by j. wesseloo, k. woodward, and j. pereira*quantitative seismology is an important tool for...

IntroductionSeismographs were first installed to monitormining-related seismicity in South Africa in1910. The first in-mine seismic systems wereinstalled in the 1960s, mainly for researchpurposes. Intensive research, development,and commercialization during the 1990s led tothe widespread implementation of real-timedigital monitoring systems and quantitativemethods of analysis (Durrheim, 2010;Mendecki et al., 1997).

Since then, the understanding of mine-induced seismicity and the quantification ofthe seismic rock mass response has formed thebasis of seismological analysis and seismichazard mitigation in South Africa. Thisprinciple of quantifying and understanding therock mass deformation and failure mechanismhas been introduced in other countries indifferent forms and focused on differentmining environments. Potvin and Wesseloo(2013) point out that the mines in Australia,Canada, and Sweden tend to have morecomplex three-dimensional orebodies and aregenerally smaller and much more containedcompared to the seismically active SouthAfrican mines. For that reason it is easier toinstall a more sensitive three-dimensionalarray to cover the mine volume. With the moresensitive systems in Australia, Canada, and

fSweden, local rock engineers tend to focus onthe overall rock mass response to miningbased on accurate source location and theanalysis of populations of seismic events withmagnitude of completeness as small as ML-2.A grid-based spatial analysis of seismic datawas developed to improve and simplify quanti-tative seismological interpretation within thisenvironment. The methods, however, havebroader application.

Funk et al. (1997) presented work on thevisualization of seismicity that resulted in thesystems for generating contours andisosurfaces of seismic parameters. The workpresented here generalizes and extends theconcepts used in that work.

Analysis with spatial filtersSeismic events tend to cluster at the locationsof active seismic sources where some form ofdynamic failure process occurs. The identifi-cation and understanding of seismic sources isimportant in seismic risk management in thatthey may be, or become in the future, thecause of significant seismic hazard. Inparticular, small events may start to formclusters at an early stage of extraction, with arelatively small stress change. When theseismic system is sensitive enough to capturesmall events, this can assist in the timelyidentification of seismic sources and allow forthe tracking of how these sources respond tomining and, more specifically, how seismichazard related to these sources evolves asextraction progresses.

Grid-based analysis of seismic databy J. Wesseloo*, K. Woodward*, and J. Pereira*

SynopsisQuantitative seismology is an important tool for investigating mine-induced seismicity and the quantification of the seismic rock massresponse. The spatio-temporal interpretation of seismic data within anever-changing three-dimensional mining environment provides somechallenges to the interpretation of the rock mass response. A grid-basedapproach for the interpretation of spatial variation of the rock massresponse provides some benefits compared with approaches based onspatial filters. This paper discusses a grid-based interpretation of seismicdata. The basic methods employed in the evaluation of the parametervalues through space are discussed and examples of applications todifferent mine sites given.

Keywordsquantitative seismology, mine induced seismicity, rock mass response,spatial evaluation.

* Australian Centre for Geomechanics, TheUniversity of Western Australia, Australia.

© The Southern African Institute of Mining andMetallurgy, 2014. ISSN 2225-6253. This paperwas first presented at the, 6th Southern AfricanRock Engineering Symposium SARES 2014,12–14 May 2014, Misty Hills Country Hotel andConference Centre, Cradle of Humankind,Muldersdrift.

815The Journal of The Southern African Institute of Mining and Metallurgy VOLUME 114 OCTOBER 2014 ▲

Grid-based analysis of seismic data

f fInterpretation of the character of seismic sources in spaceand time is generally done by analysis of some spatially sub-filtered data-set. This sub-filtering is often simply based on athree-dimensional volume in space (polygon). In Australia,Canada, and Sweden, the data is often sub-filtered onclusters.

