Basic Analysis of Spatial Patterns

Transcription

1 Basc Analss of Spaal Paerns Deparmen of Mnng and Mneral Engneerng The Unvers of Leeds Januar 003

2 Sochasc Modellng of Fracures n Roc Masses, 003. Inroducon The frs sep owards spaal paern modellng s he spaal randomness analss. A rch source of mehods can be found n he leraure bu n general erms he mehods can be classfed no hree caegores: Dsance analss Quadra coun analss Second momen analss Dsance analss uses he dsances beween evens or he dsances beween evens and seleced pons. Quadra coun analss uses he number of evens fallng no he quadras of ceran shape, eher locaed randoml or accordng o a pre-arranged paern. Second momen analss manl deals wh he K-funcon analss hough he second order nens and he covarance dens also fall no hs caegor. In hs repor, each caegor of he analss s dscussed n urn and some examples are gven.. Dsance analss As llusraed n Fgure., we use, and x o represen hree dfferen measures of dsances. s he dsance beween an par of evens and s referred as nereven dsance. s he dsance beween an even and s neares one and s named he neares neghbour dsance. x s he dsance beween a seleced pon n he regon and s neares even and s referred o as he pon o neares even dsance. For he laer wo pes of dsances, f he neres s concerned wh he -h neares even, he dsances are hen called he -h neares neghbour dsance or he -h pon o neares even dsance. Fgure. Defnons of dsance measures. Dsance dsrbuon heor

3 Sochasc Modellng of Fracures n Roc Masses, 003 For a pon process n R d, under he assumpon of homogeneous Posson process, he probabl ha here s no even whn dsance of an arbrar even can be obaned b he ner-even proper of he Posson process and can be expressed as: λv P[ no furher even whn dsance of an even] e where V s he d-dmensonal volume of a sphere wh radus and s gven below: d V π d Γ + Therefore he probabl dsrbuon funcon of measure and s dens funcon are: λv G e probabl funcon d d λ π d λv g e dens funcon d Γ + For wo dmensonal case, λπ G e probabl funcon λπ g πλ e dens funcon If we consder up o he h neares even of an arbrar even, he jon dsrbuon funcon for,,, can be derved as follows: Based on he ner-even probabl proper of a Posson process, he followng saemens can be wren for a seleced even: λv P[ no furher even whn dsance ] e no furher even whn dsance λ V V P[ ] e on he condon ha here s an even a no furher even whn dsance 3 λ V V P[ ] e on he condon ha here are evens a, no furher even whn dsance λ V [ ] V P e on he condon ha here are even a,,, The dsrbuon funcons for hese evens are: d 3

4 Sochasc Modellng of Fracures n Roc Masses, ],,, [ ], [ ] [ ] [ V V V V V V V e a even are here condon ha on he whn dsance even furher no g e a evens are here condon ha on he whn dsance even furher no g e a an even s here condon ha on he whn dsance even furher no g e whn dsance even furher no g λ λ λ λ λ λ λ λ Therefore he jon dsrbuon dens funcon s gven b: V e g λ λ,,, and he probabl funcon s: 0,,, * 0 * * 0 * * * * * * * * * * wh V e e V V dv dv dv e G V V V V V V V V λ λ λ λ λ To accoun for he edge effec, s suffcen o replace * V n he equaon b R * * * V V, where * V s he d-dmensonal sphere volume wh radus * and R s he regon under consderaon. The same heor can be appled o he dsrbuon of x, he neares even dsance o a randoml seleced pon,.e., he followng relaons can be obaned: + Γ funcon dens e x d d x f funcon probabl e x F x x V d d V λ λ π λ For wo dmensonal case, funcon dens e x x f funcon probabl e x F x x λπ λπ πλ and for he neares evens: V e x x x f λ λ,,, 0,,, * 0 * * * * * * * V wh e e V V x x x F V V λ λ λ For he ner-even dsance, Barle [] derved he dsrbuon dens n 964 for homogeneous Posson process whn a wo dmensonal square, whch s gven n he followng equaon:

5 Sochasc Modellng of Fracures n Roc Masses, < < arcsn L L L L L L L L L L L L h π where L s he sze of he square. Dggle [] gves he dsrbuon funcon for a un square as follows: < arcsn H π. Implemenaon ssues For, and x dsance analss, he dsrbuon can be calculaed n he forms of eher hsogram or cumulave hsogram. For cumulave hsogram calculaon, he can be expressed as: pons seleced of number m m m x x x F pons daa of number n n n G pons daa of number n j n j n n H j,,,, #,,,, #,,,,,, # For hsogram calculaon, whch s analogous o dens dsrbuon funcons, he ranges of, or x are dvded no C number of classes frs and he relave frequenc dsrbuon dens values are gven b: + < + < + < m m x x x x f n n g j n j n n h j,,, #,,, #,,,,, # where,j and m have he same meanngs as he las equaon, s he dsrbuon class d and s he nerval dvdng he ranges of, and x. To compare he resuls wh heor, however, edge effec has o be aen no accoun. For he calculaon of G and F, he edge effec can be correced b he followng relaons:

6 Sochasc Modellng of Fracures n Roc Masses, 003 ~ # j, d > H n n ~ #, d > G # d > ~ # x x, d > x F x # d > x, j,,, n, n,,, n,,, m # d > where d s he dsance of he seleced even ou of n evens or he seleced pon ou of m pons o he neares border edge of he regon R beng consdered. Ths reamen s equvalen o mposng a safe-guardng area or volume around he edge of he regon see Upon [] and dscard an even or pon fallng whn. Noe he guardng area volume changes n sze accordng o he dsance, or x beng consdered. Anoher approach o edge correcon specfcall n wo-dmensonal case when he regon beng consdered R s a recangle s o vruall jon he oppose sde of he recangle and urn he regon no a orus. Dsances are hen calculaed on he bass of hs vrual regon. For example, he neares even of an even locaed a he boom lef corner of he regon can be an even locaed a he op rgh corner of he regon. The random pons seleced for he calculaon of F or F ~ values can be chosen randoml or from a fxed grd. Dggle [] suggess usng a regular grd and ncreasng unl F or F ~ effecvel sablzes hroughou s range..3 Mone Carlo reference smulaon An effecve approach o es f a pon paern s a parcular realsaon from a pon process s b he help of null hpohess es based on a ceran number of Mone Carlo reference smulaon. To proceed wh hs es, he sascs of he daa, s d, s calculaed frs. Then number of ndependen smulaons based on he model defnng he pon process are conduced and he correspondng sascs for each smulaon, s, s,, s, are calculaed. s d, s, s,, s are hen rearranged n ascendng order. Then under he null hpohess wh a sgnfcance level α, he ranng of s d whn he sequence mus be: j + α for upper al es j > + α for lower al es + α j + α for wo al es where j s he ranng of s d n he sequence. An j value no honourng he above condon wll lead o he rejecon of he hpohess. For example, f he ner-even dsance of a pon process s beng analsed, one of he obvous choces of sascs for null hpohess esng s he squared dfferences beween he calculaed H and he heorecal H,.e., s d H H d where H s he probabl dsrbuon of ner-even dsances of he pon model o be esed. ndependen realsaons are generaed and each of he squared 6

7 Sochasc Modellng of Fracures n Roc Masses, 003 dfferences beween H, H,, H and H are calculaed. The above crera s hen appled o es he hpohess. In hs case, s an upper al es. For nsance, f 99 smulaons are used and he sgnfcance level s 5%, hen an ranng of s d above 96 wll rejec he hpohess. Noe f he heorecal value of H s unnown, however, can be approxmaed b he average value derved from he smulaons,.e., H H H The above example s an overall es of H for he whole range of and onl gves he pcure of average behavour. The es can also be esed on dfferen bass. In hs case, sd H d, and s H, s H,, s H, he same crera can hen be appled. In hs case, however, s a wo al es as oo small or oo large s d means he deparure of he sascs from he model a ha parcular dsance. As he es depends on value and herefore s far more comprehensble o use graph o dspla he es resuls. Based on he smulaons, f we plo he hpohess accepance envelope of H agans he whole range of, and same as he H d calculaed from he daa n he same graph, he rejecon of he hpohess can be concluded f an par of H d goes ousde he hpohess accepance envelope. Ths es s more robus and gves more deals abou he dscrepanc beween he pon model and he daa and herefore wll be adaped for curren research. The above descrpons can appl smlarl o oher sascs such as G or F. Noe, however, he es resul s correc onl f a rejecon concluson s reached. A parcular esng no rejecng he hpohess does no necessarl means he accepance of he model for he daa se. To accep a model wh confdence for a pon paern, a few dfferen sascs should normall be rgorousl esed. Noe also n he above argumens, he pon model can refer o an nown models. If he homogeneous Posson process model s used, he es wll be agans he dscrepanc beween he daa pons and a complee spaal random CSR pon paern. In oher words, he es s agans he deparure of he daa from CSR. Wh respec o he edge effec ssue, can eher be aen no accoun or gnored f sascs from daa are onl o be compared o hose from Mone Carlo smulaon usng he same reamen. If he resuls are o be compared wh he heorecal values, however, edge correcons mus be consdered. See examples below..4 Dsance analss on some generaed paerns Ths secon smpl dsplas some dsance analss resuls for some arfcall generaed paerns. The underlng models for hese paern are full defned..4. Homogeneous Posson pon paern Fgure. s a realsaon of a homogeneous Posson process for a recangular regon R0,00 0,00 wh dens λ0.0. The dsance sascs H, h, G, g, F x and f x are shown n he followng graph from Fgure.3 o Fgure.5. The Mone Carlo smulaon resuls are also ploed based on 7

