SEQUENTIAL GAUSSIAN SIMULATIONSEMINAR REPORTSubmitted byParag Jyoti DuttaRoll No: 094060!in partial fulfillment for the award of the degreeofMASTER O" TE#$NOLOG%INGEOE&PLORATIONAtDEPARTMENT O" EART$ S#IEN#ESINDIAN INSTITUTE O" TE#$NOLOG% 'OM'A%MUM'AI ( 4000)6NO*EM'ER !0091Seminar ContentsChapter 1 Introduction.......................................................................................................................2Chapter 2 Estimation versus simulation ..... 3Reproducing model statistics by simulationUsing the spatial uncertainty modelChapter 3 Monte-Carlo Simulation ....... 7 Modeling spatial uncertaintyChapter 4 The MultiGaussian ! Model.......... 9 Normal core !rans"ormChapter " The Se#uential Simulation Genre........ 12 Remar#s$mplementationChapter $ Se#uential Gaussian Simulation ........ 1% &imitationsBibliography 2Seminar Chapter 1Introductionpatial interpolationconcernsho'toestimatethe(ariableunder studyat anun)sampledlocationgi(ensampleobser(ationsat nearbylocations. !hisprocesso" estimation*#riging+aims at computing the minimum error (ariance *optimal+ estimate o" the un#no'n (alue and theassociated error (ariance at the unsampled location. $n many applications, ho'e(er, 'e are more interested in modeling the uncertaintyabout theun#no'nrather thanderi(ingasingleestimate. Uncertaintyismodeledthroughconditionalprobability distributions. !he distribution "unction)} ( | ) ( { Prob )) ( | ; ( n z x Z n z x F =madeconditional to thein"ormation a(ailable *n+"ully models that uncertainty in the sense thatprobability inter(als can be deri(ed, such as )) ( | ; ( )) ( | ; ( )} ( | ] , ( ) ( { Prob n a x F n b x F n b a x Z = .$t is 'orth noting that these probability inter(als are independent o" any particular estimate z*(x) o"the un#no'n (alue z(x). $ndeed uncertainty depends on in"ormation a(ailable *n+, and not on theparticular optimality criterion retained to de"ine an estimate.uch a model o" -local. uncertaintyallo's one to e(aluate the ris# in(ol(ed in any decision)ma#ing process, such as delineation o"rich /ones o" minerali/ation 'here a drill)core sampling programme needs to be planned. 0romthe model o" uncertainty, one can also deri(e estimates optimal "or di""erent criteria, customi/edto the speci"ic problem at hand, instead o" retaining the least)s1uares error *#riged+ estimate.2ach conditional cumulati(e distribution "unction *ccd"+)) ( | ; ( n z x Fpro(ides a measure o"localuncertainty relating to a speci"ic location x. 3o'e(er, a series o" single)point ccd"s do not pro(ideanymeasureo" multiple)point orspatialuncertainty,suchastheprobabilitythat astringo"locations 4ointly e5ceed a gi(en threshold (alue. Most applications re1uire a measure o" the 4ointuncertainty about attribute (alues at se(eral locations ta#en together. uch spatial uncertainty ismodeled by generating a set o" multiple e1uiprobable reali/ations {z(x)(l), xA}, l =1, 2,.., L}o" the 4ointdistribution o" attribute (alues in space, a process #no'n as stochastic simulation.!heset o" alternati(ereali/ationspro(idesa(isual and1uantitati(emeasure*amodel+ o"spatialuncertainty. 6llo" these reali/ations reasonably match the same sample statistics ande5actly match the conditioning data. 2ach reali/ation reproduces the (ariability o" the input data 3Seminar in the multi(ariate sense7 hence said to represent the geological te5ture or true spatial (ariabilityo" the phenomena. Chapter 2Estimation versus Simulation!he ob4ecti(e o" estimation is to pro(ide, at each point x, an estimator z*(x) 'hich is as close aspossible to the true un#no'n (alue o" the attribute z0(x). !he criteria "or measuring the 1uality o"estimation are unbiasedness and minimal estimation variance } )] ( * ) ( {[2x Z x Z . !here is noreason, ho'e(er, "or these estimators to reproduce the spatial (ariability o" the true (alues 8z0(x)9.