Rock joint coefficients and their computerized classification

Rock join coe icien s and hei compu e ized classi ica ion Tomáš Ficke Facul y o Ci il Enginee ing, B no Uni e si y o Technology, 60200 B no, Czech Republic a icle in o A icle his o y: Recei ed 24 July 2017 Recei ed in e ised o m 8 No embe 2017 Accep ed 8 July 2019 A ailable online 12 July 2019 Keywo ds: Rock join s Shea s eng h Join ock coe icien s Nume ical indica o s Compu e ized assessmen abs ac A compu e ized me hod o de e mining ock join coe icien s is p esen ed. Two ela i e simila i y indi- ca o s a e in oduced o classi y su ace mo phology o ock join s. The classi ica ion enables o compa e in es iga ed and da abase ock join s. Such a compa ison aims a inding he couple o su aces ha a e dis inguished by he highes dynamical con o mi y. The i s absolu e indica o esul s om he Fou ie ma ix and e alua es wa y shapes o su aces. The second absolu e indica o quan i ies he heigh s o su ace elie s and is de ined as he oo mean squa e heigh o he su ace ou line. Nume ical eliabili y o hese indica o s is es ed wi hin he su ace analysis o a se ies o limes one specimens. Besides he compu e ized assessmen , 25 people ha e pe o med isual assessmen o hese limes one specimens. The esul s o isual assessmen s ha e been s a is ically p ocessed and compa ed o he esul s ecei ed om he compu e ized p ocedu e. The newly in oduced absolu e indica o s ha e p o ed o be p ospec- i e nume ical ools o e alua ing join ock coe icien s. Ó2019 Published by Else ie B.V. on behal o China Uni e si y o Mining & Technology. This is an open access a icle unde he CC BY-NC-ND license (h p://c ea i ecommons.o g/licenses/by-nc-nd/4.0/). 1. In oduc ion Recen ly, a new me hod o oughness assessmen o ock join s has been ou lined [1]. The me hod has been designed as a ully compu e ized p ocedu e whose scena io esembled he isual me hod published by Ba on and Choubey [2,3]. In ha compu e - ized me hod, isual assessmen was eplaced by nume ical assess- men ealized by wo simila i y indica o s. In ha me hod, bo h he simila i y indica o s we e ep esen ed by he ela i e uni less a ios ha howe e we e occasionally nume ically uns able and, in addi ion, he used ela i e o ms p o ed o be concep ually p oblema ic. In he p esen con ibu ion, a new e sion o he compu e ized me hod is de eloped. In his me hod, he absolu e simila i y indi- ca o s a e employed ins ead o he ela i e ones. The absolu e indi- ca o s p o ide a non-p oblema ic concep ual solu ion and show be e nume ical s abili y. The p esen s udy illus a es unc ional- i y o he new absolu e indica o s and desc ibes all necessa y s eps ha p ecede, accompany and accomplish he new compu a ional p ocedu e. The ha dwa e and so wa e o he new p ocedu e ha e been modi ied o be capable o p ocessing la ge ock species, whose linea dimensions equal hose implemen ed in he Ba on and Choubey s anda d isual me hod [2,3]. The con inual in e es in oughness coe icien s o ock join s is caused by he ac ha he su ace oughness o ock join s consid- e ably in luences he mechanical s abili y o ock masses. Su icien s abili y o ock masses is a p e equisi e o he mechanical s abil- i y o la ge ci il enginee ing s uc u es, such as unnels, dams o b idges si ua ed in egions whe e he ock masses con ain discon- inui ies. Join s, aul s, shea zones, bedding su aces o olia ions a e some o he discon inui ies ha may cause s uc u al weakness o ock masses. In he majo i y o cases, he s ess le el due o g a - i y loads is smalle han he comp essi e s eng h o ocks; hus, he e is li le endency o in ac ocks o ac u e. Howe e , he discon inui ies si ua ed wi hin s eep slopes show a endency o sliding. Shea s eng h a he han comp essi e s eng h is he e- o e decisi e o s abilizing ock masses. Fo his eason, he assess- men o he shea s eng h o ock discon inui ies is c ucial. Se e al ac o s in luence shea s eng h: geome y o discon inui ies, su - ace i egula i ies, physical p ope ies o adjacen ocks, in illing ma e ials, and g oundwa e a e some o hem. I is