8 Sochasc Modellng of Fracures n Roc Masses, smulaons. Three sascs are dsplaed from he Mone Carlo, he average smulaed value, he mnmum and he maxmum accepance envelope values wh 95% confdence based on a wo als es descrbed n he above secon. As can be seen n he homogeneous case, he calculaed sascs agree well wh he Mone Carlo es resuls n all occasons. Fgure. Homogeneous Posson pons Fgure.3 a H vs dsance Fgure.3 b h vs dsance Fgure.4 a G vs dsance Fgure.4 b g vs dsance 8

9 Sochasc Modellng of Fracures n Roc Masses, 003 Fgure.5 a F x vs dsance x Fgure.5 b f x vs dsance x.4. Non-homogeneous Posson pon paern Fgure.6 s a realsaon of a non-homogeneous Posson process for he same regon bu wh he dens funcon defned as: uv 00 λ u, v 0. e where u and v are he horzonal and vercal coordnaes. The dsance sascs H, h, G, g, F x and f x are shown n Fgure.7 o Fgure.9 respecvel. As can be seen from H and h analss, he dsrbuons sar o depar from he sascs obaned from homogeneous smulaon. The degree of deparure depends on he daa bu can no easl be quanfed b he analss. The G and g analss does no even sugges he non-homogene for he paern. F x and f x onl show a ver modes degree of deparure. In general for non-homogeneous case, h wll end o be negave sewed as here wll alwas be some pon aggregaon compared wh he average dsrbuon. Ths feaure s clearl vsble from he fgure. More dscusson abou hs pon wll follow n he nex secon. Fgure.6 Non-homogeneous Posson pons 9

10 Sochasc Modellng of Fracures n Roc Masses, 003 Fgure.7 a H vs dsance Fgure.7 b h vs dsance Fgure.8 a G vs dsance Fgure.8 b g vs dsance Fgure.9 a F x vs dsance x Fgure.9 b f x vs dsance x.4.3 Posson cluser pon paern Fgure.0 s a realsaon of a Posson cluser process for he same regon. The paren process n hs example s a homogeneous Posson process wh dens λ Each paren produces a fxed number of 0 daughers. Daugher pons are 0

11 Sochasc Modellng of Fracures n Roc Masses, 003 unforml dsrbued around her paren whn a crcle of radus of 5 and cenred a her paren locaon. The realsaon consss of daugher pons onl. The dsance sascs H, h, G, g, F x and f x for he realsaon are shown from Fgure. o Fgure.3. All fgures sugges a ceran degree of deparures of he pon paern from homogeneous case. Onl h, however, demonsraes some neresng resuls. h shows mul-modes characerscs. The frs mode peas a around 5, whch suggess from he hsogram analss pon of vew ha here s a pon aggregaon wh he average ner-even dsance nsde he aggregaon a abou 5. Ths s exacl he sze of he cluser we specfed when generang he paern. There are also some oher modes ha sugges aggregaon a dfferen level. The overall rend of h, however, follows more or less he curve for homogeneous process, whch suggess ha he underlng paren process s homogeneous. Recall from he las secon for non-homogeneous case, he dsrbuon of h wll end o be negave sewed. Fgures of oher dsance analss alwas also demonsrae consderable amoun of devaon of hs paern from homogeneous case, bu he connecon beween he resuls and he defned cluser process s no obvous. Fgure.0 Posson cluser process pons Fgure. a H vs dsance Fgure. b h vs dsance

12 Sochasc Modellng of Fracures n Roc Masses, 003 Fgure. a G vs dsance Fgure. b g vs dsance Fgure.3 a F x vs dsance x Fgure.3 b f x vs dsance x To ac as a furher example, anoher realsaon of a Posson cluser process s analsed. In hs realsaon, he daugher pons are dsrbued accordng o he followng bnormal dsrbuon: h x x, e π 3.5 wh varance σ 3.5. The paren process agan s homogeneous. The pons mage s shown n Fgure.4. Dsance analss resuls are dsplaed n Fgure B comparng hese fgures wh hose of Fgure..3, s no dffcul o conclude ha grea smlares n he resuls beween he wo pon realsaons. The average dsance beween pons from he same paren s whch s correcl denfed n he h analss resul b he pea of he frs mode.

13 Sochasc Modellng of Fracures n Roc Masses, 003 Fgure.4 Posson cluser process pons Fgure.5 a H vs dsance Fgure.5 b h vs dsance Fgure.6 a G vs dsance Fgure.6 b g vs dsance 3

14 Sochasc Modellng of Fracures n Roc Masses, 003 Fgure.7 a F x vs dsance x Fgure.7 b f x vs dsance x These are no seleced examples. In fac clusered pons paern alwas show he same characerscs,.e., mul-modes n he h graph and he overall rend mplcl represen he paren process. In oher words, f h calculaed from a parcular daa se shows he same or smlar characerscs, a cluser process can hen be used as he underlng model. If here s no evdence of mul-mode h, a nonhomogeneous model ma be more suable. Ths, however, onl serves as a general rule of humb and should no be reaed as a fxed relaon. For example, n wo exreme cases, f he number of clusers aggregaed pons cloud s small compared o he whole pon populaon, or f he dsrbuon of daughers s more spread ou, resulng n greaer mxures beween daughers from dfferen paren, he mul-mode judgemen based on h ma no be suable, or ma even provde false drecons. Oher lmaons nclude he applcaons of he rule n he case of ansoropc dsrbuon of daugher process. In hs suaon, dreconal dsance analss ma help resolve he problem. Noe he rule descrbed here can onl help o judge f he pon paern has clusers and help o esablsh he mos lel cluser sze f has. I does no reveal an nformaon abou he dsrbuon of daugher process self..4.4 Cox pon paern Fgure.8 s a realsaon of a Cox process for he same regon. The Cox model s defned as a normal dsrbuon wh mean and varance defned as follows: uv 00 mean u, v 0. e uv 00 varance u, v 0.05 e where u and v are he horzonal and vercal coordnaes. The mean bascall defnes a mean dens feld whch s analogous o ha used n he example of Fgure.6. A each locaon, he mean ogeher wh he varance ha s abou 5% he mean value, defne a normal dsrbuon for he dens a ha locaon. A random value s hen generaed from hs dsrbuon o serve as he realsaon of he dens feld a he 4

15 Sochasc Modellng of Fracures n Roc Masses, 003 locaon. B comparng he mage wh ha of Fgure.6, s obvous he realsaon of Cox process expresses more randomness whch renders he defned underlng model such as he mean less obvous. Ths s ceranl due o he exra random componen whch conrols he realsaon of he random dens feld. Ths feaure of Cox process obvousl mposes exra dffcules o denf an effecve paramerc model for he modellng exercse. The dsance sascs H, h, G, g, F x and f x are shown n Fgure.9 o Fgure. respecvel. Though he resuls show ver smlar feaures as he non-homogeneous pon paern, as compared wh Fgure.7 o Fgure.9, he end o ge closer o CSR paern because of he reason saed above. Fgure.8 Cox Posson pons Fgure.9 a H vs dsance Fgure.9 b h vs dsance 5

16 Sochasc Modellng of Fracures n Roc Masses, 003 Fgure.0 a G vs dsance Fgure.0 b g vs dsance Fgure. a F x vs dsance x Fgure. b f x vs dsance x.5 Dsance analss of wo example pon daase In hs secon, wo praccal examples are analsed. The pons daa paerns are derved from wo fracure race mages whch are also dsplaed. For each of he wo daases, here are clearl wo dsnc ses of fracures whch are almos perpendcular o each oher. Ths s clearl he demonsraon of wo dfferen geologcal formaons and ma be benefcal o analse he wo fracure ses n each example separael. The separaon of fracure ses and her dsnc modellng are he fundamenal seps oward he herarch modellng descrbed b Lee & Ensen []. 6

17 Sochasc Modellng of Fracures n Roc Masses, 003 Fgure. a Example fracure races Fgure. b Example Pons daa se Fgure.3 a Example fracure race Fgure.3 b Example - fracure se Fgure.4 a Example fracure race Fgure.4 b Example - fracure se 7

18 Sochasc Modellng of Fracures n Roc Masses, 003 Fgure.5 a Example fracure races Fgure.5 b Example Pons daa se Fgure.6 a Example fracure race se Fgure.6 b Example fracure race se Fgure.7 a Example fracure race se Fgure.7 b Example fracure race se Snce cumulave hsograms for neares even dsance and pon o neares even dsance x do no acuall gve clear pcures abou he dsance characerscs and herefore n he followng analss he wll no be presened o save some space of he repor. Fgure.8 shows he hree dsance analss hsogram for he example for fracure race se and combned. Analss for race se onl s dsplaed n Fgure.9, whle for race se n Fgure.30. For he second praccal example, he correspondng analss resuls are gven n Fgure.30 o Fgure.3. 8

19 Sochasc Modellng of Fracures n Roc Masses, 003 Fgure.8 a h for E. F. + F. Fgure.8 b h for E. F. + F. Fgure.8 c g for E. F. + F. Fgure.8 d fx for E. F. + F. Fgure.9 a h for E. F. Fgure.9 b h for E. F. 9

20 Sochasc Modellng of Fracures n Roc Masses, 003 Fgure.9 c g for E. F. Fgure.9 d fx for E. F. Fgure.30 a h for E. F. Fgure.30 b h for E. F. Fgure.30 c g for E. F. Fgure.30 d fx for E. F. 0

21 Sochasc Modellng of Fracures n Roc Masses, 003 Fgure.3 a h for E. F. + F. Fgure.3 b h for E. F. + F. Fgure.3 c g for E. F. + F. Fgure.3 d fx for E. F. + F. Fgure.3 a h for E. F. Fgure.3 b h for E. F.

