no 07-10114 DeMaris A Logit modelingndashpractical applications (BeverleyHills Sage 1992) 87 p Sage University Paper Series on QuantitativeApplications in the Social Sciences no 07-08615 Holtsberg A Stixboxmdasha statistics toolbox for Matlab andOctave Logitfitm Loddsm and Loddsinvm Available athttpwwwmathslthsematstatstixboxcontentshtml 1999 (cited10200)16 Diederichs M S and Kaiser P K Rock instability and riskanalyses in open stope mine design Can Geotech J 33 1996431ndash9

Appendix

Using the Mathews stability graph method

The use of the stability graph for stability assessment in open-stope design is outlined here in an example from CSA minein Cobar taken from the work of Stewart and Forsyth8 Theexample has been extended to illustrate the use of isopro-bability contours in quantifying the uncertainty of stabilitywithin the stope design process

An open stope is planned at a depth of 1000 m in an ore-body that is 25 m wide and dips at 80deg Because ofoperational constraints the preferred stope length is 30 m and

the stope height is 75 m The geometry of the planned openstope is shown in Fig 1 below

The average unconfined compressive strength of the intactrock is 120 MPa The principal joint set is flat-dipping andclosely spaced (average spacing is 10 cm) The joint surfacesare unaltered but display surface staining Rock mass qualitydata have been collected and are summarized in AppendixTable 1 to the requirements of the NGI Q classificationsystem2

The first consideration when looking at the use of theMathews stability graph in excavation design is to considerhow current conditions compare with the data on which thetechnique was based The non-rigorous and empirical natureof the Mathews method must be considered and the strengthand predictive capability of the method relies accordingly onthe extent of the stability database the method should not beused to predict stability outside the range of experience fromwhich it was developed

The first step in use of the Mathews method at the designstage is to determine the S and N values for each surface ofthe excavation In this example four surfaces are examinedmdashthe footwall hanging-wall crown (or roof) and end-walls ofthe stope As the principal joint set is flat-lying the two end-walls of the stope are identical cases however if this were notso the stability of each end-wall would need to be investi-gated separately

A35

Fig 1 Diagram of stope layout After Stewart and Forsyth8

Table 1 Rock mass quality data used to determine Q cent value

Item Description Value

Rock quality Good RQD = 85Joint sets One joint set and random Jn = 3Joint roughness Rough or irregular undulating Jr = 3Joint alteration Unaltered with surface staining Ja = 1

way in which the joint orientation factor and the gravityadjustment factor were calculated This variant of the originalMathews method is referred to as Potvinrsquos modified stabilitygraph method Potvin also extended the method to considersupported excavations a development that was taken furtherby Nickson5 Nickson5 and Hadjigeorgiou and co-workers6

added further supported and unsupported case histories tothe stability database using Potvinrsquos modified method andwere the first to consider statistical zone definition

Stewart and Forsyth8 updated the Mathews stability graphin 1995 and proposed four stability zonesmdashpotentially stablepotentially unstable potential major failure and potential cav-ing zonesmdashseparated by three transitions (Fig 4) The zoneof potential caving approximated by Stewart and Forsyth wasbased on Laubscherrsquos caving stability graph10 No data wereavailable to validate the position of the caving line on theupdated Mathews graph which is meant to represent the truecaving situation as opposed to Potvinrsquos caved zone4 As partof an international study of block and panel caving theauthors have back-analysed a number of caving case historiesand delineated a caving zone on the Mathews graph thatreflects the true caving case defined as continuous cave pro-pagation At this stage the results remain confidential tosponsors of the project

Significant developments and modifications have beenmade to the stability graph method extending its use intocable-supported excavations4ndash7 and more recently to takeaccount of fault-driven instability11 Potvinrsquos modifiedapproach as applied to unsupported stopes was comparedwith the original Mathews method for 180 case histories fromMt Charlotte9 The variability of the results the apparentpredictive accuracy and the accuracy of the defined stabilityzones (in terms of misclassification) of the two methods wereinvestigated From these case studies Trueman et al9 con-cluded that Potvinrsquos modifications to the method ofcalculating the stability number resulted in no appreciabledifference in the predictive capability of the technique forunsupported excavationsmdasha conclusion that was in generalagreement with that of Stewart and Forsyth8 The originalmethod of determining the Mathews stability number and the

adjustment factors as proposed by Mathews et al1 hastherefore been followed here

Historically the stability zones on the Mathews graph havebeen defined by eye14 As new data were added the natureand placement of the stability zones were modified accord-ingly Nickson5 was the first to attempt to determine theposition of the boundaries statistically He applied a discrimi-nant analysis to the three-dimensional multivariate stabilitydatabase and utilized Mahalanobisrsquo distance to separate thedata into two groups Nickson derived a linear separationbetween stable and caved unsupported histories using a loga-rithmic transformation Again it should be noted that theterm lsquocavedrsquo does not represent true caving No unstablecases were considered in the analysis and separation linesbetween stable and unstable or unstable and caving zoneswere not determined Nickson compared his statisticallydetermined stablendashcaved boundary with Potvinrsquos proposedtransition zone Nickson recommended on the basis of hisresults that Potvinrsquos transition zone be used for the design ofunsupported stope surfaces Hadjigeorgiou and co-workers6

collected further stability data repeated the discriminantanalysis and obtained similar results Again no changes toPotvinrsquos transition zone were proposed The transition casehistories were not included in either statistical analysis andthe calculated boundary was only between the stable andcaved case histories

The position of the stability zones on the stability graph iscritical to the reliability of the method and a technique wassought to optimize their positioning Mathematical calcula-tion of the width and position of the stability zones is animportant step towards improving the reliability of theMathews method Removal of the subjectivity involved inzone definition is important in maximizing the value of themethod as a design tool

Extension of the Mathews method

Back-analyses of 180 open-stope surfaces at Mt Charlotte bythe Mathews method confirmed the validity of using the tech-nique as a predictive tool9 These data when combined withother Australian case histories in 1998 resulted in a com-bined stability database for more than 400 stope surfacesThe additional case histories included much larger stopesthan had previously been documented for the method andthey extend the Mathews graph to a hydraulic radius of 55 mcompared with the previous maximum hydraulic radius of23 m8 The expanded database covers a broader spectrum ofrock mass and in-situ stress conditions with stability numbersin the range from 0005 to 700 The background to the MtCharlotte data has already been given by Trueman et al9 thepurpose of the present contribution is to communicate a newapproach for the statistical treatment of stability data and forthe calculation of stability zones and isoprobability contourson stability graphs

A logistic regression analysis12 was carried out on theextended database to delineate the zones of stability statisti-cally and to determine isoprobability contours for stableminor failure and major failure scenarios All the case histo-ries adhere to the original Mathews method of determiningthe adjustment factors By the use of logistic regression theuncertainties in the application of the Mathews method canbe quantified over a larger range of stope geometries and rockmass conditions than was previously possible The extendedMathews stability graph (Fig 5) contains the new data andstatistically determined stability zones A logndashlog graph hasbeen used rather than the traditional logndashlinear plot of theMathews method as it was found to give a clearer picture ofthe zoning The logndashlog plot has linear stability boundaries

A29

Fig 4 Mathews stability graph as modified by Stewart andForsyth8

and the shape factor range of the case histories is separatedbetter on a log shape factor x-axis The new logndashlog stabilitygraph enables individual cases to be clearly distinguished aswell as presents the increased range of shape factors capturedwithin the stability database

Logistic regression analysisLogistic regression12 provides a method of calculating theprobability that a particular stope of a given dimension andgeometry will be stable It also shows how the probability ofstability is altered by a change in rock mass quality or inducedstress The regression analysis allows statistical definition ofthe different zones of stability by optimizing the number ofdata points that report to the correct zones and minimizingmisclassification

Logistic regression analyses are typically used to analysetrue or false values of the dependent variable but can easilybe extended to include estimates of the proportion truefalseA logistic regression analysis was undertaken because prob-lems with ordered outcomes are not easily modelled bytraditional regression techniques and ordinary linear regres-sion does not consider the interval nature of the dependentvariable13

The stability data were specified in terms of three variablesshape factor (or hydraulic radius) Mathews stability numberand stability (whether it was stable a failure or a majorfailure) Logistic regression of these values fits a maximumlikelihood model calculates a separation line of best fitand produces a predicted stability value The maximumlikelihood method is used to maximize the value of the log-likelihood function by estimating the value of the parametersDetermination of the maximum likelihood estimates for thelogistic regression model is an iterative process12 The stabil-ity data do not separate perfectly into each stability zone and

hence the actual values of the dependent variable (stability)differ from the logit values The logit probability values areused to optimize the boundaries of the three stability zones onthe stability graph The predicted stability values can be com-pared with the original values and the nature of the residualscan be analysed to delineate the stability zones and to con-sider misclassification

One of the benefits of logistic regression is that it providesthe predicted probabilities of event occurrence based on thelogit model By using the predicted probabilities of eventoccurrence calculated for each stability class isoprobabilitycontours of risk can be generated for the stability dataIsoprobability contours are an important component of theMathews stability graph method as they represent the esti-mated probability of event occurrence on the stability graph

Logit modelSeveral models can be applied to the data when logisticregression is undertaken For the stability data a modifiedbinary logit model was used in which the logit function is atype of probability distribution13 For the interpretation ofprobabilities the logit model is more appropriate than the tra-ditional linear model given by ordinary least squares121314

Logit analysis produces probabilities after a nonlinear trans-formation for a categorical response for the stability data twomain discrete categories can be identified (stable and majorfailure)14 and in-between probability values can be inter-preted as the failure zone The logit probability value is thenatural logarithm of the odds where the odds indicate therelative probability of falling into one or two categories onsome variable of interest14 The logit model of the stabilitydata was produced in MATLAB (version 51) using a routinedeveloped by Holtsberg15 The regression procedure enabledthe optimization of the stability zone boundaries and the gen-

A30

Fig 5 Extended Mathewsrsquo stability graph based on logistic regression

eration of isoprobability contours by use of the predicted logitprobability values determined from the logit model

A three-level logit model was used in the investigation ofthe Mathews stability data to reflect the three stability classesLogit models are typically applied to yesno outcomes (givinga binary model) but for the present analysis a third inter-mediate category was used The three-level logit modelproduced two separation lines between the three stabilityclasses The fact that the two lines were parallel is a functionof the three-level model The use of a three-level logit modelto investigate the stability data was verified by running dualbinary logit models to determine the stablendashfailure and thefailurendashmajor failure boundaries separately These boundarieswere found to be very close to parallel

For a binary logit model the dependent variable is assigneda value of either zero or one With three separate stability cat-egories an intermediate value needed to be determinedA value of 05 was used for the failure region which was con-sidered intermediate between stable and major failure Thelogit model was run for different values of the dependent vari-able categories and the distribution of the logit values soobtained was examined closely to consider overlap and spac-ing between the stability classes The values assigned to eachstability class as a result are stable 06ndash1 failure 04ndash06and major failure 0ndash04

Initial values of 1 05 and 0 were used to define the threestability categories and for each point a logit probability valuewas calculated The boundaries between the stability classescan then be determined at an appropriate probability forgiven site conditions or design requirements Alternativelygiven the equal probability lines actual counts of the numberof points in each class can be made giving a distribution ofpoints versus the probability value A graph of these results isa useful way of determining the stability zone boundaries

The inclination position and probabilities of the twoboundaries determined from the two separate binary logitmodels were compared with the two boundaries determinedfrom the single three-level logit model The observed differ-ences in the position and inclination of the boundaries wereminor and the three-level model was adopted

The general form of the logit function can be seen inequation 3 The logit model is similar to the traditional linearregression model or the general linear model for ANOVAexcept that the response is the log odds rather than the metricdependent variable14 In logit modelling the conditional log

odds of the dependent variable are expressed as a linear func-tion of a set of explanatory variables (equations 3 and 4)14 Inthe logit model of the stability data the probability of stabilityis expressed as a linear function of the shape factor theMathews stability number and a constant (equation 5)

To fit the logit model the unknown parameters a b 1 b 2 hellipb k (in equation 3) are estimated by the maximum likelihoodmethod With the use of the estimated values for a b 1 and b 2(equation 5) the estimated probability or predicted risk canbe determined (equation 4) The general form of the logitfunction is

z = a + b 1X1 + b 2X2 + b 3X3 + hellip + b kXk= a + aring b kXk (3)

(4)

where z is predicted log odds value a is a constant b 12 arenumerical coefficients and ƒ(z) is predicted logit probabilityvalue For the case of open-stope stability equation 3becomes

z = a + b 1 lnS + b 2 lnN (5)

The maximum likelihood statistic is used to define the incli-nation of the stability zone boundaries By examination of thecumulative and probability density functions of the logit valuesfor each class the position of the lines delineating each of thestability zones can be optimized according to the needs of par-ticular applications The criteria used to place the zones can bespecified and the resulting probabilities for each stability classwill then be known By using the cumulative density functionsfor each class as determined in Fig 7 the position of the sta-bility zones can be optimized for certain criteriamdashfor exampleminimized misclassification (either numerically or propor-tionally) or similarly a more conservative limit of zeromisclassification This method is a significant improvement onearlier methods of delineating stability zones in that the prob-able stability outcomes for a boundary at any position of thestability graph can be quantified statistically The probabilitydensity functions provide data on the degree of misclassifica-tion of the stability data which is an important consideration

f ze z

( ) =+( )-

1

1

A31

Fig 6 Cumulative frequency graph of logit values for stable failure and major failure points

when evaluating the confidence of a stability estimate and thepotential for misclassification

Delineation of stability zonesA cumulative distribution curve can be plotted for the logitprobability values of the data for each stability class (Fig 6)The inverse cumulative distribution curve of each class is alsoplotted The point of intersection of the stable line and theinverse failure line is termed the crossover point Thecrossover points on the cumulative distribution graph repre-sent the logit probability value that will define the separationline that will have the same proportion of misclassified pointson either side of the line The cumulative frequency graph inFig 6 shows that for a logit probability of 0865 there is a20 mismatch of stable and failure cases This means that20 of the stable points lie below the separation line definedby a logit value of 0865 (in either the failure or the majorfailure zone) and 20 of the failures lie above the line in thestable zone

