Direct Sequential Co-simulation with Joint Probability Distributions

Transcription

1 Math Geosci (2010) 42: DOI /s x Direct Sequential Co-simulation with Joint Probability Distributions Ana Horta Amílcar Soares Received: 13 May 2009 / Accepted: 3 January 2010 / Published online: 5 February 2010 International Association for Mathematical Geosciences 2010 Abstract The practice of stochastic simulation for different environmental and earth sciences applications creates new theoretical problems that motivate the improvement of existing algorithms. In this context, we present the implementation of a new version of the direct sequential co-simulation (Co-DSS) algorithm. This new approach, titled Co-DSS with joint probability distributions, intends to solve the problem of mismatch between co-simulation results and experimental data, i.e. when the final biplot of simulated values does not respect the experimental relation known for the original data values. This situation occurs mostly in the beginning of the simulation process. To solve this issue, the new co-simulation algorithm, applied to a pair of covariates Z 1 (x) and Z 2 (x), proposes to resample Z 2 (x) from the joint distribution F(z 1,z 2 ) or, more precisely, from the conditional distribution of Z 2 (x 0 ), at a location x 0, given the previously simulated value z (l) 1 (x 0) (F(Z 2 Z 1 = z (l) 1 (x 0)). The work developed demonstrates that Co-DSS with joint probability distributions reproduces the experimental bivariate cdf and, consequently, the conditional distributions, even when the correlation coefficient between the covariates is low. Keywords Geostatistics Joint sequential simulation Bivariate distribution 1 Introduction The use of geostatistical models based on stochastic simulation algorithms is a reliable option for addressing problems in environmental and earth sciences if the purpose is to assess the spatial distribution of a certain attribute as well as spatial uncertainty. A. Horta ( ) A. Soares Center for Natural Resources and Environment, Instituto Superior Técnico, Technical University of Lisbon, Lisbon, Lisbon, Portugal ahorta@ist.utl.pt

2 270 Math Geosci (2010) 42: To reproduce the spatial relation of more than one interdependent attribute or variable, joint simulation algorithms have been proposed for different research fields: environment (Goovaerts 1997), petroleum (Caers 2005; Deutsch 2002), and mining (Journel and Kyriakidis 2004). To reproduce joint spatial patterns (given by covariograms and bi-histograms), the following simulation algorithms can be used: sequential multi-gaussian co-simulation (Verly 1993), multi-gaussian co-simulation with collocated co-kriging (Almeida and Journel 1994), co-simulation with LU decomposition method (Myers 1988), and simulation of autocorrelation factors (Desbarats and Dimitrakopoulos 2000; Boucher and Dimitrakopoulos 2009). However, for most practical applications, the common stochastic co-simulation algorithm applied to a set of correlated variables is based on a sequential approach where each variable is simulated in turn with the other previously simulated variables, as secondary information (Almeida and Journel 1994). This is the principle for the sequential Gaussian co-simulation (Almeida and Journel 1994) and the direct sequential co-simulation (Soares 2001). The main advantages of using the sequential approach are its simplicity, efficiency, and the possibility of choosing which variable will condition the next covariate. Normally, this variable (called the primary) is chosen due to its evident spatial continuity or given its relevance for the case study (Almeida and Journel 1994). Let us consider the direct co-simulation (Soares 2001) of just two variables, Z 1 (x) and Z 2 (x), where Z 1 (x) is the primary variable. This joint simulation algorithm can be summarized in two steps. First, the simulation of the value z (l) 1 (x u) (where x u is a node in a regular grid) using the direct sequential simulation (DSS) algorithm (Soares 2001). Second, assuming the previously simulated Z 1 (x) as the secondary variable, the DSS algorithm is again applied to simulate Z 2 (x). Collocated simple co-kriging is used to calculate z 2 (x u ) and σsk 2 (x u) conditioned to nearby data z 2 (x α ) and the collocated datum z 1 (x u ) (Goovaerts 1997). By using direct sequential co-simulation (Co-DSS), we perform the joint simulation of a set of N v variables without any prior or subsequent transformation. Also, the method ensures the reproduction of marginal cdfs (F 1 (z 1 ) and F 2 (z 2 )), variograms (γ 1 (h) and γ 2 (h)) and the joint spatial pattern characterized by the covariograms γ 1,2 (h). The use of Co-DSS in different case studies revealed that, for some situations, the method did not reproduce the bivariate cdf, i.e. the experimental bi-histogram was not respected. As an example of this practical fact, see the case study presented by Soares (2005). Figure 1 shows the experimental scatterplot of two variables, copper (Z 1 (x)) and tin (Z 2 (x)), from a sulfide copper deposit. Figure 2 presents the resulting scatterplot of co-simulated values. By comparing real and simulated biplots, we notice that high simulated values of Z 2 (x) are collocated to low values of Z 1 (x), unlike the experimental data plot. Nevertheless, the joint simulation method ensures the reproduction of the variograms of Z 1 (x) and Z 2 (X), the marginal histograms and the correlation coefficient. Hence, for a particular class of values of Z 1 (x) = z, the simulated values z (l) 2 (x) must not exceed the higher and lower limits of the experimental data class to which z (l) 1 (x) belongs, and should reproduce the experimental conditional histogram of Z 2 (x), givenz 1 (x) = z. This can be considered a limitation of the existing sequential co-simulation algorithm (including sequential Gaussian co-simulation Co-SGS and direct sequential co-simulation). For those methods, after estimating the conditional mean and variance by simple kriging and

