Estimation of cross-sectional kriging for modeling and evaluation of ore reserve (Case study: Emarat Pb-Zn deposit)

Introduction
Modeling the spatial distribution of grade and reserve estimation are the most important issues and the main goal of exploration operation. This kind of modeling depending on the amount, type and method of carried out exploration works and available exploration information, is performed using various methods. In the following situations, underground exploratory works such as exploration tunnels are carried out: 1- vertical or steep slope mineral deposits, 2- mountainous geomorphology with high altitudes and rough topography, 3- difficult or impossible situation of drilling, and 4- very high thickness or volume of overburden on the mineral deposit. In fact, in such conditions, for the most of mineral deposits there is continuity in the vertical direction whereas the changes are mostly along the horizontal surfaces. In the present research, block modeling and geostatistical reserve estimation of the Emarat Pb-Zn deposit located in the Markazi province have been carried out using log-kriging method with a different approach and a delicate technique. To achieve the goal, firstly, processing, modeling and geostatistical estimation of total Pb-Zn assay data obtained from exploratory tunnels drilled at the various elevation levels were performed as two-dimensional, separately. Then, using the gained results, three-dimensional estimation process for whole of the deposit was done. Such an approach for geostatistical estimation of a deposit grade and reserve discovered by underground exploratory works, has not been reported in any research before.
 
Materials and Methods
Study deposit and available exploration data
The Emarat Pb-Zn deposit is located about 45 km southwest of Arak city, between longitudes 49°30' to 49°45' East and latitudes 33°45' to 34°0' North, in an area with the altitude of 2,180 m above sea level. The topography of the Emarat region is very uneven. Uniform stratigraphy, severe folding, absence of igneous rocks and strati-banding are the geological characteristics of the region, with the strike of folding conforming to the trend of the Zagros folding. The Emarat Pb-Zn deposit is located on the Sanandaj-Sirjan tectonic zone and the Malayer-Isfahan lead and zinc metallogenic belt. The geological units of the region are generally carbonate, shale, and marl sedimentary units from the Jurassic to Cretaceous periods, with folding and faulting, which are composed of steep slopes. In general, the Emarat Pb-Zn deposit is a synclinorium trending northwest-southeast, with a length of 1.5 km and a width ranging from 250 to 850 m. In the Emarat deposit, lead and zinc minerals are located within a calcareous siliceous layer interface of a dark gray massive limestone band in the footwall and a Cretaceous shale layer in the hanging-wall. This formation belongs to the lower or middle Cretaceous.
In the Emarat Pb-Zn deposit, many activities containing drilling of horizontal exploratory- exploitational tunnels and crosscuts with a total length of 11,000 m in six elevation levels with a height difference of about 10 m have been carried out, whereas 1,238 Pb-Zn assay data of samples taken from the tunnels are available.
 
Cross-sectional kriging method
From the dimensional viewpoint, in the case of two-dimensional acquisitioned data, such as assay data of surface samples, processing operation and geostatistical estimation with Kriging method is done as two-dimensional. There are also cases in which, although the data acquisition is three-dimensional, it is better to perform the geostatistical estimation first on two-dimensional surfaces, then the estimation results obtained from different levels are combined with each other and the final three-dimensional model is produced. This type of kriging estimation is called cross-sectional kriging, which can be done along a series of horizontal or vertical sections. Of course, its use in horizontal sections is more logical and desirable. Since in such cases there is practically no sample and consequently no data in the gap between two consecutive adjacent surfaces, and all processed and estimated data is related to samples taken from two-dimensional surfaces, therefore, the distance and spatial relationship of the data at each level is higher. So that the continuity of the deposit is better revealed at each level.
Log-Kriging method
If the statistical distribution of the used data is not normal, linear kriging estimation methods cannot be employed, because in this case, there will be a correlation between variance and mean and pseudo-anisotropy appears in the strike variograms. In such conditions, it is better to normalize the data with a suitable transformation method such as logarithmic so that linear methods can be used for estimation. Next, the estimation operation is applied on the logarithm of data with the ordinary (or simple) kriging method, and then the estimated values are converted to real values with an inverse transformation. This non-linear kriging method is called log-kriging.
 
