Abstract
This work addresses the problem of the cosimulation of cross-correlated variables with inequality constraints. A hierarchical sequential Gaussian cosimulation algorithm is proposed to address this problem, based on establishing a multicollocated cokriging paradigm; the integration of this algorithm with the acceptance–rejection sampling technique entails that the simulated values first reproduce the bivariate inequality constraint between the variables and then reproduce the original statistical parameters, such as the global distribution and variogram. In addition, a robust regression analysis is developed to derive the coefficients of the linear function that introduces the desired inequality constraint. The proposed algorithm is applied to cosimulate Silica and Iron in an Iron deposit, where the two variables exhibit different marginal distributions and a sharp inequality constraint in the bivariate relation. To investigate the benefits of the proposed approach, the Silica and Iron are cosimulated by other cosimulation algorithms, and the results are compared. It is shown that conventional cosimulation approaches are not able to take into account and reproduce the linearity constraint characteristics, which are part of the nature of the dataset. In contrast, the proposed hierarchical cosimulation algorithm perfectly reproduces these complex characteristics and is more suited to the actual dataset.
Highlights
These approaches, which are based on either a linear models of coregionalization or decorrelation, are restricted to linear bivariate relationships and are not able to handle complexities such as heteroscedasticity and nonlinearity in the variables. Some ways around this issue involve other geostatistical factorization approaches, such as Stepwise Conditioning Transformation (SCT) (Leuangthong and Deutsch 2003), flow anamorphosis (Van den Boogaart et al 2017), and projection pursuit multivariate transform (Barnett et al 2014, 2016). It is a common practice in multivariate geostatistics to address other types of complexity between variables, such as geological constraints, that lead to the introduction of an inequality constraint in their bivariate relations, which is different from heteroscedasticity and nonlinearity
Several techniques have been developed to reproduce this type of constraint, such as quadratic programming (Mallet 1980), constrained interpolation functions (Dubrule and Kostov 1986), Stepwise Conditional Transformation (Leuangthong and Deutsch 2003; Hosseini and Asghari 2015) and changing to variables that are free of inequality constraints (Emery et al 2004; Abildin et al 2019)
The layout of Hematite and Itabirite is first simulated by Sequential Indicator Simulation (Deutsch and Journel 1998), and Silica and Iron are simulated in each rock type by using only the data that belong to the rock type of interest
Summary
The multivariate stochastic modeling of continuous variables and the quantification of uncertainty at unsampled locations are of paramount importance in various disciplines in earth sciences, such as mineral resource classification (Battalgazy and Madani 2019a, b; Adeli et al 2018; Hosseini and Asghari 2018), environmental studies (Eze et al 2019; De Benedetto et al 2010), geochemistry (Madani and Carranza 2020), geometallurgy (Abildin et al 2019), geophysics (Castrignanoet al. 2017), and hydrogeology (Linde et al 2015; Mariethoz et al 2010). 2017), and hydrogeology (Linde et al 2015; Mariethoz et al 2010) In this respect, the existence of inherent correlations among the variables motivates one to use cosimulation approaches. This work presents an algorithm based on an updated version of the hierarchical sequential Gaussian cosimulation proposed by Almeida and Journel (1994), integrated with an acceptance–rejection technique to cosimulate variables with inequality constraints or any type of marginal distribution. The proposed methodology in a real case study involves the derivation of an equation based on a robust regression analysis that introduces an inequality constraint on the inferred function rather than on the sample points alone. The paper proceeds as follows: (1) The methodology is described, including the concept of inequality constraints, derivation of the linear regression function, acceptance– rejection technique, and description of the proposal for updated hierarchical cosimulation. The paper proceeds as follows: (1) The methodology is described, including the concept of inequality constraints, derivation of the linear regression function, acceptance– rejection technique, and description of the proposal for updated hierarchical cosimulation. (2) The application of the proposed approach to a real case study from an Iron deposit is presented. (3) The validation of the results and comparison with other common techniques of cosimulation are carried out, and (4) a conclusion is given
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