The most widely used clustering method is probably thatimplemented in the software mXrap (formerly known as ms-rap) based on the Comprehensive Seismic Event Clustering(CSEC) technique described by Hudyma (2008). The CSECmethod is a semi-automated two-pass approach, which wasdeveloped on the basis of generic clustering techniques:CLINK and SLINK (Jain et al., 1999, Romesburg, 2004). Thefirst pass of clustering using CLINK is totally automated, andclusters together events based exclusively on spatial criteria.The CLINK clusters are then submitted to a second pass ofprocessing, where clusters are selectively grouped into’cluster groups’ representing individual seismic sources. Thiscluster grouping is a manual process that requires interpre-tation of the likely seismic sources at the mine and a soundknowledge of the geology and the induced stress conditions.Cluster grouping is generally based on the similarity of sourceparameters, the spatial proximity of clusters, and on thecorrelation of the location with known geological or geometricfeatures.

The cluster grouping process can be seen as building aseismic source model by using the generated clusters as basicbuilding blocks. In this sense it is similar to analysing anarea with the use of polygons, as the polygons become thebasic units within the seismic source model. The use of aspatial filter (cluster groups or polygons) to provide a basicspatial unit for quantitative seismic analysis is a common andpractical approach. This, however, introduces a bias ofinterpretation towards the pre-defined polygon as thepolygon is originally chosen by the analyst based on pre-conceived ideas. This process is, per se, subjective, and thevvalue depends to a large degree on the understanding andtraining of the person performing this analysis. Subjectivity,however, is part of geotechnical engineering and attemptingto eliminate subjectivity from geotechnical analysis is a futileexercise. With the grid-based approach, however, we can aimto reduce interpretation bias by providing a spatial interpre-tation of the data that is independent of any chosen spatialfilter.

Having said this, one has to recognize that the grid-basedinterpretation is not free of user influence as it also isinfluenced by the chosen analysis parameters. It is ourconviction, though, that the nature of the analysisparameters, and the ease of testing the influence on theanalysis parameters on the results, leads to a systematicreduction in personal bias.

Grid-based analysis of seismic dataIn the grid-based approach, the seismic source parameters areassessed through space by interpolating the sourceparameters. This approach allows for anomalies to beidentified without prior selection of groups or polygons. Thisis illustrated in Figure 1 and Figure 2.

Figure 1 shows the spatial distribution of b-values at anAustralian mine. High values occur around the stopingvvolumes while low b-values occur at the lower abutment.

ff ffThese differences can be related to the difference in thesource mechanism in these areas; the higher b-valuescorrespond with stress fracturing seismicity, and the lowerb-values relate to a shear mechanism.

The same plot is combined with a spatial distribution ofapparent stress in Figure 2. The colouring of each grid pointin space is the same as that in Figure 1, but in Figure 2 eachgrid-point marker is scaled by the geometric mean of theapparent stress. The apparent stress is proportional to themean shear stress at the source of the event (McGarr, 1994).The areas of the mine showing higher b-values generallyshow a low apparent stress, while the lower abutment areashows a high apparent stress state corresponding with low b-values. In this example, the distribution of b-valuesthrough space can be obtained without a pre-defined modelof rock mass response.

▲

816 OCTOBER 2014 VOLUME 114 The Journal of The Southern African Institute of Mining and Metallurgy

Figure 2—Grid-based interpretation of seismicity showing the distri-bution of b-value by colour. The size of the grid point markers arescaled by apparent stress. Blue and red highlighted areas enclose thestoping and abutment areas respectively

Figure 1—Grid based interpretation of the spatial distribution of b-values in a mining area at an Australian mine. Blue and red highlightedareas enclose the stoping and abutment areas respectively

General principleThe general principle behind the grid-based approach forquantitative mine seismology can be summarized as follows:

Assign a representativep seismic parameter value to gridpoints in space based on the events in its neighbourhoodgin order to extract information from the variation of thesedifferent parameter values in space.Added value is obtained from this approach when the

different parameters are retained on each grid point, whichallows for the analysis of these different parameters togetheras shown later.

What can be regarded as ‘representative’ and ‘theneighbourhood’ depends on the purpose of the analysis, thedensity and quality of the seismic data, and the type ofparameter interpolation that is performed i.e. obtaining thecumulative parameter assessment, interpolation of the meanvvalue, or obtaining the b-value.

