A New Approach to Direct Sequential Simulation that Accounts for the Proportional Effect: Direct Lognormal Simulation

Transcription

1 A ew Approach to Direct eqential imlation that Acconts for the Proportional ffect: Direct ognormal imlation John Manchk, Oy eangthong and Clayton Detsch Department of Civil & nvironmental ngineering University of Alberta Abstract Direct kriging and simlation avoids the common assian transform and permits nbiased integration of mltiscale data. Crrently, kriging is performed on assian data since all conditional distribtions are assian and flly described by the kriging estimate and homoscedastic kriging variance. Application of kriging to data in original nits will almost certainly lead to a variance that is incorrect since real data exhibit heteroscedastic featres, that is, a proportional effect. This paper shows that the naïve form of direct seqential simlation, where the kriging variance is taken as homoscedastic, leads to poor reslts. A method of D is developed that corrects the kriging variance to accont for the proportional effect. A mltivariate lognormal distribtion is developed since it has an analytical model of the proportional effect. Introdction Direct seqential simlation D [,, 3] has been applied becase of its ability to accont for data of varios spport volmes and poplate nstrctred grids with data at different spport. Kriging with the available data in original nits is the essential idea of D. Althogh kriging provides a valid estimate and variance for a conditional distribtion, the reslting homoscedastic variance poses a significant problem when original data nits are considered; the ncertainty in low-valed areas is over stated and the ncertainty in high-valed areas is nderstated. Real data often exhibit a classical heteroscedastic relationship between the local mean and variance, commonly referred to as the proportional effect [4]. With kriging as the main engine in D, the reslting simlated vales do not reprodce a heteroscedastic featre; a method mst be developed to accont for the proportional effect inherent in original data nits. imple kriging K is important in D/simlation becase of its ability to reprodce the covariance even if the conditional distribtions are not assian []. Covariance reprodction sing K can be easily demonstrated Appendix B. Reprodction of the covariance only holds if the variance of the data is homoscedastic, bt in the case of lognormal data the variance is clearly heteroscedastic. This paper proposes a soltion to the homoscedastic kriging variance problem of D. This is a particlarly interesting case since the mathematical relationship between the lognormal and the commonly sed normal distribtion is well known, as are the eqations that describe the proportional effect of lognormal data [5]. Knowing these relations, the kriging variance can be calibrated to honor the heteroscedasticity inherent in lognormal data. This well posed case provides valable insight into the natre of D. 04-

2 imlation is commonly performed sing data that has ndergone a assian/normal transform and simlated vales are then back-transformed to original nits to prodce a set of realizations. In normal space, all local distribtions are also normal and flly defined by the local mean and the kriging variance; however, in original space this is not the case. For direct simlation, the local distribtions mst be determined in original space -space. An example of lognormal data will be sed; below we will see how lognormal data provides a very good fit to real data. Theoretical Backgrond A key reqirement in the derivation of /D see Appendix B is that the variance be independent of the conditional mean. The lognormal model is niqe in that this reqirement is not met; it is a different model. In order to move from simlation in normal space to simlation in original space, an idea of how the two methods are linked wold be sefl. Many natral variables have an approximately lognormal histogram. There is an analytical relationship between normal and lognormal space providing a method to transform data between the two. Transformations involve the distribtions, the data vales, and the variograms. All transformations are analytical. ormal to ognormal Transformation qations exist that describe a lognormal distribtion and its relation to a normal distribtion. ince these eqations are known, applying direct kriging on lognormal data is possible. A random variable, z>0, is lognormal with a mean m and standard deviation if the natral logarithm of, X ln is normally distribted with mean and standard deviation. Knowing the relation between logm, and X,, one can transform a assian distribtion, Y ~ 0,, into a lognormal distribtion. qations and show the relation between X, Y, and, where Y is a standard normal distribtion. qations 3 and 4 show the relation between m and of the lognormal with and of the normal distribtion. X Y bstitting qation into qation. yields qation : Y e X e. ln m 3 ln 4 m qations 5 and 6 describe the normal and lognormal probability distribtion crves, which are qite similar in arrangement. Figre shows the change in the distribtion shapes as Y is converted into X and as X is transformed into. 04-