22 Sochasc Modellng of Fracures n Roc Masses, 003 Fgure.3 c g for E. F. Fgure.3 d fx for E. F. Fgure.33 a h for E. F. Fgure.33 b h for E. F. Fgure.33 c g for E. F. Fgure.33 d fx for E. F. For example, analss for all hree cases fracure race se onl, fracure race se onl or fracure race se + se show ver smlar resuls. A bref ls of some of he common feaures are gven below:. Serous deparure from homogeneous Posson process.. Negave sewed hsograms for he ner-even dsance analss sugges n general a non-homogeneous pe of Posson process should be used for he modellng.

23 Sochasc Modellng of Fracures n Roc Masses, Onl sngle mode can be observed n h for all hree cases whch sugges here s no cluser feaures n he pon paerns. 4. Compared wh pons from fracure race se, pons from fracure race se show less devaon from homogeneous case whch reveals ha for hs se of fracures, here s less aggregaon,.e., fracures end o be more evenl dsrbued. The degree of devaon can be obaned b nspecng he hsogram. Greaer sew of he se sugges more serous pons aggregaon. 5. A logcal geologcal exenson o he above pon s ha fracure race se was creaed before se. When se fracures were beng formed a laer geologcal acv, ceran par of he roc was weaened more serousl han oher pars b he exsng se fracures and herefore arac more new fracures beng creaed. More fracure aggregaon would be he resuls of hs acon. Ths argumen, of cause, s onl a speculaon based on he dsance analss resuls. More verfcaons need o be done from dfferen angles, especall from he geologcal hsor of he se. 6. A common feaure for non-homogeneous pon paerns s he sgnfcanl hgh proporon of small neares even dsances compared wh homogeneous case. Ths s so as more pons aggregaon means more small ner-even dsance pon pars and more endenc oward non-homogene. Ths feaure s clearl vsble from g analss of he hree cases. 7. Snce a regular grd coverng he whole area of he regon beng suded s used o calculae he pon o neares even dsance x, he dsrbuon fx can hen be vewed as ceran measure for pons dsrbuon across he whole area. For he case when he whole area s covered b pons realsed from a homogeneous pon process, fx wll show dencal characerscs as g. For cases when pons onl occup ceran area of he whole regon, he dsrbuon of fx wll end o unform. An exreme case s shown n Fgure.34 below. Fgure.34 a An exreme case 3

24 Sochasc Modellng of Fracures n Roc Masses, 003 Fgure.34 b h for he case Fgure.34 c h for he case Fgure.34 d g for he case Fgure.34 e fx for he case Analss resuls show pcal feaures of non-homogeneous pon processes. The mos neresng resul s he dsrbuon of fx shown n Fgure.34 e. As mos of he area n he regon beng suded s no covered b daa or o be precsel, covered b daa wh zero pon dens value, fx ends o be unforml dsrbued. As for he example daase, consderable amoun of space n he regon s no covered b daa pons and herefore fx ends o sew o he rgh compared wh he smulaed resuls derved b he Mone Carlo smulaons. As a resul, he degree of unformal of fx ma be used as an ndex o descrbe he proporon of he space beng occuped b he pon cloud. More consderaon ma be gven o hs speculaon a laer sage. For example daa se, all analss bascall sugges smlar feaures o hose of example daa se. Non-homogeneous process s agan suggesed for he fracure race se onl, fracure race se onl or he combned fracure races. Wheher or no he wo fracure races form a herarch depends on some rgorous es of dependenc beween he ses. Relaons beween dfferen pon ses wll be covered n laer research. 4

25 Sochasc Modellng of Fracures n Roc Masses, Quardra coun analss Quadra coun analss s a nd of varance analss. I uses he measures of number of pons nsde quadras locaed nsde he regon. There s no specal resrcons for he shape and sze of a quadra provded ha he sze s reasonable compared o he volume area of he regon. For a parcular analss, however, he shape of sze of quadras are fxed. 3. Theorecal bacground From s orgnaon, quadra coun analss s o es samples from Posson dsrbuon. For a Posson dsrbuon wh mean m, he probabl of obanng sample value N s gven as: N m m P N e N! For an volume V locaed a X n a Posson pon feld, he mean of he Posson dsrbuon a he locaon wll be: m X λ X dx v For homogeneous Posson process, m V X λ V s no dependen on locaon X, bu on he sze of he volume V onl. Therefore, f we defne a quadra wh sze V and use he quadra o sample he regon, ndependenl, we should oban a seral of samples from a sngle Posson dsrbuon wh mean l V. V Suppose ha n number of samples mared as N, N,, N n, are obaned b he samplng. The mos obvous es o see f he are from a Posson dsrbuon s o derve a hsogram, hence a dsrbuon, from he samples and use he goodness-of-f esng echnque. There s anoher quc and effecve wa o reach he same or smlar concluson whch uses he equal proper beween mean and varance of a Posson dsrbuon. We do no, however, now he mean and varance of he Posson dsrbuon we are esng and herefore he can onl be esmaed from he samples. If N s he mean value of he sample, he rao beween he varance and he mean of he samples, whou an dsrbuon assumpon, s: n r N N n N If a Posson dsrbuon s assumed, r should be close o for he hpohess o hold. Though hs onl ess he samples obaned, f he hpohess holds, he logcal exenson wll be ha samples are from a homogeneous Posson pons feld, provded samplng s represenave and s done ndependenl. If he hpohess beng esed does no hold and we are sll assumng Posson pons feld, he onl mplcaon s ha he Posson process s no homogeneous. In hs case, he rao r helps o descrbe he degree of deparure of he pon paern from homogene. I s for hs reason, r n quadra coun analss s gven a specal name, he ndex of dsperson,.e., n ndex of dsperson IOD N N n N 5

26 Sochasc Modellng of Fracures n Roc Masses, 003 There are abou half a dozen of dfferen ndces, such as ndex of cluser sze, ndex of pachness, ndex of mean crowdng. For a more deal lsng, see Upon []. These ndces are derved more or less based on IOD and do no acuall reveal more nformaon abou he pon paern and herefore he wll no be dscussed here. Based on he defnon, sgnfcanl large IOD ndcaes large varaons for he number of pons nsde he quadras whch mples pon aggregaon. Sgnfcanl small IOD means he varaons beween quadra coun s small and he pon paern s more regular. Dggle [] sugges ha IOD es s powerful agans pon aggregaon, bu wea agans pon regular. 3.. Goodness-of-f es There s also an alernave wa o loo a IOD. For a homogeneous Posson process, he expeced number of pons whn he quadras locaed anwhere n he regon s a consan l V, where V s he volume of he quadra. Therefore, condoned on he oal pon number, he Pearson s goodness-of-f creron can be used o es he deparure of he samples from hs consan dsrbuon,.e., n N N ℵ N whch can be approxmaed b n χ on he condon ha n > 6 and N >. When hs creron s used, lower al and upper al ess are possble. Lower al sgnfcanl small ℵ value s used o es agans regular and upper al sgnfcanl large ℵ value s used o es agans pon aggregaon. Noe ℵ n- IOD. In oher words, hese wo analses are equvalen. 3.. Consderaons for choce of quadras and scheme of samplng There s no resrcon on he shape of quadras hough n general crcles and recangles are used for D regon and sphere and cubod are used for 3D regon. As for he sze of quadras, dfferen research are rng o derve he opmal quadra sze see Upon [] bu no agreemen can be reached. Ths obvousl s an applcaon dependen ssue bu as a rule of humb, he sze of quadras should be chosen such ha he mean pon coun should be a leas. There are wo ssues for he samplng scheme: he number of quadras o use and he locaons of quadras. Obvousl here are a grea range of choces here bu he man concern for he selecon s o sasf he condons of ndependen and represenave samplngs. For ndependen samplng, regularl spaced muuall exclusve quadras can serve he purpose and for represenave samplng, randoml locaed quadras ma be a beer choce. I s also possble o mpose he muuall exclusve condon o he random quadras hough he performance ma become an ssue when large number of quadras are used. Fgure 3. shows he dfference beween he random quadra samplng and conguous samplng scheme. As can be seen, random samplng ma no be able o produce oall ndependen samples because of he overlappng of some quadras. If, however, he oal number of quadras used s no oo hgh, he effec of sample dependenc on he analss resuls ma no be severe. There s research n dsance samplng analss abou he upper lm of he number of samples for ndependen 6

27 Sochasc Modellng of Fracures n Roc Masses, 003 samplng, whch s n 0. N, where N s he oal number of Posson pons Dggle []. For quadra coun analss, no smlar value s repored o m readngs. Some research effor can be dreced o he nvesgaon no he lmaon f became an ssue. For he conguous quadra samplng, he grd can floa n he regon o ge he bes represenave samples. If performance s no an ssue, coverng he whole regon wh he samplng grd ma be a safer choce. Random selecon of a ceran number of quadras from a conguous grd can also be used n some cases. Random quadras Conguous quadras Fgure 3. Quadras used for quadra coun analss 3..3 Agglomerave quadra coun analss Ths s anoher nd of varance analss oher han IOD, whch was nroduced b Greg-Smh based on he conguous quadra daa. The mehod sars b paronng he whole regon no m m regular quadras. Neghbourhood quadras are agglomeraed no blocs. A he frs sep, blocs and quadras are dencal,.e., number of quadras n each bloc q. A he second sep, each bloc conans wo quadras, q, whch can be done b horzonal or vercal agglomeraon. A he hrd sep, each bloc conans four quadras, q 4, and so on. A each sage, he squares of number of pons nsde he blocs are summed o form quan T q, expressed as follows: q T q N and he Greg-Smh varance s hen calculaed as: G T T q, q q q, 4, 8,... For homogeneous Posson pon paern, G q wll be more or less consan. If here s clusers, however, G q s clamed o be able o reach a pea a a value of q whch ndcaes he cluser sze. In m opnon, hs clam ma no necessarl be rue as peang n hs case also depends on spaal arrangemen of pon paern. Examples wll be gven n laer secon for hs argumen. In our analss, n order o elmnae he dreconal effec mposed b he orgnal proposal, nermedae blocs showng dfferences when horzonal or vercal 7

