A New MPS Simulation Algorithm Based on Gibbs Sampling

Transcription

1 A New MPS Smulaton Algorthm Based on Gbbs Samplng Steven Lyster, Clayton V. Deutsch, and Thes Dose 2 Centre for Computatonal Geostatstcs Edmonton, Alberta 2 RWE Dea Aktengsellschaft Hamburg, Germany The major heterogenety n most reservors and deposts s often characterzed by faces or rock types. Petrophyscal propertes are modeled wthn each faces or rock type. Multple-pont geostatstcal methods are beng used ncreasngly to reproduce curvlnear geologc features. A number of multple-pont smulaton methods have been put forward n the past; ths paper proposes a novel mplementaton usng a Gbbs sampler algorthm. Ths algorthm uses the concept of multple-pont events n an effort to account for hgh-order structure n the presence of arrangements of faces whch are nconsstent wth the statstcs nferred from the tranng mage. Introducton Multple-pont statstcs (MPS) methods are used to model geologc phenomena n more realstc and robust ways than tradtonal geostatstcs. Rather than consderng lnear estmates based on ndvdual condtonng ponts, MPS use several ponts smultaneously to determne condtonal probabltes. These hgher-order relatons more accurately descrbe complex features than the standard Gaussan model of spatal structure. Several methods for reproducng MPS have been explored prevously: the sngle normal equaton (SNE) approach frst proposed by Guardano and Srvastava (993) and further expanded by Strebelle and Journel (2000, 200); smulated annealng (Deutsch, 992; Lyster et al, 2004a); Gbbs samplng (Srvastava, 992); a neural network teratve scheme (Caers and Journel, 998; Caers, 200); and ntegraton of runs wth ndcator smulaton for reproducton of hgher-order relatons n contnuous data (Ortz, 2003). MPS methods have prmarly focused on categorcal ndcators and modelng rock types; ths s probably due to the relatve ease wth whch tranng mages may be created for geologc structures, as opposed to the dense samplng necessary to nfer hgh-order moments for contnuous data. Wthn the famly of categorcal methods, most use teratve approaches to reproduce MPS. Ths s lkely because of the relatve reducton n dmensonalty of the statstcs that must be calculated and stored when usng a template n an teratve method, nstead of havng to characterze the many combnatons of both ponts and faces needed for a sequental method. The approach proposed n ths paper s smlar to several of the above-mentoned methods n that t s an teratve smulaton algorthm meant to be used for modelng of geologc faces or rock types. Whle a Gbbs sampler method has been nvestgated before, a novel technque for determnng the condtonal probabltes wll be presented. Several updatng schemes to enforce proper unvarate statstcal reproducton and locally varyng proportons wll also be dscussed. Two examples usng complex tranng mages and comparng two-pont wth MPS smulaton results wll be shown. A comparson of the proposed algorthm wth several other MPS smulaton methods wll also be undertaken; the tranng mage for ths comparson s a threedmensonal model of a fluval petroleum reservor. 0-