3 Math Geosci (2010) 42: Fig. 1 Experimental biplot (Sn vs. Cu) Fig. 2 Simulation biplot using usual Co-DSS (co-simulated Sn vs. simulated Cu)

4 272 Math Geosci (2010) 42: simple collocated co-kriging, z (l) 1 (x) and z(l) 2 (x) are drawn, respectively, from its local Gaussian transform (Co-SGS) or from the marginal global distributions (Co-DSS). Although the marginal histograms of Z 1 (x) and Z 2 (x) are reproduced (as required for both methods), high conditional variances, common at the very beginning of the sequential simulation process, can produce simulated values that do not respect the conditional histograms. This can lead to erroneous conclusions, mostly when nonlinear cost functions (used in environmental and mining applications) are applied to the set of co-simulated values. In this paper, we propose a new approach for Co-DSS based on the idea of resampling Z 2 (x) from the joint distribution F(z 1,z 2 ) or, more precisely, from the conditional distribution of Z 2 (x 0 ), at a location x 0, given the previously simulated value z (l) 1 (x 0) (F(Z 2 Z 1 = z (l) 1 (x 0)). This method succeeds in reproducing the experimental bivariate cdf and, consequently, the conditional distributions. 2 Direct Sequential Co-simulation with Joint Probability Distributions 2.1 Methodology The problem posed by the implementation of the Co-DSS algorithm consists in the fact that the final bi-distribution of simulated and co-simulated values does not reproduce the same experimental distribution of the original data values. Hence a conditional distributions F [Z 2 (x) Z 1 (x)] are not necessarily reproduced at the end of the simulation. The direct sequential co-simulation procedure is based on the application of the Bayes rule in a successive sequence of steps. A first variable Z 1 (x), chosen as the most important for the case study or with the most continuous spatial pattern, is simulated for the entire area. The second variable Z 2 (x) is then co-simulated taking into account the experimental data z 2 (x α ) and previously simulated values z (l) 1 (x) as secondary data. At a given location x 0 of the sequential simulation path, we estimate the local mean and variance using collocated co-kriging, conditioned to the experimental data z 2 (x α ) and previously simulated data z (l) 1 (x 0). Based on the local mean and variance, the simulated value (z (l) 2 (x 0)) is resampled from the global cdf F [Z 2 (x)]. If the local estimated variance (σ 2 [Z 2 (x 0 ) Z 2 (x α ), Z (l) 1 (x 0)]) is sufficiently high in the beginning of simulation process, most probably the values z (l) 2 (x 0), drawn from the global cdf F [Z 2 (x)], will be outside of the boundaries determined by the conditional distribution F [Z 2 (x) Z 1 (x)]. Consequently, the bi-histograms of simulated values (z (l) 2 (x), z(l) 1 (x)) will have higher conditional variances than the corresponding estimates with experimental data, with the same correlation coefficient and marginal histograms. The solution proposed in this paper is based on the use of conditional distributions (estimated from experimental data) to simulate Z 2 (x), given the known Z 1 (x): Z 2 (x) is simulated from F [Z 2 (x) Z 1 (x) = Z (l) 1 (x)], and the known realiza- (x) as secondary data, the conditional tion z (l) 1 (x) at each location x. Assuming Z(l) 1