3 ln x exp f x 5 x π x µ exp g x 6 π The remaining theoretical relationships are easier to appreciate when referenced to the model and data sed for application of direct seqential simlation. For this prpose, an nconditional assian model was initially prodced with dimensions of 56 by data points sing a spherical variogram with zero ngget effect and a range of 3. The simlated assian vales were then transformed sing qations to 4 to give a lognormal model with arbitrarily chosen mean and standard deviation of 00 Figre shows the reslting nconditional models. From this model, 65 vales were sampled and sed for kriging, see Figre. Variogram Transformation If the variogram in assian space is known, it can be converted to the variogram in lognormal space throgh the se of qation 7. This actally relates the correlograms, bt these are easily converted to the variogram as shown with qation 7. m h h e r ρ 7 γ ρ h, rh γ Y where rh is the correlation in Y-space and ρh is the correlation in -space. Figre 3 shows the variogram sed to create the nconditional model along with the theoretical variogram for the reslting lognormal model. The difference between the variograms becomes somewhat significant as half the range is approached. Inference of Conditional Distribtions Or goal is to calclate a conditional distribtion conditional to some nmber of local data for simlation. In a mltivariate assian case we transform the data, infer the parameters in assian nits, and then back transform the reslt. The transform and back transform are particlarly easy when the data are lognormal. In fact, the shapes of all conditional distribtions in original nits are lognormal when the original global histogram is lognormal see Appendix A. The classic approach consists of kriging in assian nits; then, the distribtion in original nits is inferred. The key idea of D is to krige in original nits, bt we mst establish the correct variance, which is heteroscedastic, that is it depends on the magnitde of the data and estimate. 04-3

4 The heteroscedasticity is atomatically acconted for in the back transform. There is no back transform in D. We wold have to bild in some form of correction. Homoscedastic Variance Correction We cold imagine a linear estimate that is heteroscedastic e.g. lognormal. The kriging variance is often assmed to be homoscedastic, which is inconsistent with data that exhibit a proportional effect. With lognormal data, an eqation exists for correcting the variance sing the mean or estimate and it can be derived from qation 4: ln m e m e m where is the homoscedastic kriging variance in X nits and variance from kriging original data. m and are the estimate and ince m is the estimate from kriging it can be denoted by z. From qation A in Appendix A it is shown that the local vale can be determined from the kriging variance in Y-space and the global vale the global variance in X nits. bstitting these reslts into the eqation arrived at above yields qation 8: Y [ z ] e 8, C Where, C is the corrected variance, is the local variance in normal space, and is the Y global variance of ln. To experimentally show this relation between assian data and its lognormal transform, kriging was performed on both the normal and lognormal data and the AM program [6] was rn on the reslts at a specified lag half the range was sed and the Yh vales were extracted. plitting the reslts into 50 qantiles and determining the mean and standard deviation of each qantile shows that the variance is homoscedastic for assian data and the variance depends on the mean with lognormal data. For comparison, the analytical lines were plotted on Figre 4. A major implication from qation 8 is that kriging wold have to be performed twice; once to get the kriging variance in assian space and again to get the estimate in lognormal space Y z. To avoid this, a relation was fit to the ncorrected -space kriging variance and the Y- space kriging variance. For this particlar data set, the reslting kriging variance vales are shown in Figre 5 and fitted sing qation 9. The fit is very good with an R-sqared vale of and a standard error of estimate eqal to