2 Gbbs Sampler The Gbbs sampler s a statstcal algorthm that was frst proposed by Geman and Geman (984); t s a specal case of the Metropols algorthm (Metropols et al, 953). It s a method used for drawng samples from complcated jont or margnal dstrbutons, wthout requrng the densty functons of the dstrbutons (Casella and George, 992). In the context of geologcal modelng, a sample from the jont dstrbuton s a smulated realzaton. In a Gbbs sampler only the condtonal dstrbutons are requred to sample from the jont dstrbuton, whch s why ths approach s attractve as a MPS smulaton method; all necessary condtonal probabltes may be determned from a tranng mage. The basc dea of a Gbbs sampler s that of resamplng ndvdual varables condtonal to others n the same sample space. Drawng new values condtonal to all others, and repeatng ths process many tmes, results n an approxmaton of the jont (and margnal) dstrbuton(s). For example, to determne the densty functon for f(x,y,z), one would start wth an ntal state (values) for X, Y, and Z and then sequentally draw values from the condtonal dstrbutons f(x Y,Z), f(y X,Z), and f(z X,Y). When a number of condtonal values have been drawn, the state of (X,Y,Z) approxmates a sample from the jont dstrbuton f(x,y,z). Repeatng ths process enough tmes allows the densty (or hstogram) of the jont dstrbuton to be constructed emprcally. In geostatstcs generatng many samples from the jont dstrbuton s analogous to creatng many smulated realzatons. The Gbbs sampler s a Markov chan Monte Carlo (MCMC) method (Gelfand and Smth, 990), meanng that each new value drawn from a condtonal dstrbuton s assumed to be dependent only on the current state of the varables. Whle theoretcally the new value should be dependent on all pror states, because the current state s dependent on the prevous state and the prevous dependent on the one before that; thus, usng only the current state to determne the new value drawn accounts for all earler states and therefore the theory may be smplfed to gnore all past states and use only the current. Ths property s known as condtonal ndependence, and allows the use of a MCMC method n a geoscence applcaton to be greatly smplfed. An advantage of the Gbbs sampler workflow for geospatal modelng s that there s no strct specfcaton for how the condtonal dstrbutons need to be determned. Prevous use of the Gbbs sampler (Srvastava, 992) focused prmarly on usng krgng and two-pont statstcs; a sngle normal equaton usng MPS was also explored brefly. Besdes a MPS template, t would be possble to ntegrate any desred statstcs nto the condtonals, such as locally varyng means, secondary data, and lower-order statstcs to ensure ther proper reproducton. The results of the smulaton wll be dependent on the selecton of the condtonal dstrbuton chosen. Estmaton of Condtonal Probabltes To fnd the condtonal probabltes of faces at a pont, any desred method could be used. However, the qualty of the resultng realzatons wll be determned by the condtonal probabltes selected. For example, usng only the unvarate hstogram as a condtonal dstrbuton would guarantee that the faces proportons are matched exactly, but would not gve any structure n the results; usng the varograms and krgng to determne condtonal probabltes would reproduce the drect covarance structure but nothng more; addng the crossvarograms n a lct LMC would reproduce the relatons between faces as well. The use of MPS should hopefully reproduce hgher-order statstcs n the end results. 0-2

3 The most basc method of usng MPS n the condtonal dstrbutons for a Gbbs sampler would be to use Bayes Law to drectly nfer the probabltes of each faces drectly from the tranng mage. An example of a problem n usng ths approach s llustrated n Fgure. The central pont n the gven MPS template, and therefore the pont for whch the condtonal probabltes are needed, s marked wth an X. Suppose faces at the two ponts, marked wth dots, next to the central locaton are a msmatch for the tranng mage; such a msmatch occurs often when begnnng wth an ntal mage whch does not honour MPS. The standard procedure would be to drop the farthest away pont from the central locaton, and recalculate the condtonal dstrbutons. However, because the msmatch occurs so close to the central pont, most of the ponts n the template would need to be dropped n order for the condtonal probabltes to be nferred from the tranng mage. Fgure : Possble MPS template. Dvdng the template nto several separate sub-templates s the proposed soluton to ths problem. Rather than consderng the surroundng ponts around the locaton of nterest as a sngle event, dvdng the template nto M dscrete multple pont events (MPEs) can allow hgh-order statstcs to be used whle preservng the two-pont geostatstcal concept of accountng for the closeness and redundancy of separate data. Wth ths approach, the estmated condtonal probablty of faces k at the locaton of nterest may be expressed as: where P * ( k) M * P k I E P E P k = () k ( ) = λ ( ) ( ) + ( ) s the estmated probablty of faces k at the current pont; to MPE for the probablty of k; k λ s the weght gven I s the E s one of the MPEs near the current pont; ( E ) ndcator of the MPE; t s for all events consdered; ( E ) denoted as f j ; and P ( k) s the global probablty of k, also denoted as P k. P s the global probablty of E, also Equaton s a lnear estmate, but uses ndcators of MPEs as data nstead of sngle-pont ndcators. Because each of the M events has N ponts, there are theoretcally K N data classes for every MPE; usng only those events that actually occur around the locaton of nterest reduces the 0-3

