Truncated Gaussian Kriging as an Alternative to Indicator Kriging

Truncated Gaussian simulation (TGS) and plurigaussian simulation (PGS) are widely accepted methods for generating realisations of geological domains (lithofacies) that reproduce contact relationships. The realisations can be used to evaluate transfer functions related to the lithofacies occurrence, the simplest ones of which are the probability of occurrence of each lithofacies and the most probable lithofacies at each location of the deposit. In order to get the probability of occurrence of a lithofacies, the simulation approach can be time consuming. A shortcut method (truncated Gaussian kriging, or TGK) is proposed, based on the truncated Gaussian simulation model and the well-known multi-Gaussian kriging method. In this method, the variogram analysis stage and the definition of the truncation rule remain the same as in the traditional truncated Gaussian simulation approach. The formulation of the method is halfway between spatial estimation and simulation. The key point is to apply the truncation rule to the local distribution of the underlying Gaussian random field used in the TGS approach. Because the relationship between the lithofacies indicators and this Gaussian random field is not one-to-one, the latter is simulated at the data locations conditionally to the available indicator data. The local distributions of the Gaussian random field at the target locations are then obtained by considering the simple kriging estimates and simple kriging variances, as it is done in the multi-Gaussian kriging approach. TGK can be used as a step previous to simulating lithofacies, or as an alternative to indicator kriging, when the lithofacies exhibit a hierarchical spatial disposition or when such a disposition is a desirable feature. The proposed method is naturally extensible to plurigaussian simulation. INTRODUCTION Currently, numerical models of the spatial distribution of geological domains (lithofacies) can be generated by geostatistical methods. Two approaches are commonly used to achieve this goal: stochastic simulation and local uncertainty models. Stochastic simulation consists of creating multiple realisations of the lithofacies of the deposit. The realisations can be used to evaluate transfer functions related to the lithofacies occurrence, the simplest of which is the probability of occurrence of each lithofacies. In contrast, local uncertainty models directly provide the probabilities of occurrence of the lithofacies without generating several realisations. The main methods associated with stochastic simulation are truncated Gaussian (TGS), plurigaussian (PGS) and sequential indicator (SIS) simulation, whereas the most common method for local uncertainty models is indicator kriging (IK). This paper presents a local-stochastic approach to obtain the probability of occurrence of each lithofacies based on truncated Gaussian simulation and multi-Gaussian kriging (MGK). OVERVIEW OF CURRENT METHODS Indicator Kriging (IK) Indicator kriging [1, 2] is a non-parametric technique to calculate the conditional cumulative distribution function (CCDF) of a set of indicators, which are a binary coding of a categorical variable representing the lithofacies. It basically consists of estimating the indicator values using a kriging or cokriging of indicator data. The estimated values of each indicator are interpreted as the probability density function of the lithofacies, generating a local model of the probabilities of occurrence of lithofacies. Sequential Indicator Simulation (SIS) Sequential indicator simulation [2] rests on the sequential estimation of the CCDF associated with the lithofacies coded as indicators. The estimation is performed by indicator kriging using the sample data and previously simulated nodes as conditioning information. From the CCDF a lithofacies is drawn by Monte Carlo simulation at each node. The main advantages of the method are: its auto-conditional nature, the simple incorporation of soft data and the possibility to express spatially highly continuous patterns. As a counterpart, IK and consequently SIS suffer from order relation violations in the CCDF, among others problems [3]. Truncated Gaussian Simulation (TGS) The truncated Gaussian simulation method [4] relies on the truncation of a single Gaussian random field (GRF) in order to generate realisations of lithofacies. The main feature is the reproduction of the indicator variograms associated with the lithofacies and the hierarchical contact relationship among them. This method is adequate for deposits where the lithofacies exhibit a hierarchical spatial distribution, such as depositional environments or sedimentary formations. The procedure to obtain lithofacies realisations using TGS is described as follows: Establish the lithofacies proportions and their contact relationships. Summarise this information in a truncation rule (flag). Using the truncation rule, perform variography of the lithofacies indicators through the determination of the covariance function of the underlying GRF. Simulate the GRF at the data locations conditionally to the lithofacies coding. This step is performed using the Gibbs sampler algorithm [5]. As the relationship between the lithofacies indicators and GRF is not one-to-one, several realisations should be considered for the next steps. Simulate the GRF at the target locations using the values generated at the previous step as conditioning data. Truncate the realisations according to the truncation rule. Plurigaussian Simulation (PGS) Plurigaussian simulation [6, 7] is an extension of truncated Gaussian simulation that incorporates two or more Gaussian random fields and a set of truncation rules. The use of several GRFs allows reproducing complex contact relationships between the lithofacies. The workflow of PGS is similar to TGS. Multi-Gaussian Kriging (MGK) Multi-Gaussian kriging [8] is a method to calculate the conditional distribution of a GRF at a point support. It has been used to establish the risk of exceeding or falling short of a threshold for a continuous (not necessarily Gaussian) variable. It relies on the application of the multiGaussian hypothesis and the property of orthogonality of simple kriging. The key property of the multi-Gaussian model is that the multivariate distributions of a GRF are fully defined by its firstand second-order moments: mean and covariance function. The orthogonality property is that the simple kriging estimator is not correlated with any linear combination of the data. Therefore, it can be shown that the conditional distribution of a GRF is Gaussian, with mean equal to the simple kriging estimate and variance equal to the simple kriging variance. The workflow of the application of multi-Gaussian kriging to get the conditional distribution of a continuous variable is described below: Transform the raw variable into a Gaussian variable. Store the transformation table. Perform simple kriging of the Gaussian variable. At each target location, the conditional distribution is fully defined by the simple kriging estimate and simple kriging variance. Perform numerical integration at each target location: Sample the conditional Gaussian distribution using Monte Carlo simulation Back-transform every sampled Gaussian value according to the transformation table The distribution of back-transformed values is an approximation to the distribution of the original variable conditional to the available data. From this distribution, several measures can be derived, e.g. expected value (mean of the distribution), conditional variance (variance of the distribution), probability to exceed a given threshold, or confidence intervals. PROPOSED APPROACH: TRUNCATED GAUSSIAN KRIGING (TGK) The proposed method is based on the following aspects to get the probability of occurrence of a lithofacies: To generate realisations of lithofacies, TGS use Gaussian simulations that rely on the multi-Gaussian hypothesis. The conditional distributions of the underlying GRF used in TGS can be obtained by the multi-Gaussian kriging approach. The truncation rule can be interpreted as a particular transformation from a categorical variable (lithofacies) to a continuous variable (GRF). This transformation is similar to the one used to get the conditional distribution of a continuous variable in MGK, except that the truncation rule is not one-to-one. Therefore, it is possible to calculate the probability of occurrence of each lithofacies without simulating the GRF in the domain. Instead the multi-Gaussian approach can be used to obtain the conditional distribution at each target location. As the truncation rule is not one-to-one, we will need several independent realisations of the GRF at the data locations as conditioning data (see TGS workflow). Therefore we will have to truncate several conditional Gaussian distributions with different mean values but with the same kriging variance; recall that the simple kriging variance does not depend on the data values. The workflow of the proposed method is presented only for the stationary case, i.e., when the proportions of the lithofacies remain constant over the domain under study. Consider as contiguous lithofacies present in the deposit. The indicators associated with these lithofacies are defined as: