Scaling Multiple Point Statistics for Non-Stationary Geostatistical Modeling

Transcription

1 Scaling Multile Point Statistics or Non-Stationary Geostatistical Modeling Julián M. Ortiz, Steven Lyster and Clayton V. Deutsch Centre or Comutational Geostatistics Deartment o Civil & Environmental Engineering University o Alberta Multile-oint statistics are used in geostatistical simulation to imrove orecasting o resonses that are highly deendent on the reroduction o comlex eatures o the henomenon that cannot be catured by conventional two-oint simulation methods. Inerence o multile-oint statistics is oten based on a training image that deicts the eatures that rovide the character to the geological event being modelled. One limitation o this aroach is that the univariate distribution o categories (acies or rock tyes) in the training image may not match the statistics o the inal model. The question o scaling multile-oint statistics arises, the idea being to take the statistics rom the training image and scale them in a reeatable manner and honouring the univariate roortions o categories. An iterative scaling aroach based on the exression or scaling multile-oint statistics in a urely random case is roosed. A multi-gaussian alternative is discussed, which shows some roblems due to the symmetry o the multivariate Gaussian distribution. The imlementation is illustrated through an examle where it is shown that the roosed method lies in between two extreme cases or a Boolean simulation, namely, the change in size o the obects and the change in their number o occurrences. A second examle is resented to illustrate the otential use o this scaling rocedure or non-stationary multile-oint geostatistical simulation. Introduction Multile-oint statistics can be used in an attemt to imrove geostatistical modelling o variables distributed in sace. The relationshis between several oints at a time are estimated rom training data and imosed during the simulation rocess, in order to achieve a numerical model that correctly reresents these intricate relationshis that conventional simulation cannot cature. Figure shows a reerence image where the relationshis between the dierent geological units or acies cannot be correctly catured by conventional simulation methods. Exhaustive images such as the one deicted here can be used as an analogue to the henomenon that is being modelled; multile-oint statistics can be extracted by scanning the image and comuting the requency with which acies arranged in seciic atterns occur. The training inormation requires abundant data located over a regular grid o oints, in order to have enough relicates o each articular multile-oint event. This is oten solved by utilizing a training image (Guardiano and Srivastava, 992; Deutsch, 992) or by using available seudoregularly saced roduction data, as is the case in mining alications (Ortiz, 2003; Ortiz and Deutsch, 2004; Ortiz and Emery, 2005). Multile-oint geostatistical simulation can be erormed using any o the available methods. The single normal equation simulation roosed by Strebelle and Journel (2000) estimates the -

2 conditional distribution at every location given a multile-oint coniguration by calculating the requency with which the indicator at the location being simulated is one given that the multileoint event occurs in the training image. Alternatively, simulated annealing (Deutsch, 992) can be used to match the multile-oint requencies extracted rom a training image into a simulated numerical model. Again, the multile-oint statistics are read rom the training image as requencies o articular events occurring. Other methods such as neural networks also rely on the use o a training image to extract and reroduce the multile-oint statistics (Caers and Journel, 998). There is an imlicit assumtion o stationarity during the exorting o multile-oint statistics rom the training image to the simulated model. A stationarity assumtion stricter than the usual second order stationarity is required, since the use o conigurations o several oints also locks all lower order statistics. For instance, i our-oint conigurations are matched during the multile-oint simulation rocess, then all three-oint and two-oint statistics whose coniguration is included in the our-oint coniguration originally used are imlicitly matched ( Figure 2). Most imortantly, the histogram (one-oint statistic) is also locked when higher order statistics are honoured. This aer addresses the issue o scaling multile-oint statistics, which may be required in two distinct cases. First, consistency roblems may arise when the training image does not have exactly the same one-oint and two-oint distribution as the underlying henomenon. This consistency roblem occurs requently, since training images usually are not available with the exact same histogram o the variable that is being modelled (roortions o acies). Direct use o the multile-oint statistics extracted rom the training image will distort the histogram o the simulated realizations generating a bias in the roortions. A basic requirement or the simulated model to be acceted as a lausible reresentation o the true henomenon is that it honours a given histogram. Scaling o multile-oint statistics is then required to allow the use o the training image in a consistent manner with the data and reserving its character, hence generating accetable numerical models. Another relevant use o a scaling rocedure or multile-oint statistics is to imose nonstationary eatures, but reserving the character rovided by the training image. One could devise the use o a single training image to model a ield with locally varying roortions o the acies. We resent a methodology or scaling multile-oint statistics to generate consistent results rom simulation methods that account or this inormation and to consider non-stationary eatures during the modeling rocess. The methodology is general and could be adated to be used with continuous variables, as long as the data are coded as indicators by deining classes through a set o thresholds. In what ollows, the roblem is resented in the case o a categorical variable, which is where simulation accounting or multile-oint statistics has seen a aster develoment. Problem setting Consider a training image that deicts the satial arrangement o K categories or acies. The global roortions with which these categories are resent in the training image are denoted: k, k =,..., K. The general aearance o the training image is deemed aroriate to model a given geological setting, hence the modeler decides the training image is to be used or inerence o the multileoint statistics that a given simulation algorithm will imose to a set o realizations. The multile-oint statistics to be considered are deined by a satial arrangement o nodes and by the combination o acies values in these nodes. I we consider the case, where an N-oints -2