4 sze of the lnear estmate from MK N terms to M. Ths reducton n dmensonalty should not have a sgnfcant mpact, as the weghts for such a very large lnear estmate would be relatvely small and the global probabltes of MPEs also qute small; therefore, those MPEs whose ndcators are should have by far the greatest mpact on the estmated condtonal probabltes. Mnmzng the varance of the estmated condtonal probabltes gven n Equaton leads to: 2 2 M M 2 k k P = E I( E) P( E) I( k) P( k) E I( E) P( E) I( k) P( k) = = σ λ λ M M M k k k = E λ λj I( E) P( E) I( Ej) P( Ej) 2 E λ I( E) P( E) I( k) P( k) = j= = M λ P E = ( ) P( E ) k P ( k) P( k) 2 ( ) ( ) ( ) 2 ( ) ( ) ( ) ( ) ( ) M M M k k k λ λj j j λ = j= = 2 = P E E P E P E P E k P E P k + P k P k (2) To contnue wth the dervaton t becomes necessary to defne the covarance between MPEs: Cov { E E } = E{ [ I( E ) P( E )] [ I( E ) P( E )]} = P( E E ) P( E ) P( E ), (3) j Substtutng Equaton 3 nto Equaton 2, M M j j M k { E, E } 2 λ Cov{ E, k} Var{ k} 2 k k σ P = λ λ j Cov j + (4) = j= = And now takng the dervatve and settng t equal to zero, j j σ 2 P k λ = 2 M j= k λ Cov j { E, E } 2 Cov{ E, k} j =,, M M j= { E, E } = Cov{ E k} k λ j Cov j, =,, M (5) Equaton 5 s exactly the same as the standard smple krgng system of equatons, wth MPEs as data. Because tranng mages are used n MPS all necessary covarances, even those nvolvng complex MPEs, may be easly nferred. To justfy the use of the proposed approach utlzng MPEs, several checks may be made. Frstly, f each event consdered conssts of a sngle pont then the estmate n Equaton, the covarance n Equaton 3, and the krgng system n Equaton 5 all become the standard equatons used n tradtonal two-pont geostatstcs. Ths suggests that the theory behnd MPEs s sound, and s a more general form of tradtonal ndcator geostatstcs. The second check for the valdty of the proposed method s to use only a sngle event of N ponts. In ths case, solvng for the optmal weght n Equaton 5 yelds { E, k} { E, E } ( E k) P( E ) P( k) P( E ) [ P( E )] k Cov P λ = = (6) Cov Substtutng ths result nto Equaton, the estmated condtonal probabltes are then 0-4

5 ( k) ( E k) P( E ) P( k) P( E ) [ P( E )] * P P + [ I( E ) P( E )] P( k) = And snce the ndcator of the event occurrng s always, ths smplfes to ( k) ( E k) P( E ) ( E ) P( k) P( E ) P P = P (7) * P + = ( k) P( k E ) Equaton 7 s Bayes Law, whch s exactly the result that should be expected when usng a sngle event as data for determnng condtonal probabltes. Modfcatons to Condtonal Probabltes In addton to the MPE approach for estmatng the condtonal probabltes, other factors may be consdered as requred. As mentoned prevously there are no strct requrements for the condtonal probabltes used n a Gbbs sampler; there s only the practcal result that better selecton of the condtonal dstrbutons wll yeld more accurate estmates of the jont dstrbuton and hence hgher-qualty realzatons. Keepng wth ths dea, adjustments to the estmated condtonal probabltes may be made to help ensure the global target statstcs are honoured. The frst modfcaton s to help reproduce the global unvarate proportons, whch are often consdered very mportant. To gude the global proportons towards those desred, a servosystem s employed n the new algorthm (Strebelle and Journel, 200). The servosystem modfes the estmated condtonal probabltes as follows: * C P ( k) = P ( k) + P( k) P ( k) (8) where P ( k) s the modfed condtonal probablty of faces k; P * ( k) s the estmated probablty of faces k found wth Equaton ; P ( k ) s the global probablty of faces k; and P C ( k) s the proporton of faces k n the current realzaton feld Ths modfcaton leads to the faces whch are underrepresented beng gradually ntroduced more often and those whch occur too often beng subtly decreased. Whle the servosystem n Equaton 8 s unlkely to drastcally affect the condtonal probabltes, t wll prevent the unvarate proportons from devatng too far from those desred. In addton to the servosystem modfcaton, an addtonal updatng scheme s needed to honour locally varyng proportons of faces. Whle local proportons could be nserted drectly nto Equaton 8, the decson was made to use a Bayesan updatng scheme (Deutsch, 2002) nstead. Ths s an arbtrary decson and could be easly modfed n the future. The Bayesan updatng scheme for local proportons s as follows: ( k) ( k) L P P ( k) = P ( k) (9) P where P ( k) s the updated condtonal probablty of faces k; and P L ( k) of faces k. s the local probablty 0-5