$nthecaseo" #riging, "or instance, theminimi/ationo" theestimation(ariancein(ol(esasmoothingofthetruedispersions.!ypically, small (aluesareo(erestimated, 'hereaslarge(alues are underestimated. 6nother dra'bac# o" estimation is that the smoothing is not uni"orm.Rather, it dependsonthelocal datacon"iguration: smoothingisminimal closetothedatalocationsandincreasesasthelocationbeingestimatedisgets"arther a'ay"romthedatalocations. 6 map o" #riging estimates appears more (ariable in densely sampled areas than insparsely sampled areas. ;n the other hand, the simulation 8z(l)(x)9 'ithldenoting the lth reali/ation, has the same "irstt'o e5perimentally "ound moments *mean and co(arianceata@. @ 1@. @ 2@. @ 3@. @ =@. @ A@. @@. @1@. @2@. @3@. @=@. @A@. @@. @1. @@@2. @@@3. @@@=. @@@A. @@@Un#no'true"ield2astNorth@. @ A@. @@@@. @A@. @@@@. @2. @@@=. @@@%. @@@B. @@@1@. @@@0re1uencyprimary ) CorU@. @ A. @ 1@. @ 1A. @ 2@. @@. @@@@. 1@@@. 2@@@. 3@@@. =@@@. A@@3istogramun#no'nNumbero">ata 2A@@mean 2. ABstd. de(. A. 1Acoe". o"(ar 2. @@ma5imum 1@2. 7@upper1uartile 2. A%median @. 9%lo'er1uartile @. 3=minimum @. @1#rigingmap2astNorth@. @ A@. @@@@. @A@. @@@@. @1. @@@2. @@@3. @@@=. @@@A. @@@0re1uency2stimate@. @ A. @ 1@. @ 1A. @ 2@. @@. @@@@. 1@@@. 2@@@. 3@@@. =@@3istogram#rigingmapNumbero">ata 2A@@mean 2. 7Astd. de(. 2. =%coe". o"(ar @. 9@ma5imum 22. 7Aupper1uartile 2. B9median 1. 9%lo'er1uartile 1. =Bminimum @. @9imulatedreali/ation12astNorth@. @ A@. @@@@. @A@. @@@@. @2. @@@=. @@@%. @@@B. @@@1@. @@@0re1uency(alue@. @ A. @ 1@. @ 1A. @ 2@. @@. @@@@. 1@@@. 2@@@. 3@@@. =@@3istogramreal. 1Numbero">ata 2A@@mean 3. @Bstd. de(. =. B3coe". o"(ar 1. A7ma5imum 3@. @@upper1uartile 3. @2median 1. 37lo'er1uartile @. =2minimum @. @1imulatedreali/ation12astNorth@. @ A@. @@@@. @A@. @@@@. @2. @@@=. @@@%. @@@B. @@@1@. @@@0re1uency(alue@. @ A. @ 1@. @ 1A. @ 2@. @@. @@@@. 1@@@. 2@@@. 3@@@. =@@@. A@@3istogramreal. 2Numbero">ata 2A@@mean 2. A3std. de(. =. 3Acoe". o"(ar 1. 72ma5imum 3@. @@upper1uartile 2. A2median 1. @@lo'er1uartile @. 3=minimum @. @1+a,+-,+.,+/,ASeminar !i%ure 1( a &0igure 2 sho's *a+ the true "ield along 'ith the corresponding histogram *b+ #riged estimatesbased on the 29 data o" 0igure 1 *smoother than true "ield+7 the (ariance o" the #riged estimateis less than the actual (ariance, *c, d+ t'o se1uential Daussian simulations constrained to the29 data7 the histograms o" the Daussian simulations are similar to the true "ield.Figure 2 (a) !rue field and histogram, (b) kriging estimates (smoother than true field)" notice that the variance of thekriging estimate is less than the actual variance, (c,d) two se#uential $aussian simulations conditioned to the data"the histograms of the $aussian simulations are similar to the true field. %Seminar )sin% the spatial uncertaint' modelDenerating alternati(e reali/ations o" the spatial distribution o" an attribute is rarely a goal per se.Rather, these reali/ations ser(e as inputs to comple5 -trans"er "unctions. such as "lo' simulatorsin reser(oir engineering. 