The true significance of the stablendashfailure line can be deter-mined from the probability density functions for each stabilityclass shown in Fig 7 On the stablendashfailure boundary with alogit value of 0865 there is 48 probability of both stableconditions and failure and a 4 probability of a major failureThis can be directly converted into misclassification propor-tions ie if a designed surface classed as stable plottedexactly on the line there is a 52 probability that it would be

a failure or major failure ie would have been misclassifiedThe positions of the separation lines on the stability graph

were determined from the logit value at the crossover pointson the cumulative frequency plots Data of the type usedhere originating from multiple sources typically include thepossibility of outliers The location of outliers needs to beconsidered because the optimized location of the boundaryobtained from the cumulative plot does not specifically con-sider the distribution of the outliers within the other stabilityzones If a slight shift in the stable boundary would remove asignificant number of failure points from the stable zone for acorrespondingly lower number of stable points moving intothe failure zone a better boundary could exist than the opti-mized one indicated by the cumulative crossover logit valueThe accuracy of the model can be assessed through signifi-cance testing goodness-of-fit measures and by considering

the magnitude of the residuals Measures that have been pro-posed for the assessment of predictive efficiency in logisticregression have been detailed in the work of DeMaris14

Ultimately the decision on the criteria for placing the separa-tion lines will depend on the degree of confidence in theaccuracy of the data and the cost of failure for a givenscenario

The consequences of failure and major failure are highlysite-specific and the level of certainty required in a stableexcavation must be determined by considering the cost of anunplanned failure or major failure The consequences ofmisclassification and the probabilities of a given stabilityoutcome determined from the isoprobability contours willdepend on the data available First sufficient failure andmajor failure case studies must be available to define the sta-bility zone boundaries accurately on the stability graph andsecond adequate site-specific cases are required for accurateprediction of the consequences The lower number of failureand major failure case histories compared with stable caseswithin the stability database causes a range of confidence inthe accuracy of predicted stability zones Fewer case historiesreduce confidence in the results Mining (with the exceptionof caving) is focused principally on the production of stableexcavations The stable zone on the stability graph is definedby the greatest number of case histories and accordingly ourconfidence is highest for designs that utilize this region of thestability graph

Size bias is an issue for the stable failure and major failureclasses within the stability database Mining is focused onstability so the imbalance in subset sizes mirrors the realdistribution of stability cases within operational mines TheMathews stability graph method is moreover a non-rigoroustechnique The quality of the end-productmdashin this case a sta-bility estimate (including probabilities)mdashcan only be as goodas the quality of the underlying data on which the methodwas formulated One suggestion has been to apply a secondoptimization to consider size bias effects on the definition ofthe stability zones This could be undertaken on a larger data-base and investigated by considering a random subsamplefrom the stability database

The logit model allows the direct effect of moving thestability zone boundaries to be quantified in terms of risk orprobability of a specified stability outcome Once a costndashrisk

A32

Fig 7 Probability density functions for stability data determined from logit probability values

relationship has been developed for a site an appropriate levelof risk can be determined and the stability zone boundariesand isoprobability contours on the Mathews graph can betailored to the specific site A costndashrisk analysis is now possi-ble and the optimal design limits can be chosen for individualoperations

Isoprobability contoursAlthough the stability zones can be defined statistically anumber of case histories report to the wrong zones This is to

be expected given the inherent variability of rock massesdata that can be somewhat subjective and the fact that thedesign technique is non-rigorous Diederichs and Kaiser16

proposed the drawing of isoprobability contours to accountfor the uncertainties inherent in the design limitsIsoprobability contours allow the probabilities of stabilityfailure and major failure for a design surface to be obtaineddirectly off the stability graph (an example is given in theAppendix)

The probability density functions have been determined

A33

Fig 8 Isoprobability contours for stable excavation based on logistic regression

Fig 9 Isoprobability contours for failure based on logistic regression

for each of the stability classes (Fig 7) from the predictedlogit values From the probability density functions the iso-probability contours were produced for the three separatelevels of stabilitymdashstable (Fig 8) failure (Fig 9) and majorfailure (Fig 10)

Accuracy and use of extended Mathewsmethod

The Mathews stability graph encompasses a broad range ofopen-stoping experience and is essentially a self-validatingmodel Nevertheless the validity and accuracy of theMathews method depend on the quality and quantity of thestability data that it contains The value of the documentedstability data has been effectively increased by undertaking aleast-squares regression to treat the uncertainties and subjec-tivity of the data

The application of statistical methods in delineating stabil-ity zones and generating isoprobability contours has removedthe subjectivity from the delineation of the zones The appli-cation of regression techniques to a sufficiently large stabilitydatabase is currently the best option for minimizing the influ-ence of subjective data

Major advantages of statistical determination of the zoneson the stability graph are optimization of the design curvesand the ability to evaluate the accuracy of the model Theprecise nature of the risks associated with the position of thestability zone boundaries can be quantified This removes theneed for intermediate transition zones to be defined on thestability graph as general indicators of higher variability inthe stability outcomes Through the logit model the exactmeaning of the stability zone boundaries in terms of theprobability of a given stability outcome is precisely known

Continual improvement of the accuracy of the model forall subsequent modifications is critical for optimization of thefuture use of the Mathews method in stope design Throughstatistical techniques the value of the stability data can bemaximized despite the inherent subjectivity From the resultsobjective interpretations can be made to improve what isacknowledged to be an empirical non-rigorous yet increas-ingly powerful design tool

An example of how to use the Mathews stability graphmethod to assess the stability of an open stope is outlined inthe Appendix

Conclusions

The database for the Mathews method has been extended tomore than 400 case histories such that it now includes open-stoping experience for a broad range of surface geometriesand rock mass conditions The extended Mathews graph is alogndashlog plot with linear stable failure and major failure zonesrather than the traditional logndashlinear format with multiplecurvilinear stability boundaries The extension of the data-base has provided a wide-ranging data-set to which statisticalmethods can be applied

Logistic regression was used to optimize the placement ofstability zones and produce isoprobability contours for theprediction of stope surfaces that are stable or exhibit minor ormajor failures The advantage of logistic regression lies in itsability to minimize the uncertainties reflected in the methodthrough the use of maximum likelihood estimates

The application of statistics does not change the subject-ivity degree of reliability or lack of rigour inherent in theMathews method Statistics can be used to take into accountsome of the inherent variability within the data but the use ofstatistical regression must not be mistaken as adding a greaterlevel of rigour to the Mathews stability graph methodLogistic regression should be viewed as an objective means tocalculate zone boundaries and isoprobability contours for theavailable stability data Through logistic regression the risksassociated with using the technique can be quantified and thestatistical significance of the stability zones becomes known

The authors caution that both the Mathews stability graphmethod and subsequent statistical treatment rely on data thatare not precise and care must be taken not to promote over-confidence in a result without recognizing the nature of thedata underlying the method The preferred approach toincreasing the accuracy of the method is to increase the sizeand quality of the stability database and the authors are pur-suing this

The strength of the extended Mathews method lies in the

A34

Fig 10 Isoprobability contours for major failure based on logistic regression

improved stability graph with statistically determined stabilityzone boundaries and isoprobability contours The uncertain-ties inherent in the design technique can now be quantifiedover a larger range of stope surface geometries and rock massconditions than was previously the case This makes theextended Mathews method a powerful tool for risk assess-ment and optimization of open-stope design

Acknowledgement

The work reported here is a component of research fundedby the Australian Research Council The authors thankProfessor E T Brown for his review of the work

References1 Mathews K E et al Prediction of stable excavation spans formining at depths below 1000 m in hard rock Report to CanadaCentre for Mining and Energy Technology (CANMET)Department of Energy and Resources 19802 Barton N Lien R and Lunde J Engineering classification ofrock masses for design of tunnel support Rock Mechanics 6 1974189ndash2363 Bieniawski Z T Engineering rock mass classifications a completemanual for engineers and geologists in mining civil and petroleum engi-neering (New York Wiley 1989) 251 p4 Potvin Y Empirical open stope design in Canada PhD thesisUniversity of British Columbia 19885 Nickson S D Cable support guidelines for underground hardrock mine operations MAppSc thesis University of BritishColumbia 19926 Hadjigeorgiou J Leclair J and Potvin Y An update of the sta-bility graph method for open stope design Paper presented at CIMRock Mechanics and Strata Control session Halifax Nova Scotia14ndash18 May 19957 Potvin Y Hudyma M R and Miller H D S Design guidelinesfor open stope support CIM Bull 82 no 926 1989 53ndash628 Stewart S B V and Forsyth W W The Mathews method foropen stope design CIM Bull 88 no 992 1995 45ndash539 Trueman R et al Experience in Australia with the Mathewsmethod for open stope design CIM Bull 93 no 1036 2000 162ndash710 Laubscher D H A geomechanics classification system for therating of rock mass in mine design Jl S Afr Inst Min Metall 901990 257ndash7311 Suorineni F T Tannant D D and Kaiser P K Fault factorfor the stability graph method of open-stope design Trans Instn MinMetall (Sect A Min industry) 108 1999 A92ndash10612 Menard S Applied logistic regression analysis (Beverley HillsSage 1995) 98 p Sage University Paper Series on QuantitativeApplications in the Social Sciences no 07-10613 Liao T L Interpreting probability models logit probit andother generalized linear models (Beverley Hills Sage 1984) 88 pSage University Paper Series on Quantitative Applications in the SocialSciences no 07-10114 DeMaris A Logit modelingndashpractical applications (BeverleyHills Sage 1992) 87 p Sage University Paper Series on QuantitativeApplications in the Social Sciences no 07-08615 Holtsberg A Stixboxmdasha statistics toolbox for Matlab andOctave Logitfitm Loddsm and Loddsinvm Available athttpwwwmathslthsematstatstixboxcontentshtml 1999 (cited10200)16 Diederichs M S and Kaiser P K Rock instability and riskanalyses in open stope mine design Can Geotech J 33 1996431ndash9

Appendix

Using the Mathews stability graph method

The use of the stability graph for stability assessment in open-stope design is outlined here in an example from CSA minein Cobar taken from the work of Stewart and Forsyth8 Theexample has been extended to illustrate the use of isopro-bability contours in quantifying the uncertainty of stabilitywithin the stope design process

An open stope is planned at a depth of 1000 m in an ore-body that is 25 m wide and dips at 80deg Because ofoperational constraints the preferred stope length is 30 m and

the stope height is 75 m The geometry of the planned openstope is shown in Fig 1 below

The average unconfined compressive strength of the intactrock is 120 MPa The principal joint set is flat-dipping andclosely spaced (average spacing is 10 cm) The joint surfacesare unaltered but display surface staining Rock mass qualitydata have been collected and are summarized in AppendixTable 1 to the requirements of the NGI Q classificationsystem2

The first consideration when looking at the use of theMathews stability graph in excavation design is to considerhow current conditions compare with the data on which thetechnique was based The non-rigorous and empirical natureof the Mathews method must be considered and the strengthand predictive capability of the method relies accordingly onthe extent of the stability database the method should not beused to predict stability outside the range of experience fromwhich it was developed

The first step in use of the Mathews method at the designstage is to determine the S and N values for each surface ofthe excavation In this example four surfaces are examinedmdashthe footwall hanging-wall crown (or roof) and end-walls ofthe stope As the principal joint set is flat-lying the two end-walls of the stope are identical cases however if this were notso the stability of each end-wall would need to be investi-gated separately

A35

Fig 1 Diagram of stope layout After Stewart and Forsyth8

Table 1 Rock mass quality data used to determine Q cent value

Item Description Value

Rock quality Good RQD = 85Joint sets One joint set and random Jn = 3Joint roughness Rough or irregular undulating Jr = 3Joint alteration Unaltered with surface staining Ja = 1

and the shape factor range of the case histories is separatedbetter on a log shape factor x-axis The new logndashlog stabilitygraph enables individual cases to be clearly distinguished aswell as presents the increased range of shape factors capturedwithin the stability database

Logistic regression analysisLogistic regression12 provides a method of calculating theprobability that a particular stope of a given dimension andgeometry will be stable It also shows how the probability ofstability is altered by a change in rock mass quality or inducedstress The regression analysis allows statistical definition ofthe different zones of stability by optimizing the number ofdata points that report to the correct zones and minimizingmisclassification

Logistic regression analyses are typically used to analysetrue or false values of the dependent variable but can easilybe extended to include estimates of the proportion truefalseA logistic regression analysis was undertaken because prob-lems with ordered outcomes are not easily modelled bytraditional regression techniques and ordinary linear regres-sion does not consider the interval nature of the dependentvariable13

The stability data were specified in terms of three variablesshape factor (or hydraulic radius) Mathews stability numberand stability (whether it was stable a failure or a majorfailure) Logistic regression of these values fits a maximumlikelihood model calculates a separation line of best fitand produces a predicted stability value The maximumlikelihood method is used to maximize the value of the log-likelihood function by estimating the value of the parametersDetermination of the maximum likelihood estimates for thelogistic regression model is an iterative process12 The stabil-ity data do not separate perfectly into each stability zone and

hence the actual values of the dependent variable (stability)differ from the logit values The logit probability values areused to optimize the boundaries of the three stability zones onthe stability graph The predicted stability values can be com-pared with the original values and the nature of the residualscan be analysed to delineate the stability zones and to con-sider misclassification

One of the benefits of logistic regression is that it providesthe predicted probabilities of event occurrence based on thelogit model By using the predicted probabilities of eventoccurrence calculated for each stability class isoprobabilitycontours of risk can be generated for the stability dataIsoprobability contours are an important component of theMathews stability graph method as they represent the esti-mated probability of event occurrence on the stability graph

Logit modelSeveral models can be applied to the data when logisticregression is undertaken For the stability data a modifiedbinary logit model was used in which the logit function is atype of probability distribution13 For the interpretation ofprobabilities the logit model is more appropriate than the tra-ditional linear model given by ordinary least squares121314

Logit analysis produces probabilities after a nonlinear trans-formation for a categorical response for the stability data twomain discrete categories can be identified (stable and majorfailure)14 and in-between probability values can be inter-preted as the failure zone The logit probability value is thenatural logarithm of the odds where the odds indicate therelative probability of falling into one or two categories onsome variable of interest14 The logit model of the stabilitydata was produced in MATLAB (version 51) using a routinedeveloped by Holtsberg15 The regression procedure enabledthe optimization of the stability zone boundaries and the gen-

A30

Fig 5 Extended Mathewsrsquo stability graph based on logistic regression

eration of isoprobability contours by use of the predicted logitprobability values determined from the logit model

A three-level logit model was used in the investigation ofthe Mathews stability data to reflect the three stability classesLogit models are typically applied to yesno outcomes (givinga binary model) but for the present analysis a third inter-mediate category was used The three-level logit modelproduced two separation lines between the three stabilityclasses The fact that the two lines were parallel is a functionof the three-level model The use of a three-level logit modelto investigate the stability data was verified by running dualbinary logit models to determine the stablendashfailure and thefailurendashmajor failure boundaries separately These boundarieswere found to be very close to parallel