5 Y Implementation of Direct imlation To provide a set of reslts for comparing direct kriging and simlation, the crrent method of simlation sing IM was initially applied. By setting p a simple example in xcel, it was shown that the local distribtions are in fact lognormal so the same eqation sed to transform the normal model to a lognormal model as in Figre cold be sed to back transform assian vales to lognormal vales in IM. This reslt also allows the development of eqations to relate the local assian distribtions to the local lognormal distribtions. qations 0 and show how the local assian mean and variance can be transformed to the local alpha and beta vales for se in qations and 3, which are sed to determine the local lognormal mean and variance. ee Appendix A for the kriging example in xcel and derivations of qations 0 and. where m is the local normal mean, ln. where m 0 is the global mean of ln, and is the local normal variance and is the local mean of m e is the local variance of ln. m e 3 Three options of simlation were explored: Option Option Option 3 Transform a set of lognormal samples to normal space and perform kriging and MC, then back-transform to lognormal space. This is the standard/common approach. The limitation is that mltiscale data are not easily handled. Perform direct kriging with the lognormal vales with an adjsted variogram and do MC withot correcting the kriging variance. This is the pblished approach to D. The limitation is that heteroscedasticity/the proportional effect is not acconted for. Perform direct kriging on the lognormal vales with an adjsted variogram and correct the kriging variance prior to MC. This is the new approach that we are advocating in this paper. Mltiscale data can be sed in direct kriging and the proportional effect is explicitly acconted for. 04-5

6 For each option, 00 conditional realizations were prodced and the conditional mean and variance at every location was determined. Figre 6 shows the simlation reslts for all three options. To check the validity of each simlation method, reprodction of the global statistics as well as the variogram were checked. These are shown in Figre 7. In Figre 6, it is obvios that option 3 compares closely to the conventional simlation method option ; however, performing a naïve direct simlation as in option reslts in a variance with little flctation throgh ot the map Figre 6. Option shows that performing direct simlation, with no variance correction, tends to hide the proportional effect inherent in lognormal data. Regardless of this incorrect variance, all three options shold theoretically reprodce the mean, variance and variogram as shown by Figre 7. It seems that with direct simlation of lognormal data, the estimated mean vale is higher than that fond sing conventional assian simlation. If the mean is higher, qation 8 will force the variance at that location to be higher as well. These problems may lead to overestimating an area, however the differences are minimal in this case. For additional comparison, three points were chosen from the option reslts; one with low mean and low variance point ; one with high mean and low variance point ; and one with high mean and high variance point 3. For each point, a local distribtion was prodced by extracting the simlated vale from the 00 realizations. Figre 8 along with a smmary table and chart shows the distribtion for each point and for each simlation option. From Figre 8, it is clear that the distribtions for options and 3 are similar; however, the variance for option does not compare. Over the three locations, the variance for option remains relatively constant. This indicates that the proportional effect has been removed and the variance is homoscedastic as wold be expected with assian data. To check how the proportional effect was honored for all three options, the kriged mean and variance was extracted at all locations and plots of the mean verss the standard deviation were prodced and are shown in Figre 9. Conclsion The proportional effect is a common characteristic for many data sets and cannot be reprodced with kriging alone since the reslting variance is homoscedastic and mst be corrected to reprodce the proportional effect. The lognormal distribtion is closely related to the assian/normal distribtion, bt it has an analytically defined proportional effect. The reslts of conventional simlation and direct seqential simlation compare well with minor differences in the overall mean and variance between the realizations. Also, the local distribtions at varios locations closely resembled one another. These findings provide insight into a possible soltion to D when dealing with the proportional effect and lognormal data. It is apparent that the kriging variance mst be corrected sch that the proportional effect is reprodced with direct simlation. By imposing a correction to the kriging variance prior to simlation, it is possible to perform direct seqential simlation of lognormal data with correct/reasonable reslts. Correcting the proportional effect in this manner imparts a dependency between the estimate and the variance that is in direct contradiction to the simple kriging principle, which reqires independence between the estimate and the variance. The lognormal model is a trly different approach to direct simlation. 04-6

7 References [] W.X and A.. Jornel, DIM: A eneral eqential imlation Algorithm, tanford Centre for Reservoir Forecasting, tanford University. [] A. oares, Direct seqential simlation and cosimlation, Mathematical eology 33 8, 9 96, 000 [3] J. Caers, Adding local accracy to direct seqential simlation, Mathematical eology 3 7, , 000a [4] B.Oz and C.V. Detsch, A hort ote on the Proportional ffect and Direct eqential imlation, Centre for Comptational eostatistics, Report 4: 00/00. [5] A.. Jornel and Ch.J. Hijbregts, Mining eostatistics, Academic Press, 978, pp [6] C.V. Detsch and A.. Jornel, eostatistical oftware ibrary and User s ide, Oxford University Press, second edition, 998, pp [7] A. Jornel, on-parametric stimation of patial Distribtions, Math eology, 53:445-68,