3 statistic is considered, then K N ossible combinations are available. Each o these combinations occurs in the training image with a given requency. In act, as soon as N or K become relatively large, there is big chance that many o the K N combinations do not occur in the training image. Each one o the ossible combinations is identiied with an index that comletely deines the acies values within the N oints, however the ordering to identiy the oints in the attern must be deined rior to the calculation o the index o each multile-oint event (Figure 3). The index o each multile-oint coniguration is calculated as: = + N n= ( i n ) K where i n is the code o the n th node o the attern that identiies its acies. Here, it is assumed that acies are denoted by consecutive integers starting with code. N The requency o each multile-oint event in the training image is denoted:, =,..., K. Knowing the indexing and the requencies with which each multile-oint coniguration occurs, ermits calculation o the acies roortions k, k =,..., K.. Denoting by,..., N k, ; k =,..., K; = K, the roortion o acies k in the multile-oint arrangement identiied with the index, the roortions o the acies in the training image can be retrieved rom the multile-oint statistics as ollows. Consider the multile-oint index o interest is. We can calculate the value o the n th node, by taking: = i N int N K where int reresents the integer art o the division. Subsequently, the indexes o the n remaining nodes o the attern can be calculated recursively using the residual o the division (ractional art): n rac n K i = n int n K n =,..., N Knowing the acies values o the N nodes o each multile-oint index and their requencies, the univariate roortions can be easily calculated. Now, the simulated model must honor a set o statistics inerred rom a set o samles. For instance, the available data may show that the global declustered roortions o the acies in the domain are: k, k =,..., K, with k not necessarily equal to k or some k =,..., K. The goal o this aer is to roose a methodology to calculate the corrected requencies o multile-oint events that will honor the roortions, reserving the character o the training image, that is, keeing the eatures that make it distinct. These corrected requencies are denoted: *, =,..., K N. Scaling aroaches or multile-oint statistics To understand the concet o scaling multile-oint statistics, the ollowing examle illustrates ossible outcomes rom an increase in the roortion o a acies in a binary case and where the -3