For a binary logit model the dependent variable is assigneda value of either zero or one With three separate stability cat-egories an intermediate value needed to be determinedA value of 05 was used for the failure region which was con-sidered intermediate between stable and major failure Thelogit model was run for different values of the dependent vari-able categories and the distribution of the logit values soobtained was examined closely to consider overlap and spac-ing between the stability classes The values assigned to eachstability class as a result are stable 06ndash1 failure 04ndash06and major failure 0ndash04

Initial values of 1 05 and 0 were used to define the threestability categories and for each point a logit probability valuewas calculated The boundaries between the stability classescan then be determined at an appropriate probability forgiven site conditions or design requirements Alternativelygiven the equal probability lines actual counts of the numberof points in each class can be made giving a distribution ofpoints versus the probability value A graph of these results isa useful way of determining the stability zone boundaries

The inclination position and probabilities of the twoboundaries determined from the two separate binary logitmodels were compared with the two boundaries determinedfrom the single three-level logit model The observed differ-ences in the position and inclination of the boundaries wereminor and the three-level model was adopted

The general form of the logit function can be seen inequation 3 The logit model is similar to the traditional linearregression model or the general linear model for ANOVAexcept that the response is the log odds rather than the metricdependent variable14 In logit modelling the conditional log

odds of the dependent variable are expressed as a linear func-tion of a set of explanatory variables (equations 3 and 4)14 Inthe logit model of the stability data the probability of stabilityis expressed as a linear function of the shape factor theMathews stability number and a constant (equation 5)

To fit the logit model the unknown parameters a b 1 b 2 hellipb k (in equation 3) are estimated by the maximum likelihoodmethod With the use of the estimated values for a b 1 and b 2(equation 5) the estimated probability or predicted risk canbe determined (equation 4) The general form of the logitfunction is

z = a + b 1X1 + b 2X2 + b 3X3 + hellip + b kXk= a + aring b kXk (3)

(4)

where z is predicted log odds value a is a constant b 12 arenumerical coefficients and ƒ(z) is predicted logit probabilityvalue For the case of open-stope stability equation 3becomes

z = a + b 1 lnS + b 2 lnN (5)

The maximum likelihood statistic is used to define the incli-nation of the stability zone boundaries By examination of thecumulative and probability density functions of the logit valuesfor each class the position of the lines delineating each of thestability zones can be optimized according to the needs of par-ticular applications The criteria used to place the zones can bespecified and the resulting probabilities for each stability classwill then be known By using the cumulative density functionsfor each class as determined in Fig 7 the position of the sta-bility zones can be optimized for certain criteriamdashfor exampleminimized misclassification (either numerically or propor-tionally) or similarly a more conservative limit of zeromisclassification This method is a significant improvement onearlier methods of delineating stability zones in that the prob-able stability outcomes for a boundary at any position of thestability graph can be quantified statistically The probabilitydensity functions provide data on the degree of misclassifica-tion of the stability data which is an important consideration

f ze z

( ) =+( )-

1

1

A31

Fig 6 Cumulative frequency graph of logit values for stable failure and major failure points

when evaluating the confidence of a stability estimate and thepotential for misclassification

Delineation of stability zonesA cumulative distribution curve can be plotted for the logitprobability values of the data for each stability class (Fig 6)The inverse cumulative distribution curve of each class is alsoplotted The point of intersection of the stable line and theinverse failure line is termed the crossover point Thecrossover points on the cumulative distribution graph repre-sent the logit probability value that will define the separationline that will have the same proportion of misclassified pointson either side of the line The cumulative frequency graph inFig 6 shows that for a logit probability of 0865 there is a20 mismatch of stable and failure cases This means that20 of the stable points lie below the separation line definedby a logit value of 0865 (in either the failure or the majorfailure zone) and 20 of the failures lie above the line in thestable zone

The true significance of the stablendashfailure line can be deter-mined from the probability density functions for each stabilityclass shown in Fig 7 On the stablendashfailure boundary with alogit value of 0865 there is 48 probability of both stableconditions and failure and a 4 probability of a major failureThis can be directly converted into misclassification propor-tions ie if a designed surface classed as stable plottedexactly on the line there is a 52 probability that it would be

a failure or major failure ie would have been misclassifiedThe positions of the separation lines on the stability graph

were determined from the logit value at the crossover pointson the cumulative frequency plots Data of the type usedhere originating from multiple sources typically include thepossibility of outliers The location of outliers needs to beconsidered because the optimized location of the boundaryobtained from the cumulative plot does not specifically con-sider the distribution of the outliers within the other stabilityzones If a slight shift in the stable boundary would remove asignificant number of failure points from the stable zone for acorrespondingly lower number of stable points moving intothe failure zone a better boundary could exist than the opti-mized one indicated by the cumulative crossover logit valueThe accuracy of the model can be assessed through signifi-cance testing goodness-of-fit measures and by considering

the magnitude of the residuals Measures that have been pro-posed for the assessment of predictive efficiency in logisticregression have been detailed in the work of DeMaris14

Ultimately the decision on the criteria for placing the separa-tion lines will depend on the degree of confidence in theaccuracy of the data and the cost of failure for a givenscenario

The consequences of failure and major failure are highlysite-specific and the level of certainty required in a stableexcavation must be determined by considering the cost of anunplanned failure or major failure The consequences ofmisclassification and the probabilities of a given stabilityoutcome determined from the isoprobability contours willdepend on the data available First sufficient failure andmajor failure case studies must be available to define the sta-bility zone boundaries accurately on the stability graph andsecond adequate site-specific cases are required for accurateprediction of the consequences The lower number of failureand major failure case histories compared with stable caseswithin the stability database causes a range of confidence inthe accuracy of predicted stability zones Fewer case historiesreduce confidence in the results Mining (with the exceptionof caving) is focused principally on the production of stableexcavations The stable zone on the stability graph is definedby the greatest number of case histories and accordingly ourconfidence is highest for designs that utilize this region of thestability graph

Size bias is an issue for the stable failure and major failureclasses within the stability database Mining is focused onstability so the imbalance in subset sizes mirrors the realdistribution of stability cases within operational mines TheMathews stability graph method is moreover a non-rigoroustechnique The quality of the end-productmdashin this case a sta-bility estimate (including probabilities)mdashcan only be as goodas the quality of the underlying data on which the methodwas formulated One suggestion has been to apply a secondoptimization to consider size bias effects on the definition ofthe stability zones This could be undertaken on a larger data-base and investigated by considering a random subsamplefrom the stability database

The logit model allows the direct effect of moving thestability zone boundaries to be quantified in terms of risk orprobability of a specified stability outcome Once a costndashrisk

A32

Fig 7 Probability density functions for stability data determined from logit probability values

relationship has been developed for a site an appropriate levelof risk can be determined and the stability zone boundariesand isoprobability contours on the Mathews graph can betailored to the specific site A costndashrisk analysis is now possi-ble and the optimal design limits can be chosen for individualoperations

Isoprobability contoursAlthough the stability zones can be defined statistically anumber of case histories report to the wrong zones This is to

be expected given the inherent variability of rock massesdata that can be somewhat subjective and the fact that thedesign technique is non-rigorous Diederichs and Kaiser16

proposed the drawing of isoprobability contours to accountfor the uncertainties inherent in the design limitsIsoprobability contours allow the probabilities of stabilityfailure and major failure for a design surface to be obtaineddirectly off the stability graph (an example is given in theAppendix)

The probability density functions have been determined

A33

Fig 8 Isoprobability contours for stable excavation based on logistic regression

Fig 9 Isoprobability contours for failure based on logistic regression

for each of the stability classes (Fig 7) from the predictedlogit values From the probability density functions the iso-probability contours were produced for the three separatelevels of stabilitymdashstable (Fig 8) failure (Fig 9) and majorfailure (Fig 10)

Accuracy and use of extended Mathewsmethod

The Mathews stability graph encompasses a broad range ofopen-stoping experience and is essentially a self-validatingmodel Nevertheless the validity and accuracy of theMathews method depend on the quality and quantity of thestability data that it contains The value of the documentedstability data has been effectively increased by undertaking aleast-squares regression to treat the uncertainties and subjec-tivity of the data

The application of statistical methods in delineating stabil-ity zones and generating isoprobability contours has removedthe subjectivity from the delineation of the zones The appli-cation of regression techniques to a sufficiently large stabilitydatabase is currently the best option for minimizing the influ-ence of subjective data

Major advantages of statistical determination of the zoneson the stability graph are optimization of the design curvesand the ability to evaluate the accuracy of the model Theprecise nature of the risks associated with the position of thestability zone boundaries can be quantified This removes theneed for intermediate transition zones to be defined on thestability graph as general indicators of higher variability inthe stability outcomes Through the logit model the exactmeaning of the stability zone boundaries in terms of theprobability of a given stability outcome is precisely known

Continual improvement of the accuracy of the model forall subsequent modifications is critical for optimization of thefuture use of the Mathews method in stope design Throughstatistical techniques the value of the stability data can bemaximized despite the inherent subjectivity From the resultsobjective interpretations can be made to improve what isacknowledged to be an empirical non-rigorous yet increas-ingly powerful design tool

An example of how to use the Mathews stability graphmethod to assess the stability of an open stope is outlined inthe Appendix

Conclusions

The database for the Mathews method has been extended tomore than 400 case histories such that it now includes open-stoping experience for a broad range of surface geometriesand rock mass conditions The extended Mathews graph is alogndashlog plot with linear stable failure and major failure zonesrather than the traditional logndashlinear format with multiplecurvilinear stability boundaries The extension of the data-base has provided a wide-ranging data-set to which statisticalmethods can be applied

Logistic regression was used to optimize the placement ofstability zones and produce isoprobability contours for theprediction of stope surfaces that are stable or exhibit minor ormajor failures The advantage of logistic regression lies in itsability to minimize the uncertainties reflected in the methodthrough the use of maximum likelihood estimates

The application of statistics does not change the subject-ivity degree of reliability or lack of rigour inherent in theMathews method Statistics can be used to take into accountsome of the inherent variability within the data but the use ofstatistical regression must not be mistaken as adding a greaterlevel of rigour to the Mathews stability graph methodLogistic regression should be viewed as an objective means tocalculate zone boundaries and isoprobability contours for theavailable stability data Through logistic regression the risksassociated with using the technique can be quantified and thestatistical significance of the stability zones becomes known

The authors caution that both the Mathews stability graphmethod and subsequent statistical treatment rely on data thatare not precise and care must be taken not to promote over-confidence in a result without recognizing the nature of thedata underlying the method The preferred approach toincreasing the accuracy of the method is to increase the sizeand quality of the stability database and the authors are pur-suing this

The strength of the extended Mathews method lies in the

A34

Fig 10 Isoprobability contours for major failure based on logistic regression

improved stability graph with statistically determined stabilityzone boundaries and isoprobability contours The uncertain-ties inherent in the design technique can now be quantifiedover a larger range of stope surface geometries and rock massconditions than was previously the case This makes theextended Mathews method a powerful tool for risk assess-ment and optimization of open-stope design

Acknowledgement

The work reported here is a component of research fundedby the Australian Research Council The authors thankProfessor E T Brown for his review of the work

References1 Mathews K E et al Prediction of stable excavation spans formining at depths below 1000 m in hard rock Report to CanadaCentre for Mining and Energy Technology (CANMET)Department of Energy and Resources 19802 Barton N Lien R and Lunde J Engineering classification ofrock masses for design of tunnel support Rock Mechanics 6 1974189ndash2363 Bieniawski Z T Engineering rock mass classifications a completemanual for engineers and geologists in mining civil and petroleum engi-neering (New York Wiley 1989) 251 p4 Potvin Y Empirical open stope design in Canada PhD thesisUniversity of British Columbia 19885 Nickson S D Cable support guidelines for underground hardrock mine operations MAppSc thesis University of BritishColumbia 19926 Hadjigeorgiou J Leclair J and Potvin Y An update of the sta-bility graph method for open stope design Paper presented at CIMRock Mechanics and Strata Control session Halifax Nova Scotia14ndash18 May 19957 Potvin Y Hudyma M R and Miller H D S Design guidelinesfor open stope support CIM Bull 82 no 926 1989 53ndash628 Stewart S B V and Forsyth W W The Mathews method foropen stope design CIM Bull 88 no 992 1995 45ndash539 Trueman R et al Experience in Australia with the Mathewsmethod for open stope design CIM Bull 93 no 1036 2000 162ndash710 Laubscher D H A geomechanics classification system for therating of rock mass in mine design Jl S Afr Inst Min Metall 901990 257ndash7311 Suorineni F T Tannant D D and Kaiser P K Fault factorfor the stability graph method of open-stope design Trans Instn MinMetall (Sect A Min industry) 108 1999 A92ndash10612 Menard S Applied logistic regression analysis (Beverley HillsSage 1995) 98 p Sage University Paper Series on QuantitativeApplications in the Social Sciences no 07-10613 Liao T L Interpreting probability models logit probit andother generalized linear models (Beverley Hills Sage 1984) 88 pSage University Paper Series on Quantitative Applications in the SocialSciences no 07-10114 DeMaris A Logit modelingndashpractical applications (BeverleyHills Sage 1992) 87 p Sage University Paper Series on QuantitativeApplications in the Social Sciences no 07-08615 Holtsberg A Stixboxmdasha statistics toolbox for Matlab andOctave Logitfitm Loddsm and Loddsinvm Available athttpwwwmathslthsematstatstixboxcontentshtml 1999 (cited10200)16 Diederichs M S and Kaiser P K Rock instability and riskanalyses in open stope mine design Can Geotech J 33 1996431ndash9

Appendix

Using the Mathews stability graph method

The use of the stability graph for stability assessment in open-stope design is outlined here in an example from CSA minein Cobar taken from the work of Stewart and Forsyth8 Theexample has been extended to illustrate the use of isopro-bability contours in quantifying the uncertainty of stabilitywithin the stope design process

An open stope is planned at a depth of 1000 m in an ore-body that is 25 m wide and dips at 80deg Because ofoperational constraints the preferred stope length is 30 m and

the stope height is 75 m The geometry of the planned openstope is shown in Fig 1 below

The average unconfined compressive strength of the intactrock is 120 MPa The principal joint set is flat-dipping andclosely spaced (average spacing is 10 cm) The joint surfacesare unaltered but display surface staining Rock mass qualitydata have been collected and are summarized in AppendixTable 1 to the requirements of the NGI Q classificationsystem2

The first consideration when looking at the use of theMathews stability graph in excavation design is to considerhow current conditions compare with the data on which thetechnique was based The non-rigorous and empirical natureof the Mathews method must be considered and the strengthand predictive capability of the method relies accordingly onthe extent of the stability database the method should not beused to predict stability outside the range of experience fromwhich it was developed