0lo' simulators consider all locations simultaneously rather then oneat atime. !heprocessingo" input reali/ationsyieldsauni1ue(alue"oreachresponse, "ore5ample, auni1ue(alue"or theground'ater tra(el time"romonelocationtoanother orremediationcost. !hehistogramo" theLresponse(alues, correspondingtothoseLinputreali/ations pro(ides a measure o" the responseuncertainty resulting "romour imper"ect#no'ledgeo" thedistributiono" thephenomena*z+ inspace. !hat measurecanbeusedinsubse1uent ris# analysis and decision)ma#ing. $n the mining industry, simulations o" the spatialdistributiono" anattributecanbeused"orstudyingthetechnical andeconomice""ectso"comple5 mining operations7 "or instance, comple5 geometries in underground mining or testing(arious mining schedules on se(eral di""erent simulations. !hus simulations pro(ide anappropriate plat"orm to study any problem relating to (ariability, "or e5ample ris# analysis, in a'ay that estimates cannot. 7Seminar Chapter 3Monte-Carlo Simulation &et)) ; ( z|n x Fbethe conditional cumulati(e distribution "unction*ccd"+ modeling the uncertaintyabout the un#no'n z0(x), at the point x. Rather than deri(ing a single estimated (alue z*(x) "romthat ccd", one may dra' "rom it a series o" L simulated (alues z(l)(x), l = 1,, L. 2ach (alue z(l)(x)represents a possible reali/ation o" the RV Z(x) modelling the uncertainty at the location x.!he EMonte)CarloF simulation proceeds in t'o steps:1. 6 series o" L independent random numbers p(l), l = 1,, L, uni"ormly distributed in G@,1H, isdra'n.2. !he lth simulated (alue z(l)(x) is identi"ied 'ith the p(l))1uantile o" the ccd" *0ig 3+:z(l)(x) = )) ( | ; () ( 1n p x Fl l = 1,, L!he L simulated (alues z(l)(x) are distributed according to the conditional cd". $ndeed, ))} ( | ; ( rob{ P } ) ( { Prob) ( 1 ) (n p x F z x Zl l = "rom the pre(ious de"inition, I ))} ( | ; ( { Prob) (n z x F plsince)) ; ( z|n x F is monotonic increasing,= )) ; ( z|n x F since p(l) are uni"ormly distributed in G@,1H!his propertyo"ccd"reproductionallo'sone toappro5imate anymomentor1uantileo"theconditional distributionbythecorrespondingmoment or 1uantileo" thehistogramo" manyreali/ations z(l)(x))) ; ( z|n x Fz(l)(x) 0 (1alu2 BSeminar !i%ure 3 Monte-Carlo simulation *rom a conditional cd* )) ; ( z|n x FModelin% spatial uncertaint'!he basic idea is to generate a set o" e1uiprobable reali/ations o" the 4oint *spatial+ distributiono" attribute(aluesat se(eral locationsandtousedi""erencesamongsimulatedmapsasameasure o" uncertainty. Rather than modeling the uncertainty at one location, a set o" simulatedmaps {z(l)(x), = 1,., N}, l = 1,,L, can be generated by sampling the N-(ariate or N-pointccd" that models the 4oint uncertainty at the N locations x:)} ( | ) ( ,......, ) ( , ) ( { Prob )) ( | ,....., , ; ,....., , (2 2 1 1 2 1 2 1n z x Z z x Z z x Z n z z z x x x FN N N N =$n"erence o" the abo(e N-point conditional cd" re1uires #no'ledge or stringent hypothesis aboutthe spatial law *multi)(ariate distribution+ o" the R0 Z(x).Ccd"s can be modeled using either aparametric *a model is assumed "or the multi)(ariate distribution+or non)parametric*indicator+approaches. $n the parametric approach, the multi)Daussian R0 modelis commonly adoptedbecause it is one model 'hose spatial la'is "ully determined by the z-co(ariance "unction7 itunderlies se(eral simulation algorithms such asLUdecompositionalgorithm,SequentialGaussian Simulation and Turning bands imulation. ;ther Daussian)related techni1ues includetruncated Gaussian and pluriGaussian simulation algorithms.!'o shortcomings o" the parametric approach are: 1. !he spatial uncertainty assessment becomes (ery comple5 as the number o" grid nodesincreases.2. $t is cumbersome to chec# in