4 scaled models reserve the character o the original training image. A ield o 000 by 000 ixels is oulated with 0 by 0 ixels squares, where the centre o the squares are randomly located in the ield, that is they are generated through a Poisson rocess ( Figure 4). There are enough squares to cover 20% o the domain, leaving the remaining 80% as background. Scaling this training image so that the roortion o squares goes u to 40% may generate two equally valid outcomes: we can have more 0 by 0 ixels squares or have a smaller number o larger squares, say 20 by 20 ixels squares. These situations are o course extremes and since we do not know the exact multile-oint statistics o the variable and we borrow this inormation rom a training image that does not exactly match the roortions, a scaling rocedure that lays somewhere in between the two extremes resented above can be used as a modelling decision to handle the roblem. Two aroaches could be devised to scale multile-oint statistics when the univariate distribution changes: Scaling based on how a variable randomly distributed changes its multile-oint statistics when the univariate roortions o the acies change. Scaling based on how a categorical variable obtained by multile truncations o a correlated multigaussian continuous variable changes as the thresholds that deine the acies are modiied. Dimensionality is always a roblem when dealing with multile-oint statistics since the combinatorial becomes raidly very large as the number o oints in the multile-oint coniguration N or the number o acies K increase. For examle, 0 acies can be arranged on a 9 oints attern in billion ossible combinations. O course, this examle is extreme, but attern size oten increases in an exonential ashion: 4, 9, 6, 25 oints in 2D, or 8, 27, 64, 25 oints in 3D. This roblem may be artially solved by roceeding airwise, that is, considering the most relevant acies and all other acies together. Then, reezing the most imortant acies already simulated, one can simulate another relevant acies against all remaining acies on a reduced domain, and so on, until all acies have been individually taken into account. Scaling MPS based on the random case The irst aroach roosed is based on how multile-oint statistics change when the acies codes o the oints in the attern are randomly distributed. In this case, the requency with which a multile-oint event occurs can be calculated as the roduct o the robabilities o occurrence (roortions) o each acies values in its nodes. Since the acies values are considered uncorrelated, it is straightorward to scale the requency o occurrence o multile-oint statistics. The requency o a multile-oint coniguration can be calculated by: = K k = N k, k Since k, is the roortion o acies k in class, N k, is the number o occurrences o acies k in class. For examle (Figure 5), considering a case where two acies are available and the robability o acies revailing at a given location is 0.25, then the robability o having a ouroints coniguration where acies revails in two nodes and acies 2 revails in the remaining nodes would be calculated as: = =

5 Scaling the random case is quite simle; multilying the original requency by a series roduct o the desired requency over the old requency results in a simle equation: * = K k = k It is easy to see that the scaled multile-oint requencies honour the univariate roortions: * = K k = k k N k, ( K k k= = K N k, ) ( k ) k= k N N k, k, * = = Following u on the revious examle, we can see that a change in the global roortions o the acies will change the robability o occurrence o the multile-oint event deicted in Figure 5. Since the uncorrelated case is simle, we can easily calculate this robability or the new global roortions. Consider the case, the robability o acies revailing at a location goes u to 0.4, leaving a robability o 0.6 or acies 2. Considering the same our-oints coniguration, we can calculate its new robability o occurrence: new = = It can be checked that the exression rovided or scaling multile-oint statistics in the random case rovides the same result calculated above: * 3 = 3 = N,3 N , = This exression is valid only in the case o a multile-oint statistics rom an uncorrelated variable (ure nugget eect). As soon as the variable is satially correlated, multile-oint roortions cannot be directly calculated. An iterative aroach is roosed next to calculate the multile-oint requencies that honour the roortions, rom the initial multile-oint requencies inerred rom the training image. The ormula above overcorrects the multile-oint requencies. Convergence can be achieved by iterating using the ollowing modiied exression: * * = K k = k k k, N Multile-oint requencies or all indexes =,..., K must be udated. The new global roortions k, k =,..., K are recalculated or each iteration. This ormula does not require a large number o iterations, and usually, the desired multile-oint requencies can be obtained with a nearly erect match o the roortions, with less than 50 iterations, which takes only a ew seconds. -5

