Julián M. Ortiz

Multiple-point geostatistical simulation of dykes: application at Sungun porphyry copper system, Iran

Variogram-based methods are not capable of capturing high (>2) order statistics since the variogram measures the relationship between two points at a time only. Multiple-point geostatistics (MPS) has brought new insights into many geological modeling problems. The application of MPS methods has been well documented in realizing complex geological patterns. These methods have often been used in reservoir characterization since their advent in recent decades. The frequent non-linear behaviors of geologic continuity are not limited to reservoirs, but mineral deposits bear complicated formations in many cases. Relying on the power of MPS methods and considering the complexity of geological scenarios in mineral deposits, we have applied MPS in the modeling of mineral deposits. A training image (TI) is produced using geological data from upper horizons of a porphyry copper ore deposit which have been mined out during the previous mining operations. In this study, the SNESIM algorithm has been used. A number of realizations are produced using this multiple-point geostatistical method. Extensive validation steps are performed considering the TI as the reference model. These validations first show that the TI is representative for the domain under study and also illustrates some degrees of similarity between the TI and the realizations. Despite simplifications made to the problem, the application of MPS in mineral deposit modeling still faces many challenges.

Verifying the high-order consistency of training images with data for multiple-point geostatistics

Parameter inference is a key aspect of spatial modeling. A major appeal of variograms is that they allow inferring the spatial structure solely based on conditioning data. This is very convenient when the modeler does not have a ready-made geological interpretation. To date, such an easy and automated interpretation is not available in the context of most multiple-point geostatistics applications. Because training images are generally conceptual models, their preparation is often based on subjective criteria of the modeling expert. As a consequence, selection of an appropriate training image is one of the main issues one must face when using multiple-point simulation. This paper addresses the development of a geostatistical tool that addresses two separate problems. It allows (1) ranking training images according to their relative compatibility to the data, and (2) obtaining an absolute measure quantifying the consistency between training image and data in terms of spatial structure. For both, two alternative implementations are developed. The first one computes the frequency of each pattern in each training image. This method is statistically sound but computationally demanding. The second implementation obtains similar results at a lesser computational cost using a direct sampling approach. The applicability of the methodologies is successfully evaluated in two synthetic 2D examples and one real 3D mining example at the Escondida Norte deposit. The method allows ranking TI?s based on their relative high-order consistency with data.The method also allows obtaining an absolute consistency measure.The CPU time scales considerably down when a direct sampling approach is used.Applicability is successfully evaluated in different real and synthetic case studies.FORTRAN code of the program is presented and a practical example is attached.

Scaling multiple-point statistics to different univariate proportions

Multiple-point statistics are used in geostatistical simulation to improve forecasting of responses that are highly dependent on the reproduction of complex features of the phenomenon. Often, complex features cannot be captured by conventional two-point simulation methods, based on the variogram. Inference of multiple-point statistics requires a training image that depicts the geological features of the geological setting being modelled. The proportions of facies in the training image may not match the target statistics of the final model. This is a problem because taking multiple point statistics from a training image also takes the univariate proportions, that is, the multiple point statistics contain all lower order statistics. There is a need to scale multiple-point statistics to different target univariate proportions. In other cases, locally varying facies proportions must be honored, but a single training image is available. The multiple-point statistics from the training image are scaled to the appropriate target univariate proportions of facies. An iterative scaling approach based on the expression for scaling multiple-point statistics in a purely random case is proposed. The implementation is illustrated through an example where it is shown that the proposed method lies between two extreme cases for a Boolean simulation, namely, the change in size of the objects and the change in their number of occurrences. A second example is presented to illustrate the potential use of this scaling procedure for nonstationary multiple-point geostatistical simulation.

Integrating Multiple-point Statistics into Sequential Simulation Algorithms

Most conventional simulation techniques only account for two-point statistics via the modeling of the variogram of the regionalized variable or of its indicators. These techniques cannot control the reproduction of multiple-point statistics that may be critical for the performance of the models given the goal at hand (flow simulation in petroleum applications, planning and scheduling for mining applications). Multiple-point simulation is a way to deal with this situation. It has been implemented for categorical variables, yet the demand of large data sets (training images) to infer the multiple-point statistics has impeded its use in the case of continuous variables. We propose a method to incorporate multiple-point statistics into sequential simulation of continuous variables. Any sequential algorithm can be used. The method proceeds as follows. First, the multiple-point statistics are inferred from a training data set or training image with the typical indicator approach. The conditional probabilities given multiple-point data events enable to update the conditional distributions obtained by the sequential algorithm that uses the conventional two-point statistics. The key aspect is to preserve the shape of the conditional distribution between thresholds after updating the probability for the cutoffs used to infer the multiple-point statistics. Updating takes place under the assumption of conditional independence between the conditional probability obtained from the training set and the one retrieved from the conditional probability defined by the sequential method. The algorithm is presented in generality for any sequential algorithm and then illustrated on a real data set using the sequential indicator and Gaussian simulation methods. The advantages and drawbacks of this proposal are pointed out.

Shortcomings of multiple indicator kriging for assessing local distributions

Abstract This paper examines several drawbacks and limitations of indicator kriging applied to continuous variables, in the scope of the lognormal random function model. In particular, it focuses on precision problems, inconsistencies of the results when one sample is added or removed, and biases generated by the post-processing steps (tail extrapolation and change of support) and by the use of a non-bias condition in the kriging system. The selectivity of the local distributions is shown to be systematically overestimated when performing a change of support based on the global variance reduction factor, and underestimated when using an ordinary kriging instead of a simple kriging. This situation may lead to strong biases in the evaluation of the resources and reserves in ore deposits. To solve the change-of-support problem, a local variance reduction factor is given for the lognormal case; the proposed approach may be generalised to other models so as to improve the estimates.