8 ormal and ognormal Distribtion Transition ormal 0,} ormal, } fx ognormal m, } Figre : ormal and corresponding lognormal distribtions. The lognormal distribtion was calclated from the assian distribtion by sing the transformation eqations to 4 above. The lognormal distribtion has a mean of 6 and a standard deviation of 3. x 04-8

9 Figre : assian model and corresponding lognormal model along with their distribtions. The set of 65 samples is also shown bottom. 04-9

10 ..0 γ ognormal assian Difference Distance Figre 3: Variogram model sed to generate the nconditional model. The assian model is spherical with no ngget effect and a range of 3. The corresponding variogram of the lognormal variable is shown with the difference between the two fnctions. 04-0

11 Figre 4: catterplots of Y verss Yh for assian pper left and lognormal pper right data. assian data showing the variance is homoscedastic lower left and ognormal data displaying the proportional effect lower right. The analytical line in the lower right plot was determined sing qation 8. Becase the global variance in Y-space is and is eqal to ln see qation 4 in normal,} space, the slope of the theoretical line on the lower right graph is one. 04-

12 Original Data with Regression Polynomial Y estimation variance estimation variance Figre 5: A fit of the estimation variance in -space verss the estimation variance in Y-space prior to correcting the lognormal kriging variance. Bllets represent data and the black crve displays qation

13 Figre 6: The mean and variance taken over 00 realizations for all three simlation approaches. Top option ; traditional method of data transformation prior to kriging and simlation, then back transformation to get lognormal reslts. Middle Option ; naïve direct simlation with no variance correction. Bottom Option 3; direct simlation of lognormal data with a variance correction to accont for the proportional effect. 04-3

14 Figre 7: A check for mean, variance and variogram reprodction for the 00 realizations. Top mean reprodction; both options and 3 reslt in a slightly higher mean than option. Middle variance reprodction; the three option compare well. Bottom variogram reprodction two directions shown; all options are close to following the analytical variogram model solid line. Bllets represent the average of the 00 variograms. 04-4

15 Options Points 3 mean st dev mean st dev mean st dev tandard Deviation Mean and tandard Deviation of 3 points Option Option Option Mean Figre 8: Distribtions of three points for simlation options. eft point ; low grade, low variance. Centre point ; high grade, low variance. Right point 3; high grade, high variance. A smmary table lower left and associated graph lower right of the mean and standard deviation at each location are also provide. 04-5

16 Figre 9: ocal mean verss standard deviation at every estimated location for options left, middle, and 3 right. Options and 3 show the proportional effect and compare nicely. Option shows a homoscedastic variance since no correction was applied. 04-6

17 Appendix A Kriging of assian Data and Determination of ocal Distribtions To check if the local distribtions at each location being estimated are lognormal, a simple kriging example was set p in xcel with 4 known data and 3 locations to be seqentially estimated, see Figre A. Y Data ocations Y Y Y X ocation m Y Y Y Figre A: Data configration for kriging top and kriging reslts prior to transformation to lognormal space bottom. olid bllets are known data and circles are the points to be estimated. The data vales are also shown, both assian above and lognormal below. To generate the local distribtions corresponding to the global lognormal data with a mean and standard deviation of 00, a set of 99 qantiles was chosen ranging from to and the vale corresponding to each for the local normal distribtions was fond. Using Y e to transform the vales to -space and plotting the reslts revealed that the local distribtions are lognormal. To check if eqations A and A below are correct, they were sed to find the local alpha and beta vales and then the OIV fnction in M xcel was sed to determine the corresponding vale for each qantile. Both methods gave eqal reslts, see Figre A. 04-7

18 .0 ocal ognormal Distribtions Freqency Vale Figre A: ocal ognormal Distribtions at each estimated location. Using both a transformation method with tables as well as an analytical method gave eqal reslts. Derivation of qations A and A tarting with the eqation for transforming assian data, Y ~ 0,, to normal X-space data, X,: X Y Replacing Y with the kriged mean and X with the local mean of ln, we get qation A, which relates the local mean in X-space to that in assian space. m A Where is the local mean in X-space, is the global mean of X, is the global variance of X, and m is the kriged mean in assian space. To derive qation A, the qation for transforming Y-space vales to X-space along with the eqation defining the variance of a data set was sed. The local normal kriging variance can be defined by the following eqation: n n i i m Where is the variance in assian space, i is the vale at location i, and m is the mean of all i, i n. n 04-8

19 To determine the local variance of ln we need to know the vales of ix and m X that correspond to i and m in assian space. qation A can be sed to perform this transformation: m ix i m X bstitting these into the eqation for the variance in normal space X-space, the local variance of ln can be solved for: n ix mx n i n i m n i n n i i [ m ] n i m n i A Where is the local variance of ln, normal variance in Y-space. is the global variance of ln, and is the local 04-9

20 Appendix B The imple Kriging Principle imple kriging is the key to direct seqential simlation de to the property of covariance reprodction even if the conditional probability distribtions are not assian. Reprodcing the covariance only holds if the conditional variance is independent of the data vales homoscedastic. A proof of the covariance reprodction is provided with the following assmptions: the data stationary variable z has a mean and variance of 0 and, respectively. The conditional distribtions are flly described by the kriging mean and variance. qations B, B, and B3 describe the kriging mean and variance along with the simple kriging system considering previos data: K z λ z B λ ρ B λ ρ ρ,,..., B3 Where z is the simple kriging mean, K is the simple kriging variance, and λ,,, are the kriging weights. A random vale R can be drawn from a distribtion described by a mean of zero and a variance eqal to the kriging variance K. The kriged mean and R are added together to get the simlated vale for the location,. An important aspect of R is that its vale is chosen independent of the mean. R B4 ow that one location has been simlated, there are data vales for simlation of the next node which will be denoted. The simple kriging mean and variance at are given by qations B5 and B6 along with the kriging system shown in qations B7 and B8: ' λ ρ ' λ ' λ z λ s B5 K ρ ' B6 λ ρ λ ρ ρ ',,..., B7 04-0

21 04- ' ρ λ ρ λ B8 Where and ' K are the simple kriging mean and variance at location respectively. ote that the weights λ,,, are not the same as the weights λ,,, in qation B to B3. Once and ' K are known, a random vale R can be drawn from a distribtion with a mean of zero and a variance eqal to ' K. The simlated vale at is calclated as follows: ' ' ' R B9 et s calclate the covariance between the two simlated vales: '} } ' '} '} '} R R R R B0 Where } ' R and } ' R R are zero since and ' R are independent of each other and R and ' R are also independent. The remaining portions of the right hand side are non zero since the kriged means depend on one another and also becase the kriged mean at the second location depends on the randomly drawn vale at the first location. } ' '} '} R B xpanding and simplifying the first term from the right hand side of qation B: } } ' λ λ ρ λ λ B ow, expanding the first term of qation B and recalling qation B3: ρ λ ρ λ λ B3 xpanding the second term of qation B: } } } R B4

22 Knowing that R } 0 and } ρ : λ } λ ρ K B5 bstitting B3 and B5 back into B: ' [ ] } λ ρ λ K B6 ow, let s go back to qation B and expand and simplify the second term of the right hand side: ' R } λ R } λ R } B7 The λ R } term is zero since and R are independent. By expanding the R } portion of the second term in qation B7, it can be shown that it is eqivalent to the simple kriging variance: R } R } R } B8 ince R s } is zero the variance is homoscedastic. If the variance was heteroscedastic, R } 0 and the covariance wold not be reprodced: s R bstitting B9 back into qation B7: bstitting B6 and B0 into B and simplifying: '} λ s } R } B9 K ' R } λ K B0 ρ λ [ ] λ K '} λ ρ λ '} ρ ' B By working throgh from qation B to qation B, the covariance is correct. The marginal covariance is reprodced. K 04-