28 Sochasc Modellng of Fracures n Roc Masses, 003 agglomeraon s used are no o be used. Ths leaves he Greg-Smh varance beng calculaed b: G 4 T T4 q, 4, 6, 64,... q q q 3. Quadra coun analss of smulaed pon paerns In hs secon, we are gong o presen some quadra coun analss for four dfferen pe of smulaed pon paerns, homogeneous, non-homogeneous, cluser and Cox Posson processes. The nenson here s rng o mae he connecon beween he pon paerns and he expeced quadra coun analss resuls. 3.. Homogeneous Posson pons Fgure 3. a shows a realsaon of a homogeneous Posson process. Fve dfferen pes of quadra coun analss are conduced. Fgure 3. b s he analss b random quadra mehod where 50 random quadras are generaed for each quadra sze. Quadra szes n hs case are measured n relave scale,.e., percenage of he ranges of he regon n horzonal or vercal drecons. Quadras wh oo small sze wll resul n oo low value of mean couns n he quadras. Quadras wh oo large sze wll ncrease he neracon beween quadras and hence he correlaon beween he samples n he random quadras case, or resul n oo few oal number of quadras usable for samplng n he case of regular grd quadras. Boh of he cases should be avoded n he analss. In he resuls presened below, he szes of quadras range from 0.0 % o % of he sze of he regon. Fgure 3. a Homogeneous pons Fgure 3. b B random quadras 8

29 Sochasc Modellng of Fracures n Roc Masses, 003 Fgure 3. c B regular quadras - Fgure 3. d B regular quadras - Fgure 3. e B regular quadra - 3 Fgure 3. f B Greg-Smh grd As can be seen from Fgure 3. b, he ndex of dsperson IOD remans more or less consan a he value of around.0 for dfferen quadra szes. I also follows he curve of he average IOD based on he Mone Carlo smulaon. The χ values for dfferen quadra szes are also calculaed and shown as he green curve n he Fgure. The number of degrees of freedom n hs case s 49 and he 95% confdence crcal value s 67. As can be seen from he fgure, mos of he χ values are no sgnfcan. Onl he χ value for he quadra sze of abou 0. exceeds he crcal value. Ths phenomenon, however, s no conssen, as s no he case durng oher runnng sessons.e., wh dfferen random quadras and herefore he χ values for dfferen quadra szes should be consdered no sgnfcan. In he fgure, mean coun and quadra coun varance are also calculaed and dsplaed. For homogeneous case, hese wo sascs wll have quadrac ncrease wh he sze of quadras, as can be seen from he fgure. For quadra coun analss usng regular qurdra, hree dfferen opons are used. In he examples shown here, he sze of he grd covers 80% of he regon. The frs opon s for he grd o be fxed a ceran locaon, for example, sarng a 0. and endng a 0.9 relave scales, and he resul s shown n Fgure 3. c. The second opon s for he whole grd o be locaed randoml n he regon and he resul s shown n Fgure 3. d. The las opon s o fx he grd a ceran locaon bu onl a ceran proporon of he quadras nsde he grd are seleced randoml for he 9

30 Sochasc Modellng of Fracures n Roc Masses, 003 analss and he resul for hs opon s shown n Fgure 3. e. All resuls n hese hree fgure show ver smlar feaures. Compared wh random quadra analss, wo neresng dfference are obvous. Frsl, he χ values for small quadra sze are exremel sgnfcan mplng ha n small scales he pon process s no homogeneous. Ths s rue as an pon process can be vewed as non-homogeneous f he scale used s small enough. The reason hs feaure does no show up n random quadra analss s consdered o be due o he fxed number of random quadras used 50 compared wh abou 6000 number of quadras used n regular grd quadra analss and n he case of small quadra sze, 50 samples ma no be represenave. Secondl, noe he dfference beween he shapes of he 95% confdence envelopes for he homogeneous Mone Carlo smulaon. The dfferences beween he upper and lower 95% envelope values are vanshng and should be as he quadra sze decreases n regular grd quadra case bu hs s no so n random quadra case. Nonrepresenave samples or sample correlaons ma be he reason behnd. These feaures sugges ha n general, regular grd quadra analss should gve a more relable analss resul. For he Greg-Smh analss, he resul s presened n Fgure 3. f. There s no sgnfcan varaons n he varance excep for he large quadra szes. For large quadra szes, however, he number of quadras avalable for calculang he varances s normall small and he values are consdered less relable han he smaller quadras cases. In hs example, for quadra szes less han 5% of he sze of he regon, he varances are more or less consan, whch mples no pon aggregaons deeced accordng o he nenons of he analss. Noe also he emprcal varances follow que well he varances from Mone Carlo smulaons. 3.. Non-homogeneous Posson pons For non-homogeneous Posson process, we use he same example used n he dsance analss. Fgure 3.3 a shows a realsaon of a non-homogeneous Posson process wh he dens defned as: uv 00 λ u, v 0. e The correspondng quadra coun analss resuls are shown from Fgure 3.3 b o f. Fg. 3.3 a Non-homogeneous pons Fgure 3.3 b B random quadras 30

31 Sochasc Modellng of Fracures n Roc Masses, 003 Fgure 3.3 c B regular quadras - Fgure 3.3 d B regular quadras - Fgure 3.3 e B regular quadra - 3 Fgure 3.3 f B Greg-Smh grd From hese fgures, s no dffcul o conclude from IOD values ha serous deparure from homogeneous pon process has been deeced b he analss. The χ values are all exremel sgnfcan whch also mples he deparure of he pon paern from CSR. Greg-Smh varances also show he dscrepanc from CSR bu no sgn of pon aggregaons. One of he neresng pons shown b hese fgures s for he cases of analss when he quadra szes are small. For hese cases, he resuls do no acuall sugges deparure of he pon paern from CSR. Ths can be consdered as one of he wea pons b quadra coun esng. In oher words, he quadra coun analss s no sensve for small quadra szes Posson cluser pons We wll agan use he same cluser process used for dsance analss. Fgure 3.4 a s a realsaon of a Posson cluser process where he paren process s a homogeneous Posson process wh dens λ0.005, each paren produces a fxed number of 0 daughers and daugher pons are unforml dsrbued around her paren whn a crcle of radus of 5 and cenred a her paren locaon. The realsaon consss of daugher pons onl. Fgure 3.4 b f dspla he resuls of he quadra coun analss. Serous deparure from CSR s agan evden n all resuls. 3

32 Sochasc Modellng of Fracures n Roc Masses, 003 Fg. 3.4 a Posson cluser pons Fgure 3.4 b B random quadras Fgure 3.4 c B regular quadras - Fgure 3.4 d B regular quadras - Fgure 3.4 e B regular quadra - 3 Fgure 3.4 f B Greg-Smh grd Compared wh he analss resuls for non-homogeneous pons, Fgure 3.3, he consderable dfference s he IOD values or G-S varances for small quadra szes. The greaer values n hese fgures sugges here s more pon aggregaons n smaller scale n hs example compared wh he non-homogeneous case presened above. Ths s rue as he cluser process creaes pon aggregaons n he scale of 0 or 0. n relave scale. From hese fgures, s onl possble o conclude ha pon aggregaons happen n he scale roughl less han 0 or 0. relave, bu nohng more deals can be obaned. 3

33 Sochasc Modellng of Fracures n Roc Masses, 003 Quadra coun analss s consdered o be good a deecng cluser szes bu based on our examples hs s no generall he case. Several facors conrbue o he effecveness of hs deecon. The mos mporan facor s ha he quadras used are arranged n such a wa ha concdes wh he locaons of clusers. An dealsed example s gven n Fgure 3.5 a. In hs example, he quadras used happen o be n such a wa ha he man pars of mos of pon clusers are conaned whn he quadras. In hs case, here wll be a pea value for IOD for hs quadra sze whch s 0, as can be seen from Fgure 3.5 b. The cluser sze n hs case can easl denfed from such a quadra coun analss. However, n pracce, hs nd of quadra arrangemen s unlel o be alwas he case n cluser pon paern analss and herefore deecng cluser szes b quadra coun analss s no alwas relable. For example, he same pon paern as Fgure 3.5 a s analsed agan usng he same regular grd, bu reposoned n a slghl dfferen locaon as shown n Fgure 3.5 c, he analss resul s dsplaed n Fgure 3.5 d. As can be seen, he pea value of IOD mplng he cluser sze dsappears all ogeher. The smlar resul s obaned b random quadra coun analss as dsplaed n Fgure 3.5 e and f. The cluser sze n he laer few cases fals o be deeced. Grd used Fg. 3.5 a Specal case of clusers Fgure 3.5 b IOD resuls Grd used Fg. 3.5 c Dfferen grd locaon Fgure 3.5 d IOD resuls 33

34 Sochasc Modellng of Fracures n Roc Masses, 003 Fg. 3.5 e IOD b random quadras Fgure 3.5 f IOD b random quadras Grd used Fg. 3.5 g Specal case of clusers Fgure 3.5 h IOD resuls Fg. 3.5 G-S varance b grd Fg. 3.5 j G-S varance b grd If we jus rearrange a few of he pon clusers a dfferen locaons, as shown n Fgure 3.5 g, he pea value of IOD wll also dsappear even we use he same grd as used n Fgure 3.5 a, as shown n Fgure 3.5 h. Ths reveal a ver serous dsadvanage of usng quadra coun analss: s no a relable analss n he sense ha feaures can alwas be deeced. I s far oo sensve o boh he locaons of clusers and he quadras used for he analss. Boh facors mus concde wh each oher for he analss o reall reveal he cluser feaures. 34