5 Math Geosci (2010) 42: mean is computed for every location x 0, identified with the simple collocated cokriging estimator (Z 2 (x 0 ) ) [ z2 (x 0 ) ] sck = α λ α,x0 [ z2 (x α ) m 2 ] + λx0 [ z1 (x 0 ) m 1 ] + m2 (1) where λ is the co-kriging weights, z 2 (x α ) is the experimental conditioning data, z 1 (x 0 ) is the collocated datum, and m 1 and m 2 are the means of z 1 and z 2 data, respectively. Also, the conditional variance corresponds to the simple collocated cokriging variance. The value z (l) 2 (x 0) is drawn from the conditional distribution F [ Z 2 (x 0 ) Z1 (x 0 ) = z1 l (x 0) ] = prob { Z 2 (x 0 )<z Z1 (x 0 ) = z (l) 1 (x 0) } K (2) rather than from the global cdf of Z 2 (x) as in the usual Co-DSS. The proposed algorithm continues to ensure that simulated realizations of Z 1 (x) and Z 2 (x) reproduce the variograms (γ 1 (h) and γ 2 (h)), the marginal cdfs (F(Z 1 (x) and F(Z 2 (x)) and the conditional cdf (F(Z 2 (x) Z 1 (x)). The proposed methodology of Co-DSS with joint probability distributions can be summarized in the following sequence, beginning with the estimation of the global bi-distribution from experimental data, followed by the simulation of the first covariate Z 1 (x) using direct sequential simulation. Realizations of Z (l) 1 (x) reproduce the variogram γ 1 (h) and marginal cdf F(Z 1 (x)). Then the co-simulation of Z 2 (x) is calculated at each location x 0, and an estimation of the local mean and variance is made, identified with estimated simple collocated co-kriging and corresponding estimation variance. Then, based on previously simulated z (l) 1 (x 0), computation of the conditional cdf F [Z 2 (x) Z 1 (x) = z (l) 1 (x 0)] from the bi-distribution F [Z 1 (x), Z 2 (x)] occurs. Finally, resampling of the simulated value z (l) 2 (x 0) from the conditional cdf F [Z 2 (x) Z 1 (x) = z (l) 1 (x 0)] as in the usual direct sequential simulation procedure (Soares 2001) is made. 2.2 Corrections for Local Bias and Local Means The corrections for local bias and local means applied in DSS and Co-DSS (Soares 2001) are now applied in the framework of resampling from the conditional histogram. When resampling a histogram F [Z 2 (x) Z 1 (x) = z (l) 1 (x 0)], with little data, mostly in low frequency classes, the following bias can occur: E{z (l) 2 (x 0)} z 2 (x 0 ), where z 2 (x 0 ) is obtained by (1) (see Soares 2001 for details). Therefore, in each sequential step, the following estimator of E{z (l) (x 0 )} is calculated: Z (l) 2 (x 0) = 1 N i Ni i=1 z(l,i) 2 (x 0 ), by drawing, for example, N i values of z (l,i) 2 (x 0 ) at each iteration. The simulated value z (l) (x 0 ) is corrected by the deviation z 2 (x 0 ) z (l) 2 (x 0) z (l) 2 (x 0) = z (l) 2 (x 0) + [ z 2 (x 0 ) z (l) 2 (x 0) ] (3) Note the fact that, in each sequential step, N i values (say N i = 50) are drawn to calculate z (l) 2 (x 0), which does not have a significant impact in terms of computing