6 Usng Bayesan updatng as n Equaton 9 should have a greater mpact on the condtonal probabltes of faces wth low global proportons than an addtve servosystem. For example, f a faces has a global P k = 0. and the local P k = 0.2, an addtve modfcaton would add 0. to the estmated condtonal probablty; ths may not be sgnfcant enough to overcome the global proporton term n Equaton. Multplyng as n Equaton 9 would double the estmated probablty for ths faces, whch should have a greater effect. Alternatvely to the updatng methods suggested here, a purely multplcatve approach could be used; ths would result n the estmated condtonal probabltes beng multpled by the local probabltes over the current realzaton proportons. Ths method was found to not be strct enough n matchng the desred global proportons; as a faces s eroded away from the realzaton ts probabltes become correspondngly lower, and multplyng a very low probablty by some factor stll results n a low probablty. A purely addtve approach was also explored, but was found to not honour the local means for those faces whch have low global proportons. The proposed method s suggested as a compromse between the two alternatves. Proposed Algorthm Usng the concepts dscussed above, the proposed workflow for the new MPS smulaton method s as follows:. Scan the tranng mage and calculate all of the necessary statstcs. Selecton of the tranng mage wll not be dscussed here. The number of MPEs to be used, M, and the ponts per event, N, must be selected carefully. Usng more events wll account for more of the surroundng nformaton, but wll also slow the algorthm down and put less weght on those events whch are truly mportant. Too many ponts n each event wll result n the statstcs not beng well enough nformed from the tranng mage; too few wll not fully characterze complex features. Expermentaton wth the method has typcally yelded the best results wth M rangng from 4 to 2 and N of 4 to 8. The actual values used depend on the sze and complexty of the tranng mage, the number of faces beng smulated, and the desred sze and number of realzatons. 2. Read n the ntal mage. The mage tself should honour some of the desred features or statstcs, such as the varogram; ths speeds up the algorthm sgnfcantly and also mproves the fnal results. If a random mage s used there s lkely to be very lttle useful nformaton contaned n the MPEs at each locaton, and therefore convergence to a sample from the jont dstrbuton (e, a realzaton) wll be qute slow. 3. Develop a random path, spralng away from the condtonng data. The data locatons wll not be vsted, and therefore the data wll be explctly honoured. Spralng away from the data should ad n the reproducton of condtonng nformaton wthout dscontnutes. 4. Follow the path, drawng new values. Estmate the condtonal probabltes wth Equaton and update wth Equatons 8 and 9. Draw a new faces value at each locaton from the updated condtonal dstrbuton. Ths s the Gbbs sampler part of the algorthm. Drawng random values at every locaton has been found to often leave random nose and dscontnutes n the resultng realzatons, even though the overall structure s reproduced. Therefore, a verson of maxmum a-posteror selecton (MAPS) (Deutsch, 998) has been mplemented n the fnal loop of each realzaton to clean up the mage. 0-6

7 MPS are stll used to determne the condtonal probabltes, however. The only modfcaton to the estmaton of the condtonal dstrbuton s the omsson of the global mean values P(k) n Equaton ; workng only wth resduals, the updatng s performed wth Equatons 8 and 9, then the faces wth the maxmum probablty s selected. Ths results n the mage beng cleaned qute well wthout dscardng many of the MPS honoured. 5. Fnsh the realzaton. After some number of loops over all locatons (ncludng the fnal MAPS loop), wrte out the results and repeat from step 2 for as many realzatons as are desred. The number of loops s also a modelng choce whch must be made for each ndvdual case. Example Channel Tranng Image, Two Faces Shown at the top of Fgure 2 s a two-dmensonal tranng mage wth two faces: channel and non-channel. The mage s 256 pxels square for a total of 65,536 cells. The channel structures seen n the tranng mage have snuosty and curvlnearty that s not possble to capture usng two-pont statstcs. In the lower left of Fgure 2 are two realzatons created usng sequental ndcator smulaton wth the covarance calculated drectly from the tranng mage. Whle some of the general shape of the features s reproduced, such as the long east-west contnuty and dagonal connectvty, n general there s very lttle reproducton of actual channels and potental flow paths are not seen n the realzatons. The lower rght of Fgure 2 shows two realzatons produced wth the proposed Gbbs sampler algorthm. Ten MPEs of sx ponts each were used for determnng the condtonal probabltes; the two-pont smulatons shown on the left were used as ntal mages; and ten loops were performed ncludng a sngle MAPS cleanng loop at the end. Smulatng ten realzatons took 8:3 usng these parameters, ncludng calculaton of all relevant statstcs. The computer used was a dual processor 3.2GHz PC wth 3GB of RAM. Comparng these realzatons to the SIS results, t s clear that the MPS enforce sgnfcantly more structure than two-pont relatons. Curvlnear bodes, long-range connectvty, and snuosty n the channels are all evdent. In the current mplementaton of the method, condtonal probabltes cannot be determned for the faces values at the very edge of the grd. For now ths problem s smply beng gnored, and the ntal values at the boundary of the realzaton are left n place unchanged; n Fgure 2 ths may be seen as edge effects. Ths decson does not appear to have had too sgnfcant of an mpact on the results n the rest of the grd, but a better method for dealng wth edge effects s desrable. Example 2 Complex Tranng Image, Fve Faces A complex tranng mage wth fve faces and very complex relatons between the dfferent codes s shown at the top of Fgure 3. The mage s 200 pxels by 00 pxels, a total of 20,000 cells. The lower left of Fgure 3 shows three realzatons whch were produced usng all relevant two-pont statstcs derved drectly from the tranng mage, ncludng all drect and cross covarances. Whle some of the structure was properly reproduced (n partcular the green dagonal bandng), and many of the relatons between faces were honoured (blue/red are nsde/on top of lght blue), the realzatons look very lttle lke the tranng mage. 0-7