practice the (alidity o" the Daussian assumption, and datasparsity pre(ents us "rom per"orming such chec#s "or more than t'o locations at a time. 9Seminar Chapter 4The MultiGaussian ! Model!he spatial la' o" the R0 Z(x) as deri(ed by the assumed model must be congenial enough sothatall ccd"s)) ; ( z|n x F,, A x ha(e the sameanalyticale5pressionand are"ully speci"iedthrough a "e' parameters. !he problem o" determining the ccd" at location xthus reduces tothat o" estimating a "e' parameters, say, the mean and (ariance. !he multi(ariate Daussian R0model is most 'idely used because it.s e5tremely congenial properties render the in"erence o"the parameters o" the ccd" straight"or'ard. !he approach typically re1uires a prior normal scoretransformo" data to ensure that at least the uni(ariate distribution *histogram+is normal. !henormal score ccd" then undergoes a bac#)trans"orm to yield the ccd" o" the original (ariable.$" {Y(x),A x } is a standard multi(ariate Daussian R0 'ith co(ariance "unction), (h CYthen the"ollo'ing are true *Doo(aerts, 1997+:1. 6ll subsets o" that R0, e.g., {Y(x),A D x }, are also multi(ariate normal.2. !he uni(ariate cd" o" any linear combination o" R? components is normal:) (1 x Y Un==is normally distributed, "or any choice o" n locations A x and any set o" 'eights .3. !hebi(ariatedistributiono" anypairso" R?sY(x)andY(x+h)isnormal and"ullydetermined by the co(ariance "unction) (h CY.=. $" t'o R?s ) (x Yand ) (x Y are uncorrelated, i.e., i", 0 )} ( ), ( Cov{ = x Y x Ythey are alsoindependent.A. 6ll conditional distributions o" any subset o" the R0) (x Y, gi(en reali/ations o" any othersubsets o" it, are *multi)(ariate+ normal. $n particular, the conditionaldistribution o" thesingle (ariable) (x Ygi(en the ndata ) (x yis normal and "ully characteri/ed by its t'o 1@Seminar parameters, meanand(ariance, 'hicharetheconditional meanandtheconditional(ariance o" the R? ) (x Ygi(en the in"ormation *n+:=)} ( | ) ( Var{)} ( | ) ( E{))] ( | ; ( [n x Yn x y yG n y x G'here (.) Gis the standard normal cd".Under themultiDaussianmodel, themeanand(arianceo" theccd" at anylocationxareidentical to the simple #riging *J+ estimate ) (*x ySKand J (ariance ) (2xSKobtained "rom the ndata ) (x y(Kournel and 3ui4bregts, 197B+. !he ccd" is then modelled as =) () ())] ( | ; ( [**xx y yG n y x GSKSKSK'ith )] ( ) ( [ ) ( ) (1* x m x y x m x ynSKSK + ==) ( ) 0 ( ) (12x x C C xnSKSK == +ormal Score Trans*orm!he multiDaussian approach is (ery con(enient: the in"erence o" the ccd" reduces to sol(ing asimple #riging system at any location x. !he trade)o"" cost is the assumption that data "ollo' amultiDaussian distribution, 'hich implies "irst that the one point distribution o" data *histogram+is normal. 3o'e(er, many (ariables in earth sciences sho' an asymmetric distribution 'ith a"e'(erylarge(alues*positi(es#e'ness+. !husthemultiDaussianapproachstarts'ithanidenti"ication o" the standard normal distribution and in(ol(es the "ollo'ing steps:1. !he original z)data are "irst trans"ormed into y)(alues 'ith a standard normal histogram.uch a trans"ormis re"erred to as anormal score transform, and they)(alues)) ( ( ) ( x z x y =are called normal scores.2. Lro(idedthebiDaussianassumptionisnot in(alidated, themultiDaussianmodel isapplied to the normal scores, allo'ing the deri(ation o" the Daussian ccd" at anyunsampled location x:)} ( | ) ( rob{ )) ( | ; ( n y x Y n y x G =3. !he ccd" o" the original (ariable is then retrie(ed as )} ( | ) ( { Prob )) ( | ; ( n z x Z n z x F = 11Seminar

