Rotation and affinity invariance in multiple-pointgeostatistics

Tuanfeng Zhang

December, 2001

Abstract

Multiple-point stochastic simulation of facies spatial distribution requires a train-ing image depicting the geometric patterns deemed present in the study volume. Thattraining image is the equivalent of the variogram model used in traditional 2-pointstatistics-based simulation. Therefore the training image should be reasonably sta-tionary, that is invariant by any translation. However, even within a homogeneousformation, actual geological patterns may change gradually in space, e.g., in terms ofdirection and aspect ratio. Such location-dependent patterns can be simulated usinglocation-dependent linear transforms (rotation + affinity) of the stationary patternsread from the training image.

The linear transform of training patterns is discussed in this report and its appli-cation to simulation of non-stationary fan deposits is shown.

1 Introduction

Training images as source of structured data are taking an important role in geostatis-tics. Under the useful decision of stationarity, multiple-point statistics can be liftedfrom a training image and used to model a target area. The training image shouldreflect the patterns of spatial variability deemed to prevail over the target area. Thesepatterns must be repeated often enough over the training image so that they can becaptured by multiple-point statistics.

Statistics, no matter their order and the number of locations (multiple points) theyinvolve, can only represent stationary features or patterns; they reflect average pat-terns not location-specific unique features. Geostatistics, as any other spatial statis-tics, can only capture stationary patterns from a training image and can only modeland simulate these stationary patterns. The specific locations and local transforms

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of these stationary patterns can only be obtained through local data: this is the taskof conditioning and the essential difference between “non-conditional” simulationwhich can only reproduce stationary patterns and “conditional” simulation which, inaddition, also aims at locating them in space. Any unique (non repeated) pattern,such as a single fracture, is either detected by local data and can be deterministicallyreproduced, or it is not and no statistics could ever model it.

A training image, that is a representation of repeated spatial patterns deemedpresent over the target area, need not and should not carry any location-specific char-acteristic. Training patterns are defined relative to each other and their statisticsshould be independent of the coordinate system used. More precisely, they shouldbe independent of any linear transform of the coordinates. Training images can beadapted to the target area by a series of local rotations and affinity of the coordinatessystem, much like an isotropic variogram model can be made location-specific by asuccession of rotations and affinity transforms of the coordinates, Xu (1996).

Consider the training image (TI) of Figure 1a which features EW elongated chan-nels with little varying NS width. This TI can be used to model channels of anydirection, see Figures 1 b-c-d, and of any width, see Figures 2 and 3. The train-ing patterns taken from the TI are made rotation and scale-invariant, more preciselyindependent of any linear transform of the coordinates system.

The impact of stationarity of statistics on stochastic simulation is now investi-gated.

2 Multiple-point statistics

Let fS(u);u 2 D)g be the stationary random function modeling the spatial distri-bution of an attribute s(u) over an reservoir D. If S(u) has a finite number of out-comes: s1; s2; :::; sK and a conditional data set fs(u�);� = 1; 2; :::; ng is available,the probability of the variable S(u) at any unsampled location u can be obtainedby considering the correlation between the single-point event S(u) and the knownmultiple-point (mp) data event B = fS(u�);� = 1; 2; :::; ng. The outcome of themultiple-data event B conditions the probability of the single-point event S(u). Themp data is considered as a whole instead of being considered one datum location ata time. Such mp data allows capturing patterns within the data set. This is differentfrom the traditional two-point geostatistics approach, in which only the correlationbetween any two points is considered to construct the probability of S(u). Tradi-tional geostatistics only captures two-point correlation characteristics of the reser-voir attributes. In contrast, multiple-point statistics can model curvilinear and largescale structures such as undulating channels.

Consider the binary random variable IB associated with occurrence of the data

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event B:

IB =

(1 if B occurs0 if not

IB is called an event indicator variable. If the event B involves a single pointu in space, IB is the ordinary indicator variable usually denoted by I(u). If B is amultiple-point event, IB is then an indicator of the specific pattern constituted by allsingle data constituting B.

Consider now the two events Ak = fS(u) = skg and B = fS(u�);� =1; 2; :::; ng. Simple kriging can be used to estimate the probability of the unknownevent Ak given that data event B has occurred:

P (AkjB) = E(IAk = 1jiB) = E(IAk) + �[1�E(B)] (1)

A single normal equation provides the weight �:

�V arfBg = CovfAk; Bg (2)

It is easy to check that equations (1) and (2) exactly identify Bayes’ relation,Strebelle (2000):

P (AkjB) =P (Ak; B)

P (B)(3)

The equivalent relations (1) and (3) show that inference of the conditional prob-ability P (AkjB) requires knowing the probability of the joint occurrence of the twoevents Ak and B as well as that of B alone. In the usual situation of sparse data, it isnot possible to infer directly from actual data these two probabilities which are muchbeyond variograms and any other two-point statistics. An alternative is to identifythese probabilities to proportions read from an exhaustively sampled training image,assuming that this training image has the same multi-point statistics as the reservoirunder study.