8 Usng the proposed Gbbs sampler, the realzatons look much more lke the tranng mage; three realzatons are shown n the lower rght of Fgure 3. Ten realzatons were constructed by usng ten MPEs of three ponts each; twelve loops were performed, wth the fnal one usng the MAPS cleanng; and the two-pont realzatons n Fgure 3 were used as ntal mages. Smaller MPEs were used than for the frst example because, wth a smaller tranng mage and fve dfferent faces, there was less nformaton from whch to nfer even hgher-order MPS. Smulatng the ten realzatons took 2:03 on the same machne as that used for the frst example. The ellptcal bodes are much more clearly reproduced by MPS than two-pont statstcs, and the relatons wth the red and blue faces may be seen more clearly. As seen prevously, there are edge effects n the realzatons n Fgure 3. These artfacts once agan appear to have lttle effect on the rest of the realzaton results. Consderng the speed of the algorthm, t may be reasonable to consder smulatng a larger grd than s needed and trmmng the border. Comparson wth Other Smulaton Methods To analyze the potental of the proposed algorthm, a bref comparson study was performed. A complex three-dmensonal tranng mage wth fluval structures was used for the study; slces taken from ths mage are shown n Fgure 4. The tranng mage contans four faces: background floodplan (ndcator code 0), levee sand (code 2), channel sand (code 4), and sheet flood sand (code 5). The mage s 67 x 48 x 57 blocks n sze, for a total of 83,32 blocks. Sx dfferent smulaton methods were consdered n ths study:. Full ndcator cosmulaton, wth all covarances and cross covarances calculated drectly from the tranng mage. The program TISIS was used for ths method (Lyster and Deutsch, 2006). 2. Sngle normal equaton smulaton usng MPS; the SNESIM program was used (Strebelle and Journel, 2000). 3. Smulated annealng usng 2x2x2 hstograms of MPS as an objectve functon. Ths method used the MPASIM program (Lyster et al, 2004b). 4. Smulated annealng, post-processng the results of tradtonal two-pont smulaton. Agan, the MPASIM program was used. 5. A greedy varant of smulated annealng, wth the temperature set to zero; ths case also used MPASIM. 6. Smulaton usng the proposed Gbbs sampler algorthm, usng MPEs to determne condtonal probabltes. A new program, MPESIM, was wrtten for ths purpose. Four MPEs of fve ponts each were used for the smulaton. Ten realzatons were created usng each of the sx methods lsted above. The crtera used for the comparson were: tme requred for smulaton, vsual goodness of the realzatons, reproducton of the target faces proportons n the tranng mage, ndcator varogram reproducton, error n the smulated MPS hstograms, and dstrbutons of multple-pont runs (Bosvert et al, 2006). Table gves a summary of the computatonal tme requred for each of the algorthms. As expected, the two-pont method was the fastest. The sngle normal equaton smulaton took by 0-8