6 Scaling MPS based on the multigaussian case An alternative method consists on considering how a categorical variable deined by multile truncations o a multigaussian ield changes as the thresholds or truncations change. The idea can be understood quite simly, by considering a two-oint attern ( Figure 6). A generalization to N oints can be achieved, associating a multivariate Gaussian distribution o dimension N. In this simle examle, two oints searated by a vector h constitute the multile-oint attern. Initially, we can consider a binary variable, hence it can be deined by a single truncation o the Gaussian variable with a threshold y C. The two categories (acies) have robabilities: = G( y ) 2 C = C = G( y An indicator variogram can be calculated and associated with a variogram or the underlying multigaussian variable (Kyriakidis and others, 999). Given the satial arrangement o the two oints, Their satial correlation can be calculated rom the variogram model and assuming multivariate Gaussianity, all conditional robabilities can be retrieved. The robabilities o all our ossible multile-oint events can be calculated, by using conditional robabilities that only deend on two-oint statistics, that is, the variogram model. For examle, assuming acies is associated with the continuous Gaussian variable when the value is below the threshold y C : Prob ( yc i = ; i2 = 2) = g( y, y2 ) dy2 dy y where g(y,y 2 ) is the bivariate Gaussian robability density unction. This simle case can be easily extended to multile oints (N > 2), however, calculating the oint robabilities becomes cumbersome as the dimensionality increases. A simle numerical aroximation can be used to calculate the multile-oint event requencies. In the multile-oint case, all correlations between the nodes in the attern must be taken into account in order to aroximate the integration o the multivariate Gaussian distribution using the aroriate thresholds to deine the acies. A Monte-Carlo aroach to simulate multivariate Gaussian vectors using the matrix decomosition method is roosed (Davis, 987). The general rocedure is:. Given the training image and roortions k, k =,..., K and k, k =,..., K, calculate the thresholds that give those robabilities when truncating a standard Gaussian distribution: y y C, N = G M = G N i= i C C ) C, N = G M C, ( ) C, ( ) 2. Comute the indicator variograms and iner the variogram o the Gaussian variable that better its the calculated indicator variograms. 3. Given the searation vectors between the N oints in the multile-oint attern, build the N by N matrix o covariances between oints in the attern ( y y = G N i= i -6

7 4. Figure 7): C( h) = C( h 2) C M C( h N) C( h C( h C( h 5. Using the matrix decomosition simulation algorithm generate a large number o simulated multivariate vectors that honor the covariance between the oints. The algorithm roceeds as ollows: a. Decomose the covariance matrix as a roduct o a lower (L) and an uer (U) triangular matrix such that L = U T (Cholesky decomosition). b. Draw a vector o indeendent standard Gaussian deviates w. c. Generate a vector o correlated Gaussian values by multilying the lower matrix by the vector o random normal deviates: y = L w. 6. Identiy the multile-oint index o each simulated vector by alying the truncation rule deined by the thresholds, that is transorm the vectors o correlated Gaussian values to vectors o acies values, by coding the values as acies using the thresholds. Then, calculate the multile-oint index corresonding to each coniguration o acies. 7. For each multile-oint coniguration that existed in the training image, calculate the roortion o simulated vectors that match that multile-oint index when alying the thresholds calculated or the training image roortions and when the roortions are used. 8. Correct the multile-oint statistics o the training image, using a ratio o those roortions. 9. Standardize to ensure the sum o roortions rom the scaled multile-oint requencies add to one. This rocedure accounts or the correlation between the oints. It is an aroximation due to the act that the Gaussian variable is continuous, hence all multile-oint combination o indexes in the attern has a robability o occurrence, which is not the case in the training image, since this is a discrete reresentation. A correction is required to match exactly the roortions, but this cannot always be achieved, since using the multivariate Gaussian case imoses constraints with resect to the occurrence o symmetric events: or instance the resence o acies one on the right o acies two imlies that acies two must exist also on the right o acies one. For this reason, this aroach is generally not recommended. Examles First, to illustrate the use o these scaling aroaches, the small examle resented in Figure 4 is ollowed u. The multile-oint histogram or a two by two oints attern is calculated. The requency o occurrence o all 6 multile-oint events is comuted and a histogram o the requency with which each indexed event occurs is lotted as a summary. The 6 ossible events can be easily interreted in terms o the geometry o the obects (squares), as shown in Figure 8. The reerence image (ma on the let hand side o M 2 22 N2 ) ) ) L L O L C( h C( h M C( h N 2N NN ) ) ) -7