The neighbourhood is defined by assigning a maximuminfluence distance, with the addition of two more qualitychecks. For a grid point to be assigned a representativeseismic parameter, the grid point events around it must beboth close enough and dense enough. This is illustrated inFigure 3. The condition that events must be dense enough inthe vicinity of the grid-point prevents the transfer ofparameter values from areas further away, with dense data,to a grid point where the density of the events close to thegrid point does not warrant the calculation of a parametervvalue.

Grid-based analysis of seismic dataThe grid-based interpretation of seismic data requiresdifferent approaches for different types of parameters,wwhether it is the mean or cumulative value of a parameterthat is of interest, or the b-value.

It is important to note that in all the different approaches,the gridding process involves some level of smearing, thedegree of which is dependent on the analysis parameters thatare discussed in this paper. It is important that the resolutionof the interpretation should match the resolution of theoriginal input data. Sparse data-sets would require moresmearing than high-resolution dense data-sets. A simplesensitivity analysis should be performed to test thesensitivity of the outcomes to chosen input parameters.

Obtaining a grid-based interpretation of b-valueThe b-value of the frequency magnitude distribution isproportional to the mean of the magnitude and, as such, issimply a statistical parameter. The appropriate b-valuecannot be obtained without knowledge of the magnitude ofcompleteness of each subset of data. For this reason, themethod for obtaining the spatial distribution of the b-value isquite involved and is treated separately in the sister paperpublished in this volume (Wesseloo, 2014).

For current purposes it will suffice to summarize theprocess as follows. For each grid point:

➤ Obtain the closest N events➤ For these events obtain the mmin and associated b-

value➤ Retain b-values for grid points passing the quality

tests.

Mean value of all parametersIn order to obtain the mean value of parameters, thegeometric mean of the parameters of the closest events toeach grid point is calculated. The process can be summarizedas follows. For each grid point:

➤ Find all the events (if ≥ N) within a distanceNN Rmin orthe closest N events within a seach distance of Rmax

➤ Calculate the geometric/arithmetic mean for theparameter

➤ Perform quality checks (see above) and retaincalculated values for only the grid points passing thequality tests.

This approach is used for obtaining spatial distribution ofparameters like energy index (EI), apparent stress (AS), andtime of day (TOD).

Smearing cumulative parametersIn the cases where one is interested in the cumulative effectof different events, for example to obtain an event density orthe cumulative apparent volume, a smearing process is used.In contrast to the method used to obtain the mean parameterof neighbouring events (previous section), the parametervalue (intensity, I) of each event is distributed to (or‘smeared onto’) grid points within its zone of influence.

This procedure can be summarized as follows:

➤ For each event– Find all grid points within its influence zone– Distribute a portion of its value to every grid point

➤ For every grid point– Sum all the portions received from each event– Perform quality checks.

This is performed with a variable smoothing where thekernel bandwidth is linked to the event source size.

The distribution of the events’ parameter is performedwith an inverse distance weighting. The cumulativeparameter at each grid point is obtained as the sum of all thevalues of all the events registered to that grid point. This canbe expressed as follows:

[1]


817The Journal of The Southern African Institute of Mining and Metallurgy VOLUME 114 OCTOBER 2014 ▲

Figure 3—Illustration of the concept of data neighbourhood


[2]

[3]

[4]

wwhere pjp is the cumulative parameter at grid point j, and s arethe location vector of the event and grid points. The intensityvvalue, I, is the parameter value of interest and K(θ) is thekernel function. The bandwidth function h defines theinfluence zone of each event, i.

F is a correction factor that ensures that no errors areintroduced due to the discretization of the volume space, i.ethat the following condition is met:

[5]

In other words, summing the parameter values over allevents must equal the sum of the parameter value associatedwwith each of the grid points.

This process is graphically illustrated in Figure 4. Thecircles in Figure 4a represent different events with differentsource sizes. The parameter values of each of these curvesare distributed to the grid points in the influence zone withthe kernel function (Figure 4b). These values for each gridpoint sum to the final spatial distribution of the parametervvalue (Figure 4c).