The first step in use of the Mathews method at the designstage is to determine the S and N values for each surface ofthe excavation In this example four surfaces are examinedmdashthe footwall hanging-wall crown (or roof) and end-walls ofthe stope As the principal joint set is flat-lying the two end-walls of the stope are identical cases however if this were notso the stability of each end-wall would need to be investi-gated separately

A35

Fig 1 Diagram of stope layout After Stewart and Forsyth8

Table 1 Rock mass quality data used to determine Q cent value

Item Description Value

Rock quality Good RQD = 85Joint sets One joint set and random Jn = 3Joint roughness Rough or irregular undulating Jr = 3Joint alteration Unaltered with surface staining Ja = 1

eration of isoprobability contours by use of the predicted logitprobability values determined from the logit model

A three-level logit model was used in the investigation ofthe Mathews stability data to reflect the three stability classesLogit models are typically applied to yesno outcomes (givinga binary model) but for the present analysis a third inter-mediate category was used The three-level logit modelproduced two separation lines between the three stabilityclasses The fact that the two lines were parallel is a functionof the three-level model The use of a three-level logit modelto investigate the stability data was verified by running dualbinary logit models to determine the stablendashfailure and thefailurendashmajor failure boundaries separately These boundarieswere found to be very close to parallel

For a binary logit model the dependent variable is assigneda value of either zero or one With three separate stability cat-egories an intermediate value needed to be determinedA value of 05 was used for the failure region which was con-sidered intermediate between stable and major failure Thelogit model was run for different values of the dependent vari-able categories and the distribution of the logit values soobtained was examined closely to consider overlap and spac-ing between the stability classes The values assigned to eachstability class as a result are stable 06ndash1 failure 04ndash06and major failure 0ndash04

Initial values of 1 05 and 0 were used to define the threestability categories and for each point a logit probability valuewas calculated The boundaries between the stability classescan then be determined at an appropriate probability forgiven site conditions or design requirements Alternativelygiven the equal probability lines actual counts of the numberof points in each class can be made giving a distribution ofpoints versus the probability value A graph of these results isa useful way of determining the stability zone boundaries

The inclination position and probabilities of the twoboundaries determined from the two separate binary logitmodels were compared with the two boundaries determinedfrom the single three-level logit model The observed differ-ences in the position and inclination of the boundaries wereminor and the three-level model was adopted

The general form of the logit function can be seen inequation 3 The logit model is similar to the traditional linearregression model or the general linear model for ANOVAexcept that the response is the log odds rather than the metricdependent variable14 In logit modelling the conditional log

odds of the dependent variable are expressed as a linear func-tion of a set of explanatory variables (equations 3 and 4)14 Inthe logit model of the stability data the probability of stabilityis expressed as a linear function of the shape factor theMathews stability number and a constant (equation 5)

To fit the logit model the unknown parameters a b 1 b 2 hellipb k (in equation 3) are estimated by the maximum likelihoodmethod With the use of the estimated values for a b 1 and b 2(equation 5) the estimated probability or predicted risk canbe determined (equation 4) The general form of the logitfunction is

z = a + b 1X1 + b 2X2 + b 3X3 + hellip + b kXk= a + aring b kXk (3)

(4)

where z is predicted log odds value a is a constant b 12 arenumerical coefficients and ƒ(z) is predicted logit probabilityvalue For the case of open-stope stability equation 3becomes

z = a + b 1 lnS + b 2 lnN (5)

The maximum likelihood statistic is used to define the incli-nation of the stability zone boundaries By examination of thecumulative and probability density functions of the logit valuesfor each class the position of the lines delineating each of thestability zones can be optimized according to the needs of par-ticular applications The criteria used to place the zones can bespecified and the resulting probabilities for each stability classwill then be known By using the cumulative density functionsfor each class as determined in Fig 7 the position of the sta-bility zones can be optimized for certain criteriamdashfor exampleminimized misclassification (either numerically or propor-tionally) or similarly a more conservative limit of zeromisclassification This method is a significant improvement onearlier methods of delineating stability zones in that the prob-able stability outcomes for a boundary at any position of thestability graph can be quantified statistically The probabilitydensity functions provide data on the degree of misclassifica-tion of the stability data which is an important consideration

f ze z

( ) =+( )-

1

1

A31

Fig 6 Cumulative frequency graph of logit values for stable failure and major failure points

when evaluating the confidence of a stability estimate and thepotential for misclassification

Delineation of stability zonesA cumulative distribution curve can be plotted for the logitprobability values of the data for each stability class (Fig 6)The inverse cumulative distribution curve of each class is alsoplotted The point of intersection of the stable line and theinverse failure line is termed the crossover point Thecrossover points on the cumulative distribution graph repre-sent the logit probability value that will define the separationline that will have the same proportion of misclassified pointson either side of the line The cumulative frequency graph inFig 6 shows that for a logit probability of 0865 there is a20 mismatch of stable and failure cases This means that20 of the stable points lie below the separation line definedby a logit value of 0865 (in either the failure or the majorfailure zone) and 20 of the failures lie above the line in thestable zone

The true significance of the stablendashfailure line can be deter-mined from the probability density functions for each stabilityclass shown in Fig 7 On the stablendashfailure boundary with alogit value of 0865 there is 48 probability of both stableconditions and failure and a 4 probability of a major failureThis can be directly converted into misclassification propor-tions ie if a designed surface classed as stable plottedexactly on the line there is a 52 probability that it would be

a failure or major failure ie would have been misclassifiedThe positions of the separation lines on the stability graph

were determined from the logit value at the crossover pointson the cumulative frequency plots Data of the type usedhere originating from multiple sources typically include thepossibility of outliers The location of outliers needs to beconsidered because the optimized location of the boundaryobtained from the cumulative plot does not specifically con-sider the distribution of the outliers within the other stabilityzones If a slight shift in the stable boundary would remove asignificant number of failure points from the stable zone for acorrespondingly lower number of stable points moving intothe failure zone a better boundary could exist than the opti-mized one indicated by the cumulative crossover logit valueThe accuracy of the model can be assessed through signifi-cance testing goodness-of-fit measures and by considering

the magnitude of the residuals Measures that have been pro-posed for the assessment of predictive efficiency in logisticregression have been detailed in the work of DeMaris14

Ultimately the decision on the criteria for placing the separa-tion lines will depend on the degree of confidence in theaccuracy of the data and the cost of failure for a givenscenario

The consequences of failure and major failure are highlysite-specific and the level of certainty required in a stableexcavation must be determined by considering the cost of anunplanned failure or major failure The consequences ofmisclassification and the probabilities of a given stabilityoutcome determined from the isoprobability contours willdepend on the data available First sufficient failure andmajor failure case studies must be available to define the sta-bility zone boundaries accurately on the stability graph andsecond adequate site-specific cases are required for accurateprediction of the consequences The lower number of failureand major failure case histories compared with stable caseswithin the stability database causes a range of confidence inthe accuracy of predicted stability zones Fewer case historiesreduce confidence in the results Mining (with the exceptionof caving) is focused principally on the production of stableexcavations The stable zone on the stability graph is definedby the greatest number of case histories and accordingly ourconfidence is highest for designs that utilize this region of thestability graph

Size bias is an issue for the stable failure and major failureclasses within the stability database Mining is focused onstability so the imbalance in subset sizes mirrors the realdistribution of stability cases within operational mines TheMathews stability graph method is moreover a non-rigoroustechnique The quality of the end-productmdashin this case a sta-bility estimate (including probabilities)mdashcan only be as goodas the quality of the underlying data on which the methodwas formulated One suggestion has been to apply a secondoptimization to consider size bias effects on the definition ofthe stability zones This could be undertaken on a larger data-base and investigated by considering a random subsamplefrom the stability database

The logit model allows the direct effect of moving thestability zone boundaries to be quantified in terms of risk orprobability of a specified stability outcome Once a costndashrisk

A32

Fig 7 Probability density functions for stability data determined from logit probability values

relationship has been developed for a site an appropriate levelof risk can be determined and the stability zone boundariesand isoprobability contours on the Mathews graph can betailored to the specific site A costndashrisk analysis is now possi-ble and the optimal design limits can be chosen for individualoperations

Isoprobability contoursAlthough the stability zones can be defined statistically anumber of case histories report to the wrong zones This is to

be expected given the inherent variability of rock massesdata that can be somewhat subjective and the fact that thedesign technique is non-rigorous Diederichs and Kaiser16

proposed the drawing of isoprobability contours to accountfor the uncertainties inherent in the design limitsIsoprobability contours allow the probabilities of stabilityfailure and major failure for a design surface to be obtaineddirectly off the stability graph (an example is given in theAppendix)

The probability density functions have been determined

A33

Fig 8 Isoprobability contours for stable excavation based on logistic regression

Fig 9 Isoprobability contours for failure based on logistic regression

for each of the stability classes (Fig 7) from the predictedlogit values From the probability density functions the iso-probability contours were produced for the three separatelevels of stabilitymdashstable (Fig 8) failure (Fig 9) and majorfailure (Fig 10)

Accuracy and use of extended Mathewsmethod

The Mathews stability graph encompasses a broad range ofopen-stoping experience and is essentially a self-validatingmodel Nevertheless the validity and accuracy of theMathews method depend on the quality and quantity of thestability data that it contains The value of the documentedstability data has been effectively increased by undertaking aleast-squares regression to treat the uncertainties and subjec-tivity of the data

The application of statistical methods in delineating stabil-ity zones and generating isoprobability contours has removedthe subjectivity from the delineation of the zones The appli-cation of regression techniques to a sufficiently large stabilitydatabase is currently the best option for minimizing the influ-ence of subjective data

Major advantages of statistical determination of the zoneson the stability graph are optimization of the design curvesand the ability to evaluate the accuracy of the model Theprecise nature of the risks associated with the position of thestability zone boundaries can be quantified This removes theneed for intermediate transition zones to be defined on thestability graph as general indicators of higher variability inthe stability outcomes Through the logit model the exactmeaning of the stability zone boundaries in terms of theprobability of a given stability outcome is precisely known

Continual improvement of the accuracy of the model forall subsequent modifications is critical for optimization of thefuture use of the Mathews method in stope design Throughstatistical techniques the value of the stability data can bemaximized despite the inherent subjectivity From the resultsobjective interpretations can be made to improve what isacknowledged to be an empirical non-rigorous yet increas-ingly powerful design tool

An example of how to use the Mathews stability graphmethod to assess the stability of an open stope is outlined inthe Appendix

Conclusions

The database for the Mathews method has been extended tomore than 400 case histories such that it now includes open-stoping experience for a broad range of surface geometriesand rock mass conditions The extended Mathews graph is alogndashlog plot with linear stable failure and major failure zonesrather than the traditional logndashlinear format with multiplecurvilinear stability boundaries The extension of the data-base has provided a wide-ranging data-set to which statisticalmethods can be applied

Logistic regression was used to optimize the placement ofstability zones and produce isoprobability contours for theprediction of stope surfaces that are stable or exhibit minor ormajor failures The advantage of logistic regression lies in itsability to minimize the uncertainties reflected in the methodthrough the use of maximum likelihood estimates

The application of statistics does not change the subject-ivity degree of reliability or lack of rigour inherent in theMathews method Statistics can be used to take into accountsome of the inherent variability within the data but the use ofstatistical regression must not be mistaken as adding a greaterlevel of rigour to the Mathews stability graph methodLogistic regression should be viewed as an objective means tocalculate zone boundaries and isoprobability contours for theavailable stability data Through logistic regression the risksassociated with using the technique can be quantified and thestatistical significance of the stability zones becomes known

The authors caution that both the Mathews stability graphmethod and subsequent statistical treatment rely on data thatare not precise and care must be taken not to promote over-confidence in a result without recognizing the nature of thedata underlying the method The preferred approach toincreasing the accuracy of the method is to increase the sizeand quality of the stability database and the authors are pur-suing this

The strength of the extended Mathews method lies in the

A34

Fig 10 Isoprobability contours for major failure based on logistic regression

improved stability graph with statistically determined stabilityzone boundaries and isoprobability contours The uncertain-ties inherent in the design technique can now be quantifiedover a larger range of stope surface geometries and rock massconditions than was previously the case This makes theextended Mathews method a powerful tool for risk assess-ment and optimization of open-stope design

Acknowledgement

The work reported here is a component of research fundedby the Australian Research Council The authors thankProfessor E T Brown for his review of the work

References1 Mathews K E et al Prediction of stable excavation spans formining at depths below 1000 m in hard rock Report to CanadaCentre for Mining and Energy Technology (CANMET)Department of Energy and Resources 19802 Barton N Lien R and Lunde J Engineering classification ofrock masses for design of tunnel support Rock Mechanics 6 1974189ndash2363 Bieniawski Z T Engineering rock mass classifications a completemanual for engineers and geologists in mining civil and petroleum engi-neering (New York Wiley 1989) 251 p4 Potvin Y Empirical open stope design in Canada PhD thesisUniversity of British Columbia 19885 Nickson S D Cable support guidelines for underground hardrock mine operations MAppSc thesis University of BritishColumbia 19926 Hadjigeorgiou J Leclair J and Potvin Y An update of the sta-bility graph method for open stope design Paper presented at CIMRock Mechanics and Strata Control session Halifax Nova Scotia14ndash18 May 19957 Potvin Y Hudyma M R and Miller H D S Design guidelinesfor open stope support CIM Bull 82 no 926 1989 53ndash628 Stewart S B V and Forsyth W W The Mathews method foropen stope design CIM Bull 88 no 992 1995 45ndash539 Trueman R et al Experience in Australia with the Mathewsmethod for open stope design CIM Bull 93 no 1036 2000 162ndash710 Laubscher D H A geomechanics classification system for therating of rock mass in mine design Jl S Afr Inst Min Metall 901990 257ndash7311 Suorineni F T Tannant D D and Kaiser P K Fault factorfor the stability graph method of open-stope design Trans Instn MinMetall (Sect A Min industry) 108 1999 A92ndash10612 Menard S Applied logistic regression analysis (Beverley HillsSage 1995) 98 p Sage University Paper Series on QuantitativeApplications in the Social Sciences no 07-10613 Liao T L Interpreting probability models logit probit andother generalized linear models (Beverley Hills Sage 1984) 88 pSage University Paper Series on Quantitative Applications in the SocialSciences no 07-10114 DeMaris A Logit modelingndashpractical applications (BeverleyHills Sage 1992) 87 p Sage University Paper Series on QuantitativeApplications in the Social Sciences no 07-08615 Holtsberg A Stixboxmdasha statistics toolbox for Matlab andOctave Logitfitm Loddsm and Loddsinvm Available athttpwwwmathslthsematstatstixboxcontentshtml 1999 (cited10200)16 Diederichs M S and Kaiser P K Rock instability and riskanalyses in open stope mine design Can Geotech J 33 1996431ndash9