35 Sochasc Modellng of Fracures n Roc Masses, 003 I s he same sor when Gregh-Smh varance analss s used. For he pon paern dsplaed n Fgure 3.5 a, he Gregh-Smh varance analss resuls for he grd sze of s gven n Fgure 3.5. The pea value of he varance a he grd cell sze of abou 0 s obvous. However, hs resul agan s sensve o he changes n he grd used. Fgure 3.5 j gves he analss resul for he same pon paern bu usng he grd. As can be seen, he pea value presen n Fgure 3.5 dsappears. If we use he grd o analss he pon paern n Fgure 3.5 g, he pea value s no presen eher. Ths demonsraes he smlar concluson reached above ha deecng cluser sze b quadra coun analss s no a relable ool. Quadra coun analss can provde a relable resuls for deecng he deparure of he pon paern from CSR, bu no for deecng he sze of clusers Cox pon paern For hs analss, we wll use he same Cox process used for he dsance analss. The Cox model s defned as a normal dsrbuon wh mean and varance defned as follows: uv 00 mean u, v 0. e uv 00 varance u, v 0.05 e where u and v are he horzonal and vercal coordnaes. A each locaon, he mean ogeher wh he varance ha s abou 5% he mean value, defne a normal dsrbuon for he dens a ha locaon. A random value s hen generaed from hs dsrbuon o serve as he realsaon of he dens feld a he locaon. A realsaon of hs Cox process s gven n Fgure 3.6 a. Fg. 3.6 a Cox pons Fgure 3.6 b B random quadras 35

36 Sochasc Modellng of Fracures n Roc Masses, 003 Fgure 3.6 c B regular quadras - Fgure 3.6 d B regular quadras - Fgure 3.6 e B regular quadra - 3 Fgure 3.6 f B Greg-Smh grd Resuls from quadra coun analss are gven from Fgure 3.6 b f. These fgure show smlar feaures as he analss for non-homogeneous case, Fgure 3.3. Apar from he concluson ha here s serous deparure of he pon paern from CSR, no oher specfc feaures are apparen. 3.3 Quadra coun analss of wo example pon daase The wo example daa ses and her sub-ses presened n Fgure..7 are analsed agan usng he quadra coun analss dscussed n he prevous secons. The resuls are presened below: 36

37 Sochasc Modellng of Fracures n Roc Masses, 003 Fg. 3.7 a Example Fgure 3.7 b B random quadras Fgure 3.7 c B regular quadras - Fgure 3.7 d B regular quadras - Fgure 3.7 e B regular quadra - 3 Fgure 3.7 f B Greg-Smh grd 37

38 Sochasc Modellng of Fracures n Roc Masses, 003 Fg. 3.8 a Example se Fgure 3.8 b B random quadras Fgure 3.8 c B regular quadras - Fgure 3.8 d B regular quadras - Fgure 3.8 e B regular quadra - 3 Fgure 3.8 f B Greg-Smh grd 38

39 Sochasc Modellng of Fracures n Roc Masses, 003 Fg. 3.9 a Example se Fgure 3.9 b B random quadras Fgure 3.9 c B regular quadras - Fgure 3.9 d B regular quadras - Fgure 3.9 e B regular quadra - 3 Fgure 3.9 f B Greg-Smh grd 39

40 Sochasc Modellng of Fracures n Roc Masses, 003 Fg. 3.0 a Example Fgure 3.0 b B random quadras Fgure 3.0 c B regular quadras - Fgure 3.0 d B regular quadras - Fgure 3.0 e B regular quadra - 3 Fgure 3.0 f B Greg-Smh grd 40

41 Sochasc Modellng of Fracures n Roc Masses, 003 Fg. 3. a Example se Fgure 3. b B random quadras Fgure 3. c B regular quadras - Fgure 3. d B regular quadras - Fgure 3. e B regular quadra - 3 Fgure 3. f B Greg-Smh grd 4

42 Sochasc Modellng of Fracures n Roc Masses, 003 Fg. 3. a Example se Fgure 3. b B random quadras Fgure 3. c B regular quadras - Fgure 3. d B regular quadras - Fgure 3. e B regular quadra - 3 Fgure 3. f B Greg-Smh grd These fgures reveal nohng more han he concluson ha he pon paern beng analsed have serous deparure from CSR. An pon aggregaon feaures can no be concluded from hese resuls. Man of he fgures presened here are solel for he purpose of gvng a complee se of he analss. 3.5 General conclusons Quadra coun analss s an effecve ool for deecng he deparure of he pon paern from CSR. I s wea for quanfng an underlng feaures of he pon 4

43 Sochasc Modellng of Fracures n Roc Masses, 003 paern. Que ofen, characerscs such as cluser sze ma no be able o be deeced b hs mehod. 4. K-funcon analss The las paern analss ool o be dscussed n hs repor s he K-funcon analss, whch belongs o he caegor of second momen analss of pon dens. The analss s equvalen o he varogram analss n geosascal modellng and reveals some valuable spaal correlaons for he pon dens measuremen. 4. Theorecal bacground As saed, K-funcon s abou pon dens, naurall a sarng pon of he analss wll be he defnon of he pon dens, denoed as lx: E[ N dx ] λ X lm dx 0 dx where X s he locaon varable, dx s an nfnesmal volume conanng locaon X, NV s he number of pons whn volume V and E[..] s he expeced value. Smlarl he second-order pon dens, denoed as l X s defned as follows: E[ N dx N dy ] λ X, Y lm dx, dy 0 dx dy where Y s also a locaon varable. Furher, we defne a covarance dens gx,y ha s drecl relaed o lx and l X,Y, analogous o covarance defnon of wo random varables, γ X, Y λ X, Y λ X λ Y For saonar pon processes, or homogeneous pon processes, lx wll be a consan value l, ndependen of locaons. For hese cases, λ X, Y λ X Y, gx,y gx - Y,.e., l X,Y and gx,y are also locaon ndependen; If he process s also soropc, λ X, Y λ and gx,y g, where s he dsance beween locaons X and Y. Usng hese defnons, supposed lx can be evaluaed accurael a all locaons whn he regon R beng consdered, he random pons are hen ransformed no a random feld quanfed b he dens values. Tools such as geosascs can hen be used o analse and model he varable. There are, however, wo man pons whch obsruc us o go drecl n hs roue for he modellng. Frsl, he esmaon of pon dens lx s dffcul o be conduced n an objecve and accurae manner. Secondl, geosascs onl descrbes he model whch s correc n a global scale and s lac of descrpons for local deals. Ths characersc wll fnd he echnque dffcul o model sensbl some specal pon processes whch requre boh global and local models. A hand example for hs case s he cluser process whch needs he global model o descrbe he dsrbuons of clusers whn he regon, and he local model o descrbe he pon dsrbuons whn clusers. Ths argumen does no mpl ha geosascs s no applcable n he case of pon process modellng. I ma be worhwhle o do some comparson analss a laer sage. 43

44 Sochasc Modellng of Fracures n Roc Masses, 003 Meanwhle somehng suable for modellng spaal correlaons of pon dens for pon process s needed. K-funcon s such a ool. A smple and praccal defnon for K funcon s as follows: E[ number of furher evens whn dsance of an arbrar even] K λ For formal defnon of K funcon based on reduced Palm dsrbuon, please see Cresse []. The reason ha he defnon ncludes he pon dens value as s denomnaor s for normalsaon of he expresson of K b pon dens values, hence elmnang he scalng effec of he dens value. In oher words, K corresponds o he expeced number of furher evens whn dsance of an arbrar even when he pon dens s a un. Tae he saonar pon process as an example. Accordng o he defnon of pon dens lx, he expeced number of evens whn a volume of V can be expressed as: N V λ X dx V For saonar cases, NV l V, and herefore he K funcon n hese cases wll be: λ V K V λ whch s ndependen of dens value l,.e., same pon paerns wh dfference onl n dens values can be descrbed wh he same model. For wo dmensonal case, K p. Snce K funcon s also a second order measure drecl relaed o pon dens, her relaon can be formerl esablshed. Condoned on a nown arbrar even locaed a X, we can fnd he condonal probabl of anoher even a locaon Y as: P{ N dy > 0, N dx } P { N dy > 0 / N dx } P{ N dx } where dx and dy are nfnesmal volume cenered a locaon X and Y. If dy s done n such a wa ha onl one even s possble nsde he volume, we can hen have he followng relaons: E{ N dy / N dx } P{ N dy / N dx } P{ N dy / N dx } E{ N dy, N dx } P{ N dy, N dx } P{ N dy, N dx } E{ N dx } P{ N dx } P{ N dx } The condonal expecaon of he number of evens whn dy s hen: E{ N dy / N dx } P{ N dy / N dx } + 0 P{ N dy 0 / N dx } P{ N dy / N dx } P{ N dy, N dx } P{ N dx } E{ N dy, N dx } E{ N dx } E{ N dy, N dx } dx dy λ X, Y dy dy E{ N dx } lm dx, dy 0 λ X dx Noe dy s onl an nfnesmal volume cenred a locaon Y and o ge he oal expeced number of evens whn dsance of he locaon X, whch s l K accordng o defnon, dy mus be negraed over he ball cenred a X. For example, n wo dmensonal case, he negraon mus be done over he area of he crcle cenred a X,.e., 44