In the smearing process, described in the followingsection, each event has an influence zone. Our currentapproach is to define an influence based on the event sourceradius, as defined by Brune (1970). A lower cut-off valueequal to the grid spacing is imposed to ensure stability of themethod for coarser grid discretization. A limiting ceiling valueis also introduced for numerical efficiency and stability. Theresults are not sensitive to the ceiling value.

fIt is important to note that the results of the smearingprocess are not sensitive to the influence of large events, asthe parameter values of these large events are distributed tomore grid points within the larger influence zone.

Grid-based quantitative analysisThis section provides a short discussion on some of theparameters used in the grid-based analysis and examples toillustrate the application of the method. The use of thesemethods is not limited to these parameters.

Energy stress and apparent stressApparent stress is generally calculated as:

[6]

where G is the shear stiffness of the rock mass, and E andMoMM are the total radiated energy and average moment for anevent, respectively. As indicated by its name, the definition ofapparent stress relates to the stress state in the rock mass atthe occurrence of the event. The apparent stress is propor-tional to the mean shear stress at the source of the event,(McGarr, 1994), and is defined as follows:

[7]

where η is the seismic efficiency and τ and τrτ are the peakand residual shear stress, respectively.

As η is unknown, the absolute shear stress is alsounknown. The value of apparent stress is, however, a goodindicator of the relative stress state. This concept is refined byMendecki (1993), who showed that for a given slope of thelog(E)-log(M) relation, the intercept value relates to theMMstress level. A simpler way to express this relative value ofthe intercept is the log(EI), as defined by van Aswegen andIIButler (1993).

Obtaining the log(E)-log(M) trendline introduces someMMdifficulties with unsatisfactory ‘best-fit’ lines. This problemcan easily be overcome. For the sake of maintaining the focusof this paper, this will be discussed further in Appendix A.

The results of a grid-based analysis of the log(EI) at anIIAustralian mine are shown in Figure 5. The colour scale ofthe grid points reflects the values of log(EI). TheIItransparency of the grid points is also scaled with log(EI).IIThe upper 50% of the log(EI) values are more solid while theIIlower 50% are more transparent.

In this particular case, low log(EI) values occur at theIIcentre of the volume with surrounding higher log(EI) values.IIThis corresponds with lower stress areas in the immediatevicinity of the mined stopes, while the more competent rockfurther away from the stopes is under a higher stress state.The particular shape of the grid cloud is determined by thelocation of seismic data, as grid points are generated onlywhere seismic data exists.

Time of dayThe time-of-day parameter (TOD) is a measure of thetemporal differences in the seismic response. It is defined asthe ratio of the rate of seismicity occurring within a specifiedtime window(s) to the rate of seismicity occurring outside of

▲


Figure 4—Illustration of the smearing of parameters onto grid points

those time windows. As an example, where the rock massreacts strongly (in a purely temporal sense) to blasts, theTOD value will be high for a time window around blast time.Orepass noise, on the other hand, results in TOD values ofless than 1 for time windows around shift change, as noorepass activity occurs during shift change.

fGrid points with a low TOD were isolated for anAustralian mine. These grid points all located in a couple ofdistinct areas. The diurnal chart for the events reporting tothese grid points is shown in Figure 7. From this chart it isclear that the events are small and inversely correlated to theshift change, indicating that these events are human-inducednoise, in this case orepass noise. The fact that these eventsare human-induced noise is highlighted by the fact that thetime of lunch breaks during the two shifts is visible in thediurnal chart.

Figure 8 shows the b-value and TOD plots for threeadjacent areas in a mine. Area (1) has an unnaturally highb-value with a very low TOD and is the result of crushernoise. Area (2) has a very high b-value and a higher TODand is the result of a raise bore experiencing some dogearing.Area (3) has a lower, but still fairly high, b-value, with avery high TOD. The seismicity in this area generally relates tostress fracturing around development blasting temporallyconcentrated around blast time. The diurnal charts for thesethree areas area shown in Figure 9.

Cumulative damageIt is generally accepted that there is a correlation betweenhistorical seismic activity in an area and the damageaccumulated in the rock mass. Despite the work of Falmagne(2001), Cai et al. (2001), Coulson and Bawden (2008), andPfitzner et al. (2010), there appears to be no accepted way toquantify this damage accumulation from seismic data. Untilthese difficulties are solved, we propose the use of apparentvolume and the cube root of moment as proxies for damage.

Apparent volume has been linked to the amount of co-seismic strain (Mendecki, 1997), while the cube root ofmoment is proportional to the maximum displacement at theseismic source (McGarr and Fletcher, 2003).

Figure 10 shows the results of a grid-based analysis at anAustralian mine. In this example, log(EI) as a proxy forIIstress and the cube root of moment as a proxy for damageare combined. Log(EI) is represented by the colour scale andIIthe transparency varies with damage. The mean log(EI) isIIcalculated for a 6-month data period, while the damage isaccumulated over the whole history of the mine.


The Journal of The Southern African Institute of Mining and Metallurgy VOLUME 114 OCTOBER 2014 819 ▲

Figure 5—Grid-based interpretation of log(EI(( ). Both colour andIItransparency reflect the log(EI(( )II

Figure 6—TOD definition

Figure 7—TOD definition


The destressed area in this case corresponds with thearea of high historical damage, with some high-stress areason the abutments, where similar levels of damage haveaccumulated. It should be noted that the destressing did nottake place due to the damage alone but is a result of themining voids. As one would expect, the area of high damageis concentrated close to the stopes.

Event densityEvent density is conceptually a very simple parameter and isvery easy to interpret as the number of events occurring perunit volume. From a mathematical viewpoint, event density isa cumulative parameter with the intensity value, I, equal to 1(refer to Equation [5]) and for this reason is calculated withthe same method as is used for the cumulative damage.

Figure 11 shows examples from the Tasmania Mine fordifferent time periods during the life of the mine.

Calibrating numerical modelsIn mining geomechanics, the need for calibrating orconstraining the models with physical observations is wellrecognized. Seismic data provides a valuable source ofinformation on the rock mass response to mining, and forthis reason has been used by some investigators to providecalibration data for their models.

Often the calibration of numerical models with seismicdata is limited to a visual correlation between the eventlocation and strain or stress contours from the models, or thevisual correlation of event density with areas of higher plasticstrain. More recently, correlations between the energy releasemonitored within a specified cell and the modelled plasticstrain energy have been used (Levkovitch et al., 2013; Arndtet al., 2013). Both of these groups limit themselves to energyand perform basic grid calculations, simply summing theenergy of all monitored events located within a specific gridcell.

▲


Figure 9—Diurnal charts for the events shown in the three areas inFigure 8

Figure 10—Analysis results, combining log(EI(( ) results in colour withIIcumulative ‘damage’ results plotted with varying transparency

Figure 8—b-value and TOD plots for three adjacent areas in anAustralian mine

We are of the conviction that the grid-based approach tothe evaluation of seismic data presented here provides theopportunity to better utilize the seismic data to constrainnumerical models.

As the grid-based interpretation can be continuallyupdated, this provides further opportunity to continually testpredicted rock mass behaviour against experiencedbehaviour, with the possibility of flagging deviation from thepredicted behaviour.

Concluding remarksA grid-based interpretation of seismic data has beendiscussed and some examples of results obtained with themethod presented. A grid-based interpretation allows thespatial variation of seismic source parameters to be evaluatedwwithout predetermined analysis volumes. As such, it providessome buffer against biasing of interpretations towards pre-conceived ideas.

The gridding process involves some level of smearing, thedegree of which is dependent on the analysis parametersdiscussed. It is important that the resolution of the interpre-

ftation should match the resolution of the original input data.A sparse data-set would require more smearing than high-resolution dense data-sets. A simple sensitivity analysisshould be performed to test the sensitivity of the outcomes tochosen input parameters.

The grid-based analysis approach is well suited tocompare with results from numerical modelling approaches.

AcknowledgementsOur sincere appreciation to William Joughin and ProfessorRay Durrheim for reviewing the paper. The following organi-zations provided funding for this research through the MineSeismicity and Rockburst Risk Management project: BarrickGold of Australia, BHP Billiton Nickel West, BHP BillitonOlympic Dam, Independence Gold (Lightning Nickel), LKAB,Perilya Limited (Broken Hill Mine), Vale Inc., Agnico-EagleCanada, Gold Fields, Hecla USA, Kirkland Lake Gold, MMGGolden Grove, Newcrest Mining, Xstrata Copper (Kidd Mine),Xstrata Nickel Rim, and The Minerals Research Institute ofWestern Australia.

References

ARNDT, S., LOUCHNIKOV, V., WELLER, S., and O'HARE, A. 2013. Forcasting mining

induced seismicity from modelled energy release in high stress stope

extraction. 8th International Symposium on Rockbursts and Seismicity in

Mines, St. Petersburg and Moscow, Russia, 1–7 September 2013.

pp. 267–272.

BRUNE, J. 1970. Tectonic stress and the spectra of seismic shear waves from

earthquakes. Journal of Geophysical Research, vol. 75. pp. 4997–5009.

CAI, M., KAISERKK , P., and MARTIN, C.D. 2001. Quantification of rock mass damage

in underground excavations from microseismic event monitoring.

International Journal of Rock Mechanics and Mining Sciences, vol. 38.

pp. 1135–1145.

COULSON, A. and BAWDEN, W. 2008. Observation of the spatial and temporal

changes of microseismic source parameters and locations, used to identify

the state of the rock mass in relation to the peak and post-peak strength

conditions. 42nd US Rock Mechanics Symposium, San Francisco, CA, 29

June – 2 July 2008. American Rock Mechanics Association, Alexandria,

VA.

DURRHEIM, R.J. 2010, Mitigating the rockburst risk in the deep hard rock South

African mines: 100 years of research. Extracting the Science: a Century of

Mining Research. Brune, J. (ed.). Society for Mining, Metallurgy, and

Exploration, Inc. pp. 156–171. ISBN 978-0-87335-322-9,

FUNK, C.W., VANVV ASWEGEN, G., and BROWN, B. 1997. Visualisation of seismicity.

Proceedings of the 4th International Symposium on Rockbursts and

Seismicity in Mines, Kraków, Poland, 1–14 August 1997. Gibowicz, S.J.

and Lasocki, S. (eds). CRC Press, Boca Raton, FL.

FALMAGNE, V. 2001. Quantification of rock mass degradation using micro-

seismic monitoring and applications for mine design. PhD thesis, Queen's

University, Ontario.

HUDYMA, M. 2008. Analysis and interpretation of clusters of seismic events in

mines. PhD thesis, University of Western Australia, Perth.

JAIN, A.K., MURTY, M.N., and FLYNN, P.J. 1999. Data clustering: a review. ACM

Computing Surveys (CSUR), vol. 31. pp. 264 –323.


The Journal of The Southern African Institute of Mining and Metallurgy VOLUME 114 OCTOBER 2014 821 ▲

Figure 11—Event density results for different time periods at theTasmania Mine. The mine layout is representative of the mine duringthe stage represented in (d)


LEVKOVITCH, V., BECK, D., and REUSCHRR f, F. 2013. Numerical simulation of the

released energy in strain-softening rock materials and its application in

estimating seismic hazards in mines. 8th International Symposium on

Rockbursts and Seismicity in Mines, St. Petersburg and Moscow, Russia,

1–7 September 2013. pp. 259-266.

MCGARR, A. 1994. Some comparisons between mining-induced and laboratory

earthquakes. Pure and Applied Geophysics, vol. 142. pp. 467–489.

MCGARR, A. and FLETCHER, J. 2003. Maximum slip in earthquake fault zones,

apparent stress, and stick-slip friction. Bulletin of the Seismological

Society of America, vol. 93. pp. 2355–2362.