Appendix

Using the Mathews stability graph method

The use of the stability graph for stability assessment in open-stope design is outlined here in an example from CSA minein Cobar taken from the work of Stewart and Forsyth8 Theexample has been extended to illustrate the use of isopro-bability contours in quantifying the uncertainty of stabilitywithin the stope design process

An open stope is planned at a depth of 1000 m in an ore-body that is 25 m wide and dips at 80deg Because ofoperational constraints the preferred stope length is 30 m and

the stope height is 75 m The geometry of the planned openstope is shown in Fig 1 below

The average unconfined compressive strength of the intactrock is 120 MPa The principal joint set is flat-dipping andclosely spaced (average spacing is 10 cm) The joint surfacesare unaltered but display surface staining Rock mass qualitydata have been collected and are summarized in AppendixTable 1 to the requirements of the NGI Q classificationsystem2

The first consideration when looking at the use of theMathews stability graph in excavation design is to considerhow current conditions compare with the data on which thetechnique was based The non-rigorous and empirical natureof the Mathews method must be considered and the strengthand predictive capability of the method relies accordingly onthe extent of the stability database the method should not beused to predict stability outside the range of experience fromwhich it was developed

The first step in use of the Mathews method at the designstage is to determine the S and N values for each surface ofthe excavation In this example four surfaces are examinedmdashthe footwall hanging-wall crown (or roof) and end-walls ofthe stope As the principal joint set is flat-lying the two end-walls of the stope are identical cases however if this were notso the stability of each end-wall would need to be investi-gated separately

A35

Fig 1 Diagram of stope layout After Stewart and Forsyth8

Table 1 Rock mass quality data used to determine Q cent value

Item Description Value

Rock quality Good RQD = 85Joint sets One joint set and random Jn = 3Joint roughness Rough or irregular undulating Jr = 3Joint alteration Unaltered with surface staining Ja = 1

when evaluating the confidence of a stability estimate and thepotential for misclassification

Delineation of stability zonesA cumulative distribution curve can be plotted for the logitprobability values of the data for each stability class (Fig 6)The inverse cumulative distribution curve of each class is alsoplotted The point of intersection of the stable line and theinverse failure line is termed the crossover point Thecrossover points on the cumulative distribution graph repre-sent the logit probability value that will define the separationline that will have the same proportion of misclassified pointson either side of the line The cumulative frequency graph inFig 6 shows that for a logit probability of 0865 there is a20 mismatch of stable and failure cases This means that20 of the stable points lie below the separation line definedby a logit value of 0865 (in either the failure or the majorfailure zone) and 20 of the failures lie above the line in thestable zone

The true significance of the stablendashfailure line can be deter-mined from the probability density functions for each stabilityclass shown in Fig 7 On the stablendashfailure boundary with alogit value of 0865 there is 48 probability of both stableconditions and failure and a 4 probability of a major failureThis can be directly converted into misclassification propor-tions ie if a designed surface classed as stable plottedexactly on the line there is a 52 probability that it would be

a failure or major failure ie would have been misclassifiedThe positions of the separation lines on the stability graph

were determined from the logit value at the crossover pointson the cumulative frequency plots Data of the type usedhere originating from multiple sources typically include thepossibility of outliers The location of outliers needs to beconsidered because the optimized location of the boundaryobtained from the cumulative plot does not specifically con-sider the distribution of the outliers within the other stabilityzones If a slight shift in the stable boundary would remove asignificant number of failure points from the stable zone for acorrespondingly lower number of stable points moving intothe failure zone a better boundary could exist than the opti-mized one indicated by the cumulative crossover logit valueThe accuracy of the model can be assessed through signifi-cance testing goodness-of-fit measures and by considering

the magnitude of the residuals Measures that have been pro-posed for the assessment of predictive efficiency in logisticregression have been detailed in the work of DeMaris14

Ultimately the decision on the criteria for placing the separa-tion lines will depend on the degree of confidence in theaccuracy of the data and the cost of failure for a givenscenario

The consequences of failure and major failure are highlysite-specific and the level of certainty required in a stableexcavation must be determined by considering the cost of anunplanned failure or major failure The consequences ofmisclassification and the probabilities of a given stabilityoutcome determined from the isoprobability contours willdepend on the data available First sufficient failure andmajor failure case studies must be available to define the sta-bility zone boundaries accurately on the stability graph andsecond adequate site-specific cases are required for accurateprediction of the consequences The lower number of failureand major failure case histories compared with stable caseswithin the stability database causes a range of confidence inthe accuracy of predicted stability zones Fewer case historiesreduce confidence in the results Mining (with the exceptionof caving) is focused principally on the production of stableexcavations The stable zone on the stability graph is definedby the greatest number of case histories and accordingly ourconfidence is highest for designs that utilize this region of thestability graph

Size bias is an issue for the stable failure and major failureclasses within the stability database Mining is focused onstability so the imbalance in subset sizes mirrors the realdistribution of stability cases within operational mines TheMathews stability graph method is moreover a non-rigoroustechnique The quality of the end-productmdashin this case a sta-bility estimate (including probabilities)mdashcan only be as goodas the quality of the underlying data on which the methodwas formulated One suggestion has been to apply a secondoptimization to consider size bias effects on the definition ofthe stability zones This could be undertaken on a larger data-base and investigated by considering a random subsamplefrom the stability database

The logit model allows the direct effect of moving thestability zone boundaries to be quantified in terms of risk orprobability of a specified stability outcome Once a costndashrisk

A32

Fig 7 Probability density functions for stability data determined from logit probability values

relationship has been developed for a site an appropriate levelof risk can be determined and the stability zone boundariesand isoprobability contours on the Mathews graph can betailored to the specific site A costndashrisk analysis is now possi-ble and the optimal design limits can be chosen for individualoperations

Isoprobability contoursAlthough the stability zones can be defined statistically anumber of case histories report to the wrong zones This is to

be expected given the inherent variability of rock massesdata that can be somewhat subjective and the fact that thedesign technique is non-rigorous Diederichs and Kaiser16

proposed the drawing of isoprobability contours to accountfor the uncertainties inherent in the design limitsIsoprobability contours allow the probabilities of stabilityfailure and major failure for a design surface to be obtaineddirectly off the stability graph (an example is given in theAppendix)

The probability density functions have been determined

A33

Fig 8 Isoprobability contours for stable excavation based on logistic regression

Fig 9 Isoprobability contours for failure based on logistic regression

for each of the stability classes (Fig 7) from the predictedlogit values From the probability density functions the iso-probability contours were produced for the three separatelevels of stabilitymdashstable (Fig 8) failure (Fig 9) and majorfailure (Fig 10)

Accuracy and use of extended Mathewsmethod

The Mathews stability graph encompasses a broad range ofopen-stoping experience and is essentially a self-validatingmodel Nevertheless the validity and accuracy of theMathews method depend on the quality and quantity of thestability data that it contains The value of the documentedstability data has been effectively increased by undertaking aleast-squares regression to treat the uncertainties and subjec-tivity of the data

The application of statistical methods in delineating stabil-ity zones and generating isoprobability contours has removedthe subjectivity from the delineation of the zones The appli-cation of regression techniques to a sufficiently large stabilitydatabase is currently the best option for minimizing the influ-ence of subjective data

Major advantages of statistical determination of the zoneson the stability graph are optimization of the design curvesand the ability to evaluate the accuracy of the model Theprecise nature of the risks associated with the position of thestability zone boundaries can be quantified This removes theneed for intermediate transition zones to be defined on thestability graph as general indicators of higher variability inthe stability outcomes Through the logit model the exactmeaning of the stability zone boundaries in terms of theprobability of a given stability outcome is precisely known

Continual improvement of the accuracy of the model forall subsequent modifications is critical for optimization of thefuture use of the Mathews method in stope design Throughstatistical techniques the value of the stability data can bemaximized despite the inherent subjectivity From the resultsobjective interpretations can be made to improve what isacknowledged to be an empirical non-rigorous yet increas-ingly powerful design tool

An example of how to use the Mathews stability graphmethod to assess the stability of an open stope is outlined inthe Appendix

Conclusions

The database for the Mathews method has been extended tomore than 400 case histories such that it now includes open-stoping experience for a broad range of surface geometriesand rock mass conditions The extended Mathews graph is alogndashlog plot with linear stable failure and major failure zonesrather than the traditional logndashlinear format with multiplecurvilinear stability boundaries The extension of the data-base has provided a wide-ranging data-set to which statisticalmethods can be applied

Logistic regression was used to optimize the placement ofstability zones and produce isoprobability contours for theprediction of stope surfaces that are stable or exhibit minor ormajor failures The advantage of logistic regression lies in itsability to minimize the uncertainties reflected in the methodthrough the use of maximum likelihood estimates

The application of statistics does not change the subject-ivity degree of reliability or lack of rigour inherent in theMathews method Statistics can be used to take into accountsome of the inherent variability within the data but the use ofstatistical regression must not be mistaken as adding a greaterlevel of rigour to the Mathews stability graph methodLogistic regression should be viewed as an objective means tocalculate zone boundaries and isoprobability contours for theavailable stability data Through logistic regression the risksassociated with using the technique can be quantified and thestatistical significance of the stability zones becomes known

The authors caution that both the Mathews stability graphmethod and subsequent statistical treatment rely on data thatare not precise and care must be taken not to promote over-confidence in a result without recognizing the nature of thedata underlying the method The preferred approach toincreasing the accuracy of the method is to increase the sizeand quality of the stability database and the authors are pur-suing this

The strength of the extended Mathews method lies in the

A34

Fig 10 Isoprobability contours for major failure based on logistic regression

improved stability graph with statistically determined stabilityzone boundaries and isoprobability contours The uncertain-ties inherent in the design technique can now be quantifiedover a larger range of stope surface geometries and rock massconditions than was previously the case This makes theextended Mathews method a powerful tool for risk assess-ment and optimization of open-stope design

Acknowledgement

The work reported here is a component of research fundedby the Australian Research Council The authors thankProfessor E T Brown for his review of the work

References1 Mathews K E et al Prediction of stable excavation spans formining at depths below 1000 m in hard rock Report to CanadaCentre for Mining and Energy Technology (CANMET)Department of Energy and Resources 19802 Barton N Lien R and Lunde J Engineering classification ofrock masses for design of tunnel support Rock Mechanics 6 1974189ndash2363 Bieniawski Z T Engineering rock mass classifications a completemanual for engineers and geologists in mining civil and petroleum engi-neering (New York Wiley 1989) 251 p4 Potvin Y Empirical open stope design in Canada PhD thesisUniversity of British Columbia 19885 Nickson S D Cable support guidelines for underground hardrock mine operations MAppSc thesis University of BritishColumbia 19926 Hadjigeorgiou J Leclair J and Potvin Y An update of the sta-bility graph method for open stope design Paper presented at CIMRock Mechanics and Strata Control session Halifax Nova Scotia14ndash18 May 19957 Potvin Y Hudyma M R and Miller H D S Design guidelinesfor open stope support CIM Bull 82 no 926 1989 53ndash628 Stewart S B V and Forsyth W W The Mathews method foropen stope design CIM Bull 88 no 992 1995 45ndash539 Trueman R et al Experience in Australia with the Mathewsmethod for open stope design CIM Bull 93 no 1036 2000 162ndash710 Laubscher D H A geomechanics classification system for therating of rock mass in mine design Jl S Afr Inst Min Metall 901990 257ndash7311 Suorineni F T Tannant D D and Kaiser P K Fault factorfor the stability graph method of open-stope design Trans Instn MinMetall (Sect A Min industry) 108 1999 A92ndash10612 Menard S Applied logistic regression analysis (Beverley HillsSage 1995) 98 p Sage University Paper Series on QuantitativeApplications in the Social Sciences no 07-10613 Liao T L Interpreting probability models logit probit andother generalized linear models (Beverley Hills Sage 1984) 88 pSage University Paper Series on Quantitative Applications in the SocialSciences no 07-10114 DeMaris A Logit modelingndashpractical applications (BeverleyHills Sage 1992) 87 p Sage University Paper Series on QuantitativeApplications in the Social Sciences no 07-08615 Holtsberg A Stixboxmdasha statistics toolbox for Matlab andOctave Logitfitm Loddsm and Loddsinvm Available athttpwwwmathslthsematstatstixboxcontentshtml 1999 (cited10200)16 Diederichs M S and Kaiser P K Rock instability and riskanalyses in open stope mine design Can Geotech J 33 1996431ndash9

Appendix

Using the Mathews stability graph method

The use of the stability graph for stability assessment in open-stope design is outlined here in an example from CSA minein Cobar taken from the work of Stewart and Forsyth8 Theexample has been extended to illustrate the use of isopro-bability contours in quantifying the uncertainty of stabilitywithin the stope design process

An open stope is planned at a depth of 1000 m in an ore-body that is 25 m wide and dips at 80deg Because ofoperational constraints the preferred stope length is 30 m and

the stope height is 75 m The geometry of the planned openstope is shown in Fig 1 below

The average unconfined compressive strength of the intactrock is 120 MPa The principal joint set is flat-dipping andclosely spaced (average spacing is 10 cm) The joint surfacesare unaltered but display surface staining Rock mass qualitydata have been collected and are summarized in AppendixTable 1 to the requirements of the NGI Q classificationsystem2

The first consideration when looking at the use of theMathews stability graph in excavation design is to considerhow current conditions compare with the data on which thetechnique was based The non-rigorous and empirical natureof the Mathews method must be considered and the strengthand predictive capability of the method relies accordingly onthe extent of the stability database the method should not beused to predict stability outside the range of experience fromwhich it was developed

The first step in use of the Mathews method at the designstage is to determine the S and N values for each surface ofthe excavation In this example four surfaces are examinedmdashthe footwall hanging-wall crown (or roof) and end-walls ofthe stope As the principal joint set is flat-lying the two end-walls of the stope are identical cases however if this were notso the stability of each end-wall would need to be investi-gated separately

A35

Fig 1 Diagram of stope layout After Stewart and Forsyth8

Table 1 Rock mass quality data used to determine Q cent value

Item Description Value

Rock quality Good RQD = 85Joint sets One joint set and random Jn = 3Joint roughness Rough or irregular undulating Jr = 3Joint alteration Unaltered with surface staining Ja = 1

relationship has been developed for a site an appropriate levelof risk can be determined and the stability zone boundariesand isoprobability contours on the Mathews graph can betailored to the specific site A costndashrisk analysis is now possi-ble and the optimal design limits can be chosen for individualoperations