no i ial o decide which one is mos impo an . The ac is ha he su ace i egula i y o ock join s is one o he mos impo an ac o s. Rock join s a e o en plana and mos ly o m a space ne wo k embedded in o ock masses (Fig. 1). In geo echnics, he e is a long-las ing in e es in he su ace mo phology and shea s eng h o ock join s. Se e al models o calcula ing shea s eng h exis . These models ake in o accoun aspe i ies as ele an ac o s in luencing shea s eng h. Pa on [4] was p obably he i s who de eloped a shea s eng h model o ough ock join s. His model was based on expe imen s ca ied ou on saw- oo h iangula aspe i ies. La e on, Ladanyi and A chambaul [5] succeeded in de eloping a model ha inco po- h ps://doi.o g/10.1016/j.ijms .2019.07.002 2095-2686/Ó2019 Published by Else ie B.V. on behal o China Uni e si y o Mining & Technology. This is an open access a icle unde he CC BY-NC-ND license (h p://c ea i ecommons.o g/licenses/by-nc-nd/4.0/). E-mail add ess: [email p o ec ed] In e na ional Jou nal o Mining Science and Technology 29 (2019) 701–709 Con en s lis s a ailable a ScienceDi ec In e na ional Jou nal o Mining Science and Technology jou nal homepage: www.else ie .com/loca e/ijms a ed he mechanisms o sliding and shea ing o aspe i ies in o one equa ion. Kula ilake e al. [6,7] published al e na i e peak shea s eng h c i e ia o ock join s. Thei empi ical model o peak shea s eng h is mul i-pa ame ic and akes in o accoun he di ec ional dependence o he shea s eng h. Ba on’s [2] model is one o he mos equen ly used models in geo echnical p ac ice. Ba on and Choubey [3] showed expe imen- ally ha he shea s eng h o ock join s s depends s ongly on he aspe i ies o adjacen su aces, as ollows: s ¼ n g R JRC log o n  þU b  ð1Þ whe e R JRC is he so-called join oughness coe icien (JRC), n he no mal e ec i e s ess, o he e ec i e join wall comp essi e s eng h (JCS) and U b he basic ic ion angle (i.e. he ma e ial con- s an ). The complexi y o e alua ing s consis s o he complexi y o de e mining R JRC . The join oughness coe icien R JRC has no assumed an exac analy ical exp ession ha would show accep able pe o mance in all he cases whe e he join su aces may mani es hemsel es. Ba on and Choubey [3] sugges ed calcula ing R JRC as ollows: R JRC ¼ a c g s n  log o n  ð2Þ In addi ion, hey p oposed a sys em o 10 e e ence g aphical su ace pa e ns oge he wi h he co esponding join oughness coe icien s R JRC 2(0, 20). These s anda d pa e ns o known R JRC alues enable isual assessmen o su ace i egula i ies o ock join s. This isual assessmen may be a he subjec i e, bu is o en used as a apid and app oxima e ool o de e mining R JRC alues. A good o e iew o a ious me hods o measu ing and quan i- ying he join oughness coe icien s in cu en geo echnical p ac- ice has been published by Mo elli [8]. A comp ehensi e o e iew o exis ing empi ical i ing pa e ns o JRC can be ound in he wo ecen ly published pape s by Li and Zhang [9] and Li and Huang [10]. In he i s pape Li and Zhang summa ize hose pa e ns which employ a ious measu able opog aphic pa ame e s. The second pape by Li and Huang summa izes exis ing empi ical i - ing pa e ns o JRC ha a e based on he ac al dimensions D o he measu ed p o iles o join ed su aces. O he wo pape s con- ce ning beha iou o ock join s ha e appea ed only ecen ly [11,12]. Besides he join oughness coe icien s, associa ed wi h he Ba on isual me hod, he e a e a ious o he oughness coe i- cien s ha can cha ac e ize su ace i egula i ies [13–17], bu he join oughness coe icien s speci ied by Eq. (2) a e e y special indica o s de i ed om expe imen al measu emen s o shea s eng h de ined by Eq. (1). This makes hem op imally adap ed o geo echnical pu poses. Soon a e Ba on’s JRC concep [3] had been published, some ollowe s appea ed who ied o exp ess he coe icien s R JRC in a - ious analy ical o ms [18–21] using eg ession p ocedu es. Al hough many o he sugges ed eg ession pa e ns p o ide good esul s, he Ba on isual assessmen o join oughness coe icien s emains he mos equen ly used me hod in geo echnical p ac ice. The isual compa a i e me hod p oposed by Ba on [2] employs he measu ed oughness coe icien R JRC assigned o each s anda d wo-dimensional (2D) da abase p o ile. The measu ed JRC alues ep esen one g ea ad an age o his me hod, since measu ed da a a e usually close o eali y han hose gene a ed by heo e ical models. Howe e , isual compa ison o su aces wi h he s anda d pa e ns seems o be a he p oblema ic. When se e al people pe - o m isual compa isons, i can ha dly be expec ed ha hey all will come o iden ical esul s. The esul s will be s a is ically sca - e ed mo e o less a ound an a e age alue. To a ain a eliable a e age alue o R JRC , i is desi able o ask mo e people o hei judgmen s (i.e. o use a la ge s a is ical ensemble). Howe e , he e a e o he possibili ies as o how o a oid subjec i i y in he isual me hod. One o hese possibili ies is a compu e ized assessmen based on image ecogni ion, which seems o be a p omising solu ion. Employmen o he Fou ie o malism o classi ying su ace shapes is no a new idea in he ield o geo echnics. Fou ie shape desc ip o s ha e been used o classi ying mo phological ea u es o g anula ma e ials o se e al decades [22–27]. In pa icula , sedimen ologis s ha e de eloped a ious me hods o he assess- men o su ace mo phology o mic oscopic g ains. An in e es ing applica ion o he Fou ie analysis o digi al imaging o pa icle shapes has been published by We imuny and Penumady [26]. So a , he g ea majo i y o s udies ha e conce ned mic oscopic single g ains in es iga ed p e alen ly in 2D egimes. To he bes o ou knowledge, he e is a lack o Fou ie shape s udies conce ning mac oscopic 3D su ace elie s o la ge ock specimens. Since Ba - on’s isual me hod is based on he shape compa ison be ween la - ge ock su aces, i s compu e ized coun e pa should also be capable o p ocessing la ge species. This s udy aims a op imizing such a me hod. P ope nume ical indica o s o su ace simila i ies should be he co e o ha me hod. Finding such indica o s is no a s aigh o wa d ma e , and equi es ho ough es s ha a e desc ibed he e. Fig. 1. Ne wo ks o ock join s si ua ed in he o me lime qua y ‘Hády’ nea he ci y o B no in he Czech Republic. 702 T. Ficke / In e na ional Jou nal o Mining Science and Technology 29 (2019) 701–709 2. Scanning 3D su aces As has been men ioned, he shape classi ica ion o 3D objec s equi es hei scanning and con e sion in o digi al o ms. To scan digi ally 3D elie s o la ge ock samples (10 cm 6.67 cm), a spe- cial pho og aphic echnique has been employed. I uses ho izon al op ical sec ions made by he came a, mo able in he e ical di ec- ion, abo e he in es iga ed su ace (see Fig. 2). To copy he su - ace, i is necessa y o eco d many op ical sec ions (i.e. digi al images aken in disc e e e ical posi ions). A ypical numbe o sec ions amoun s o abou 200 images. Op ical sec ions se e as inpu da a o he special so wa e ha o ms he disc e e 3D elie (x i ,y i ). To supp ess he possible da a nonlinea i y and nons a ion- a i y, p e-p ocessing p ocedu es ha e been inco po a ed in o he so wa e like polynomial i ing and sub ac ing he global end om he p o ile da a along wi h emo ing noise by means o a e - aging he mul iplica i ely aken snapsho s. One may ask a ques ion as o whe he he used sec ional econ- s uc ion me hod shows a quali y compa able wi h o he digi al me hods, e.g. wi h he scanning lase sys ems. A de ailed s udy compa ing he sec ional me hod wi h he scanning lase sys em has been ecen ly epo ed [27,28]. This s udy documen ed ha he esul s o he sec ional me hod we e compa able wi h hose o he lase scanning sys em. One o he many 3D elie s o med by he sec ional me hod is shown in Fig. 3. The ha dwa e used o cap u ing op ical sec ions consis s o he pho og aphic came a Canon EOS 600D augmen ed by he objec i e EF 100 mm /2.8 Mac o USM (Fig. 2). This objec i e possesses a su - icien ly small dep h o op ical ield when i is se o /2.8. The pho- og aphic came a is moun ed on a ough aluminum s and wi h a s epping mo o , enabling disc e e mo emen in he e ical di ec- ion. The mo emen consis s o equidis an posi ions a which he came a akes snaps o he su ace. The dis ance be ween wo posi- ions has o be se so ha i may no exceed he dep h o he op ical ield o he objec i e. The dep h o he op ical ield should be a leas se e al imes smalle han he a e age heigh o he su ace p o usions, o he a e age dep h o he su ace dep essions. The de eloped ha dwa e is usable no only in labo a o ies bu also in ou doo e ain. The pho og aphic came a moun ed on an assembly o wo ipods and equipped wi h a p ope s epping mechanism may ake snaps o la ge pa s o ock elie s di ec ly in ou doo spaces as shown in Figs. 4 and 5. 3. Fou ie 3D elie s o ock join s Fo he mo phology analyses o ock join s, he Fou ie scheme is p oposed. P o ided ha a 3D su ace elie is a ailable in a dis- c e e scanned o m (x i ,y i ), i can be i ed by he Fou ie pa ial sum: ðx i ;y j ÞF N ðx i ;y j Þ ¼X N1 k;n¼0 s kn a kn cos k p x i pcos n p y j qþb kn sin k p x i pcos n p y j q  þc kn cos k p x i psin n p y j qþd kn sin k p x i psin n p y j qð3Þ whe e, X¼ x2ðp;þpÞ;y2ðq;þqÞg ð4Þ a kn ¼1 pq ZZ X ðx;yÞcos k p x pcos n p y qdxdy ð5Þ b kn ¼1 pq ZZ X ðx;yÞsin k p x pcos n p y qdxdy ð6Þ c kn ¼1 pq ZZ X ðx;yÞcos k p x psin n p y qdxdy ð7Þ d kn ¼1 pq ZZ X ðx;yÞsin k p x psin n p y qdxdy ð8Þ s kn ¼ 1k>0;n>0 1=2k¼0;n>0o k >0;n¼0 1=4k¼n¼0 8 > < > : 9 > = > ; ð9Þ The symbol Nde ines he uppe bound (N-1) o he subsc ip s n,k, and es ic s he numbe o e ms in he Fou ie pa ial sum speci ied by Eq. (3). Fig. 4. A modi ied op ical de ice adap ed o e ain scanning. Fig. 2. A labo a o y scanning de ice o cap u ing 3D su ace elie s. Fig. 3. A scanned elie o limes one su ace 10 cm 6.67 cm wi h a esolu ion o 720 pixels 480 pixels. T. Ficke / In e na ional Jou nal o Mining Science and Technology 29 (2019) 701–709 703 To de elop a co esponding ma hema ical basis o he Fou ie indica o s o 3D elie s, i is con enien o ew i e Eq. (3) as ollows: ðx i ;y j Þ X N1 k;n¼0 a k cos k p x i pþb k sin k p x i p   c n cos n p y j qþd n sin n p y j q  ð10Þ whe e, a k c n ¼ s kn a kn b k c n ¼ s kn b kn a k d n ¼ s kn c kn b k d n ¼ s kn d kn 8 > > > < > > > : ð11Þ Eq. (10) con ains a p oduc o wo e ms in b acke s. These e ms ep esen ha monic wa es wi h cha ac e is ic wa eleng hs k x =2p/kand k y =2q/n(in pixels). The unc ional p oduc s speci ied by Eq. (10) ep esen 2D wa e modes possessing indices (k,n) and wa eleng hs (k x ,k y ). A squa e ma ix M N o he Fou ie indica o s D N (k,n) may be in oduced as ollows: M N ¼ D N ð0;0ÞD N ð0;1Þ:::::::::: D N ð0;N1Þ D N ð1;0ÞD N ð1;1Þ:::::::::: D N ð1;N1Þ :::::::::: :::::::::: D N ðN1;0ÞD N ðN1;1Þ:::::::::: D N ðN1;N1Þ 0 B B B B B B @ 1 C C C C C C A ð12Þ D N ðk;nÞ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð s kn a kn Þ 2 þð s kn b kn Þ 2 þð s kn c kn Þ 2 þð s kn d kn Þ 2 qð13Þ Some o he ma ix elemen s ha e special meanings. Fo exam- ple, he i s elemen D N (0,0) = a 0,0 /4 is associa ed wi h an in ini- i ely la ge wa eleng h and ep esen s a basic heigh o he ho izon al le el. I s alue will be he la ges compa ed o all o he ma ix elemen s. F om he iewpoin o he shape (wa e) analysis, his i s diagonal elemen is i ele an and will be se o ze o. I will no be aken in o accoun in u he conside a ion. The ma ix M N is a cha ac e is ic ma ix and con ains all ele an wa e modes o su ace elie s. One example o his ma ix plo ed in g aphical o m as a unc ion o wo a iables k,nis p esen ed in Fig. 6. Na u ally, i would be possible o s udy each pa icula indica- o D N (k,n) o g oups o indica o s, and o e eal hei unc ionali y associa ed wi h local mo phological ea u es o 3D elie s. Undoub edly, he indica o s o lowe o de s (k,n< 10) bea in o - ma ion on a global wa y shape o he elie , since hey co espond o spa ial wa es o la ge wa eleng hs (i.e. small equencies). The indica o s o highe o de s (k,n> 10) would undoub edly quan i y a ine s uc u e o 3D elie s. All hese p ope ies can be obse ed on he g aphical plo o he ma ix M N in Fig. 6, whe e he indica- o s o lowe o de s