MENDECKI, A. 1993. Real time quantitative seismology in mines. Keynote

Address: 3rd International Symposium on Rockbursts and Seismicity in

Mines, Kingston, Ontario, 16-18 August 1993. Young, R. Paul (ed.).

Barnes & Noble. pp. 287–295.

MENDECKI, A. 2013. Characteristics of seismic hazard in mines. Keynote Lecture:

8th International Symposium on Rockbursts and Seismicity in Mines, St.

Petersburg and Moscow, Russia, 1–7 September 2013. pp. 275–292.

MENDECKI, A.J. 1997. Seismic Monitoring in Mines. Chapman & Hall, London;

New York.

PFITZNER, M., WESTMAN, E., MORGAN, M., FINN, D., AND BECK, D. 2010. Estimation

of rock mass changes induced by hydraulic fracturing and cave mining by

double difference passive tomography. 2nd International Symposium on

Block and Sublevel Caving, Perth, Australia. Potvin, Y. (ed.). Australiangg

Centre for Geomechanics, Perth, WA.

POTVIN, Y. and WESSELOO, J. 2013. Improving seismic risk management in

hardrock mines. 8th International Symposium on Rockbursts and

Seismicity in Mines, St. Petersburg and Moscow, Russia, 1–7 September

2013. pp. 371–386.

ROMESBURG, H.C. 2004. Cluster Analysis for Researchers. Lifetime Learning

Publications, Belmont, CA. 334 pp.

VVANVV ASWEGEN, G. and BUTLER, A.G. 1993. Application of quantitative seismology

in SA gold mines. 3rd International Symposium on Rockbursts and

Seismicity in Mines, Kingston, Ontario, 16–18 August 1993. Young,

R.P.(ed.). Balkema, Rotterdam. pp. 261–266.

WWESSELOO, J. And POTVIN, Y. 2012. Advancing the Strategic Use of Seismic Data

in Mines. Minerals and Energy Research Institute of Western Australia,

East Perth, WA.

WWESSELOO, J. 2014. Evaluation of the spatial variation of b-value. Journal of the

Southern African Institute of Mining and Metallurgy, vol. 114, no. 10.

pp.823–828.

AAppendix A

Obtaining the log(E)-log(M) trendlineIt should be recognized that a least-squares ‘best-fit’approach assumes the total radiated energy (E) to be avvariable dependent on the total moment (M). ThisMMassumption is incorrect as M and E are two independentparameters and as such the use of the least-squares best-fitmethod is invalid.

Applying the general least-squares best-fit approach tothe Log(E)-Log(M) relation underestimates the slope of thetrendline and overestimates the intercept (Figure 12).

When evaluating the log(E)-log(M) relationship we areMMnot interested in obtaining log(E) as a function of log(M) butMMin the general statistical relationship between theseparameters.

Wesseloo and Potvin (2012) suggested the use of thelog(E)-log(M) quantile-quantile (QQ) relationship for thisMMpurpose, which is the method implemented in the softwaremXrap (Figure 13).

It should be noted that this approach, although muchsimpler, is equivalent to the approach suggested by Mendecki(2013).

The QQ method for obtaining the log(E)-log(M)MMrelationship is performed as follows.

➤ Independently sort E and M, both in ascending order➤ Plot log(Ei) against log(MiMM ) for every value of i. Note

that the actual values of log(Ei) and log(MiMM ) are notfrom the same event and the only link between them isthe fact that they both represent the N

i quantile of thetwo different sets log(E) and log(M)MM

➤ For practical use a best-fit line can be fitted to the QQrelationship. ◆

▲


Figure 12—Least-squares ‘best fit’ to the log(E(( ) log(EE M(( ) dataMM

Figure 13—‘Best fit’ to the QQ plot of log(E(( )-log(EE M(( ) dataMM

by j. wesseloo*, k. woodward*, and j. pereira*quantitative seismology is an important tool for...

Documents

by j. wesseloo, k. woodward, and j. pereira*quantitative seismology is an important tool for...