Speeding up Conditional Simulation: Using Sequential Gaussian Simulation with Residual Substitution

SPEEDING UP CONDITIONAL SIMULATION: USING SEQUENTIAL GAUSSIAN SIMULATION WITH RESIDUAL SUBSTITUTION

Alejandro Cáceres

Geoinnova Consultores, Chile

Xavier Emery

Department of Mining Engineering, University of Chile, Chile

ALGES Laboratory, Advanced Mining Technology Center, University of Chile, Chile

Marcelo Godoy

Golder Associates, Chile

ABSTRACT

Sequential Gaussian simulation (SGSIM) and turning bands simulation (TBSIM) are widely used to generate realisations of mineral grades and to evaluate mineral resources. SGSIM relies on the recursive application of Bayes theorem and is designed as a direct conditional method. TBSIM is based on a stereological device that generates non-conditional realisations, which are subsequently conditioned through residual substitution kriging. SGSIM uses a random path to visit the nodes targeted for simulation. In practice, for each realisation the path has to be changed in order to avoid artefacts or artificial correlation between realisations. This is clearly the case when conditional simulation is performed. However, for non-conditional simulation with a proper implementation, the resulting realisations do not exhibit such artificial behaviour.

In this context, the proposed algorithm (SGSIM-RS) uses the sequential approach to generate non-conditional realisations, with a single controlled path. Then, a conditioning step is performed using residual substitution as it is done in the TBSIM approach. The advantage of using a single path is the possibility to generate many realisations at the same time and to condition them in a single kriging step. The CPU time reduction is considerable. For example, the time to create n realisations is equivalent to the time of one sequential simulation plus one conditioning kriging. The counterpart is the memory needed for storing the non-conditional realisations all together. However, this requirement is less demanding with actual hardware and operative systems. The proposed algorithm is presented through a case of study and its performance is compared to the traditional SGSIM and TBSIM approaches.

INTRODUCTION

Running time and management of large files involved in simulation studies could be discouraging factors for incorporating conditional simulation as a daily basis practice in the mining industry and

become critical when large simulation grids are used [1]. The most widespread algorithms for Gaussian conditional simulation are sequential Gaussian (SGSIM) [2, 3, 4, 5, 6] and turning bands (TBSIM) [7, 8] simulation, with available implementations in opensource projects and comercial software.

Both algorithms rest upon the multi-Gaussian model and the homoscedasticity property of the Gaussian distribution, together with the orthogonality of simple kriging. TBSIM directly uses those properties by separating the problem in two steps: simulating first a non-conditional Gaussian random field (Y(x)) and then conditioning to the data using the residual substitution (RS) approach [7]. RS is applicable to convert non-conditional simulations into conditional ones. In contrast, the SGSIM algorithm [3], making use of screening effects, search strategy, node migration, visiting sequence and multiple grids, among other considerations, directly derives a conditional distribution at each target location , from which a simulated value is drawn as follows:

where is the simple kriging estimate of calculated from the original data and previously simulated nodes, is the associated kriging standard deviation, and is an independent Gaussian random variable.

An attractive feature of SGSIM is its ability to directly provide conditional simulations, avoiding the two-step approach used in TBSIM [3]. However, the cost of this remarkable feature is the requirement of whole re-simulation if new data are added or removed, while TBSIM can be updated by just adding or removing the data in the conditioning step.

A method that allows faster simulation and can easily manage the update of new drilling campaigns or removing certain data would be beneficial to practitioners in the mining industry. This paper presents such a method that uses the sequential Gaussian algorithm for generating non-conditional simulations and the residual substitution approach for conditioning to sample data.

SEQUENTIAL SIMULATION ALGORITHMS

Traditional Sequential Gaussian Simulation (SGSIM)

Sequential Gaussian simulation uses a random visiting order for the nodes targeted for simulation. This visiting sequence, often called “random path”, is changed from realisation to realisation in order to avoid artificial correlation or similarity between realisations. This is clearly valid when conditional simulation is performed, because for every node the same conditioning data locations and original data values are used to determine the local distribution of the value to simulate. The implementation of the sequential Gaussian approach has therefore two sources of randomness: a theoretical one related to the Gaussian value U used in equation 1 and the random visiting sequence as an implementation aspect.

If non-conditional simulation is performed with SGSIM, the very first nodes (for which there is no conditioning data) are simulated from a Gaussian distribution without any covariance or spatial consideration. Then, simulated values are available and the procedure goes on using these first simulated values as conditioning data, i.e., non-conditional simulation in SGS becomes a simulation conditional to these first nodes.

By construction, the non-conditional values simulated at the first nodes are independent from realisation to realisation, so the use of a changing visiting sequence for each realisation can be avoided. Therefore several non-conditional realisations can be generated in a single execution of the sequential algorithm using an unique visiting sequence. This approach has already been suggested [9] and the use of deterministic or modified visiting sequences been explored [10, 11, 12, 13].

Proposed Methodology (SGSIM-RS)

The global procedure to get conditional realisations using the SGSIM-RS approach is: Normal score transformation of the raw data into Gaussian values . Variogram analysis of the normal scores data, defining a covariance model (or, equivalently, a variogram model) Non-conditional Gaussian simulation using a single-path sequential approach. Get the residuals between the Gaussian