45 Sochasc Modellng of Fracures n Roc Masses, 003 π λ X, π λ X K X, d dθ X X λ λ λ X, d Noe d here means he negraon over and s dfferen o he meanng of dy dscussed above. The above relaon can be re-arranged as: [ λ X ] λ X, K X, π For saonar case, X can be dropped from he relaon: λ λ K π For pon process n d-dmensons, smlar negraon can be done and he resul s gven below: d λ Γ + λ K d d d π Ths esablshes he formal relaon beween l and K. 4. Esmaon of K Analogous o he varogram n geosascs, K can be esmaed emprcall from pon paern realsaons. As l K s defned as he expeced number of furher evens whn dsance of of an arbrar even, a drec esmae can be obaned for a pons realsaon wh N number of evens as: N N λk M number of furher evens whn dsance of even N N To esmae M, he number of furher evens whn dsance of even, we assgn ndcaor values for all he evens excep even as follows: f d j I j where d j s he dsance beween even and even j 0 oherwse Then, M N j Ths leads o a smple esmae of K as: K λ N I j N N j j j I j where l can be replaced wh he emprcal nens N/VR. Ths esmae does no nclude par of evens for whch even j s ousde he regon R and s no observable. In oher words, he edge effec s no aen no accoun and he esmae s based. To oban an unbased esmae for K, we can use he guard volume echnques dscussed n Secon of hs repor. Ths approach, however, effecvel hrow awa a consderable amoun of valuable pons. Anoher approach s o ae no accoun he condonal probabl p j ha even j s observed gven ha he dsance beween he even and he even s d j. 45

46 Sochasc Modellng of Fracures n Roc Masses, 003 For wo dmensonal case, p j can be calculaed as he proporon of he crcumference nsde he regon R of he crcle cenred a and wh radus d j. As shown n Fgure 4., p j s he proporon of he sold crcumference lne over he whole crcumference of he crcle. When he crcle s full enclosed b he regon R, p j. In oher words, he edge does no affec he par. When he crcle s parall enclosed, p j <, whch means here are possbles some evens wh he same dsance as even j o even are ousde he regon R and herefore he calculaed ndcaor value I j mus be compensaed. To compensae I j, he wegh w j /p j s used o ncrease he ndcaor value gong no he calculaon of K. R p j l sold /l sold +l dash w j / p j R p j Ss l n /S [s l n +s l ou ] w j / p j d j l sold j s l+ s l d j s l n j l dash s l ou Fgure 4. a Edge correcon Fgure 4. b Numercal calculaon Smlar correcon o ge unbased esmae of K for d-dmensonal case can also be obaned. In hs case, surface areas of he d-dmensonal sphere can be used nsead o calculae p j. Noe he oal surface area of a d-dmensonal sphere wh radus r s: d d d π r S d Γ + The unbased esmae of K can fnall be wren as: N N K wj I j λ N j j In pracce, K s normall evaluaed for ceran number of dsances and hen he graph of K vs s ploed. There s a resrcon on he selecon of values for he evaluaons: can no go oo large. From he above dscusson abou he edge correcon, s possble as goes large, p j 0 and w j. Dggle [] sugges he upper bound of half he maxmum possble dsance wh he regon R as he maxmum value for. For a un square regon, hs wors ou o be 0.7. For analss purpose, he graph of {K-p } vs s normall used nsead of K vs. Because Kp for homogeneous Posson pon process, he plo of {K-p } vs wll be a horzonal lne wh he value of 0 for homogeneous cases. For cases oher 46

47 Sochasc Modellng of Fracures n Roc Masses, 003 han homogeneous one, he plo wll demonsrae drecl he degree of deparure of he pon paern from CSR. 4.3 Implemenaon ssues The dffcul par for unbased esmae of K les n he esmae of he edge correcon weghs, w j. For wo dmensonal case wh a recangular regon, Dggle [] gve he soluon of w j as follows: mn d, d j mn d, d j cos { + cos d j d j pj f d j d + d π p j 3 4 cos d d j + cos π d d j where d mnx, a-x, d mn, b- and a and b are he szes of he recangles n x and drecons. d and d as defned are he shorer dsance of o he wo vercal edges and he wo horzonal edge of he recangular regon. The above equaons are onl applcable when d j s n he range of [0, mn a, b ], whch que ofen ma no be adequae n praccal analss, especall when he regon s an exremel slm fla recangle.e., consderable dfference n a and b. Ths resrcon provdes onl a paral edge correcon for K evaluaon n he regon for small values, no he whole range of neres. We approach he calculaon of p j purel b numercal approxmaon descrbed below. As shown n Fgure 4. b, we dvde he crcumference of he crcle no L equal lengh segmens. The probabl p j s hen calculaed as: sl sl s nsde R L pj [ s + s ] L l sl s nsde R l sl s ousde If he value of L s large enough, he numercal mehod should provde a ver good approxmaon. Accepable L value obvousl depends on he sze of he regon. In mos of he cases we use he value of 50 for L and he dfferences beween he numercal resuls and hose calculaed b he above equaons for small d j values are neglgble. The advanage of usng numercal approxmaon s ha he p j can be calculaed for large d j value up ll he maxmum possble dsance n he regon. Ths s requred for a complee edge correcon for he evaluaon of K n he regon. As dscussed above, here s possbl ha w j becomes unbounded for oo large d j values, whch wll push he correced K values for large owards nfn. Anoher advanage of hs approach s ha he echnque can be readl adaped for hgher dmensonal cases where no analcal soluons for p j are no avalable. More on hs pon when we come o he sage o deal wh hree dmensonal problems. R f d j > d + d 47

48 Sochasc Modellng of Fracures n Roc Masses, 003 The second ssue needs dscussng s he evaluaon of second order pon dens l and he covarance dens g. From Secon 4., we undersand he evaluaons of hese wo denses nvolve he calculaon of he dervaves of K,.e., K. As onl dscree values of K are avalable, K can onl be evaluaed numercall a he where K value s avalable. The average of he forward and bacward dervaves a s used as K a ha dsance. As llusraed n Fgure 4., he forward and bac dervaves a dsance can be wren as: K + K K forward K K K bacward K K forward + K bacward K K+D K-D -D +D Fgure 4. Calculaon of K where D s he dvson ncremen used for he evaluaon of K. 4.4 K-funcon analss of generaed pon paerns We wll sar he K funcon analss on some pon daases generaed from nown pon process. Ths should help us o buld he correspondence beween he characerscs of K, g and he underlng pon process. A he curren sage of he progress, onl wo dmensonal analss s avalable and wll be presened Homogeneous Posson pon process For homogeneous Posson process, he analcal soluons for K, l and g are smple. In hs case, lx l,.e., he pon dens s ndependen of pon locaons. From he defnon and relaons, he followngs can be obaned: λ λ K π or K π 0 γ 0 As can be seen, g 0, whch mples here s no correlaon beween pon denses n dfferen locaons for hs process. 48

49 Sochasc Modellng of Fracures n Roc Masses, 003 The frs daa se generaed s from a homogeneous Posson process wh l0.0 for he same regon 0,00 0,00 used n prevous demonsraons. Fgure 4.3 a s one realsaon of he process. Fgure 4.3 a Homogeneous pons Fgure 4.3 b [K-p ] vs The funcons of K, l and g for hs pon paern are gven n Fgure 4.3 b. The K funcon values from 00 Mone Carlo smulaons are also ploed. The green and pn lnes are he 95% confdence envelope of [K-p ] when he underlng process s homogeneous. As can be seen from he fgure, [K-p ] from he daa follows more or less he average value from he Mone Carlo smulaons, and s oscllang around 0, mplng homogeneous pon paern. g s also oscllang around 0 whch s also he behavour of homogeneous Posson pon paern as dscussed above. Noe hs analss s conduced up ll he dsance of 90. To demonsrae he effec of edge correcon on he evaluaon of K, l and g, we have he followng comparson. Fgure 4.4a and b shows he emprcal values of K, l and g wh edge correcon mposed for a homogeneous Posson process. Fgure 4.4 c and d gves he evaluaed values whou edge correcon. As can be seen, K, l and g values whou edge correcon could be ver msleadng. Sgnfcan deparure from values he should have s obvous afer a ver shor dsance. All funcons sugges wrongl a non-csr pon paern. The funcon values wh edge correcon, on he oher hand, agree well wh analcal resuls and all sugges he correc pon process. Fgure 4.4 a [K-p ] vs wh edge correcon Fgure 4.4 b K vs wh edge correcon 49

50 Sochasc Modellng of Fracures n Roc Masses, 003 Fgure 4.4 c [K-p ] vs no edge correcon Fgure 4.4 d K vs no edge correcon 4.4. Non-homogeneous Posson pon process The same non-homogeneous Posson process used n he prevous analss for he same regon s also used here for he K funcon analss. The dens funcon s defned as: uv 00 λ u, v 0. e Fgure 4.5 a s one of he realsaons and Fgure 4.5 b s he evaluaed values for K, l and g wh edge correcon. Fgure 4.5 a Non-homogeneous pons Fgure 4.5 b [K-p ] vs As can be seen, all funcons [K-p ], l and g dspla sgnfcan deparures from he values for CSR case. A few neresng pons are worh lsed: [K-p ] curve s a of parabolc shape concavng upward. I ncreases as ncreases when s small, peas a a ceran dsance o be dscussed and hen decreases as ncreases for large values. From defnon, K s drecl proporonal o he expeced number of pons whn an area of p. Therefore a posve value of [K-p ] mples ha he acual number whn he area s greaer han ha o be expeced f he pons are evenl dsrbued n he regon.e., homogeneous dsrbuon of pons. In oher words, pon aggregaon occurs whn he area p defned b he dsance. Negave [K- p ], on he oher hand, sgnfes he acual number s lower han ha expeced for even dsrbuon case. Ths wll alwas happen n he cases of large 50