8 Figure 4) is used to calculate the multile-oint requencies in a 2 by 2 ixels attern and these are scaled to reach a global roortion o squares o These scaled statistics are then comared with the two extreme cases resented on the right hand side o Figure 4. Figure 9 shows the three multile-oint histograms suerimosed. It can be seen that the scaling using the random aroach rovides statistics that lie in between the two extreme cases. We can exect that these scaled u statistics reresent a case where there are more squares in the domain, but these are also slightly larger than the ones in the training image. The relevant eatures o the reerence image are then catured in the statistics. A second examle has been designed to illustrate the use o the scaling rocedure roosed. A binary two dimensional training image has been created with black horizontal stries two units thick in a white background. The training image is shown in Figure 0. The original roortions are black = 0.25 and white = Multile-oint statistics have been inerred rom the training image and unconditional realizations have been comuted or dierent roortions o black (stries) and white (background) acies. A simulated annealing rogram to match the multile-oint statistics was reared. Dierent attern sizes imly control over dierent scales o the henomenon. For examle, Figure shows two realizations considering a 4 by 4 nodes attern. Notice that with this attern size the general aearance o the resulting models is close to the training image. I a dierent attern is considered, the long range eatures may not be roerly catured. Figure 2 shows realizations made with dierent attern sizes to illustrate this eect. The scaling rocedure was used to scale u the roortion o the black acies. The initial roortions (25% black, 75% white) was changed and the multile-oint statistics were scaled. These were then used to simulated realizations with the intent o generating realizations that ket the essential eatures o the training image, but honouring the acies roortions imosed to the new models. Figure 3 shows three realizations constructed using the multile-oint statistics or a 3 by 3 nodes attern, scaled to match the ollowing acies roortions: 30% black and 70% white. These roortions were then changed to 50% or the black acies and 50% or the white background, and realizations with statistics rom 2 by 2, 3 by 3, and 4 by 4 nodes atterns considered. These results are shown in Figure 4. It can be seen rom the realizations dislayed above that, deending on the attern size, two situations can arise: the stries may become wider i the attern is small enough not to cature its entire width, or the stries may become more abundant i the attern is big enough to cature the relationshis between the width o the black acies obects and the background, as is the case when 3 by 3 or 4 by 4 atterns are used. Conclusions Scaling multile-oint statistics can imrove the current use o multile-oint geostatistical simulation techniques, by allowing locally varying roortions o acies to be reroduced still honouring the relationshis catured by multile-oint statistics. Another imortant use o a multile-oint scaling technique is the use o a reresentative training image that does not exactly match the simulation roortions. This article resents an aroach to scale multile-oint statistics by iteratively changing them in a manner similar to the change that occurs in multile-oint statistics o a urely random ield where the roortions o the acies are modiied. This aroach can be shown to rovide -8

