Characterization of High Order Correlation for Enhanced Indicator Simulation

Geostatistical simulations aim to mimic the real variations of the underlying phenomena. Traditional simulation approaches only use two point statistics such as the variogram. In presence of non linear features or large range connectivity, such traditional methods do not provide good reproduction of those features. Connectivity of high and low values critical for most studies, e.g., permeability ow paths in a petroleum reservoir, grades in a mineral deposit, and concentrations of a pollutant in a environmental study. The proposed methodology builds on the theory of runs used in statistics to characterize high order correlation in sequences of data. These \runs" are equivalent to high order indicator covariances and can be used in the context of extended normal equations or projection theory for estimation or simulation. Implementation of the extended normal equations using the information provided by runs is proposed for more realistic simulations. The methodology is discussed with exploratory examples. The examples present runs above and below di erent quantiles to characterize high order correlation in pseudo-random series and in correlated series; maps of frequency of length of runs above quantiles and maps of di erences in frequencies with respect to the random case were constructed to show the in uence of correlation. Introduction Geostatistical techniques are used to simulate realizations of regionalized variables to help in decision making; however, the realizations must correctly reproduce important aspects of the true distribution. When the heterogeneities are not well reproduced in the resulting model or the uncertainty is inaccurate, incorrect predictions and wrong decisions may be made. One feature of classical geostatistical simulation techniques, such as Gaussian and indicator approaches, is that they account only for twopoint statistics through a covariance or variogram function. This limitation is mainly due to the diÆcult inference of higher order statistics [1, 3, 12, 20]. Simulation techniques would be improved with multiple-point statistics, since the resulting realizations would share more quantitative information than two-point statistics. More realistic numerical models likely lead to better decisions. Several researchers have tried to incorporate multiple-point statistics in geostatistics, by using simulated annealing or training images [1, 8, 10, 20]. Most of them have failed in their applicability, because they are extremely CPU time consuming or because they require too many parameters to be set. Training images have been used to extract multiple-point statistics from outcrops, which supposedly represent the domain under study. The training 1 image may consist of a conceptual geological model of the site being characterized. We will never know how representative the training image is of the domain of interest. The indicator approach is based on the use of the rank order of the data. Conditional probabilities are identi ed by the conditional expectations of the indicator transforms at particular thresholds. Kriging or the normal equations are used to estimate the conditional expectations as a function of the indicator values of the data and previously simulated locations at the same threshold being estimated. Cokriging can be used to consider the cross-correlations between indicators at di erent thresholds. The products of indicators would require inference of high order correlation functions. This would lead to the best non-linear unbiased estimator. As mentioned, high correlation is generally ignored, which reduces the information space and degrades the resulting estimation or simulation. Some interesting results in number theory motivate us to study application to the indicator approach. This research aims to quantify high order correlation using the frequency of runs above and below di erent thresholds. Once high order correlation is known, the extended kriging equations may be applied in simulation with the consequent improvement on the result. The applicability of this method is seen mainly (but not restricted to) using well data in petroleum applications and drillhole data in mining applications where the number of data is enough to calculate the expected lengths of runs. Figure 1 presents a well with 22 composites (samples of equal length). The actual values are shown as a solid line, while the sample values are shown as black dots (they represent the average of the values in the sample, assuming no sampling and sample preparation errors). Given 5 thresholds, zi; i = 1; :::; 5, a run of length l can be seen as the event of having l consecutive samples with grade higher than the threshold. In the example here presented runs are represented by thicker solid lines. For z1, there is one run of length 16; for z2, there is one run of length 13; for z3, there are two runs, one of length 4 and the other of length 5; for z4 there are three runs of lengths 2, 2, and 1 respectively; nally, for z5 there are no runs in this example. The probability of having a run of length l is equivalent to the expected value of the product of l indicators corresponding to samples in a row or column. This high order moment is a multiple point statistic that would better characterize the true multivariate distribution or \spatial law". Literature Review This literature review focuses on the theory of runs, indicator formalism and extended normal equations. Theory of Runs The results presented here are based on a paper by A. M. Mood published in 1940 [16]. This work summarizes most of the work done previously by other authors, and can be considered as the basis for the majority of the subsequent statistical studies on runs (see for example [5, 6, 9, 17, 18, 21]). The author derives the \distribution of runs of given length both from random arrangements of xed numbers of elements of two or more kinds, and from binomial and multinomial populations". He also gives the limiting form of these