mani es la ge alues (peak alues), whe eas he indica o s o highe o de s quickly lose hei magni udes. How- e e , a de ailed analysis o his kind would no be p ac ical. One- alued indica o s enabling quick o ien a ion and a apid decision a e needed. To each his goal, e o mula ing he ma ix M N is nec- essa y. This ask is accomplished in he nex sec ion. 4. Classi ying su ace mo phology P io o selec ing simila i y indica o s, i mus be clea which su ace ea u es should be compa ed and wha should ac ually be unde s ood unde he no ion o ‘su ace simila i y’. Only su ace ea u es ha a e decisi e o he shea s eng h o ock join s ha e o be p e e ably aken in o accoun . Ba on and Choubey [2,3] di ided su aces o ock join s in o 10 ca ego ies acco ding o hei Fig. 6. An example o he g aphical ou pu o he Fou ie ma ix M N (k,n). Fig. 5. Rock elie scanned in e ain (A e [28,29]). 704 T. Ficke / In e na ional Jou nal o Mining Science and Technology 29 (2019) 701–709 mo phologies, s a ing om plana and smoo h su aces up o wa y and ough su aces. Likewise, Bandis [30] desc ibed a se o su aces acco ding o hei mo phologies, and men ioned h ee basic shape ca ego ies, namely plana , undula ing and s epped su - aces. He di ided each ca ego y in o h ee heigh subca ego ies, namely he subca ego ies ha he e med as slickensided, smoo h and ough. These ca ego ies and subca ego ies a e mu ually inde- penden and may be combined in o nine di e en mo phological pa e ns. Na u ally, his is a simpli ied classi ica ion and, in eali y, u he a ian s o su ace p o iles may appea . Howe e , e en om his simplis ic dis ibu ion, i is clea ha he shape and heigh o he su ace p o iles a e wo ele an su ace ea u es ha a e decisi e in he assessmen o he shea s eng h o ock join s. These cha ac e is ics also play an impo an ole in e alua ing he simila i y o ock join su aces. Howe e , unde he e m ‘simila i y’ canno be unde s ood a simple geome ical simila i y since join ed su aces show speci ic dynamical p ope ies esul ed in shea s eng h ha is di ec ion- ally dependen . Consequen ly, he co esponding shape indica o should be sensi i e o he di ec ion o shea ing, i.e., i should be o a ionally sensi i e. Such an indica o is capable o classi ying he so called ‘dynamical simila i y’ o join ed su aces. On he o he hand, he global heigh s o 3D p o iles a e independen o di ec- ions and hus he co esponding heigh indica o should be in a i- an o o a ions. I is necessa y o ind a leas wo independen indica o s, one o which will classi y shape (plana i y, wa iness, s epped shape, e c.) and will be o a ionally sensi i e whe eas he second will classi y p o ile heigh and will show in a iance o o a ion. The ma ix M N seems o be a good s a ing poin o o m- ing a one- alued shape indica o , since his ma ix con ains ‘signa- u es’ o all cha ac e is ic wa e modes o he su aces. As a heigh indica o , he so-called oo mean squa e (RMS) pa ame e R q migh be con enien , since i e alua es global heigh s o su ace p o iles. This global indica o is equen ly used when analyses o su ace oughness a e pe o med [13–17]. Fo he pu pose o compa ison o join ed su aces, he ollowing wo absolu e nume ical indica o s S, and Hwill be employed. The i s indica o Sis de i ed om he elemen s D N (k,n) o he Fou ie ma ix M N as a ue indica o o dynamical simila i y; and he sec- ond indica o Hquan i ies he global in a ian heigh s o 3D su - ace p o iles: SðNÞ¼X N1 k¼0 X N1 n¼0 jD ð0Þ N ðk;nÞD N ðk;nÞj ð14Þ HðNÞ¼jR ðoÞ q ðNÞR q ðNÞj ð15Þ R q ðNÞ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 KLX K i¼1 X L j¼1 ðx i ;y j ÞF ðlowÞ N ðx i ;y j Þ hi 2 u u ð16Þ whe e (x i ,y j ) is he scanned elie and F ðlowÞ N ðx i ;y j Þ ep esen s he Fou ie elie , which se es as a basis o compu ing heigh di e - ences. The auxilia y quan i ies R q ðNÞand R ðoÞ q ðNÞa e RMS alues ela ed o he es ed su ace and he da abase su ace, espec i ely. The symbols D N (k,n) and D ðoÞ N ðk;nÞin Eq. (14) a e associa ed wi h he es ed su ace and he da abase su ace, espec i ely. The p o- duc K·Lis a pixel