6 274 Math Geosci (2010) 42: time. Regarding the correction of local means, suppose that m z is the conditional mean of F(Z 2 (x) Z 1 (x)) to be reproduced in the final simulated maps. A deviation e s can be calculated between m z and the mean of simulated values z (l) 2 (x) at a given step s of the sequential procedure where m (l) z is the mean of all simulated values at step s m (l) z e s = m z m (l) z (4) = 1 N s N s i=1 z (l) 2 (x i) (5) N s being the number of simulated nodes up to step s. Hence, the estimated local means (1) can be corrected with the deviation e s The remaining steps of Co-DSS are unchanged. Z (l) (x 0 ) = z (l) (x 0 ) + e s (6) 2.3 Computing Conditional Distributions: Practical Implementation Based on previously simulated z (l) 1 (x 0), the conditional cdf F [Z 2 (x) Z 1 (x) = z (l) 1 (x 0)] is calculated from the bi-distribution F [Z 1 (x), Z 2 (x)] using moving classes (rather than fixed classes) of Z 2 (x). The user must define the minimum number of data N c of each class of the conditional histogram F [Z 2 (x) Z 1 (x) = z (l) 1 (x 0)]. By ranking the pairs of values (Z 1 (x i ), Z 2 (x i )) by increasing order of Z 1 (x i ), for one given conditioning value z (l) 1 (x 0) = Z 1 (x j ) we have the corresponding Z 2 (x j ),inthe position jth of ordered (Z 1 (x i ), Z 2 (x i )). Hence, F [Z 2 (x) Z 1 (x) = z (l) 1 (x 0)] is composed of the closest N c values of Z 2 (x j ) in the rank-ordered list. Figure 3 shows this procedure schematically. 3 Case Study 3.1 Data Analysis To illustrate the proposed methodology, we chose a case study of a sulfide copper deposit (Neves Corvo, Portugal). Copper (Cu) and tin (Sn) contents were analyzed in 523 boreholes (marginal histograms are presented in Figs. 4 and 5). The Cu/Sn biplot in Fig. 1 shows the existing relationship between both elements. Spatial continuity main patterns of Cu and Sn can both present a similar isotropic behavior, modeled by an exponential model. Sn shows a clear nugget effect, representing 20% of the total variance. As noted in Soares (2005), the biplot in Fig. 1 shows two populations with a different behavior regarding the correlation between Cu and Sn. Splitting the total set of samples using a Cu threshold of 10%, the values with a Cu content lower than 10% show a higher correlation coefficient (r = 0.72; Fig. 6) than the values with a Cu content higher than 10%, which do not present a significant correlation with Sn (r = 0.37; Fig. 7). This set of data was used to test both Co-DSS methods.

7 Math Geosci (2010) 42: Fig. 3 Computing moving classes to define the conditional distribution: for a given z 1 (x 0 ), the conditional distribution F(Z 2 (x) Z 1 (x)) is calculated with the N c selected values of Z 2 (x) 3.2 Co-simulation with Joint Probability Distributions of Cu and Sn Cu and Sn grades were simulated in a regular grid of points ( m). First, Cu was simulated using DSS. Then, Sn values were obtained by Co-DSS assuming previously simulated maps of Cu as a secondary variable. The two algorithms were tested, with and without the improvement proposed to account for joint distributions. Figure 8 presents the result (biplot of co-simulated Sn and simulated Cu) using the proposed Co-DSS with joint probability distributions. Figure 9 shows the result using the usual Co-DSS. The comparison with Fig. 1 (experimental biplot) reveals that the worst matching between simulated Cu and Sn is shown in Fig. 9, when the usual Co-DSS is used. In this case, low value classes of Cu are related with high simulated values of Sn, contradicting the experimental bi-histogram. Also, the marginal histograms obtained for Sn (Figs. 10 and 11) were compared with the experimental histogram (Fig. 5). These results show that the histogram is better reproduced when we use Co-DSS with joint probability distributions.