51 Sochasc Modellng of Fracures n Roc Masses, 003 dsance, hence large areas covered. These behavours descrbed are exacl he basc characerscs of non-homogeneous pon dsrbuon across he regon as a whole. The dsance when [K-p ] peas represens a balancng pon when he degree of pon aggregaons sar decreasng as he area p ncreases. Ths pon corresponds o he pon where covarance g changes sgns from posve o negave, mplng negave correlaons beween pon dens separaed beond hs dsance. Afer he balancng pon, [K-p ] value connues decreasng as ncreases and evenuall becomes negave, mplng lower number of pons han expeced. For ver large dsances, [K-p ] becomes unbounded. Ths ma be caused b he unbound proper of he edge correcon wegh w j and herefore should be dscarded. Noe n hs example, he dsance of hs balancng pon s abou 50, half of he sze of he edge of he recangular regon. As dscussed, covarance dens g wll change from posve o negave as ncreases. The dsance where g 0 represens he boundar whn whch correlaon beween he pon dens of wo locaons s posve. Dens values of wo locaons separaed more han hs dsance wll have negave correlaon beween hem. As menoned above, hs dsance corresponds o he pea value of [K-p ]. Recall from geosascs, he srucural analss alwas suggess he range of nfluence beond whch he wo random varables wll no longer correlae,.e., correlaon coeffcen 0. Ths seems no o be he case n pon dens analss. The nfluence of pon dens n one locaon on anoher locaon can eher be posve or negave excep for homogeneous cases. The reason for hs s due o he consrucon of he reference homogeneous average pon dsrbuon used for dens analss, whch aes all pons and he whole regon. The K funcon analss s acuall abou he dfference beween he acual and he reference pon paerns and herefore dens varables n he whole regon are correlaed. The shape of he curve for g ma be mporan. I ma be used o reveal he characerscs of he underlng dens model of he pon process. We wll come bac o hs pon n laer dscussons Posson cluser process Agan we wll use he same cluser process used n he prevous analss. The paren process n hs s a homogeneous Posson process wh dens λ Each paren produces a fxed number of 0 daughers. Daugher pons are unforml dsrbued around her paren whn a crcle of radus of 5 and cenred a her paren locaon. The realsaon consss of daugher pons onl. In he followng analss, I use he word bump. Please If ou can hn of a beer suggeson. Fgure 4.6 a and b are he generaed paern and he resuls for emprcal K, l and g. The mos neresng feaure of he graph s he bump characerscs. Ths s acuall he feaure unque o cluser pon processes. To demonsrae hs, we sar wh jus one cluser, such as he one shown n Fgure 4.7 a. The K, l and g for hs pon se are gven n Fgure 4.7 b and c. As can be seen, one bump s presen for [K-p ], bu none for ohers. For wo clusers, he resuls are shown n Fgure 4.8 and wo bumps are observed for [K-p ] and one for l and g. For 5

52 Sochasc Modellng of Fracures n Roc Masses, 003 hree clusers, Fgure 4.9 shows four bumps for [K-p ] and hree for l and g. In fac, for n clusers, he number of bumps presen n he curve of [K-p ] vs wll be n n + and he n n for he curves of l and g. Ths nd of behavour can be explaned below: Fgure 4.6 a Cluser pons Fgure 4.6 b [K-p ] vs A Sze» 0 Fgure 4.7 a One cluser Fgure 4.7 b [K-p ] vs Sze» 0 A Dsance» 58 B Sze» 0 Fgure 4.7 c [K-p ] vs Fgure 4.8 a Two clusers 5

53 Sochasc Modellng of Fracures n Roc Masses, Dsance» 58 0 Dsance» 58 Fgure 4.8 b K vs Fgure 4.8 c [K-p ] vs Ds. 88 B A Ds. 58 Ds. 3 C Fgure 4.9 a Three clusers Fgure 4.9 b K vs For one cluser case, Fgure 4.7 a, when s whn he scale of he cluser sze, he K funcon analss s equvalen o he analss of a small regon A conanng all he pons. The resuls wll be smlar o ha shown n Fgure 4.4 d as edge correcon s no an ssue here. As ncreases and ges over he scale of he cluser sze, K wll sa unchanged as all pons are alread ncluded. [K-p ] wll decrease bu l wll reman as zero because for Fgure 4.9 c [K-p ] vs an wo locaons separaed b hs dsance, pon dens for one of he locaons wll alwas be zero. g wll also reman consan when K reman he same. For wo cluser case, Fgure 4.8 a, when dsance s whn he scale of he cluser sze, he behavour of K, l and g are he same as n he sngle cluser case excep he absolue funcon values. As ncreases and before he area defned b spans boh cluser A and B, K and g wll also sa unchanged and l wll 53

54 Sochasc Modellng of Fracures n Roc Masses, 003 reman as zero. As he area sars nclude pons from boh clusers a he same me, values for hese funcons wll sar ncreasng unl he maxmum separae dsance n he example 58 s reached. Afer ha, K and g wll agan sa unchanged and l wll agan become zero. As for hree cluser case, Fgure 4.9 a, K, l and g have he same behavour as hose dscussed above. Onl he absolue values of he funcons are dfferen. As dsance ncrease, K, l and g wll reman unchanged before he area defned b can possbl cover pons from a leas wo ou of he hree clusers. The funcon values wll hen sar ncreasng unl he sablse a anoher level. Noe n hs case, here are hree dfferen dsances separang he clusers and herefore hree furher sablsng sages, or hree furher bumps can be observed. For he case of n clusers, smlar characerscs as hose dscussed above can be expeced for K, l and g. As menoned above, he oal number of bumps s equal o n n + for [K-p ] and n n for l and g. The acual numbers, however, ma be dfferen dependng on he cluser dsrbuons across he regon. For example, f all clusers are separaed b he same dsance onl one bump wll be observed as pons from all clusers wll come no effec a he same dsance. Anoher possble case wll be when he dfferences n dsances beween clusers are small or here are oo man clusers n he regon, wll no possble o dsngush wo or more ver close bumps. In exremel case when nf,.e., here are nfne number of clusers separaed b all possble dsances, an nfne number of bumps wll mae up he curves whch wll acuall come ou as smooh curves,.e., no bumps a all. The number of bumps for [K-p ] s alwas one more han n n. The frs bump corresponds o he behavour of pons whn clusers and he res of n n bumps are he behavours beween pons from dfferen clusers. Therefore, he frs bump provdes a hand ool for he esmaon of average sze of he clusers. For example, all he fgures gven above show correcl he cluser sze of abou 0 for he pon process. In mos cases hs esmaon s correc. I sll can be used when he dsance beween clusers s less han he cluser sze as n hs case clusers jon ogeher o form large clusers and he frs bump can sll be used o esmae averaged joned cluser sze. The onl excepon s when all clusers mx ogeher o form a smeared pcure of pons n he regon so ha pon clusers vsuall dsappears. In hs case, here ma be sll he frs lump n he [K-p ] curve, whch ma or ma no correcl denf he sze of he underlng pon clusers, or he frs bump ma no show up a all. To demonsrae hs pon, loo a he followng examples. Fgure 4.0 a s realsed from a cluser process wh cluser radus of 0 cluser sze 40 and as can be seen from Fgure 4.0 b, he cluser sze s correcl denfed from he frs bump n he [K-p ] funcon. When he number of cluser s ncreased, however, he cluser paerns become mxed up and a smeared pon paern s obaned, as shown n Fgure 4.0 c. The [K-p ] funcon shown n Fgure 4.0 d fals o denf an cluser effec a all. I onl reveals a non-homogeneous pon process. 54

55 Sochasc Modellng of Fracures n Roc Masses, Fgure 4.0 a A few clusers Fgure 4.0 b [K-p ] vs Fg. 4.0 c A few more clusers added Fgure 4.0 d [K-p ] vs The feaure of he frs bump n he [K-p ] curve dscussed above s onl broadl correc f he cluser paern s he domnan characerscs whn he regon. In pon processes, pon clusers are presen bu he domnan feaures of he whole pon paern s somehng else such as a non-homogeneous pon process. In hs case, [K-p ] curve wll sll gve he bump feaures dscussed above bu he frs bump dsappears as for he dsance n he scale of he cluser sze, non-homogeneous pon process s domnan. Two examples are gven n Fgure 4. where he paren process s a non-homogeneous process wh dens lfx. 0 or 0 daugher pons are generaed for each paren and he are dsrbued unforml whn he crcle of radus 5 and cenred a paren pon. As can be seen from 4. b and d, he frs bump of he [K-p ] curve correspondng o he cluser sze fals o show up clearl. The curves do sll preserve he bump feaures. Care should be exercsed n esmang he cluser sze when feaures oher han cluserng such non-homogene are domnan n he pon paern. 55

56 Sochasc Modellng of Fracures n Roc Masses, 003 Fg. 4. a 0 daugher pons Fgure 4. b [K-p ] vs Fg. 4. c 0 daugher pons Fgure 4. d [K-p ] vs From he heorecal sde, he K funcon for a cluser process can be expressed as follows see Dggle []: E[ S S ] H K π + ρ E[ S} where r s he paren pon dens, S s he number of daughers per paren, E[ ] s he expecaon and H [ ] s he dsrbuon funcon of he PDF h [ ] defned as: h Y h X h X Y dx and h[ ] s he PDF of daugher pons relave o her parens. For example, f each paren produces a Posson number of daugher pons and daugher pons are dsrbued around her parens accordng o he b-varae model: h u + v σ u, v e πσ s s he dsperson varance of daugher pons. The K funcon for hs process can be deduced as see Cresse []: 4σ K π + [ e ] ρ I hn hs heorecal soluon, however, onl aes no accoun he cluserng effec n he scale of he cluser sze,.e., cluserng of pons from he same paren. The cluserng or non-homogeneous effecs from pons n dfferen clusers are onl approxmaed wh a homogeneous erm p n he equaon. In oher words, he 56