An easy and fast way to store and then retrieve the proportions of occurrence ofany training pattern is provided by the concept of search tree Strebelle (2000). Thetraining image is scanned only once with a specified template and the correspondingproportions of pattern occurrence are stored. Once the search tree is created, theprobabilities attached to all patterns collected from the training image within thespecified template can be easily retrieved.

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3 Search Tree

The search tree is a powerful tool to store the number of replicates of any mp dataevent (i.e. pattern) found over the training image within the template retained. Re-lated mp probabilities are then identified to these training proportions and retrieved.The building of a search tree proceeds as follows:

� Specify the size and geometry of the template and order the corresponding gridnodes according to their distance to the center of the template. An anisotropicvariogram distance can be used. Consider a template � with n ordered nodes:

� = fh�;� = 1; 2; :::; ng

where h� = u� � u is the vector linking the center u of the template to the�-th node location u�.

� Scan the training image for all occurrences of any data event within the previ-ously specified template and any of its n sub-templates defined as:

�k = fhi; i = 1; 2; :::;Kg; K = 1; 2; :::; n

The smallest sub-template �0 corresponds to a single datum location at thecenter u of template � . The next sub-template �1 includes that central locationu and the next closest node u1. Finally:

�0 � �1 � �2 � ::: � �n = �

� Starting from the smallest �0, these sub-templates are centered at each nodeof the training image: the corresponding training data event is recorded and aproper count is incremented. A data event consists of the sub-template geom-etry and data values:

dk = fs(u+ hi) = si; i = 1; 2; :::; kg

Once scanning of the training image is completed, a search tree has been gener-ated. That tree stores the numbers of replicates of any data event found withinthe specified data template �

The search tree is divided into n + 1 levels, each level i corresponding to thesub-template �i; i = 1; 2; :::; n. The root of the search tree (level i = 0) stores thenumber of single-point occurrences of each category in the training image, that is theglobal proportion of each category. This is a single-point statistics. Level 1 of the

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search tree records the number of occurrences of each category at the second node ofthe sub-template �1. From these first two levels of the search tree, one can derive theprobability of occurrence of any data event within the sub-template �1. This is a two-point statistics. Similarly, one can extract three-point, four-point, ..., up to n-pointstatistics.

Note that not all potential n-point data configurations may exist in any specifictraining image, in such cases some branches of the search tree will stop before the(n + 1)th level. This feature significantly reduces the size of the search tree andRAM demand.

In a simulation using multi-point statistics, we need to infer the probability of aspecific event conditioned by specific data values arranged along a specific geometrictemplate. Denote this conditioning data event by:

d = fs(u+ hj) = sjjj = 1; 2; :::;mg

In most situations, this specific data template �m does not identify any one ofthe sub-templates �i used to generate the search tree. In such case, retain the largesttraining sub-template �i included in the conditioning data template �m, with �i � �m.

The probability associated to the conditioning data event d is then approximatedby that corresponding to that largest training sub-template �i.

4 Rotation and affinity transforms

The rotation and affinity (squeezing) transform of a training image is now discussed.These linear coordinates transforms are associated with the rotation and scale invari-ance of multiple-point statistics. At first, consider a transform with constant rotationangle and constant affinity factors. Next, a transform with location-dependent rota-tion angles and affinity factors is considered. Application is made to a channel fandeposit simulated from a training image with straight channels. All examples are2D, but the algorithm is easily extended to 3D. All simulations are non-conditionalto focus strictly on reproduction of the mp statistics.

4.1 Rotation

Let fT (u);u 2 DLg be the training image defined over the grid DL. Correspondingunconditional simulations should display a single main direction of continuity butpossibly different from that of the training image. Let � be the desired direction ofcontinuity measured in degrees clockwise from the continuity direction (East) of thetraining image. The simulated image is denoted by fS�(u)ju 2 Dg. The necessaryrotation can be implemented as follow:

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� Keep the original search tree. Rotate any conditioning data event by -� withinthe data template prior to looking for the number of replicates in the searchtree.