esolu ion o he used digi al images o su aces. In ou case K·L=720 pixels 480 pixels ( he colo pixel as e o he CMOS senso o he came a), which co esponds o he a ea 10 cm 6.67 cm (measu ed on he samples). The symbol (x i ,y j ) in Eq. (16) ep esen s he scanned elie . In he i s phase o he mo phology esea ch o join ed su - aces, we ied o ind nume ical indica o s independen o mea- su ing echniques in o de he esul s coming om di e en labo a o ies could be compa able. F om his iewpoin , he ela- i e indica o s in oduced as ac ions in which he nume a o s had absolu e o ms (see Eqs. (14) and (16)) whe eas he denom- ina o s con ained he da abase e ms D ðoÞ N ðk;nÞand R ðoÞ q ðNÞ, espec- i ely, seemed o be con enien [1]. Howe e , u he esea ch showed ha ela i e indica o s a e no as nume ical s able as he absolu e indica o s speci ied by Eqs. (14) and (16) and, in addi ion, he denomina o s may occasionally change he igh p edic ions o absolu e nume a o s. In o he wo ds, he ela i e indica o s a e no as eliable as he absolu e indica o s. Fo his eason, we ha e abandoned he ela i e indica o o ms in he p esen pape and in oduced he absolu e indica o s de ined by Eqs. (14) and (16) as con enien pa ame e s ha a e capable o e ealing dynamical simila i ies o join ed su aces mo e eliably. Fu he de ailed discussion conce ning unc ional beha iou o he absolu e nume ical indica o s and hei p ope ies can be ound elsewhe e [31]. I should be highligh ed ha he join ed su aces acqui ed by d illing he e ain slopes usually show some a i icial base ha is o en inclined o bended. This addi ional unwan ed pa s o join s ep esen unc ional ends which a e no inhe en pa s o join s and ha e o be emo ed. These unc ional ends can be i ed by he Fou ie p o iles wi h su icien ly low N- alues, i.e. F ðlowÞ N ðx i ;y j Þo by he wo-dimensional polynomials. These i - ed disc e e unc ions se e as e e ence le els and a e employed o co ec ing he scanned p o iles ( elie s) (x i ,y j ) as shown in Eq. (16). When compu ing he Fou ie ma ix speci ied by Eq. (12), he co ec ed scanned p o ile has o be used as well, i.e. Fig. 7. Eigh limes one species o a ious su ace ex u es selec ed o simila i y es s. T. Ficke / In e na ional Jou nal o Mining Science and Technology 29 (2019) 701–709 705 ½ ðx i ;y j ÞF ðlowÞ N ðx i ;y j Þ ! ðco ec edÞ ðx i ;y j Þ. This co ec ed scanned unc ion is hen used o compu ing he accu a e ma ix D N (k,n), bu his ime i is necessa y o pe o m he Fou ie expansion wi h a su icien ly high N- alue o ai h ully desc ibe all he su ace i egula i ies including wa y shapes and aspe i y s uc u es. Since he oo -mean-squa e heigh indica o His calcula ed acco ding o Eqs. (15) and (16) as a di e ence be ween he e - e ence le el F ðlowÞ N ðx i ;y j Þand he scanned elie (x i ,y j ), i.e. as a wo-le el di e ence, he indica o His independen o o a ion. The equi ed sensi i i y o he indica o S o he ope a ions o o a ion dese es a de ailed discussion and can be ound in Re . [31]. Bo h he indica o s Sand H ul ill he necessa y p op- e ies o co ec assessing he dynamical con o mi y o h ee- dimensional elie s. Eqs. (14)–(16) should be capable o iden i ying such a da abase elie ha shows he highes dynamical con o mi y wi h he elie unde in es iga ion. When wo iden ical elie s a e compa ed, a ull con o mi y (dynamical simila i y) mus be indica ed, and he wo indica o s ha e o show ze o alues (S=0,H= 0). Such a si ua ion can only a ely occu in p ac ice when wo independen elie s a e compa ed. In he g ea majo i y o si ua ions, only incomple e dynamical simila i y will be ound, which will be indica ed by some minimum posi i e alues o shape and heigh indica o s, i.e. 0 < S, 0<H. In addi ion, heigh and geome ical shape a e wo mo e o less independen p ope ies and may show a a ie y o combina- ions. Acco dingly, he indica o s Sand Hmay show di e en esul s in some si ua ions and in his case he esul associa ed wi h a smalle alue o JRC should be accep ed since his alue is a he side o highe sa e y. An analogous p oblem may a ise when a g oup