)} ( | ) ( { Prob n y x Y =

)) ( | ) ( ; ( n z x G =under the condition that the trans"orm "unction(.) is monotonic increasing!henormal scoretrans"orm"unction(.) canbederi(edthroughagraphical correspondencebet'een the uni(ariate cd"s o" the original and standard normal (ariables *0igure =+.&et ) (z F and ) ( y Gbe the stationary uni(ariate cumulati(e density "unctions *cd"+ o" the originalR0) (x Zand the standard normal R0) (x Y

} ) ( { Prob ) ( z x Z z F =} ) ( { Prob ) ( y x Y y G =!he trans"orm that allo's one to go "rom a R0) (x Z'ith cd") (z Fto a R0 ) (x Y'ith standardDaussian cd" ) ( y Gis depicted by arro's in 0igure = and is 'ritten as ))] ( ( [ )) ( ( ) (1x Z F G x Z x Y= = 'here (.)1 Gis the in(erse Daussian cd" or 1uantile "unction o" the R0 ) (x Y!(z) "(#)z ) ( # z = 0(1alu23y(1alu23!i%ure 4:Draphicalprocedure "or trans"orming the cumulati(e distribution o" originalz-(alues into thestandard normal distribution o" original y)(alues called normal scores.$n practice, the normal score trans"orm proceeds in three steps: 12Seminar 1. !he original data {z(x), = 1,., N}are ran#ed in ascending order. ince the normalscore trans"orm is monotonic, ties in z)(alues must be bro#en.2. !hesamplecumulati(edistribution"unctiono" theoriginal data(ariablez(x),iscalculated.3. !henormal scoretrans"ormo" thez)datum'ithran#kismatchedtothe*kp)1uantile o" the standard normal cd":) ( ))] ( ( * [ ) (* 1 1kp G x z F G x y = = Chapter 5The Se#uential Simulation Genre!he 'ide class o" simulation algorithms #no'n under the generic name sequential simulation isessentiallybasedonthesameunderlyingtheory: insteado" modelingtheN)(ariateccd", auni(ariateccd" ismodeledandsampledat eacho" theNnodes(isitedalongarandomse1uence. !oensurereproductiono" thez)co(ariancemodel, eachuni(ariateccd" ismadeconditional not only to the originaln data but also to all (alues simulated at pre(iously (isitedlocations. &et } ,......, 1 ), ( { Nx Z

= be a set o" random (ariables de"ined at N locations

x 'ithin the studyareaA. !heselocations neednot begridded. !heob4ecti(eis togeneratese(eral 4ointreali/ations o" these N R?s:} ,......, 1 ), ( {) (Nx z

l= l = 1,, L, conditional to the data set} ,......, 1 ), ( { n x z = &et us consider the 4oint simulation o"z)(alues at t'o locations only, say,1xand2x. 6 set o"reali/ations)} ( ), ( {2) (1) (x z x zl l , l = 1,, L, can be generated by sampling the bi(ariate ccd":)} ( | ) ( , ) ( { Prob )) ( | , ; , (2 2 1 1 2 1 2 1n z x Z z x Z n z z x x F = 6nalternati(eapproachispro(idedbyMayes. a5iom, 'herebyanybi(ariateccd" canbee5pressed as a product o" t'o uni(ariate ccd"s:)) ( | ; ( )) 1 ( | ; ( )) ( | , ; , (1 1 2 2 2 1 2 1n z x F n z x F n z z x x F + = 13Seminar 'here E|(n$1)F denotes conditioning to the n data ) (x z, and to the reali/ation ). ( ) (1) (1x z x Zl = !heabo(edecompositionallo'sonetogeneratethepair)} ( ), ( {2) (1) (x z x zl l int'osteps: the(alue) (1) (x zlis "irst dra'n "romthe ccd")) ( | ; (1 1n z x F ,then the ccd" at location2xisconditioned to the reali/ation ) (1) (x zlin addition to the original data *n) and its sampling yieldsthe correlated (alue ) (2) (x zl. !he idea is to trade the sampling hence modeling o" the bi(ariateccd" "or the se1uential sampling o" t'o uni(ariate ccd"s easier to in"er, hence the generic namesequential simulation algorithm. !he se1uential principle can be generali/ed to more than t'o locations. My recursi(e applicationo" the Mayes. a5iom, the N)(ariate ccd" can be 'ritten as the product o" N uni(ariate ccd"s:.. .......... )) 2 ( | ; ( )) 1 ( | ; ( )) ( | ,...... ; ,...... (1 1 1 1 + + = N n z x F N n z x F n z z x x FN N N N N N.)) ( | ; ( )) 1 ( | ; (1 1 2 2n z x F n z x F + 'here, "or e5ample,)) 1 ( | ; ( + N n z x FN Nis the ccd" o") (Nx Z gi(en the set o"noriginal data(alues and the) 1 ( Nreali/ations 1 ,......, 1 ), ( ) () ( = = Nx z x Z

