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.
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9%lo'er1uartile @. 3=minimum @. @1#rigingmap2astNorth@. @ A@. @@@@.
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=@@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*=+, [email protected], 6.D., 199=. Modeling
uncertainty: ome conceptual thoughts. Geostatistics for the !e"t
#entury, pages 3@P=3. Jlu'er, >ordrecht. 2@Seminar