o people pe o ms isual assessmen s o ock join s. The es s in he ollowing sec ion a e aimed a he unc ionali y o he indica o s Sand H. 5. Tes o simila i y indica o s Eigh agmen s aken om he o me limes one qua y ‘Hády’ nea he ci y o B no in he Czech Republic (Fig. 1) ha e been ga h- e ed. Thei su aces (Fig. 7) ha e di e se mo phologies s a ing om e y coa se (No. 3 in Fig. 7) up o e y ine (No. 6 in Fig. 7). The agmen s do no ep esen eal ock join s, bu hei su ace mo phologies co e almos he en i e ange o Ba on’s s anda d p o iles [2]. These eigh su aces ha e been subjec ed o simila i y es s. A pa icula su ace has been compa ed o he emaining se en su aces ha play he ole o da abase su aces. This p oce- du e was epea ed eigh imes, i.e., each su ace was compa ed o he es o he o he da abase su aces. The compa a i e p ocedu e Table 1 Visual and compu e assessmen s. In es iga ed su ace Visual assessmen Compu e assessmen The mos simila da abase su aces The mos simila da abase su aces SH 14 44 2 3 and 8 8 8 32 22 41 11 58 88 67 77 76 66 8 5 and 2 5 5 Fig. 8. Values o S-indica o s compu ed o couples o su aces. The S-indica o s ha e o show ze o alues when wo iden ical su aces a e compa ed. The couples o iden ical su aces a e as ollows: 1–1, 2–2, 3–3, 4–4, 5–5, 6–6, 7–7, and 8–8. These simple es s illus a e he co ec unc ionali y o he so wa e used. 706 T. Ficke / In e na ional Jou nal o Mining Science and Technology 29 (2019) 701–709 was pe o med bo h isually (subjec i ely) and au oma ically (objec i ely) on he compu e . Simila i y compa isons o su aces a e p oblema ic, since he e a e no ools enabling a decision wi h absolu e ce ain y. Fo his eason, we pe o med a double compa ison (i.e. isual and compu - e ized) in o de ha he esul s may be con on ed. 5.1. Visual assessmen To ob ain s a is ically ele an esul s, we asked 25 people o classi y he su aces acco ding o he s anda d p o iles o Ba on [2,3]. In his way, he co esponding JRC- alues ha e been asc ibed o su aces and he couples o he mos dynamically simila su - aces ha e been speci ied. Na u ally, hese indi idual decisions we e no uni ied bu a he dispe sed. Since i was ha dly possible o weigh s a is ically each decision, we calcula ed common a i h- me ic a e ages o JRC alues and he couples o he mos simila su aces ha e been sa ed in Table 1. I should be highligh ed ha he isual assessmen s o ock su - aces based on human judgmen s a e a he p oblema ic, since he esul s a e a ec ed by s a is ical a iabili y caused by indi idual human di e ences. 5.2. Compu e assessmen The second ype o simila i y compa ison has been pe o med on he compu e by using indica o s Sand H. Scanned su aces No. 1-No. 8 (Fig. 7) we e con e ed in o he co ec ed Fou ie elie s, om which he desc ip o ma ices M N (Eqs. (12) and (13)) we e de i ed, and a e wa ds he alues o he global shape indica o Sand he heigh indica o Hwe e e alua ed. To e i y he unc ionali y o he so wa e, a simple es was ca ied ou . The es was based on compa isons be ween one cho- sen su ace and he eigh ‘da abase’ su aces among which he cho- sen su ace was also p esen . Wi hin such a compa ison, whe e wo iden ical su aces can be me , he simila i y indica o s Sand Hha e o show ull dynamical simila i y (ze o alues). Figs. 8 and 9 show he esul s o such es s. The g aphs in hese plo s con- i m ha he so wa e co ec ly iden i ied all eigh couples o iden- ical su aces. The main es s a e, howe e , ocused on simila i y compa isons o non-iden ical su aces. An in es iga ed su ace was compa ed o he emaining se en da abase su aces. The e a e eigh such a angemen s: (i) su ace No. 1 e sus Nos. 2–8, (ii) su ace No. 2 e sus Nos. 1 and 3–8, (iii) su ace No. 3 e sus Nos. 1, 2 and 4– 8, (i ) su ace No. 4 e sus Nos. 1–3 and 5–8, ( ) su ace No. 5 e - sus Nos. 1–4 and 6–8, ( i) su ace No. 6 e sus Nos. 1–5 and 7, 8, ( ii) su ace No. 7 e sus Nos. 1–6 and 8, ( iii) su ace No. 8 e sus Nos. 1–7. All hese nume ical compa isons we e accomplished, and he co esponding g aphs a e p esen ed in Figs. 10 and 11. In hese g