9 far the longest tme to perform ten smulatons; ths could be due to the tme needed to buld large search trees needed for the large template and four faces. The annealng smulatons took less tme than the normal equatons approach, but stll orders of magntude more than the two-pont ndcator smulaton. The proposed Gbbs sampler algorthm was much faster than the other MPS methods, and wthn an order of magntude of the two-pont smulaton tme. Though computatonal requrements are an mportant consderaton, f a faster method produces worse results t may be less desrable despte ths advantage. Method Tme for 0 Realzatons Full ndcator cosmulaton :55 Sngle normal equatons 7:03:07 Multple-pont smulated annealng 3:06:02 Post processng usng MP SA 3:4:5 Post processng usng greedy crtera :53:38 Gbbs sampler algorthm usng MPEs 7:24 Table : Tme requred for the sx dfferent smulaton methods to produce ten realzatons. Vsual nspecton of realzatons s the easest way to judge f a method s unacceptable, but ths measure s very subjectve. Fgures 5 through 0 show slces of realzatons produced by each of the methods. Vsually, the two-pont method (Fgure 5) does not reproduce the channel structure at all, though the relatons between channel and levee sand do appear somewhat correct. The sngle normal equaton approach (Fgure 6) does seem to reproduce the short-range structure, though long-range contnuty s somewhat lackng. Ths s probably an mplementaton ssue (not enough grds used for smulaton). The annealng smulaton (Fgure 7) appears to be qute good n the cross sectonal slces, though the curvlnear channel features are not seen n the aeral vews. The post processng and greedy annealng approaches (Fgures 8 and 9 respectvely) appear smlar to the unconstraned annealng approach, though wth better reproducton of the levee sand features along the edges of the channel sand. In the Gbbs sampler results (Fgure 0), the relatons between the channel sand and levee sand faces are reproduced. The long-range channel structure s not seen n the aeral vew; however, the Gbbs algorthm does appear to replcate the very long-range contnuty of the flood plan faces better than any of the other methods. The edge effects seen prevously were crcumvented by smulatng an addtonal three blocks on all sdes of the feld, then trmmng these extra cells; ths reduced, but dd not completely remove, the effects of the unperturbed nodes. A more objectve measure of a method s the reproducton of unvarate statstcs; n ths case, the proportons of dfferent faces. Fgure shows the unvarate CDFs of each of the sx methods along wth the reference tranng mage CDF (shown n red). A notceable devaton from the target statstcs are the annealng and annealng post-processng results sgnfcantly underrepresentng the flood plan faces n every realzaton, n favour of channel sand. Also notable s the systematc msmatch to the target proportons of the proposed Gbbs sampler algorthm. All of the faces are ether over- or under-reproduced, wth close to the same results n every realzaton. Ths suggests the servosystem approach n Equaton 8 s not strong enough to ensure reproducton of the target unvarate statstcs. Fgures 2 through 5 show the ndcator varograms for the four faces, for all sx smulaton methods consdered as well as the reference tranng mage varograms. Surprsngly, the twopont smulaton dd not reproduce the varograms well at all; ths s probably due to the excessve 0-9

10 randomness caused by large krgng matrces. The dfferent flavours of annealng appear to best match the target varograms, whch s surprsng consderng the small template used for the objectve functon. The very long-range horzontal contnuty of the sheet flood faces was not reproduced by most of the methods; only the annealng and annealng post-processng methods came close to matchng the varograms, and both of those methods were sgnfcantly skewed n the reproducton of the unvarate proporton of the sheet flood. The proposed Gbbs sampler reproduced the structures of each varogram (wth the excepton of the sheet flood), but the slls are wrong due to the bas n the unvarate proportons whch was mentoned above. As the smulaton methods beng compared utlze MPS (wth one excepton), comparsons usng multple-pont statstcs are desrable. The MPS hstograms, usng 2x2x2 statstcs, were compared to those calculated from the tranng mage; the dstrbutons of runs were also calculated and compared. The average error (devaton from reference) for the hstograms and runs for each of the smulaton methods s shown n Table 2, along wth the rankngs of each method. As should be expected, the two-pont smulaton performed the worst by far when measured by MPS crtera. The sngle normal equaton algorthm performed better n runs than n the hstogram measure, and was mddle of the pack overall. The annealng methods performed very well, partcularly n the hstogram measure; however, the MPS hstogram could be based towards the annealng because a 2x2x2 statstc was used for both the objectve functon and the hstogram error calculaton. The proposed Gbbs sampler method lagged somewhat behnd n both MPS measures, though the results are not bad enough to dscourage further development of the algorthm. Method MPH Error Rank Runs Error Rank Full ndcator cosmulaton Sngle normal equatons Multple-pont smulated annealng Post processng usng MP SA Post processng usng greedy crtera Gbbs sampler algorthm usng MPEs Table 2: Comparsons of multple-pont statstcs. Average devatons of multple-pont hstograms and runs dstrbutons from the tranng mage are shown, as well as rankngs of the dfferent methods. Comparng the MPE Gbbs sampler methodology to other faces smulaton technques, there s certanly room for mprovement. The proposed method dd not produce the best results by any objectve measure. However, gven the great savngs n tme requred for performng the smulaton, future modfcatons to the algorthm could make t a very attractve possblty. Conclusons As a MPS smulaton method, the proposed Gbbs sampler algorthm shows promse. Further refnement s needed to properly reproduce long-range features, to elmnate the edge effects, and to ensure condtonng data are honoured. Better reproducton of the desred MPS s also a goal, though ths would lkely be a sde effect of the other mprovements mentoned. 0-0