The coordinates rotation is written:

unew� = R�u

old� u = (x; y; z) (4)

where unew� and uold� are the old and new coordinates of each single datum com-ponent � in a data event. R� is the rotation matrix.

In 2D with u = (x; y), this is written:(xnew� = xold� cos� � yold� sin�

ynew� = xold� sin� + yold� cos�(5)

Because the rotated location (xnew� ; ynew� ) does not usually fall on a grid node ofthe training image, it is relocated to the nearest grid node. Note that this relocationentails changing somewhat the original data geometry, see Appendix.

A 250 � 250 training image (Figure 1a) and a square data template with 7 � 7= 49 grid nodes are used. Only two facies are considered: channel in black withtraining proportion 0.3 and shale in white with proportion 0.7. Three rotation angles� = 0Æ; 45Æ, and 90Æ are considered. The corresponding unconditional realizationsof size 150�150 are displayed in Figures 1b to 1d. The realizations do reproduce thedesired direction of maximum continuity. The lack of full channel continuity acrossthe simulated area is not related to the rotations done. It is a consequence of using atemplate of limited size (7� 7)

4.2 Affinity

We can also squeeze the given training image (an affinity transform) to change theaspect ratio of its geometric patterns. In a fluvial sedimentary environment, thisamounts to change the width and sinuosity of the training image channels.

Two affinity factors ax, ay along the two desired major and minor directionsof continuity control the magnitude of the affinity. If the channel elongation in thetraining image is horizontal (x-direction), then ay > 1 leads to thinner channels andax > 1 leads to more sinuous channels.The corresponding coordinates transform iswritten:

unew� = Au

old� u = (x; y; z) (6)

where unew� and uold� are the old and new coordinates of each single datum loca-tion � of the mp data event. A is the affinity matrix:

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A =

0B@ ax 0 0

0 ay 00 0 az

1CA

In 2D with u = (x; y), this is written:

xnew� = axxold�

ynew� = ayyold�

(7)

Using the 250 � 250 training image of Figure 1a, the following affinity factorswere considered: ax =1.0, ay = 2.0, 3.0, 4.0, 5.0. The corresponding four realizationsare displayed in Figure 2. The simulated channels are increasingly thinner as ayincreases.

The four realizations corresponding to ax =1.0, ay = 0.9, 0.8, 0.7, 0.6, are dis-played in Figure 3. The simulated channels are increasingly wider as ay decreases.

If ay is fixed, increasing ax leads to increase channel sinuosity, see Figure 4.

4.3 Full linear transform

We can simultaneously rotate and squeeze the patterns present in the training image.The corresponding coordinate transform is written:

unew� = AR�u

old� u = (x; y; z) (8)

where unew� and uold� are the old and new coordinates of each single datum loca-tion � of the mp data event. R� and A are the rotation and affinity matrices.

In 2D, this is written:(xnew� = ax[x

old� cos� � yold� sin�]

ynew� = ay[xold� sin� + yold� cos�]

(9)

Figure 5a shows the sequence of rotation and affinity.The simulated realization, corresponding to a 30Æ rotation angle and affinity fac-

tors ax = 1:0, ay = 3:0, is displayed in Figure 5b. That realization is similar to thatof Figure 2b except for the 30Æ rotation. With a 45Æ rotation and ax = ay = 3:0, therealization is displayed in Figure 5c; it is seen to be similar to that of Figure 4b.

4.4 Location-dependent transform

Stationarity is a statistics requirement rarely exactly met by natural phenomena. In-stead, geological patterns typically evolve gradually from one area to another; this

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gradual deformation can be modeled by a series of location-dependent linear trans-forms (rotations and affinities). If these location-dependent transforms can be evalu-ated, e.g. from seismic data, the stationary channel patterns of the training image canbe gradually transformed to deliver the locally variable patterns desired. For exam-ple, a fan deposit may be seen as consisting of three parts: upper fan, middle fan andlower fan. A main feeder channel (or a few feed channels) split and produce manysmaller braided channels as they move forward from the upper fan to the lower fan.Hence, the channels tend to deviate from the feeder direction and be increasinglysmaller. The location-dependent transform is written similarly to expression (8):

unew� = A(u0)R�(u0)u

old� u = (x; y; z) (10)

where unew� and uold� are the old and new coordinates of each single datum loca-tion � of the mp data event; u0 is the center of that data event; R�(u0) and A(u0)are the rotation and affinity matrices which are location u0 dependent.