8 276 Math Geosci (2010) 42: Fig. 4 Cu marginal histogram Fig. 5 Sn marginal histogram

9 Math Geosci (2010) 42: Fig. 6 Experimental biplot (Sn vs. Cu with content lower than 10%) Fig. 7 Experimental biplot (Sn vs. Cu with content higher than 10%)

10 278 Math Geosci (2010) 42: Fig. 8 Simulation biplot using Co-DSS with joint probability distributions (co-simulated Sn vs. simulated Cu) Fig. 9 Simulation biplot using usual Co-DSS (co-simulated Sn vs. simulated Cu)

11 Math Geosci (2010) 42: Fig. 10 Marginal histogram of Sn co-simulated values using Co-DSS with joint probability distributions Fig. 11 Marginal histogram of Sn co-simulated values using usual Co-DSS

12 280 Math Geosci (2010) 42: Another test was done using the two subsets of Cu samples, one corresponding to the classes of Cu values lower than 10% (Fig. 6) and the other to the classes of Cu values equal to or greater than 10% (Fig. 7). Co-simulation results are plotted in Fig. 12(a) and (b) (Co-DSS with joint probability distributions) as well as Fig. 13(a) and (b) (Co-DSS). When compared with the experimental biplot (Figs. 6 and 7), we conclude that even for different correlation coefficients, Co-DSS with joint probability distributions succeeds in better reproducing the experimental bivariate relation. Also, conditional histograms were calculated corresponding to F(Sn Cu < 5%), F(Sn Cu < 7.5%), F(Sn Cu < 10%), F(Sn 5% < Cu < 10%) and F(Sn 10% < Cu < 15%) (see Fig. 14(a c), Fig. 15(a c), Fig. 16(a c), Fig. 17(a c), and Fig. 18(a c)). For these examples, Sn conditional histograms obtained with Co- DSS with joint probability distributions and with the usual Co-DSS were compared with the conditional histogram obtained from the experimental data. The evidence is that conditional histograms are better reproduced with the improved Co-DSS algorithm. It is worth noting that, for low classes of Cu values (less than 5%), there is an excessive frequency of zero values of Sn that are not observed in the experimental conditional histogram. Finally, we compare, in Fig. 19(a) and (b), co-simulated images obtained by both methods. Using low threshold of Sn values (for example, 0.2%) (Fig. 19(c) and (d)) it is noticeable that Co-DSS tends to overestimate low Sn values since it is not conditioned to the experimental bi-histogram. 4 Conclusions The work presented in this paper contributes to the improvement of the Co-DSS algorithm to ensure that simulated covariates reproduce the bivariate distributions as measured or estimated by the experimental data. This was not yet implemented in Co-DSS or any sequential co-simulation algorithm. The advantages of the proposed approach are related with the importance of conditional distributions of simulated covariates for several reasons, such as environmental applications, where the definition of critical areas in contaminated sites, based on the joint concentration of multi-elements, is provided by the product of impact costs of different pollutant concentrations above a certain remediation level. Therefore, the correct characterization of conditional distributions is extremely important, mostly when impact costs are non-linear functions of contaminant contents and consequently can magnify the existing bias of the conditional distributions. This approach can also benefit earth science applications, where the classification of mining reserves of multi-element ore bodies must be based on the joint distributions, mostly when treatment and commercial costs are non-linear functions of the grades and risk of magnifying the costs of misclassification errors. Concerning the practical implementation of the new algorithm, there are no significant differences between both algorithms in what concerns CPU efficiency, although for a large data set, Co-DSS with joint probability distributions could have a slower performance. Other practical issues can be pointed out as limitations in the use of Co-DSS with joint probability distributions. It can be used when little experimental data is available but is advisable to use an optimization algorithm of smoothing

13 Math Geosci (2010) 42: (a) (b) Fig. 12 Simulation biplot using Co-DSS with joint probability distributions: (a) co-simulated Sn vs. simulated Cu with content lower than 10%; (b) co-simulated Sn vs. simulated Cu with content higher than 10%

14 282 Math Geosci (2010) 42: (a) (b) Fig. 13 Simulation biplot using usual Co-DSS: (a) co-simulated Sn vs. simulated Cu with content lower than 10%; (b) co-simulated Sn vs. simulated Cu with content higher than 10%

15 Math Geosci (2010) 42: (a) (b) Fig. 14 Conditional histogram of Sn given Cu content less than 5%: (a) experimental data; (b) Co-DSS with joint probability distributions; (c)co-dss