Results and Discussion
Based on the statistical processing of total Pb-Zn assay data for the elevation levels of 2032, 2024, 1998, 1988, 1978 and 1964-1968 m separately, data distribution was known lognormal, normalized with two and three parameters logarithmic transformation. Also, plotting standard deviation against the total Pb-Zn assay data of the various elevation levels revealed dependence of the variance with mean. To analyze the spatial structure of the region, horizontal strike variograms with azimuths ranging from 25 to 135 degrees were drawn for each elevation level separately, as well as two better horizontal variograms in perpendicular directions were selected. Due to the lognormal distribution of the assay data, to avoid the appearance of pseudo-anisotropy, the variography was performed for the transformed total lead and zinc assay data. All theoretical variogram models fitted over empirical variograms are spherical type, and strike variograms in various directions have the same sill but different ranges. Therefore, the studied area has geometric anisotropy. Using variography of the different elevation levels and determination of ellipse search radii, grade of each elevation level was estimated with geostatistical cross-sectional log-kriging method for 10 by 10 m blocks as well as isograde map from estimation process was drawn. According to the isograde maps at most elevation levels of the Emarat Pb-Zn deposit, the mineral deposit grade in the eastern half is higher than the western part. Afterward, the estimation process was validated through cross-validation approach with the well-known jackknife method and kriging estimation error maps of each elevation level. In such validation method based on the variogram model, each time one of the input data (known) is estimated through the Kriging method using neighboring samples around that sample, while the estimated values are compared with the actual ones. In other words, any known data is estimated by assuming that its value is unknown. In this regard, the determination coefficient of regression between actual and estimated grade values in all cases except elevation level of 2024 is more than 0.5, indicating good correlation between the data. Thus, the estimation has a good validity.
To create block model of the deposit, a prototype grid model with size of 1110×530×265 m, cell size of 10×10×10 m and estimated assay data of different elevation levels through the advanced inverse distance weighted (AIDW) algorithm, was used. The size of blocks was set to 10*10*10 m based on the size and extension area of mineral deposit in each elevation level, as well as distance of the elevation levels. In AIDW algorithm, it is possible to weight the distance with different powers in various directions. In this case, weights of two and one were assigned to the data in the horizontal (exploration tunnels at each elevation level) and vertical directions (distance between different elevation levels) respectively, due to greater variety in the horizontal direction. At the end, different categories of proven, probable and prospected reserves were determined for seven cut-off grades of 4, 4.5, 5, 5.5, 6, 6.5 and 7 percent.
According to the tonnage-grade diagram of the total deposit for different categories of proven, probable and prospected reserves, the proven reserve and related average grade diagrams reveal little changes for various cut off grades locating a short distance from each other. The low changes and almost horizontal state of both two graphs are due to the high total grade of lead and zinc in the proven reserve category, discovered using more exploration activities with less estimation error.
 
Conclusion
In this research, a new approach was used for geostatistical estimation of grade and ore reserve, in the Emarat Pb-Zn deposit. To achieve the goal, based on data mining of different elevation levels of the deposit, first a 2D geostatistical estimation was carried out using the cross-sectional log-kriging method for each elevation level, separately. Afterward, 3D block model of the deposit was created. In fact, in this study, the exploration data available on horizontal surfaces at the different elevation levels of the Emarat Pb-Zn deposit were generalized to 3D space through variography and 2D geostatistical estimation, by creating block model. The results of the research show that in some cases according to the conditions of the studied deposit and type of carried out exploration activities, modeling and estimation of ore reserve is possible more simply with optimal accuracy through the relatively new and accurate geostatistical methods. This scenario was performed by the approach of dimension reduction of estimation space and then converting the estimation space from two to three dimensions. The results of this research will be useful for all earth sciences users, including geologists, mining exploration and exploitation engineers.