11 Long-range features may be ntroduced n the ntal mages, rather than n the algorthm tself. In general for teratve algorthms, better ntal mages wll lead to better results. The edge effects could smply be trmmed, the border of the realzatons could use a dfferent method for selecton of condtonal probabltes, or the grd could be wrapped. All of these possbltes have ther own postve and negatve ponts. Honourng condtonng data s the most mportant mplementaton aspect that needs to be developed. The major weakness of many teratve methods s that hard data are explctly honoured, but wth a dscontnuty. Ths problem can easly present tself n the proposed Gbbs sampler. Several steps may be taken to try and prevent ths: ensure the ntal mages honour the condtonng data, use locally varyng proportons of faces to reduce the lkelhood of sudden dscontnutes, or modfy the condtonal probabltes to explctly account for nearby hard data. Overall mprovement to the technque could be accomplshed through greater sophstcaton. Herarchcal smulaton of faces, ntegraton of lower-order (but longer-range) statstcs to the condtonal probabltes, and other modfcatons are all possble because of the great speed of the algorthm; there s the possblty for mprovements to be made, even f smulaton proceeds slower because of t. Developments of these aspects, as well as other mplementaton ssues, wll need to be further researched. Any method needs to be qute robust before t can be accepted and used n quantfcaton of uncertanty. The advantages of MPS over tradtonal two-pont methods are offset by the extent to whch krgng-based algorthms have been developed. Havng a dverse range of technques whch utlze MPS s benefcal to the feld as each method may have ts own area n whch the algorthm works best. Development of each dea wll progress over tme, and as problems are solved for one method the same soluton may be applcable to others and the robustness of MPS wll be ncreased. References Bosvert, J.B., Pyrcz, M.J., and Deutsch, C.V. (2006) Choosng Tranng Images and Checkng Realzatons wth Multple Pont Statstcs. Centre for Computatonal Geostatstcs, No. 8, awatng publcaton. Caers, J. (200) Geostatstcal Reservor Modellng Usng Statstcal Pattern Recognton. Journal of Petroleum Scence and Engneerng, Vol. 29, No. 3, May 200, pp Caers, J. and Journel, A.G. (998) Stochastc Reservor Smulaton Usng Neural Networks Traned on Outcrop Data. SPE Annual Techncal Conference and Exhbton, New Orleans, Oct. 998, pp SPE # Casella, G. and George, E.I. (992) Explanng the Gbbs Sampler. The Amercan Statstcan, Vol. 46, No. 3, Aug. 992, pp Deutsch, C.V. (992) Annealng Technques Appled to Reservor Modelng and the Integraton of Geologcal and Engneerng (Well Test) Data. Ph.D. Thess, Stanford Unversty, 304 p. Deutsch, C.V. (998) Cleanng Categorcal Varable (Lthofaces) Realzatons Wth Maxmum A-Posteror Selecton. Computers & Geoscences, Vol. 24, No. 6, pp Deutsch, C.V. (2002) Geostatstcal Reservor Modelng. Oxford Unversty Press, New York, 376 p. 0-