In 2D, this is can be developed as:(xnew� (u) = ax(u0)[x

old� cos(�(u0))� yold� sin(�(u0))]

ynew� (u) = ay(u0)[xold� sin(�(u0)) + yold� cos(�(u0))]

Consider the simulation of a fan deposit in which the feeder channels are knownto move from northwest to southeast. A large stationary 500 � 500 training image,twice larger than the simulated grid size of 250� 250 is used, see Figure 6a. At eachsimulated grid node, a different set of rotation angle and affinity factors is used toimpose local characteristics of that fan deposit. The maps of local rotation angles andaffinity factors used are shown in Figures 6b and 6c. Figure 6d gives one resultingsimulated realization.

The format of the data file containing rotation angles and affinity factors as wellas the corresponding comments are given in Appendix of this report.

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Assume the following notation:

� A is the binary (indicator) random variable associated to the occurrence of statesk at location u.

� B is the binary random variable associated to the occurrence of the data eventconstituted by the n hard conditioning data S(u+ h�) = sk� , � = 1; 2; :::; n,considered jointly.

� C is the binary random variable associated to the occurrence of the data eventconstituted by the nY soft conditioning data Y (u+h

0

�) = y� , � = 1; 2; :::; nY ,considered jointly.

If seismic information can be calibrated to provide the partial conditional probabilityP (AjC), it can be combined with the other partial probability P (AjB) obtainedfrom hard data to give the full conditional probability P (AjB;C), Journel(2000).Two realizations corresponding to different combinations of P (AjB) and P (AjC)are shown in Figure 7; the second (Figure 7b) gives more influence to the seismiccomponent P (AjC).

A more complex example is shown in Figure 8. In this case, a composite of fandeposits is considered in which the continuity direction and width of the channelsare different in different lobes of the composite fan. The spatial distributions oflocal rotation angles and affinity factors are given in Figure 8b and 8c. The resultingsimulated realization is displayed in Figure 8d.

4.5 Non-stationary examples

A non-stationary representation of a delta fan cannot be used as a training imagebecause it is too location-specific and its patterns are not repeated over the train-ing area. Figure 9 shows one such non-stationary training image of a delta fan andone resulting simulated realization. The characteristics of the original training deltafan are not reproduced. Indeed, if a non-stationary training image is scanned, thelocation-dependent patterns of this training image are averaged out by the multiple-point statistics and results in the “average” stationary patterns seen on the simulatedrealization. This fundamental remark is true for all statistics including the variogrammodels of 2-point geostatistics.

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5 Preliminary conclusions

From the above discussion and preliminary test runs, the following conclusions canbe made:

� Training patterns should be stationary with rotation and affinity invariance.

� Location-specific patterns associated to a specific reservoir can be reproducedthrough coordinates linear transform (rotation and affinity). The multiple-pointstatistics can be read from a modular stationary training image.

� Very different facies distributions, although all related to the same geologicalsedimentary style, can be generated from one single modular training image.

Reference

[1] S. Strebelle., Sequential simulation drawing structures from training images.SCRF report, 2000, and PhD thesis.

[2] W. Xu., Conditional curvilinear stochastic simulation using pixel-based algo-rithms. Mathematical Geology, 28(7).

[3] C.V.Deutsch and A.G.Journel., GSLIB - Geostatistical Software Library andUser’s Guide. Oxford University Press, 1992

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a- Training image

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Figure 1: Training image (a) and simulated realizations with different rotation angles (b),(c), and (d)

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a- Realization with ax=1.0, ay=2.0

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b- Realization with ax=1.0, ay=3.0

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Figure 2: Realizations with decreasing channel width (ay > 1)

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a- Realization with ax=1.0, ay=0.9

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b- Realization with ax=1.0, ay=0.8

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c- Realization with ax=1.0, ay=0.7

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d- Realization with ax=1.0, ay=0.6

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Figure 3: Realizations with increasing channel width (ay < 1)

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a- Realization with ax=2.0, ay=3.0

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b- Realization with ax=3.0, ay=3.0

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c- Realization with ax=4.0, ay=3.0

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Figure 4: Realizations with increasing channel sinuosity (ax > 1)

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1u

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Figure 5: Realizations with different rotation angles and affinity factors

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a- Large training image

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Figure 6: Training image for a delta fan (a), maps of rotation angles (b) and affinity factors(c), one simulated realization (d).

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a- Large training image

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Figure 7: Training image (a), seismic data given as P(AjC) (b), and two realizations withdifferent emphasis given to the C-seismic data (c) and (d).

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a- Large training image

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Figure 8: Training image (a), rotation angles (b), affinity factors (c), and one realization(d).

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Training image

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Figure 9: Non-stationary delta fan used as a training image and one resulting simulatedrealization

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