l

!he abo(e decomposition allo's one to generate a reali/ation o" the random(ector} ,......, 1 ), ( { Nx Z

= in N successi(e steps: Model the cd" at the "irst location1x, conditional to the n original data ) (x z:)} ( | ) ( { Prob )) ( | , (1 1n z x Z n z x F = >ra'"romthat cd" areali/ation), (1) (x zl'hichbecomesaconditioningdatum"orallsubse1uent dra'ings.,,,,, 6t the !th node!x (isited, model the conditional cd" o" ) (!x Z gi(en the n original data andall the (! %1) (alues) () (

lx z simulated at pre(iously (isited locations 1 ,....., 1 , = !x

1=Seminar )} 1 ( | ) ( { Prob )) 1 ( | , ( + = + ! n z x Z ! n z x F! ! >ra' "rom that ccd"a reali/ation ), () (!lx z 'hich becomes a conditioning datum "or allsubse1uent dra'ings. Repeat the t'o pre(ious steps until all the N nodes are (isited and each has been gi(ena simulated (alue.!he resulting set o" simulated (alues} ,......, 1 ), ( {) (Nx z

l= represents 4ust one reali/ation o"the R0} ), ( { A x x Z o(er theNnodes

x. 6ny numberLo" such reali/ations} ,......, 1 ), ( {) (Nx z

l= ,l= 1,,L,can be obtained by repeating L times the entire se1uentialprocess 'ith possibly di""erent paths to (isit the N nodes.emar-s(1. !hese1uential simulationalgorithmre1uiresthedeterminationo" aconditional cd" ateachlocationbeingsimulated.!'oma4orclasseso" se1uential simulationalgorithmscan be distinguished, depending on 'hether the series o" conditional cd"s aredetermined using the multi)Daussian or the indicator "ormalisms.2. e1uential simulationensuresthat dataarehonoredat their locations*conditional+.$ndeed,at any datum locationx, the simulated (alue is dra'n "rom a /ero)(ariance,unit step ccd" 'ith mean e1ual to the z)datum ) (x zitsel". $" large measurement errorsrender 1uestionable the e5act matching o" data (alues, one should allo' the simulated(aluestode(iatesome'hat "romdataat their locations. $" theerrorsarenormallydistributed, the simulated (alue could be dra'n "rom a Daussian ccd" centered on thedatum (alue and 'ith a (ariance e1ual to the error (ariance. 3. !he se1uential principle can be e5tended to simulate se(eral continuous or categoricalattributes.Implementation 1ASeminar Search strategies!he se1uential simulation algorithm re1uires the determination o" N successi(e conditional cd"s)) 1 ( | ; ( , )),....... ( | ; (1 + N n z x F n z x FN, 'ithanincreasingle(el o" conditioningin"ormation.Correspondingly, thesi/eo" the#rigingsystem*s+ tobesol(edtodeterminetheseccd"sincreases and becomes 1uic#ly prohibiti(e as the simulation progresses. !he data closest to thelocation being estimated tend to screen the in"luence o" more distant data. !hus, in the practiceo" se1uential simulation, only the original data and those pre(iously simulated (alues closest tothe location xbeing simulated are retained. Dood practice consists o" using the semi)(ariogramdistance ) ( x x so that the conditioning data are pre"erentially selected along the direction o"ma5imum continuity.6s the simulation progresses, the original data tend to be o(er'helmed by the large number o"pre(iously simulated (alues, particularly 'hen the simulation grid is dense. 6 balance bet'eenthe t'o types o" conditioning in"ormation can be preser(ed by separately searching the originaldata and the pre(iously simulated (alues *t'o part search+: at each locationx, a "i5ed number) (x n o" closest original data are retained no matter ho' many pre(iously simulated (alues arein the neighborhood o"x.Visiting sequence$n theory, the N nodes can be simulated in any se1uence. 3o'e(er, because only neighboringdata are retained, arti"icial continuity may be generated along a deterministic path (isiting the Nnodes. 3ence, a random se1uence or path is recommended. Nhen generating se(eral reali/ations, the computational time can be reduced considerably by#eeping the same random path "or all reali/ations. $ndeed, the N #riging systems, one "or eachnode