12 Gelfand, A.E. and Smth, A.F.M. (990) Samplng-Based Approached to Calculatng Margnal Denstes. Journal of the Amercan Statstcal Assocaton, Vol. 85, No. 40, June 990, pp Geman, S. and Geman, D. (984) Stochastc Relaxaton, Gbbs Dstrbutons, and the Bayesan Restoraton of Images. IEEE Transactons on Pattern Analyss and Machne Intellgence, No. 6, Nov. 984, pp Guardano, F.B. and Srvastava, R.M. (993) Multvarate Geostatstcs: Beyond Bvarate Moments. Soares, A., Edtor, Geostatstcs Troa 92, Vol., pp Lyster, S., Leuangthong, O., and Deutsch, C.V. (2004a) Smulated Annealng Post Processng for Multple Pont Statstcal Reproducton. Centre for Computatonal Geostatstcs, No. 6, 5 p. Lyster, S., Ortz, J.M, and Deutsch, C.V. (2004b) MPASIM: A Multple Pont Annealng Smulaton Program. Centre for Computatonal Geostatstcs, No. 6, 8 p. Lyster, S. and Deutsch, C.V. (2006) TISIS: A Program to Perform Full Indcator Cosmulaton Usng a Tranng Image. Centre for Computatonal Geostatstcs, No. 8, awatng publcaton. Metropols, N., Rosenbluth, A.W., Rosenbluth, M.N., Teller, A.H., and Teller, E. (953) Equatons of State Calculatons by Fast Computng Machnes. Journal of Chemcal Physcs, Vol. 2, No. 6, pp Ortz, J.M. (2003) Characterzaton of Hgh Order Correlaton for Enhanced Indcator Smulaton. Ph.D. Thess, Unversty of Alberta, 255 p. Srvastava, M. (992) Iteratve Methods for Spatal Smulaton. Stanford Center for Reservor Forecastng, No. 5, 24 p. Strebelle, S.B. and Journel, A.G. (2000) Sequental Smulaton Drawng Structures From Tranng Images. Klengeld, W.J. and Krge, D.G., Edtors, 6 th Internatonal Geostatstcs Congress, 2 p. Strebelle, S.B. and Journel, A.G. (200) Reservor Modelng Usng Multple-Pont Statstcs. SPE Annual Techncal Conference and Exhbton, New Orleans, Oct. 200, p. SPE #

13 Fgure 2: Top: TI wth two faces and channel features. Left: Realzatons produced usng twopont statstcs. Rght: Realzatons usng the proposed Gbbs sampler algorthm. 0-3

14 Fgure 3: Top: Complex TI wth fve faces. Left: Realzatons usng only covarances and crosscovarances. Rght: Realzatons usng the proposed Gbbs sampler algorthm. 0-4

15 Fgure 4: Slces from the fluval tranng mage used n the comparson of smulaton methods. Top left: X-Y; Top rght: Y-Z; Bottom: X-Z. 0-5

16 Fgure 5: Slces from a realzaton produced by full ndcator cosmulaton. Top left: X-Y; Top rght: Y-Z; Bottom: X-Z. 0-6

17 Fgure 6: Slces from a realzaton produced by multple-pont sngle normal equaton smulaton. Top left: X-Y; Top rght: Y-Z; Bottom: X-Z. 0-7

18 Fgure 7: Slces from a realzaton produced by multple-pont smulated annealng. Top left: X- Y; Top rght: Y-Z; Bottom: X-Z. 0-8

19 Fgure 8: Slces from a realzaton produced by multple-pont smulated annealng, post processng realzatons produced by full ndcator cosmulaton. Top left: X-Y; Top rght: Y-Z; Bottom: X-Z. 0-9

20 Fgure 9: Slces from a realzaton produced by multple-pont smulated annealng wth temperature set to zero (e, a greedy approach), post processng realzatons produced by full ndcator cosmulaton. Top left: X-Y; Top rght: Y-Z; Bottom: X-Z. 0-20

21 Fgure 0: Slces from a realzaton produced by the Gbbs Sampler MPE approach, usng 4 statstcs of 5 ponts each. Top left: X-Y; Top rght: Y-Z; Bottom: X-Z. 0-2

22 Fgure : Unvarate CDFs for all realzatons of each of the smulaton methods compared. The reference TI CDF s shown n red. 0-22

23 Fgure 2: Varogram reproducton for each method for faces 0 (background floodplan). The sold lnes are the reference TI varograms; the dotted lnes are realzatons. Red s n the X drecton, black s the Y drecton, and blue s the Z drecton. 0-23

24 Fgure 3: Varogram reproducton for each method for faces 2 (levee sand). The sold lnes are the reference TI varograms; the dotted lnes are realzatons. Red s n the X drecton, black s the Y drecton, and blue s the Z drecton. 0-24

25 Fgure 4: Varogram reproducton for each method for faces 4 (channel sand). The sold lnes are the reference TI varograms; the dotted lnes are realzatons. Red s n the X drecton, black s the Y drecton, and blue s the Z drecton. 0-25

26 Fgure 5: Varogram reproducton for each method for faces 5 (sheet flood). The sold lnes are the reference TI varograms; the dotted lnes are realzatons. Red s n the X drecton, black s the Y drecton, and blue s the Z drecton. 0-26