Isoprobability contoursAlthough the stability zones can be defined statistically anumber of case histories report to the wrong zones This is to

be expected given the inherent variability of rock massesdata that can be somewhat subjective and the fact that thedesign technique is non-rigorous Diederichs and Kaiser16

proposed the drawing of isoprobability contours to accountfor the uncertainties inherent in the design limitsIsoprobability contours allow the probabilities of stabilityfailure and major failure for a design surface to be obtaineddirectly off the stability graph (an example is given in theAppendix)

The probability density functions have been determined

A33

Fig 8 Isoprobability contours for stable excavation based on logistic regression

Fig 9 Isoprobability contours for failure based on logistic regression

for each of the stability classes (Fig 7) from the predictedlogit values From the probability density functions the iso-probability contours were produced for the three separatelevels of stabilitymdashstable (Fig 8) failure (Fig 9) and majorfailure (Fig 10)

Accuracy and use of extended Mathewsmethod

The Mathews stability graph encompasses a broad range ofopen-stoping experience and is essentially a self-validatingmodel Nevertheless the validity and accuracy of theMathews method depend on the quality and quantity of thestability data that it contains The value of the documentedstability data has been effectively increased by undertaking aleast-squares regression to treat the uncertainties and subjec-tivity of the data

The application of statistical methods in delineating stabil-ity zones and generating isoprobability contours has removedthe subjectivity from the delineation of the zones The appli-cation of regression techniques to a sufficiently large stabilitydatabase is currently the best option for minimizing the influ-ence of subjective data

Major advantages of statistical determination of the zoneson the stability graph are optimization of the design curvesand the ability to evaluate the accuracy of the model Theprecise nature of the risks associated with the position of thestability zone boundaries can be quantified This removes theneed for intermediate transition zones to be defined on thestability graph as general indicators of higher variability inthe stability outcomes Through the logit model the exactmeaning of the stability zone boundaries in terms of theprobability of a given stability outcome is precisely known

Continual improvement of the accuracy of the model forall subsequent modifications is critical for optimization of thefuture use of the Mathews method in stope design Throughstatistical techniques the value of the stability data can bemaximized despite the inherent subjectivity From the resultsobjective interpretations can be made to improve what isacknowledged to be an empirical non-rigorous yet increas-ingly powerful design tool

An example of how to use the Mathews stability graphmethod to assess the stability of an open stope is outlined inthe Appendix

Conclusions

The database for the Mathews method has been extended tomore than 400 case histories such that it now includes open-stoping experience for a broad range of surface geometriesand rock mass conditions The extended Mathews graph is alogndashlog plot with linear stable failure and major failure zonesrather than the traditional logndashlinear format with multiplecurvilinear stability boundaries The extension of the data-base has provided a wide-ranging data-set to which statisticalmethods can be applied

Logistic regression was used to optimize the placement ofstability zones and produce isoprobability contours for theprediction of stope surfaces that are stable or exhibit minor ormajor failures The advantage of logistic regression lies in itsability to minimize the uncertainties reflected in the methodthrough the use of maximum likelihood estimates

The application of statistics does not change the subject-ivity degree of reliability or lack of rigour inherent in theMathews method Statistics can be used to take into accountsome of the inherent variability within the data but the use ofstatistical regression must not be mistaken as adding a greaterlevel of rigour to the Mathews stability graph methodLogistic regression should be viewed as an objective means tocalculate zone boundaries and isoprobability contours for theavailable stability data Through logistic regression the risksassociated with using the technique can be quantified and thestatistical significance of the stability zones becomes known

The authors caution that both the Mathews stability graphmethod and subsequent statistical treatment rely on data thatare not precise and care must be taken not to promote over-confidence in a result without recognizing the nature of thedata underlying the method The preferred approach toincreasing the accuracy of the method is to increase the sizeand quality of the stability database and the authors are pur-suing this

The strength of the extended Mathews method lies in the

A34

Fig 10 Isoprobability contours for major failure based on logistic regression

improved stability graph with statistically determined stabilityzone boundaries and isoprobability contours The uncertain-ties inherent in the design technique can now be quantifiedover a larger range of stope surface geometries and rock massconditions than was previously the case This makes theextended Mathews method a powerful tool for risk assess-ment and optimization of open-stope design

Acknowledgement

The work reported here is a component of research fundedby the Australian Research Council The authors thankProfessor E T Brown for his review of the work

References1 Mathews K E et al Prediction of stable excavation spans formining at depths below 1000 m in hard rock Report to CanadaCentre for Mining and Energy Technology (CANMET)Department of Energy and Resources 19802 Barton N Lien R and Lunde J Engineering classification ofrock masses for design of tunnel support Rock Mechanics 6 1974189ndash2363 Bieniawski Z T Engineering rock mass classifications a completemanual for engineers and geologists in mining civil and petroleum engi-neering (New York Wiley 1989) 251 p4 Potvin Y Empirical open stope design in Canada PhD thesisUniversity of British Columbia 19885 Nickson S D Cable support guidelines for underground hardrock mine operations MAppSc thesis University of BritishColumbia 19926 Hadjigeorgiou J Leclair J and Potvin Y An update of the sta-bility graph method for open stope design Paper presented at CIMRock Mechanics and Strata Control session Halifax Nova Scotia14ndash18 May 19957 Potvin Y Hudyma M R and Miller H D S Design guidelinesfor open stope support CIM Bull 82 no 926 1989 53ndash628 Stewart S B V and Forsyth W W The Mathews method foropen stope design CIM Bull 88 no 992 1995 45ndash539 Trueman R et al Experience in Australia with the Mathewsmethod for open stope design CIM Bull 93 no 1036 2000 162ndash710 Laubscher D H A geomechanics classification system for therating of rock mass in mine design Jl S Afr Inst Min Metall 901990 257ndash7311 Suorineni F T Tannant D D and Kaiser P K Fault factorfor the stability graph method of open-stope design Trans Instn MinMetall (Sect A Min industry) 108 1999 A92ndash10612 Menard S Applied logistic regression analysis (Beverley HillsSage 1995) 98 p Sage University Paper Series on QuantitativeApplications in the Social Sciences no 07-10613 Liao T L Interpreting probability models logit probit andother generalized linear models (Beverley Hills Sage 1984) 88 pSage University Paper Series on Quantitative Applications in the SocialSciences no 07-10114 DeMaris A Logit modelingndashpractical applications (BeverleyHills Sage 1992) 87 p Sage University Paper Series on QuantitativeApplications in the Social Sciences no 07-08615 Holtsberg A Stixboxmdasha statistics toolbox for Matlab andOctave Logitfitm Loddsm and Loddsinvm Available athttpwwwmathslthsematstatstixboxcontentshtml 1999 (cited10200)16 Diederichs M S and Kaiser P K Rock instability and riskanalyses in open stope mine design Can Geotech J 33 1996431ndash9

Appendix

Using the Mathews stability graph method

The use of the stability graph for stability assessment in open-stope design is outlined here in an example from CSA minein Cobar taken from the work of Stewart and Forsyth8 Theexample has been extended to illustrate the use of isopro-bability contours in quantifying the uncertainty of stabilitywithin the stope design process

An open stope is planned at a depth of 1000 m in an ore-body that is 25 m wide and dips at 80deg Because ofoperational constraints the preferred stope length is 30 m and

the stope height is 75 m The geometry of the planned openstope is shown in Fig 1 below

The average unconfined compressive strength of the intactrock is 120 MPa The principal joint set is flat-dipping andclosely spaced (average spacing is 10 cm) The joint surfacesare unaltered but display surface staining Rock mass qualitydata have been collected and are summarized in AppendixTable 1 to the requirements of the NGI Q classificationsystem2

The first consideration when looking at the use of theMathews stability graph in excavation design is to considerhow current conditions compare with the data on which thetechnique was based The non-rigorous and empirical natureof the Mathews method must be considered and the strengthand predictive capability of the method relies accordingly onthe extent of the stability database the method should not beused to predict stability outside the range of experience fromwhich it was developed

The first step in use of the Mathews method at the designstage is to determine the S and N values for each surface ofthe excavation In this example four surfaces are examinedmdashthe footwall hanging-wall crown (or roof) and end-walls ofthe stope As the principal joint set is flat-lying the two end-walls of the stope are identical cases however if this were notso the stability of each end-wall would need to be investi-gated separately

A35

Fig 1 Diagram of stope layout After Stewart and Forsyth8

Table 1 Rock mass quality data used to determine Q cent value

Item Description Value

Rock quality Good RQD = 85Joint sets One joint set and random Jn = 3Joint roughness Rough or irregular undulating Jr = 3Joint alteration Unaltered with surface staining Ja = 1

for each of the stability classes (Fig 7) from the predictedlogit values From the probability density functions the iso-probability contours were produced for the three separatelevels of stabilitymdashstable (Fig 8) failure (Fig 9) and majorfailure (Fig 10)

Accuracy and use of extended Mathewsmethod

The Mathews stability graph encompasses a broad range ofopen-stoping experience and is essentially a self-validatingmodel Nevertheless the validity and accuracy of theMathews method depend on the quality and quantity of thestability data that it contains The value of the documentedstability data has been effectively increased by undertaking aleast-squares regression to treat the uncertainties and subjec-tivity of the data

The application of statistical methods in delineating stabil-ity zones and generating isoprobability contours has removedthe subjectivity from the delineation of the zones The appli-cation of regression techniques to a sufficiently large stabilitydatabase is currently the best option for minimizing the influ-ence of subjective data

Major advantages of statistical determination of the zoneson the stability graph are optimization of the design curvesand the ability to evaluate the accuracy of the model Theprecise nature of the risks associated with the position of thestability zone boundaries can be quantified This removes theneed for intermediate transition zones to be defined on thestability graph as general indicators of higher variability inthe stability outcomes Through the logit model the exactmeaning of the stability zone boundaries in terms of theprobability of a given stability outcome is precisely known

Continual improvement of the accuracy of the model forall subsequent modifications is critical for optimization of thefuture use of the Mathews method in stope design Throughstatistical techniques the value of the stability data can bemaximized despite the inherent subjectivity From the resultsobjective interpretations can be made to improve what isacknowledged to be an empirical non-rigorous yet increas-ingly powerful design tool

An example of how to use the Mathews stability graphmethod to assess the stability of an open stope is outlined inthe Appendix

Conclusions

The database for the Mathews method has been extended tomore than 400 case histories such that it now includes open-stoping experience for a broad range of surface geometriesand rock mass conditions The extended Mathews graph is alogndashlog plot with linear stable failure and major failure zonesrather than the traditional logndashlinear format with multiplecurvilinear stability boundaries The extension of the data-base has provided a wide-ranging data-set to which statisticalmethods can be applied

Logistic regression was used to optimize the placement ofstability zones and produce isoprobability contours for theprediction of stope surfaces that are stable or exhibit minor ormajor failures The advantage of logistic regression lies in itsability to minimize the uncertainties reflected in the methodthrough the use of maximum likelihood estimates

The application of statistics does not change the subject-ivity degree of reliability or lack of rigour inherent in theMathews method Statistics can be used to take into accountsome of the inherent variability within the data but the use ofstatistical regression must not be mistaken as adding a greaterlevel of rigour to the Mathews stability graph methodLogistic regression should be viewed as an objective means tocalculate zone boundaries and isoprobability contours for theavailable stability data Through logistic regression the risksassociated with using the technique can be quantified and thestatistical significance of the stability zones becomes known

The authors caution that both the Mathews stability graphmethod and subsequent statistical treatment rely on data thatare not precise and care must be taken not to promote over-confidence in a result without recognizing the nature of thedata underlying the method The preferred approach toincreasing the accuracy of the method is to increase the sizeand quality of the stability database and the authors are pur-suing this

The strength of the extended Mathews method lies in the

A34

Fig 10 Isoprobability contours for major failure based on logistic regression

improved stability graph with statistically determined stabilityzone boundaries and isoprobability contours The uncertain-ties inherent in the design technique can now be quantifiedover a larger range of stope surface geometries and rock massconditions than was previously the case This makes theextended Mathews method a powerful tool for risk assess-ment and optimization of open-stope design

Acknowledgement

The work reported here is a component of research fundedby the Australian Research Council The authors thankProfessor E T Brown for his review of the work

References1 Mathews K E et al Prediction of stable excavation spans formining at depths below 1000 m in hard rock Report to CanadaCentre for Mining and Energy Technology (CANMET)Department of Energy and Resources 19802 Barton N Lien R and Lunde J Engineering classification ofrock masses for design of tunnel support Rock Mechanics 6 1974189ndash2363 Bieniawski Z T Engineering rock mass classifications a completemanual for engineers and geologists in mining civil and petroleum engi-neering (New York Wiley 1989) 251 p4 Potvin Y Empirical open stope design in Canada PhD thesisUniversity of British Columbia 19885 Nickson S D Cable support guidelines for underground hardrock mine operations MAppSc thesis University of BritishColumbia 19926 Hadjigeorgiou J Leclair J and Potvin Y An update of the sta-bility graph method for open stope design Paper presented at CIMRock Mechanics and Strata Control session Halifax Nova Scotia14ndash18 May 19957 Potvin Y Hudyma M R and Miller H D S Design guidelinesfor open stope support CIM Bull 82 no 926 1989 53ndash628 Stewart S B V and Forsyth W W The Mathews method foropen stope design CIM Bull 88 no 992 1995 45ndash539 Trueman R et al Experience in Australia with the Mathewsmethod for open stope design CIM Bull 93 no 1036 2000 162ndash710 Laubscher D H A geomechanics classification system for therating of rock mass in mine design Jl S Afr Inst Min Metall 901990 257ndash7311 Suorineni F T Tannant D D and Kaiser P K Fault factorfor the stability graph method of open-stope design Trans Instn MinMetall (Sect A Min industry) 108 1999 A92ndash10612 Menard S Applied logistic regression analysis (Beverley HillsSage 1995) 98 p Sage University Paper Series on QuantitativeApplications in the Social Sciences no 07-10613 Liao T L Interpreting probability models logit probit andother generalized linear models (Beverley Hills Sage 1984) 88 pSage University Paper Series on Quantitative Applications in the SocialSciences no 07-10114 DeMaris A Logit modelingndashpractical applications (BeverleyHills Sage 1992) 87 p Sage University Paper Series on QuantitativeApplications in the Social Sciences no 07-08615 Holtsberg A Stixboxmdasha statistics toolbox for Matlab andOctave Logitfitm Loddsm and Loddsinvm Available athttpwwwmathslthsematstatstixboxcontentshtml 1999 (cited10200)16 Diederichs M S and Kaiser P K Rock instability and riskanalyses in open stope mine design Can Geotech J 33 1996431ndash9