x, needbesol(edonlyoncesincetheNconditioningdatacon"igurationsremainthesame "rom one reali/ation to another. !he trade)o"" cost is the ris# o" generating reali/ations thatare t'o similar. !here"ore, it is better to use a di""erent random path "or each reali/ation. 1%Seminar Multiple grid simulation !heuseo" asearchneighborhoodlimitsreproductiono" theinput co(ariancemodel totheradiuso" that neighborhood.6nother obstacletoreproductiono" long)rangestructureisthescreening o" distant data by too many data closer to the location being simulated. !hemultiple)gridconcept *theattribute(aluesare"irst simulatedonacoarsegridandthencontinueona"iner grid+ allo'sonetoreproducelong)rangecorrelationstructures'ithoutha(ing to consider large search neighborhoods 'ith too many conditioning data. !he pre(iouslysimulated (alues on the coarse grid are used as data "or simulation on the "ine grid. 6 randompath is "ollo'ed 'ithin each grid. !he procedure can be generali/ed to any number o"intermediate grids7 the number depends on the number o" structures 'ith di""erent ranges to bereproduced and the "inal grid spacing.Chapter 6Se#uential Gaussian Simulation $mplementation o" the se1uential principle under the MultiDaussian R0 model is re"erred to assequentialGaussiansimulation*sDs+. e(eral algorithmse5ist: algorithms"or simulatingasingle attribute using only (alues o" that attribute, 'ith modi"ications to account "or secondaryin"ormation as 'ellas "or 4ointsimulation o" se(eralcorrelated attributes.3ere, only the "irstcase, i.e., accounting "or a single attribute, is considered.&et us consider thesimulationo" thecontinuous attributezatNnodes

xo" agrid*notnecessarily regular+ conditional to the data set} ,......, 1 ), ( { n x z = .e1uential Daussian simulation proceeds as "ollo's:1, 0irst step: chec# the appropriateness o" the multiDaussian R0 model, 'hich calls "or aprior trans"ormo"z)dataintoy)data'ithastandardnormal cd" usingthenormal score 17Seminar trans"orm. Normalityo" thebi(ariatedistributiono" theresultingnormal score(ariable)) ( ( ) ( x Z x Y =is then chec#ed. $n practice, i" indicator semi(ariograms or ancillaryin"ormationdonot in(alidatethebiDaussianassumption, themultiDaussian"ormalismisadopted.2, $" themultiDaussian R0model is retained"or they)(ariable,se1uentialDaussiansimulation is per"ormed on the y)data: >e"ine a random path (isiting each node o" the grid only once. 6t each nodex, determine the parameters *mean and (ariance+ o" the Daussianccd")) ( | ; ( n y x G using J 'ith the normal score (ariogram model) (hY. !heconditioning in"ormation (n)consists o" a speci"ied number) (x n o" both normalscoredata ) (x yand (alues ) () (

lx y simulated at pre(iously (isited grid nodes. >ra' a simulated (alue ) () (x yl"rom that cd", and add it to the data set. Lroceed to the ne5t node along the random path, and repeat the t'o pre(ioussteps. &oop until all N nodes are simulated.3, !he "inal step consists o" bac#)trans"orming the simulated normal scores} ,...., 1 ), ( {) (Nx y

l= into simulated (alues "or the original (ariable, 'hich amounts toapplying the in(erse o" the normal score trans"orm to the simulated y)(alues:)) ( ( ) () ( 1 ) (

l

lx y x z = N,....., 1 ='ith(.)), ( (.)1 1G F = 'here (.)1 Fis the in(erse cd" or 1uantile "unction o" the (ariable Z,and G(.) is the standard Daussian cd". !hat bac#)trans"orm allo's one to identi"y the originalz)histogram) (z F. $ndeed,} )) ( ( Prob{ } ) ( Prob{) ( ) (z x Y z x Zl " l = I )} ( ) ( Prob{ z x Y#l$ since (.) is monotonic increasing 1BSeminar I ) ( )] ( [ z F z G = "rom the de"inition o" normal score trans"orm;ther reali/ations}, ,......, 1 ), ( {) (Nx z

l= , l l are obtained by repeating steps 2 and 3 'ith adi""erent random path. !he basic steps o" the sDs algorithm are illustrated in the "ollo'ing "lo' chart.Non)stationary beha(iors could beaccounted "or using algorithms other than simple #riging to estimate the mean o" the Daussianccd": ordinary riging or universal riging of the order . 3o'e(er, Daussian theory re1uires thatthe simple #riging (ariance o" normal scores be used "or (ariance o" the Daussian ccd" *Kournel,19B@+..imitations(?arious limitations and shortcomings can be attributed to se1uential Daussian simulation:1. sDs relies on the assumption o" multi)(ariate Daussianity, an assumption that can ne(er be"ully chec#ed in practice, yet al'ays seems to be ta#en "or granted. Multi)Daussianity leads 19Seminar to simulated reali/ations that ha(e ma5imally disconnected e5tremes *ma5imum entropy+, aproperty that o"ten con"licts 'ith geological reality.2. sDs re1uires a trans"ormation into Daussian space be"ore simulation and a correspondingbac#)trans"ormation a"ter simulation is "inished. 3o'e(er, o"ten the primary (ariable to besimulated has to be conditioned to a secondary (ariable that is a linear or non)linear (olumea(erage o" the primary (ariable. Normal)score trans"orms are non)linear trans"orms, hencethey destroy the possible linear relation that e5ists bet'een primary and secondary (ariable,or, they change the non)linearity i" that relation is non)linear.3. sDsreproduces, bytheory,onlythenormal score(ariogram, not theoriginal (ariogrammodel. Usuallyreproductiono" thenormal score(ariogramentailsreproductiono" theoriginaldata (ariogram i" the data histogram is not too s#e'ed. 3o'e(er in case o" highs#e'ness, the reproduction o" the (ariogrammodel a"ter bac#)trans"ormation is notguaranteed at all.6ctually, reproductiono" theco(ariancemodel) (h CYdoesnot re1uirethesuccessi(eccd"modelstobeDaussian7 they can beo"any typeaslongastheir means and (ariancesaredetermined by simple #riging *Kournel, 199=+. !his result leads to an important e5tension o" these1uential simulationparadigm'herebytheoriginalz)attribute(aluesaresimulateddirectly'ithout any normal score trans"orm. !his algorithm is called direct sequential simulation *dssim+.$n the absence o" a normal score trans"orm and bac#)trans"orm, there is, ho'e(er, no control onthe histogram o" simulated (alues. Reproduction o" a target histogram can be achie(ed by postprocessing the dssim reali/ation. /i&lio%raph' Doo(aerts, L., 1997. Deostatistics "or Natural Resources 2(aluation. ;5"ord Uni(. Lress, Ne' Oor#, A12 pp.Kournel, 6.D., 3ui4bregts, C.K., 197B. Mining Deostatistics. 6cademic Lress, Ne' Oor#, %@@ pp.Kournel, 6.D., 19B@. !he lognormal approach to predicting local distributions o" selecti(e mining unit grades. Mathematical Geology, 12*=+, 2BAP3@3.Kournel, 6.D., 199=. Modeling uncertainty: ome conceptual thoughts. Geostatistics for the !e"t #entury, pages 3@P=3. Jlu'er, >ordrecht. 2@Seminar