A methodology to construct training images for vein-type deposits

Fracture models of vein formation can produce realistic training images (TIs) for use in multiple-point geostatistics. Vein formation is modeled by applying flow simulation to a fracture model to mimic the flow of an ore-bearing fluid through fractured rock. TIs are generated by assuming that veins form in areas of high flow where there would be preferential deposition of the mineral of interest. We propose a methodology to simulate mineralized veins by constructing a fracture model within the deposit, modeling the permeability and simulating the flow of ore-bearing fluids. The veins are defined by considering the areas of high flow. The methodology is implemented with a fracture model of the Whiteshell area in Manitoba. To assess the reasonableness of the TIs, comparisons are made to geological models of gold deposits in Quebec and Nova Scotia that display similar geometric characteristics such as braiding, thickening and thinning. A FORTRAN program TIGEN, based on GSLIB program formats is included and can be used to generate TIs from fracture models.

Urban Dynamic Estimation Using Mobile Phone Logs and Locally Varying Anisotropy

In telecommunications, the billing data of each telephone, denoted call detail records (CDRs), are a large and rich database with information that can be geo-located. By analyzing the events logged in each antenna, a set of time series can be constructed measuring the number of voice and data events in each time of the day. One question that can be addressed using these data involves estimating the movement or flow of people in the city, which can be used for prediction and monitoring in transportation or urban planning. In this work, geostatistical estimation techniques such as kriging and inverse distance weighting (IDW) are used to numerically estimate the flow of people. In order to improve the accuracy of the model, secondary information is included in the estimation. This information represents the locally varying anisotropy (LVA) field associated with the major streets and roads in the city. By using this technique, the flow estimation can be obtained with a better quantitative and qualitative interpretation. In terms of storage and computing power, the volume of raw information is extremely large; for that reason big data technologies are mandatory to query the database. Additionally, if high-resolution grids are used in the estimation, high-performance computing techniques are necessary to speed up the numerical computations using LVA codes. Case studies are shown, using voice/data records from anonymized clients of Telefonica Movistar in Santiago, capital of Chile.

Multiple-Point Geostatistical Simulation Based on Genetic Algorithms Implemented in a Shared-Memory Supercomputer

Multiple-point geostatistical simulation aims at generating realizations that reproduce pattern statistics inferred from some training source, usually a training image. The most widely used algorithm is based on solving a single normal equation at each location using the conditional probabilities inferred during the training process. Simulated annealing offers an alternative implementation that, in addition, allows to incorporate additional statistics to be matched and imposing constraints based, for example, on secondary information. Another class of stochastic simulation algorithms, called genetic algorithms (GA), allows to incorporate additional statistics in the same way as simulated annealing. This paper focuses on a sequential implementation of a genetic algorithm to simulate categorical variables to reproduce multiple-point statistics, and also the details concerning its parallelization and execution in a shared-memory supercomputer. Examples are provided to show the simulated images with their objective functions and running times.

Two approaches to direct block-support conditional co-simulation

Change of support is a common issue in the geosciences when the volumetric support of the available data is smaller than that of the blocks on which numerical modeling is required. In this paper, we present two algorithms for the direct block-support simulation of cross-correlated random fields that are monotonic transforms of stationary Gaussian random fields. The first algorithm is a variation of sequential Gaussian co-simulation, in which each block value is simulated in turn, conditionally to the original data and to the previously simulated block values, while the second algorithm is based on spectral co-simulation in the framework of the discrete Gaussian change-of-support model. These two algorithms are implemented in computer programs and applied to a synthetic case study and to a mining case study. Their properties and performances are compared and discussed.

Multiple Point Geostatistical Simulation with Simulated Annealing: Implementation Using Speculative Parallel Computing

Multiple-point geostatistical simulation aims at generating realizations that reproduce pattern statistics inferred from some training source, usually a training image. The most widely used algorithm is based on solving a single normal equation at each location using the conditional probabilities inferred during the training process. Simulated annealing offers an alternative implementation that, in addition, permits to incorporate additional statistics to be matched and imposing constraints based, for example, on secondary information. This paper focuses on an innovative implementation of simulated annealing to simulate categorical variables, reproducing multiple-point statistics. It is based on a well known paradigm in computer science, namely, speculative computing. In simulated annealing, categories are initially randomly distributed. Nodes are visited iteratively and a perturbation is proposed to approach the distribution of the categories to some target statistics. A decision is made to accept or conditionally reject the change, depending on an objective function that must approach zero to match the target statistics. Rejection will occur with a probability that changes during the simulation process, as defined in the annealing schedule. Speculative computing consists of using multiple processes in parallel to pre-calculate the next step in the simulation in both situations: accepting or rejecting the change. While the decision is made in the first process, a second level of two processes is used to calculate the two possible cases and subsequent levels can also be initiated. Once the decision is made, processes that do not conform to this decision are dropped and speculations about other possible perturbations at the current simulation stage are initiated. This implementation of simulated annealing can speed up the process significantly, hence making this algorithm a reasonable alternative to current methods. An example using a geologic data set is provided to demonstrate the improvements achieved and the potential this method has for larger models. Some future work is also proposed.