Appendix

Using the Mathews stability graph method

The use of the stability graph for stability assessment in open-stope design is outlined here in an example from CSA minein Cobar taken from the work of Stewart and Forsyth8 Theexample has been extended to illustrate the use of isopro-bability contours in quantifying the uncertainty of stabilitywithin the stope design process

An open stope is planned at a depth of 1000 m in an ore-body that is 25 m wide and dips at 80deg Because ofoperational constraints the preferred stope length is 30 m and

the stope height is 75 m The geometry of the planned openstope is shown in Fig 1 below

The average unconfined compressive strength of the intactrock is 120 MPa The principal joint set is flat-dipping andclosely spaced (average spacing is 10 cm) The joint surfacesare unaltered but display surface staining Rock mass qualitydata have been collected and are summarized in AppendixTable 1 to the requirements of the NGI Q classificationsystem2

The first consideration when looking at the use of theMathews stability graph in excavation design is to considerhow current conditions compare with the data on which thetechnique was based The non-rigorous and empirical natureof the Mathews method must be considered and the strengthand predictive capability of the method relies accordingly onthe extent of the stability database the method should not beused to predict stability outside the range of experience fromwhich it was developed

The first step in use of the Mathews method at the designstage is to determine the S and N values for each surface ofthe excavation In this example four surfaces are examinedmdashthe footwall hanging-wall crown (or roof) and end-walls ofthe stope As the principal joint set is flat-lying the two end-walls of the stope are identical cases however if this were notso the stability of each end-wall would need to be investi-gated separately

A35

Fig 1 Diagram of stope layout After Stewart and Forsyth8

Table 1 Rock mass quality data used to determine Q cent value

Item Description Value

Rock quality Good RQD = 85Joint sets One joint set and random Jn = 3Joint roughness Rough or irregular undulating Jr = 3Joint alteration Unaltered with surface staining Ja = 1

improved stability graph with statistically determined stabilityzone boundaries and isoprobability contours The uncertain-ties inherent in the design technique can now be quantifiedover a larger range of stope surface geometries and rock massconditions than was previously the case This makes theextended Mathews method a powerful tool for risk assess-ment and optimization of open-stope design

Acknowledgement

The work reported here is a component of research fundedby the Australian Research Council The authors thankProfessor E T Brown for his review of the work

References1 Mathews K E et al Prediction of stable excavation spans formining at depths below 1000 m in hard rock Report to CanadaCentre for Mining and Energy Technology (CANMET)Department of Energy and Resources 19802 Barton N Lien R and Lunde J Engineering classification ofrock masses for design of tunnel support Rock Mechanics 6 1974189ndash2363 Bieniawski Z T Engineering rock mass classifications a completemanual for engineers and geologists in mining civil and petroleum engi-neering (New York Wiley 1989) 251 p4 Potvin Y Empirical open stope design in Canada PhD thesisUniversity of British Columbia 19885 Nickson S D Cable support guidelines for underground hardrock mine operations MAppSc thesis University of BritishColumbia 19926 Hadjigeorgiou J Leclair J and Potvin Y An update of the sta-bility graph method for open stope design Paper presented at CIMRock Mechanics and Strata Control session Halifax Nova Scotia14ndash18 May 19957 Potvin Y Hudyma M R and Miller H D S Design guidelinesfor open stope support CIM Bull 82 no 926 1989 53ndash628 Stewart S B V and Forsyth W W The Mathews method foropen stope design CIM Bull 88 no 992 1995 45ndash539 Trueman R et al Experience in Australia with the Mathewsmethod for open stope design CIM Bull 93 no 1036 2000 162ndash710 Laubscher D H A geomechanics classification system for therating of rock mass in mine design Jl S Afr Inst Min Metall 901990 257ndash7311 Suorineni F T Tannant D D and Kaiser P K Fault factorfor the stability graph method of open-stope design Trans Instn MinMetall (Sect A Min industry) 108 1999 A92ndash10612 Menard S Applied logistic regression analysis (Beverley HillsSage 1995) 98 p Sage University Paper Series on QuantitativeApplications in the Social Sciences no 07-10613 Liao T L Interpreting probability models logit probit andother generalized linear models (Beverley Hills Sage 1984) 88 pSage University Paper Series on Quantitative Applications in the SocialSciences no 07-10114 DeMaris A Logit modelingndashpractical applications (BeverleyHills Sage 1992) 87 p Sage University Paper Series on QuantitativeApplications in the Social Sciences no 07-08615 Holtsberg A Stixboxmdasha statistics toolbox for Matlab andOctave Logitfitm Loddsm and Loddsinvm Available athttpwwwmathslthsematstatstixboxcontentshtml 1999 (cited10200)16 Diederichs M S and Kaiser P K Rock instability and riskanalyses in open stope mine design Can Geotech J 33 1996431ndash9

Appendix

Using the Mathews stability graph method

The use of the stability graph for stability assessment in open-stope design is outlined here in an example from CSA minein Cobar taken from the work of Stewart and Forsyth8 Theexample has been extended to illustrate the use of isopro-bability contours in quantifying the uncertainty of stabilitywithin the stope design process

An open stope is planned at a depth of 1000 m in an ore-body that is 25 m wide and dips at 80deg Because ofoperational constraints the preferred stope length is 30 m and

the stope height is 75 m The geometry of the planned openstope is shown in Fig 1 below

The average unconfined compressive strength of the intactrock is 120 MPa The principal joint set is flat-dipping andclosely spaced (average spacing is 10 cm) The joint surfacesare unaltered but display surface staining Rock mass qualitydata have been collected and are summarized in AppendixTable 1 to the requirements of the NGI Q classificationsystem2

The first consideration when looking at the use of theMathews stability graph in excavation design is to considerhow current conditions compare with the data on which thetechnique was based The non-rigorous and empirical natureof the Mathews method must be considered and the strengthand predictive capability of the method relies accordingly onthe extent of the stability database the method should not beused to predict stability outside the range of experience fromwhich it was developed

The first step in use of the Mathews method at the designstage is to determine the S and N values for each surface ofthe excavation In this example four surfaces are examinedmdashthe footwall hanging-wall crown (or roof) and end-walls ofthe stope As the principal joint set is flat-lying the two end-walls of the stope are identical cases however if this were notso the stability of each end-wall would need to be investi-gated separately

A35

Fig 1 Diagram of stope layout After Stewart and Forsyth8

Table 1 Rock mass quality data used to determine Q cent value

Item Description Value

Rock quality Good RQD = 85Joint sets One joint set and random Jn = 3Joint roughness Rough or irregular undulating Jr = 3Joint alteration Unaltered with surface staining Ja = 1

Q cent valueApplication of equation 1 of the main text to the rock massquality data in Table 1 yields a Q cent value of 85

Shape factorThe shape factor is the ratio of the area of the stope surface tothe length of the perimeter of the surface The S values calcu-lated for the stope surfaces are given in Table 2

Stress factorTo determine the stress factor for a surface the inducedstresses at the mid-point of the surface must first be calcu-lated The induced stresses are rarely measured and thusmust usually be estimated If virgin in-situ stress measure-ments have been undertaken these should be used as thebasis for calculation of the induced stresses If this is not thecase the in-situ stress state must be estimated from regionalconditions The induced stresses can be determined fromelastic formulae stress distribution plots design curves or byrunning a simple elastic model in an appropriate numericalmodelling package

In this example the in-situ stresses have not been measuredso the stress values must be estimated As the stope is at adepth of 1000 m the vertical stress is estimated to be27 MPa The ratio of the average horizontal to vertical stressK is estimated to be 14 This means that the average hori-zontal stress is calculated to be 38 MPa The in-situ stressesused in determining the stress factor are shown in Fig 2

The next step after estimation of the in-situ stresses beforethe stope is extracted is to determine the induced stresses foreach surface once the stope has been mined (Fig 2) Themagnitude of the induced stresses relative to the unconfinedcompressive strength of the rock is an important componentof the stability assessment and is used to determine the rockstress factor

CrownConsidering the top of the mid-stope vertical plane

s V = 27 MPa s H2= 38 MPa

K = s H2 s V = 14

Surface height = 75 Surface span = 25Height to span ratio = 3

With a height to span ratio of 3 and a K value of 14 (fromFig 3) s 1s V is estimated at 26 From this relationship it canbe calculated that

s 1 = 26 27 = 70 MPa

Once s 1 is known the ratio of the unconfined compressivestrength of the rock to the induced stresses can be calculatedin this case

s cs 1 = 12070 = 17

A36

Table 2 Calculated S values

Stope surface Area m2 Perimeter m Shape factor S m

Crown 750 110 68Hanging-wall 750 210 107Footwall 2250 210 107Endwall 1875 200 94

Fig 2 In-situ (virgin) and induced stress diagrams showing mid-stope planes used to calculated rockstress factors for each surface After Stewart and Forsyth8

Fig 3 Curves for estimation of induced stresses in backs and end-walls After Stewart and Forsyth8

The final stage in determining the magnitude of the stressfactor consists in applying the value of s cs 1 to Fig 2 andreading off the rock stress factor which in this case is 01

End-wallConsidering the strike end of the mid-stope horizontal plane

s H1= 38 MPa s H2

= 38 MPaK = s H2

s H1= 1

Surface height = 30 Surface span = 25Height to span ratio = 12

For a height to span ratio of 12 and a K value of 1 s 1s H1is

estimated at 10 (from Fig 3) Accordingly

s 1 = 10 38 = 38 MPa

By reference to Fig 2

s cs 1 = 12038 = 32

and the rock stress factor for the end-wall is 025

Hanging-wall and footwallThe next step is to determine the induced stresses in thehanging-wall and footwall considering the vertical and hori-zontal mid-stope planes In the case where two estimates ofthe stress factor A are obtained (downdip and along strike)the lower value is used

Considering the mid-stope vertical plane (downdip)

s V = 27 MPa s H2= 38 MPa

K = s H2 s V = 14

Surface height = 75 Surface span = 25Height to span ratio = 3

For a height to span ratio of 3 and a K value of 14 (from Fig4) s 1s V is estimated at ndash01 As the value is negative s 1 s Vis set to zero and s 1 becomes zero This makes s c s 1 greaterthan 10 and therefore the stress factor is equal to 1

Note that the horizontal joints intersecting the hanging-wall will open because the induced stress at the centre of thehanging-wall span is tensile

Considering the mid-stope horizontal plane (along strike)

s H1= 38 MPa s H2

= 38 MPaK = s H2

s H1= 1

Surface height = 30 Surface span = 25Height to span ratio = 12

For a height to span ratio of 12 and a K value of 1 s 1s H1is

estimated to be 075 (from Fig 4) Accordingly

s 1 = 075 38 = 278 MPa

From Fig 2

s cs 1 = 120278 = 43

and the rock stress factor is 035The hanging-wall and footwall are both in compression in

the direction of strike and in tension near the mid-span in thedirection of dip This gives two values on induced stress andtwo corresponding A values Since the lower value is adoptedthe rock stress factor for the hanging-wall and footwall of thestope is 035 The rock stress factors determined for eachexcavation surface are summarized in Table 3

Joint orientation factorThe relative dip of the principal joint set and the excavationsurface is used to determine the orientation factors (Table 4)for the individual surfaces (cf Fig 2) The principal joint setis defined as the feature that predominantly influences stabil-ity In this case the dominant structure set has been identifiedas flat-dipping

Surface orientation factorThe orientation of the stope surface influences the stabilityand accordingly a gravity adjustment factor C (Table 5) isdetermined (cf Fig 2)

A37

Fig 4 Curves for estimation of induced stresses in hanging-wallsAfter Stewart and Forsyth8

Table 3 Rock stress factors A determined for each excava-tion surface

Stope surface A value

Crown 01Hanging-wall 035Footwall 035End-wall 025

Table 4 B values

Stope surface Orientation degrees B value

Crown 0 05Hanging-wall 100 10Footwall 80 10End-wall 90 10

Table 5 C values

Stope surface Dip of stope surface C valuedegrees from C = 8 ndash7 cos (Dip horizontal from horizontal)

Crown 0 1Hanging-wall 80 68Footwall +90 80End-wall 90 80

Mathews stability numberThe Mathews stability number for the exposed surfaces isdetermined from

N = Q cent acute A acute B acute C

Predicted stability outcome from stability graphOnce the shape factor and stability number are known thesurfaces are plotted on the Mathews graph (Fig 5) All thestope walls plot high in the stable zone but the stope crownlies within the failure zone This illustrates the quick preli-minary use of the stability chart a more detailed assessmentof the predicted risk of instability for each surface can bedetermined from isoprobability contours

Estimation of probability of stability from isoprob-ability contoursOnce the stability number and shape factor for the stope sur-faces have been determined the probability of stability can beestimated from the plots of isoprobability contours for the

A38

Fig 5 Positions of surfaces can be plotted on stability graph to determine in which stability zonesthey lie

Fig 6 Positions of surfaces can be plotted on stable isoprobability contours to determine probabilityof stope surface being stable

Table 6 N values

Stope surface N value

Crown 43Hanging-wall 200Footwall 240End-wall 170

different stability outcomes In this way a measure of theconfidence in a stable estimate can be incorporated into thedesign process

From consideration of the plotted position of the hanging-wall footwall end-wall and crown on the isoprobabilitycontours in Figs 8 9 and 10 it is possible to estimate theprobability of stability for the respective stope surfaces Thehanging-wall footwall and end-walls have an estimated 100probability of being stable (Fig 6) In contrast the crown ofthe stope has a high risk of failuremdashthe surface has 18 prob-ability of being stable 64 probability of being a failure and18 probability of being a major failure

Discussion of probability resultsFor this example the stability of the stope walls is high Thepredicted instability of the crown of the excavation is consid-erable and acceptability of the risk of failure of the crown willdepend on the consequences of failure The predicted costand consequences of a crown failure are highly site-specificand should be developed from historical records for a givensite

The probabilities of instability determined for the crowncan drive the design process in several directions If 82probability of failure or major failure is acceptable the stopedesign can be implemented If 82 probability of crownfailure is unacceptable the design of the stope needs to bereaddressed One option would be to reduce the stope heightto lessen the roof stress or reduce the crown area Previousexperience will indicate whether a more conservative redesignof the stope or acceptance of the risk of crown failure is thebetter option in terms of safety economic and operationalissues A second case may arise where the concern is not withfailure but with a major failure in the crown In this case the18 probability of major failure may be unacceptable andcould necessitate a redesign of the stope This could be thecase if a major failure were predicted to have consequencesfor the whole operation

ConclusionsKnowing the probability of stability of a designed excavationallows the engineer to consider the level of risk that is accept-able on the basis of the probable extent and cost of failureEstimation of the probabilities of stability enables a riskanalysis to be undertaken and if the cost of the varyingdegrees of failure can be approximated by reference to histor-ical events the design of the excavation can be optimizedeconomically with true regard for the probability and costoutcomes