9 reasonable values o the statistics that lie in between two extreme cases that are easy to show in the case o obects: an increase in the number o obects or an increase in the size o the obects. The scaling rocedure has the otential o being incororated in a non-stationary simulation algorithm that honors the eatures rovided by a training image, through the local udating o multile-oint statistics. Acknowledgements The authors would like to acknowledge the industry sonsors o the Centre or Comutational Geostatistics at the University o Alberta and the Chair in Ore Reserve Evaluation at the University o Chile sonsored by Codelco Chile or suorting this research. Reerences Caers, J. and Journel, A. G., Stochastic reservoir simulation using neural networks trained on outcro data, in 998 SPE Annual Technical Conerence and Exhibition, New Orleans, LA. Society o Petroleum Engineers. SPE aer # 49026, 998, Davis, M. W., Production o conditional simulations via the LU triangular decomosition o the covariance matrix. Mathematical Geology, vol. 9, no. 2, 987, Deutsch, C. V., Annealing Techniques Alied to Reservoir Modeling and the Integration o Geological and Engineering (Well Test) Data. PhD thesis, Stanord University, Stanord, CA, 992. Guardiano, F. and Srivastava, M., Multivariate geostatistics: Beyond bivariate moments, in A. Soares, editor, Geostatistics Tróia 92. Kluwer, Dordrecht, vol., 993, Kyriakidis, P. C., Deutsch C. V., and Grant M. L., Calculation o the normal scores variogram used or truncated Gaussian lithoacies simulation: theory and FORTRAN code. Comuters & Geosciences 25 (2): 6-69, March 999. Ortiz, J. M., Characterization o High Order Correlation or Enhanced Indicator Simulation. PhD thesis, University o Alberta, Edmonton, AB, Canada, Ortiz, J. M. and Deutsch, C. V., Indicator simulation accounting or multile-oint statistics. Mathematical Geology, vol. 36, no. 6, 2004, Ortiz, J. M. and Emery, X., Integrating multile-oint statistics into sequential simulation algorithms, in O. Leuangthong and C. V. Deutsch, editors, Geostatistics Ban Kluwer Academic, Dordrecht, The Netherlands, Vol. 2, 2005, Strebelle, S. and Journel, A. G., Sequential simulation drawing structures rom training images, in 6th International Geostatistics Congress, Cae Town, South Arica, Geostatistical Association o Southern Arica,

10 Figure : an exhaustive image showing intrincated relationshis o our acies in multile-oint atterns that cannot be easily catured by conventional simulation techniques. Figure 2: A our-oints attern and all lower order conigurations that are imlicitly matched by honouring the our-oint statistics: three-oints statistics in our conigurations; two-oints statistics or the corresonding lag distances searating the centres o the nodes (these reresent indicator variogram values); and the one-oint statistics that corresonds to the histogram. Nodes are reresented by their surrounding squares or illustration. -0

11 Figure 3: A our-oint coniguration, the order or considering the nodes and codes o the acies. Three examles o calculation o the multile-oints indexes are illustrated. Figure 4: Two ossible uscaled models rom the training image on the let. The to right ma shows more squares o the same dimension as in the reerence image; the bottom right ma shows larger squares. Both mas have a roortion o ixels belonging to squares o

12 Figure 5: examle o our-oints coniguration with two acies to calculate the robability o occurrence o an event in the random case. Figure 6: An illustration o the relationshi between global roortions, deined by the thresholds that truncate the multigaussian distribution, and the multile-oint requencies. A two-oints attern o nodes searated by a lag vector h allows calculation o the correlation coeicient rom the normal scores variogram γ(h). This deines the bivariate Gaussian distribution that is truncated so that the robability o being below y C is and comletely deines the multile-oint robabilities. These robabilities change in a tractable manner when the global roortions are changed by modiying the threshold. -2

13 Figure 7: Four-oints coniguration and the corresonding lag searation vectors used to build the covariance matrix to simulate multivariate Gaussian vectors. Figure 8: Geometric interretation o the dierent multile-oint conigurations or the examle o drawing squares randomly in the domain. -3

14 Frequency Multile-Point Index scaled 0 x 0 20 x 20 Figure 9: Multile-oint histogram or the scaled statistics and the two extreme cases: increased number o 0 by 0 squares and increased size o the squares to 20 by 20. A logarithmic scale has been used or the requency axis to better show that he scaled statistics lie in between the requencies or the extreme cases. Figure 0: Reerence training image. Black stries are two nodes wide and searated by 6 nodes in the north direction. -4

15 Figure : Two realizations that honor the original roortions, made considering a 4 by 4 nodes attern. Figure 2: Two realizations that honor the original roortions, made considering statistics extracted rom a 2 by 2 (let) and a 3 by 3 nodes attern (right). -5

16 Figure 3: Three realizations considering the statistics rom a 3 by 3 nodes attern and scaled roortion o 30% or the black acies and 70% or the white background. Figure 4: Three realizations with multile-oint statistics scaled to 50% black acies and 50% white background, with increasing attern size: 2 by 2 (let), 3 by 3 (middle), and 4 by 4 (right). -6