57 Sochasc Modellng of Fracures n Roc Masses, 003 equaon s onl correc up ll he dsance equal o he average sze of he clusers. Tae he example shown n Fgure 4.6. The heorecal curve based on he above equaon s onl roughl correc up o he dsance 0, whch s he average sze of clusers whn he regon, as shown n Fgure 4. below. 0 Theorecal model Fgure 4. [K-p ] vs of he example of Fgure Cox pon paern The same Cox process for he same regon used n prevous analss s also used here for K funcon analss.. The Cox model s defned as a normal dsrbuon wh mean and varance defned as follows: uv 00 mean u, v 0. e uv 00 varance u, v 0.05 e Fgure 4.3 a s a realsaon of he process and Fgure 4.3 b shows he correspondng resuls for K, l and g. The frs mpresson he fgure gves s ha hese curves loo ver smlar o hose derved for non-homogeneous process. In fac, he core elemen of hs Cox process s he same as he non-homogeneous dens model used for he example of Fgure 4.5. The Cox model, n addon, add anoher random componen o he process and herefore push he pon paern oward more homogeneous. Ths can be proved b he vsuall nspecng he pon paern of Fgure 4.3 a and Fgure 4.5 a, or can also be proved b he absolue [K-p ] values of he wo pon paern. The dscrepanc beween [K-p ] value of he Cox pon paern and he homogeneous case shown n Fgure 4.3 b s smaller compared o he value n Fgure 4.5 b, and hus reveals a more homogeneous paern for Fgure 4.3 a. The same concluson can also be reached b comparng he covarance value n Fgure 4.3 b and Fgure 4.5 b. The dfference beween he absolue g value wh 0 for homogeneous case of he Cox process s smaller compared o ha of he non-homogeneous process. Noe he g values dsplaed n Fgure 4.3 b and Fgure 4.5 b nclude he scalng effec of l. To mae he wo fgures comparable for g, he g curve mus be dvded b he scalng facor l whch wll be dfferen n hese wo cases. For he Cox process Fgure 4.3, l and l For he non-homogeneous process Fgure 4.5, l 0.03 and l The g curve shown n Fgure 4.3 b should herefore be scaled down b a facor of 3.8 o be comparable o he g curve n Fgure 4.5 b. 57

58 Sochasc Modellng of Fracures n Roc Masses, 003 Fgure 4.3 a Cox pons Fgure 4.3 b [K-p ] vs Fgure 4.5 a and b are reproduced here for eas comparson Fgure 4.5 a Non-homogeneous pons Fgure 4.5 b [K-p ] vs 4.5 K-funcon analss of wo example pon daases We now urn o he K funcon analss of he wo acual daases used n he prevous analss. See he reference for he sources of he daases Daa se The K, l and g funcon resuls for he whole of daase and he wo subses are gven n Fgure 4.4 below. 58

59 Sochasc Modellng of Fracures n Roc Masses, 003 Fgure 4.4 a Whole daase - Fgure 4.4 b [K-p ] vs Fgure 4.4 c Subse - Fgure 4.4 d [K-p ] vs Fgure 4.4 e Subse - Fgure 4.4 f [K-p ] vs These fgures suppor he obvous suggeson ha he process s non-homogeneous. The also reveal ha he sub-se pons are more homogeneous han sub-se pons,.e., here s more pon aggregaon n sub-se, whch can be concluded from he K values of he wo sub-ses. Ths s also obvous from he drec vsual nspecon of he pon paern Daa se 59

60 Sochasc Modellng of Fracures n Roc Masses, 003 The K, l and g funcon resuls for he whole of daase and he wo subses are gven n Fgure 4.5 below. [K-p ] Fgure 4.5 a Whole daase - Fgure 4.5 b [K-p ] vs Fgure 4.5 c Sub-se Fgure 4.5 d [K-p ] vs Fgure 4.5 e Sub-se Fgure 4.5 f [K-p ] vs Agan, hese fgures show feaures of non-homogeneous process. Ths example, however, demonsrae he effec of boundar surroundng he pons. B nspecng vsuall Fgure 4.5 a and e, s no dffcul o sugges ha b some polgonal boundar he pon process ma be homogeneous, such as one shown n Fgure

61 Sochasc Modellng of Fracures n Roc Masses, 003 e, bu he overall behavour of he pon paern s non-homogeneous. Ths ma be a oall mspercepon bu neverheless s worh some more nvesgaon. 4.6 General dscussons The K funcon analss presened n hs secon s amng for wo purposes: Buldng he correspondence beween he funcon characerscs and he nown pon paern Proposng he mos suable pon process model based on emprcal funcons. So far we onl demonsrae clear correspondence for wo pon processes: homogeneous and cluser processes, whch n general show clear characerscs n he second order funcons. For homogeneous process, [K-p ] and g are varng around zero. For cluser process, he bump feaures can be expeced for [K-p ], l and g and he frs bump s of parcular mporance as can be used o esmae he average cluser sze n he process. For non-homogeneous and Cox processes, second order funcons show ver smlar feaures whch mpl ha boh processes are all non-homogeneous n naure. In oher words, an nhomogeneous pon paern can be eher modelled b non-homogeneous Posson model or b he Cox model. Boh, f modelled accurael, should sascall gve he same answer on average. Apar from he exra modellng componen freedom provded b Cox process, Cox modellng s no dfference o he nonhomogeneous modellng. To llusrae hs pon, we use a one dmensonal example. In Fgure 4.6 a, a non-consan pon dens s o be modelled b a nonhomogeneous process. In hs case, he model fx seleced should concde wh he acual lx for accurae modellng. The specfcaon of fx ma prove dffcul n praccal suaons and herefore hs leads o anoher choce ou of large number of possble approaches. We can model he rend of lx b a smple model gx and hen model he resduals b a smple nose model g X. gx and g X are seleced n such a wa ha her accumulave effec sascall reproduce correcl he orgnal lx, he process s llusraed n Fgure 4.6 b. The frs modellng echnque s he drec non-homogeneous modellng and he second one s he Cox modellng. lx l X rend model fx gx X X sascall gx+g X fx l X g X nose model X Fg. 4.6 a Non-homogeneous modellng Fgure 4.6 b Cox modellng 6

62 Sochasc Modellng of Fracures n Roc Masses, 003 Neverheless non-homogeneous model s he foundaon of boh he modellng processes. I wll be necessar a some sage o buld he possble correspondence beween he characerscs of second order funcon and some now lx such as lnear or quadrac nens relaons. In oher words, f smlar feaures are presen n he [K-p ], l and g funcons, he correspondng nens funcon can hen be used for he modellng. The second possble mprovemen for modellng process can be acheved b coordnae ransformaon. Tae he sub-se of he frs example daase for nsance. If he coordnae ssem s roaed 40 o clocwse and shfed as shown n Fgure 4.7 a below. The [K-p ], l and g funcons calculaed for he roaed case are shown n Fgure 4.7 b. B comparng hs fgure wh Fgure 4.4 d, s neresng o noce ha he pon process ends more homogeneous. In some cases, s possble o ransform a non-homogeneous pon paern o a homogeneous case smpl b coordnae ransformaon and hen he process can be modelled more easl and more accurael b he homogeneous process. Ths seems o conradc he fundamenal assumpon we have for he pon process: saonar, whch saes ha afer smple coordnae ransformaon characerscs of he process should sa he same. As a maer of fac, here s no conradcon here. The coordnae ransformaon we are usng here s smpl o ge rd of he whe spaces where pons do no occup. Wh hose blan spaces nsde he regon, he pon paern s nhomogeneous n he scale of he whole regon. Wh whe spaces aen ou, her effec of nroducng non-homogene dsappears. Smple coordnae ransformaons do no change he nernal srucures of he pon paern. For example, he ransformaon does no change he pon aggregaon shown n Fgure 4.4 e. Ths s demonsraed n Fgure 4.7 c and d below. Fgure 4.7 a Transformed daase Fgure 4.7 b [K-p ] vs 6

63 Sochasc Modellng of Fracures n Roc Masses, 003 Fgure 4.7 c Transformed daase Fgure 4.7 d [K-p ] vs Smple coordnae ransformaon does no cu ou all he whe spaces occuped whn he regon. Ths suggess anoher possble approach o mprove he modellng. If a polgonal boundar s defned around he regon of neres, s possble o use a smpler pon process for easer and more accurae modellng of he pon paern whn he polgonal regon. Two examples are gven n Fgure 4.8 below, where a s for sub-se of example daa se and b s for whole daase. The program we have has no e mplemened hs echnque e. However, onl b vsual nspecon, s no dffcul o sugges pons whn he defned polgons can be modelled b homogeneous pon process. Fgure 4.8 a Polgonal daase - Fgure 4.7 b Polgonal daase - 5. References. Cresse, N. A. C., Sascs for spaal daa, John Wle & Sons, Inc., New Yor, Dggle, P, Sascal analss of spaal pon paerns, Academc press, Lee, J. S., Ensen, H. H. & Venezano, D., Sochasc and cenrfuge modellng of joned roc, Par III sochasc and opologcal fracure geomer model, echncal repor MIT CE R-90-5, MIT, Upon, G and Fngleon, B, Spaal daa analss b example, John Wle & Sons, van Leshou, M. N. M., Marov pon processes and her applcaons, Imperal College Press, London,

64 Sochasc Modellng of Fracures n Roc Masses,

65 Sochasc Modellng of Fracures n Roc Masses, 003 In he followng fgures, we r o buld he correspondence beween he characerscs of second order funcons and some now lx. In oher words, n smlar feaures are presen n he [K-p ], l and g funcons, he correspondng dens funcon can hen be used for modellng. Fgure 4.5 e Sub-se Fgure 4.5 f [K-p ] vs Fgure 4.5 e Sub-se Fgure 4.5 f [K-p ] vs Fgure 4.5 e Sub-se Fgure 4.5 f [K-p ] vs 65