Authors

C Mawdesley graduated in geological engineering from the RoyalMelbourne Institute of Technology She has worked in mining geo-logy rock mechanics and mine planning principally at Mount IsaMines Ltd In 1998 she commenced PhD studies in the field ofblock caving at the Julius Kruttschnitt Mineral Research Centre(JKMRC) University of Queensland Australia

Address Julius Kruttschnitt Mineral Research Centre Isles RoadIndooroopilly Brisbane Queensland 4068 Australia

R Trueman is currently Principal Research Fellow in mining at theJKMRC He gained his PhD in mining engineering from theUniversity of Wales He has worked previously as a rock mechanicsengineer for the Anglo American Corporation in South Africa as aproduction mining engineer for the National Coal Board and assenior lecturer in rock mechanics at the Camborne School of Minesin the United Kingdom and as research group manager for CSIROAustralia

W J Whiten joined the JKMRC in 1966 and gained his PhD in1972 from the University of Queensland He is currently ChiefScientist at the JKMRC

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The final stage in determining the magnitude of the stressfactor consists in applying the value of s cs 1 to Fig 2 andreading off the rock stress factor which in this case is 01

End-wallConsidering the strike end of the mid-stope horizontal plane

s H1= 38 MPa s H2

= 38 MPaK = s H2

s H1= 1

Surface height = 30 Surface span = 25Height to span ratio = 12

For a height to span ratio of 12 and a K value of 1 s 1s H1is

estimated at 10 (from Fig 3) Accordingly

s 1 = 10 38 = 38 MPa

By reference to Fig 2

s cs 1 = 12038 = 32

and the rock stress factor for the end-wall is 025

Hanging-wall and footwallThe next step is to determine the induced stresses in thehanging-wall and footwall considering the vertical and hori-zontal mid-stope planes In the case where two estimates ofthe stress factor A are obtained (downdip and along strike)the lower value is used

Considering the mid-stope vertical plane (downdip)

s V = 27 MPa s H2= 38 MPa

K = s H2 s V = 14

Surface height = 75 Surface span = 25Height to span ratio = 3

For a height to span ratio of 3 and a K value of 14 (from Fig4) s 1s V is estimated at ndash01 As the value is negative s 1 s Vis set to zero and s 1 becomes zero This makes s c s 1 greaterthan 10 and therefore the stress factor is equal to 1

Note that the horizontal joints intersecting the hanging-wall will open because the induced stress at the centre of thehanging-wall span is tensile

Considering the mid-stope horizontal plane (along strike)

s H1= 38 MPa s H2

= 38 MPaK = s H2

s H1= 1

Surface height = 30 Surface span = 25Height to span ratio = 12

For a height to span ratio of 12 and a K value of 1 s 1s H1is

estimated to be 075 (from Fig 4) Accordingly

s 1 = 075 38 = 278 MPa

From Fig 2

s cs 1 = 120278 = 43

and the rock stress factor is 035The hanging-wall and footwall are both in compression in

the direction of strike and in tension near the mid-span in thedirection of dip This gives two values on induced stress andtwo corresponding A values Since the lower value is adoptedthe rock stress factor for the hanging-wall and footwall of thestope is 035 The rock stress factors determined for eachexcavation surface are summarized in Table 3

Joint orientation factorThe relative dip of the principal joint set and the excavationsurface is used to determine the orientation factors (Table 4)for the individual surfaces (cf Fig 2) The principal joint setis defined as the feature that predominantly influences stabil-ity In this case the dominant structure set has been identifiedas flat-dipping

Surface orientation factorThe orientation of the stope surface influences the stabilityand accordingly a gravity adjustment factor C (Table 5) isdetermined (cf Fig 2)

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Fig 4 Curves for estimation of induced stresses in hanging-wallsAfter Stewart and Forsyth8

Table 3 Rock stress factors A determined for each excava-tion surface

Stope surface A value

Crown 01Hanging-wall 035Footwall 035End-wall 025

Table 4 B values

Stope surface Orientation degrees B value

Crown 0 05Hanging-wall 100 10Footwall 80 10End-wall 90 10

Table 5 C values

Stope surface Dip of stope surface C valuedegrees from C = 8 ndash7 cos (Dip horizontal from horizontal)

Crown 0 1Hanging-wall 80 68Footwall +90 80End-wall 90 80

Mathews stability numberThe Mathews stability number for the exposed surfaces isdetermined from

N = Q cent acute A acute B acute C

Predicted stability outcome from stability graphOnce the shape factor and stability number are known thesurfaces are plotted on the Mathews graph (Fig 5) All thestope walls plot high in the stable zone but the stope crownlies within the failure zone This illustrates the quick preli-minary use of the stability chart a more detailed assessmentof the predicted risk of instability for each surface can bedetermined from isoprobability contours

Estimation of probability of stability from isoprob-ability contoursOnce the stability number and shape factor for the stope sur-faces have been determined the probability of stability can beestimated from the plots of isoprobability contours for the

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Fig 5 Positions of surfaces can be plotted on stability graph to determine in which stability zonesthey lie

Fig 6 Positions of surfaces can be plotted on stable isoprobability contours to determine probabilityof stope surface being stable

Table 6 N values

Stope surface N value

Crown 43Hanging-wall 200Footwall 240End-wall 170

different stability outcomes In this way a measure of theconfidence in a stable estimate can be incorporated into thedesign process

From consideration of the plotted position of the hanging-wall footwall end-wall and crown on the isoprobabilitycontours in Figs 8 9 and 10 it is possible to estimate theprobability of stability for the respective stope surfaces Thehanging-wall footwall and end-walls have an estimated 100probability of being stable (Fig 6) In contrast the crown ofthe stope has a high risk of failuremdashthe surface has 18 prob-ability of being stable 64 probability of being a failure and18 probability of being a major failure

Discussion of probability resultsFor this example the stability of the stope walls is high Thepredicted instability of the crown of the excavation is consid-erable and acceptability of the risk of failure of the crown willdepend on the consequences of failure The predicted costand consequences of a crown failure are highly site-specificand should be developed from historical records for a givensite

The probabilities of instability determined for the crowncan drive the design process in several directions If 82probability of failure or major failure is acceptable the stopedesign can be implemented If 82 probability of crownfailure is unacceptable the design of the stope needs to bereaddressed One option would be to reduce the stope heightto lessen the roof stress or reduce the crown area Previousexperience will indicate whether a more conservative redesignof the stope or acceptance of the risk of crown failure is thebetter option in terms of safety economic and operationalissues A second case may arise where the concern is not withfailure but with a major failure in the crown In this case the18 probability of major failure may be unacceptable andcould necessitate a redesign of the stope This could be thecase if a major failure were predicted to have consequencesfor the whole operation

ConclusionsKnowing the probability of stability of a designed excavationallows the engineer to consider the level of risk that is accept-able on the basis of the probable extent and cost of failureEstimation of the probabilities of stability enables a riskanalysis to be undertaken and if the cost of the varyingdegrees of failure can be approximated by reference to histor-ical events the design of the excavation can be optimizedeconomically with true regard for the probability and costoutcomes

Authors

C Mawdesley graduated in geological engineering from the RoyalMelbourne Institute of Technology She has worked in mining geo-logy rock mechanics and mine planning principally at Mount IsaMines Ltd In 1998 she commenced PhD studies in the field ofblock caving at the Julius Kruttschnitt Mineral Research Centre(JKMRC) University of Queensland Australia

Address Julius Kruttschnitt Mineral Research Centre Isles RoadIndooroopilly Brisbane Queensland 4068 Australia

R Trueman is currently Principal Research Fellow in mining at theJKMRC He gained his PhD in mining engineering from theUniversity of Wales He has worked previously as a rock mechanicsengineer for the Anglo American Corporation in South Africa as aproduction mining engineer for the National Coal Board and assenior lecturer in rock mechanics at the Camborne School of Minesin the United Kingdom and as research group manager for CSIROAustralia

W J Whiten joined the JKMRC in 1966 and gained his PhD in1972 from the University of Queensland He is currently ChiefScientist at the JKMRC

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Mathews stability numberThe Mathews stability number for the exposed surfaces isdetermined from

N = Q cent acute A acute B acute C

Predicted stability outcome from stability graphOnce the shape factor and stability number are known thesurfaces are plotted on the Mathews graph (Fig 5) All thestope walls plot high in the stable zone but the stope crownlies within the failure zone This illustrates the quick preli-minary use of the stability chart a more detailed assessmentof the predicted risk of instability for each surface can bedetermined from isoprobability contours

Estimation of probability of stability from isoprob-ability contoursOnce the stability number and shape factor for the stope sur-faces have been determined the probability of stability can beestimated from the plots of isoprobability contours for the

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Fig 5 Positions of surfaces can be plotted on stability graph to determine in which stability zonesthey lie

Fig 6 Positions of surfaces can be plotted on stable isoprobability contours to determine probabilityof stope surface being stable

Table 6 N values

Stope surface N value

Crown 43Hanging-wall 200Footwall 240End-wall 170

different stability outcomes In this way a measure of theconfidence in a stable estimate can be incorporated into thedesign process

From consideration of the plotted position of the hanging-wall footwall end-wall and crown on the isoprobabilitycontours in Figs 8 9 and 10 it is possible to estimate theprobability of stability for the respective stope surfaces Thehanging-wall footwall and end-walls have an estimated 100probability of being stable (Fig 6) In contrast the crown ofthe stope has a high risk of failuremdashthe surface has 18 prob-ability of being stable 64 probability of being a failure and18 probability of being a major failure

Discussion of probability resultsFor this example the stability of the stope walls is high Thepredicted instability of the crown of the excavation is consid-erable and acceptability of the risk of failure of the crown willdepend on the consequences of failure The predicted costand consequences of a crown failure are highly site-specificand should be developed from historical records for a givensite

The probabilities of instability determined for the crowncan drive the design process in several directions If 82probability of failure or major failure is acceptable the stopedesign can be implemented If 82 probability of crownfailure is unacceptable the design of the stope needs to bereaddressed One option would be to reduce the stope heightto lessen the roof stress or reduce the crown area Previousexperience will indicate whether a more conservative redesignof the stope or acceptance of the risk of crown failure is thebetter option in terms of safety economic and operationalissues A second case may arise where the concern is not withfailure but with a major failure in the crown In this case the18 probability of major failure may be unacceptable andcould necessitate a redesign of the stope This could be thecase if a major failure were predicted to have consequencesfor the whole operation

ConclusionsKnowing the probability of stability of a designed excavationallows the engineer to consider the level of risk that is accept-able on the basis of the probable extent and cost of failureEstimation of the probabilities of stability enables a riskanalysis to be undertaken and if the cost of the varyingdegrees of failure can be approximated by reference to histor-ical events the design of the excavation can be optimizedeconomically with true regard for the probability and costoutcomes

Authors

C Mawdesley graduated in geological engineering from the RoyalMelbourne Institute of Technology She has worked in mining geo-logy rock mechanics and mine planning principally at Mount IsaMines Ltd In 1998 she commenced PhD studies in the field ofblock caving at the Julius Kruttschnitt Mineral Research Centre(JKMRC) University of Queensland Australia

Address Julius Kruttschnitt Mineral Research Centre Isles RoadIndooroopilly Brisbane Queensland 4068 Australia

R Trueman is currently Principal Research Fellow in mining at theJKMRC He gained his PhD in mining engineering from theUniversity of Wales He has worked previously as a rock mechanicsengineer for the Anglo American Corporation in South Africa as aproduction mining engineer for the National Coal Board and assenior lecturer in rock mechanics at the Camborne School of Minesin the United Kingdom and as research group manager for CSIROAustralia

W J Whiten joined the JKMRC in 1966 and gained his PhD in1972 from the University of Queensland He is currently ChiefScientist at the JKMRC

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different stability outcomes In this way a measure of theconfidence in a stable estimate can be incorporated into thedesign process

From consideration of the plotted position of the hanging-wall footwall end-wall and crown on the isoprobabilitycontours in Figs 8 9 and 10 it is possible to estimate theprobability of stability for the respective stope surfaces Thehanging-wall footwall and end-walls have an estimated 100probability of being stable (Fig 6) In contrast the crown ofthe stope has a high risk of failuremdashthe surface has 18 prob-ability of being stable 64 probability of being a failure and18 probability of being a major failure

Discussion of probability resultsFor this example the stability of the stope walls is high Thepredicted instability of the crown of the excavation is consid-erable and acceptability of the risk of failure of the crown willdepend on the consequences of failure The predicted costand consequences of a crown failure are highly site-specificand should be developed from historical records for a givensite

The probabilities of instability determined for the crowncan drive the design process in several directions If 82probability of failure or major failure is acceptable the stopedesign can be implemented If 82 probability of crownfailure is unacceptable the design of the stope needs to bereaddressed One option would be to reduce the stope heightto lessen the roof stress or reduce the crown area Previousexperience will indicate whether a more conservative redesignof the stope or acceptance of the risk of crown failure is thebetter option in terms of safety economic and operationalissues A second case may arise where the concern is not withfailure but with a major failure in the crown In this case the18 probability of major failure may be unacceptable andcould necessitate a redesign of the stope This could be thecase if a major failure were predicted to have consequencesfor the whole operation

ConclusionsKnowing the probability of stability of a designed excavationallows the engineer to consider the level of risk that is accept-able on the basis of the probable extent and cost of failureEstimation of the probabilities of stability enables a riskanalysis to be undertaken and if the cost of the varyingdegrees of failure can be approximated by reference to histor-ical events the design of the excavation can be optimizedeconomically with true regard for the probability and costoutcomes

Authors

C Mawdesley graduated in geological engineering from the RoyalMelbourne Institute of Technology She has worked in mining geo-logy rock mechanics and mine planning principally at Mount IsaMines Ltd In 1998 she commenced PhD studies in the field ofblock caving at the Julius Kruttschnitt Mineral Research Centre(JKMRC) University of Queensland Australia

Address Julius Kruttschnitt Mineral Research Centre Isles RoadIndooroopilly Brisbane Queensland 4068 Australia

R Trueman is currently Principal Research Fellow in mining at theJKMRC He gained his PhD in mining engineering from theUniversity of Wales He has worked previously as a rock mechanicsengineer for the Anglo American Corporation in South Africa as aproduction mining engineer for the National Coal Board and assenior lecturer in rock mechanics at the Camborne School of Minesin the United Kingdom and as research group manager for CSIROAustralia

W J Whiten joined the JKMRC in 1966 and gained his PhD in1972 from the University of Queensland He is currently ChiefScientist at the JKMRC

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