aphs, he co esponding mos simila da abase su aces a e indi- ca ed by bold a ows. Fig. 10 shows he esul s o simila i y compa isons pe o med by means o he S-indica o s. Fo example, he chosen su ace No. 1 is compa ed o he da abase su aces Nos. 2–8; i s da abase coun e pa No. 4 was ound as i s mos simila pa ne among he emaining da abase su aces. Acco dingly, in Fig. 10, he bold a ow poin s o he da abase su ace No. 4. Since he compa isons a e no pe o med be ween iden ical su aces, he alues o he Fig. 9. Values o H-indica o s compu ed o couples o su aces. The H-indica o s ha e o show ze o alues when wo iden ical su aces a e compa ed. The couples o iden ical su aces a e as ollows: 1–1, 2–2, 3–3, 4–4, 5–5, 6–6, 7–7, and 8–8. These simple es s illus a e he co ec unc ionali y o he so wa e used. T. Ficke / In e na ional Jou nal o Mining Science and Technology 29 (2019) 701–709 707 Fig. 11. Compu e ized simila i y es s o non-iden ical su aces. Values o H-indica o s a e compu ed o he ollowing couples: 1 e sus 2–8; 2 s. 1, 3–8; 3 s. 1, 2, 4–8; 4 s. 1–3, 5–8; 5 s. 1–4, 6–8; 6 s. 1–5, 7, 8; 7 s. 1–6, 8; 8 s. 1–7. The a ows poin o he mos simila da abase su aces. Fig. 10. Compu e ized simila i y es s o non-iden ical su aces. Values o S-indica o s a e compu ed o he ollowing couples: 1 e sus 2–8; 2 s. 1, 3–8; 3 s. 1, 2, 4–8; 4 s. 1–3, 5–8; 5 s. 1–4, 6–8; 6 s. 1–5, 7, 8; 7 s. 1–6, 8; 8 s. 1–7. The a ows poin o he mos simila da abase su aces. 708 T. Ficke / In e na ional Jou nal o Mining Science and Technology 29 (2019) 701–709 S-indica o s canno be equal o ze o, bu hey show small posi i e non-ze o alues, which ep esen he o al minima o he S-g aphs. The o he in es iga ed su aces (Nos. 2–8) we e ea ed analogously. Fig. 11 o e s esul s achie ed by means o he heigh H-indica o s. The a angemen is he same as in Fig. 10, and he bold a ows again poin o he mos simila da abase su aces. All he esul s o he H-indica o s shown in Fig. 11 ag ee wi h hose o he S-indica o s p esen ed in Fig. 10. The esul s o he simila i y assessmen s achie ed by he wo indica o s Sand Ha e collec ed in Table 1. 5.3. Compa ing isual and compu e assessmen s Table 1 con ains he esul s o bo h he isual and compu e assessmen s. Bo h hese assessmen s a e in good ag eemen . The absolu e indica o s Sand Hseem o be con enien o he compu - e ized simila i y assessmen s o ock join s. They cons i u e p inci- pal ools in he de eloped so wa e code. Al hough hei p edic i e esul s ha e been iden ical in all ou p esen ed es s and in many o he s so a no published, i should be epea edly s essed ha hese wo indica o s may some imes sligh ly di e in hei p edic- ions especially when heigh and shape p ope ies o su ace elie s a e no su icien ly co ela ed. In such cases he smalle JRC alue should be p e e ably accep ed as a pa ame e ha gua an ees highe sa e y. The pe o med es s ha e shown ha he absolu e o ms o he indica o s Sand Hha e succeeded in hei ole o simila i y indica- o s inding ock join coe icien s associa ed wi h he da abase elie s. The modi ied ha dwa e and so wa e we e capable o co - ec ly p ocessing la ge ock su aces whose linea sizes we e ully compa able o hose o he Ba on s anda d JRC p o iles. 6. Conclusions The isual assessmen o he oughness o ock join s, o en used in geo echnical enginee ing, may be eplaced by he ully compu - e ized p ocedu e based on he absolu e heigh Hand shape Sindi- ca o s. The absolu e unc ional o ms o hese indica o s ha e p o ed o be nume ically mo e s able compa ed o he ela i e indi- ca o s used p e iously. The compu e ized p ocedu e based on com- pa ison wi h he da abase su aces o known JRC yields wo kinds o he ock join coe icien s, namely JRC H and JRC S , which a e e med acco ding o he indica o s Hand S. The alues JRC H and JRC S a e usually iden ical, especially when he e is a highe co ela ion be ween he heigh and wa y shapes o he su ace p o iles. Ne e - heless, a weake co ela ion may cause a sligh di e ence be ween hese wo coe icien s. 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