Coated granular materials, whether naturally occurring or synthetically produced in laboratories, offers substantial potential for various engineering applications. This study focuses on the use of dimethyldichlorosilane as a coating solution, inducing water repellency for granular materials, a property of interest in the development of advanced materials and structures. Although prior research suggests that established methods for natural granular materials analysis are generally applicable to coated materials, there remains an inherent stochasticity and probabilistic dependence in the properties of coated materials, influenced by variable extents of coating damage contingent on stress level. The deterministic empirical regression relationship alone is insufficient to represent the significant uncertainty evident in the experimental observations. In addressing these uncertainties, this study presents a probabilistic analysis approach, underpinned by copula theory, to define the probabilistic dependence structures of the coated materials. Compared to traditional measures of correlation, copula theory can reflect the various nonlinear probabilistic dependencies among multiple variables. The study utilizes lognormal probability density functions (PDFs) to assess the critical stress ratios for natural, thin-coated, and thick-coated materials. The results of natural and thin-coated materials indicate a congruity between the critical stress ratios derived from PDFs and those obtained through linear regression, implying the viability of the proposed probabilistic approach. Notably, for thick -coated granular material, significant uncertainties in the critical stress ratios emerge, correlating robustly with the imposed stress level. We thus propose a conditional PDF between the stress level and the corresponding critical stress ratios to better predict shearing behavior. To delineate the probabilistic dependence among the three key soil properties - initial state parameter, peak friction angle, and peak point dilatancy - multiple copula density functions are applied. The analysis highlights an increased variability and dispersion for thick -coated materials, attributable to the unpredictable nature of coating damage. Overall, the findings underscore the potential of probabilistic methodologies in the study of coated granular materials, leading to a more nuanced understanding of their behavior, particularly during shearing stages with different stress level. This approach can yield more accurate forecasts and superior engineering solutions, contributing significantly to the development and study of similar artificially -created materials.

Classification of water ice region on lunar surface with Mini-SAR data is quite challenging. Therefore, a probability density function (pdf) based pattern analysis approach has been applied to classify lunar surface. This paper represents the pattern analysis approach to fit data points to a distribution function for understanding the distribution behaviour of Mini-SAR data which helps in developing a method based on density functions to differentiate two types of craters namely icy (type-I) and non-icy (type-II) craters. Circular polarization ratio (CPR) is a very important parameter in study of lunar surface. More specifically, the criterion CPR>1 is used to determine possible presence of water-ice deposits on lunar surface So, it's important to study distribution behaviour of CPR pixels and to determine best fitted distribution function representing this behaviour. Therefore, in this paper, pattern analysis techniques have been applied to differentiate two crater types based on the distribution behaviour of CPR. The best fitted function for CPR has been obtained as Generalized Extreme Value function which clearly differentiate type-I and type-II craters.