This reduction in variability is known as smoothing. As cited by Goovaerts , all interpolation algorithms tend to smooth the spatial variability of the attribute. This effect is characterized by underestimation of high values and overestimation of low values. Smoothing can be easily observed to comparing the histogram of a database with the result of their kriging estimation.
As shown by Olea , the simulation was the solution adopted to solve the smoothing problem of kriging. But, according to the author, the gain in overall accuracy causes the reduction of local accuracy. In fact, the realizations of the simulation scenarios are not free from errors of reproducing reality and, on average, the mistakes are higher than kriging.
Therefore, simulations should not be thought of as a substitute method for kriging, but as a variability and uncertainty verification tool involved in kriging estimation, since kriging is still the best unbiased estimator.
Presented initially by Matheron , the stochastic conditional simulation techniques allowed the variability and uncertainty involved in the estimation of mineral deposits to be quantitatively evaluated. Until that moment the kriging variance was the only existing way to evaluate the estimates.
But Journel and, later, Brus and Gruijter began to question the use of this parameter as an estimative quality index. According to Deutsch and Journel , the absence of an estimation error component R x 0 provides smoothing of kriging. The equation presented below shows that the real value of a particular attribute in a point x 0 can be written as the sum of its estimate with the estimated error.
Thus, to have access to real variability of the random function Z x 0 , the random function R x 0 should be simulated many times and added to the estimated value.
Each ith simulated value in each of the simulation scenarios can be written as the sum of the kriging value with the l-th realization of the random function R x 0. Existing simulation methods seek to randomly determine the error component based on the known method of Monte Carlo.
Thus, as the process is random, the realizations will be different, but honoring the sample histogram and the sample variogram model. Histogram and variogram reproduction is classified as an overall accuracy. For the result of interpolation, the same sample statistic used in the estimation was maintained. Actually kriging, classified as a local precision method, does not produce the histogram and sample variogram, but shows a high correlation between estimated and used samples. Among the many methods of stochastic simulation, the algorithms of sequential simulation methods are the most used to reproduce the spatial distribution and uncertainty of different variables Soares, , p.
The multiplicand presented in Equation 3 summarizes the theoretical basis of the sequential simulation methods, where each simulation generates a new point that is used to update the conditional cumulative distribution function.
Thus, the conditional cumulative distribution function is always updated by a subsequent simulated value and the n sampling points. The main and most common method used of sequential simulation on deposit modeling is the sequential Gaussian simulation method SGS because of its simplicity, flexibility and reasonable efficiency Deutsch, , p. To use this method, one should work with a normal distribution with null mean and unit variance, so data must receive prior to processing.
The block model of the deposit, with a total of The estimation method used by the simulation was simple kriging. In Figure 1 , there can be seen three different scenarios of the same section of the block model plan where the warmer colors represent blocks with higher grades. After performing the simulations, it is important to verify that the scenarios obtained as a result satisfactorily reproduce the distribution of sample data and used variogram models. The 50 scenarios reproduce quite acceptably the sample distribution histograms and spatial continuity variograms.
The figure below is a variogram of all scenarios were well adherent to the variogram model in three main directions. Obtaining the optimal pit was achieved by using the three dimensional optimization algorithm of Lerchs-Grossman , by NPV Scheduler 4 software. In the following table are shown the values and parameters used. The sale price considered for copper was 2. Figure 3 presents the mathematical pit obtained by optimizing the model estimated by ordinary kriging.
The optimization of simulated scenarios follows the same premises adopted for the kriging model Table 1.
The only difference is that instead of using the estimated copper value by ordinary kriging as product, the value obtained by simulation in each scenario was used. Due to the great demand of time and the amount of information generated, not all scenarios were optimized. In order to maintain the representation of all the simulations without the need to calculate the 50 pits, a criterion was adopted to select the simulations that participated in the study.
In this criterion, the optimal extraction sequence OES obtained by optimizing the kriging model was used to calculate the NPV of the 50 scenarios. Thus, the 50 NPV values found were ranked in increasing order and 11 scenarios were chosen to participate in the evaluation of the pit Table 2 , which divided the entire population into a range of 5 in 5 NPV values. The figure 4 shows the limits in the plant of 11 pits for the selected scenarios compared to the pit limit of the kriging model red line. It can be seen that in certain regions the kriging model pit presents an optimistic behavior, being located externally to the simulated limits.
While in other regions, the kriging pit has a more conservative behavior in relation to the simulations. Such behavior can also be viewed in depth as shown in the following Figure 5. Areas with large variation limits in plant and depth can be used to define potential targets for additional drilling Godoy, Thus, this great fluctuation limits results from a high variability provided by the absence or small amount of information. In the following Figure 6 is shown the amount of ore contained in each of the incremental pits obtained by optimizing the kriging model and selected simulated scenarios.
By analyzing the previous figure, it is easy to see that the kriging model pit stands out from the others by having a greater amount of ore and, consequently, a lower strip ratio. Such disparity was provided by the known smoothing effect of ordinary kriging, where low values were overestimated and high values were underestimated. By performing smoothing, the kriging brought a certain amount of material that should be below the cutoff grade for higher grade values, transforming part of the low grade material to ore.
This explains the greater amount of ore presented in the graphic of the previous figure. The lack of low values reproduction for the estimate was evidenced by the contained ore graphic, but the lack of high values reproduction can easily be seen in the following figure 7 , where the absence of very high grade blocks responsible for a large increase in the value of the NPV provides a low NPV for the pit of the model estimated by ordinary kriging compared to the pits of simulated models.
Approaching the geological uncertainty with the application of geostatistical simulation techniques proved to be an essential and indispensable tool in the evaluation of long term mine planning projects. Although it is not widely used, quantifying the geological uncertainty with the use of geostatistical simulation can considerably reduce the possible risks associated with some project factors. There are also a number of dangerous natural hazards, e.
It is of paramount interest to study them all, to describe them, to understand their origin and - if possible - to predict them.
He has over ten years teaching and academic research experience, and over twenty eight years in the oil and gas industry. This course aims to cover many important aspects of geology, geophysics and prospect evaluation whilst illustrating how geologists deal with uncertainty and risk during exploration stages. This is a fun session, but involves a real investment decision. See our disclaimer. The aim is to determine which of the following has the highest expected profitability; ordering components on the crucial path the genset before first drilling, after first successful drilling or after second successful drilling.
While uncertainties, geological risks and natural hazards are often mentioned in geological textbooks, conferences papers, and articles, no comprehensive and systematic evaluation has so far been attempted. This book, written at an appropriately sophisticated level to deal with complexity of these problems, presents a detailed evaluation of the entire problem, discussing it from both, the geological and the mathematical aspects. Read more Read less.
K; ed. Review From the reviews: "The last decade has seen the emergence of a large body of literature devoted to the quantification, analysis, and management of risks and uncertainties. No customer reviews. Share your thoughts with other customers. Write a customer review. Discover the best of shopping and entertainment with Amazon Prime. Prime members enjoy FREE Delivery on millions of eligible domestic and international items, in addition to exclusive access to movies, TV shows, and more.
Evaluation of Uncertainties and Risks in Geology. New Mathematical Approaches for their Handling. Authors: Bardossy, György, Fodor, János. Free Preview. Evaluation of Uncertainties and Risks in Geology. New Mathematical Approaches for their Handling. Authors; (view affiliations). György Bárdossy; János Fodor.