16 284 Math Geosci (2010) 42: (c) Fig. 14 (Continued) (a) Fig. 15 Conditional histogram of Sn given Cu content less than 7.5%: (a) experimental data; (b)co-dss with joint probability distributions; (c)co-dss

17 Math Geosci (2010) 42: (b) Fig. 15 (Continued) (c)

18 286 Math Geosci (2010) 42: (a) (b) Fig. 16 Conditional histogram of Sn given Cu content less than 10%: (a) experimental data; (b) Co-DSS with joint probability distributions; (c)co-dss

19 Math Geosci (2010) 42: (c) Fig. 16 (Continued) (a) Fig. 17 Conditional histogram of Sn given Cu content between 5% and 10%: (a) experimental data; (b) Co-DSS with joint probability distributions; (c)co-dss

20 288 Math Geosci (2010) 42: (b) Fig. 17 (Continued) (c)

21 Math Geosci (2010) 42: (a) (b) Fig. 18 Conditional histogram of Sn given Cu content between 10% and 15%: (a) experimental data; (b) Co-DSS with joint probability distributions; (c)co-dss

22 290 Math Geosci (2010) 42: (c) Fig. 18 (Continued) (a) Fig. 19 Co-simulated Sn: (a) Co-DSS with joint probability distributions; (b) Co-DSS; (c) Snvalues lower than 0.2% using Co-DSS with joint probability distributions; (d) Sn values lower than 0.2% using Co-DSS

23 Math Geosci (2010) 42: (b) (c) Fig. 19 (Continued) bi-histograms, for example as proposed by Deutsch and Journel (1998). The purpose is to obtain a more representative (or less erratic fluctuations) bi-histogram. Also, the proposed method is based on a sequential approach where each variable is simulated in turn having the others, previously simulated, as secondary information: F [Z 1,Z 2,...,Z N ]=F [Z 1 ] F [Z 2 Z 1 ]...Then, the joint simulation of multielements reproduces the conditional distributions for the pair of covariates which are being co-simulated. The case study presented refers to the particular case for the joint simulation of only two variables. The influence of each covariate in each cosimulation was not evaluated in this work.

24 292 Math Geosci (2010) 42: (d) Fig. 19 (Continued) Acknowledgements The company Somincor Sociedade Mineira de Neves Corvo is gratefully acknowledged for giving permission to publish the data. This paper was produced in the context of Project Soil Contamination Risk Assessment (PTDC/CTE-SPA/69127/2006) (financed by FEDER through the National Operational Science and Innovation Program 2010 with the support of the Foundation for Science and Technology). References Almeida A, Journel A (1994) Joint simulation of multiple variables with a Markov-type coregionalization model. Math Geol 26(5): Boucher A, Dimitrakopoulos R (2009) Block simulation of multiple correlated variables. Math Geosci 41: Caers J (2005) Petroleum geostatistics. Society of Petroleum Engineers, Richardson Desbarats AJ, Dimitrakopoulos R (2000) Geostatistical simulation of regionalized pore-size distributions using min/max autocorrelation factors. Math Geol 32(8): Deutsch CV (2002) Geostatistical reservoir modeling. Oxford University Press, New York Deutsch CV, Journel AG (1998) Gslib: geostatistical software library and user s guide. Oxford University Press, New York Goovaerts P (1997) Geostatistics for natural resources evaluation. Oxford University Press, New York Journel AG, Kyriakidis P (2004) Evaluation of mineral reserves a simulation approach. Oxford University Press, New York Myers DE (1988) Vector conditional simulation. In: Armstrong M (ed) Geostatistics. Kluwer, Dordrecht, pp Soares A (2001) Direct sequential simulation and cosimulation. Math Geol 33(8): Soares A (2005) Classification of mining reserves using direct sequential simulation. In: Leuangthong O Deutsch CV (eds) Geostatistics Banff 2004, vol 1. Springer, Dordrecht, pp Verly GW (1993) Sequential Gaussian cosimulation: A simulation method integrating several types of information. In: Soares A (ed) Geostatistics Troia, vol 1. Kluwer, Dordrecht, pp