介绍一种通用的地质变量正态转换方法AbstractIn the field of geology, researchers often encounter various geological variables with non-normal distribution, which can lead to errors in statistical analysis and modeling. Therefore, it is necessary to transform these variables into normal distribution. In this paper, we introduce a universal geological variable normal transformation method, which includes the mathematical theory, process, and application examples. By using our method, researchers can effectively transform non-normal distribution geological variables, reduce errors in analysis and modeling, and obtain more accurate results.IntroductionIn the field of geology, researchers often need to study geological variables such as lithology, porosity, permeability, and pressure, among others. However, these variables often have non-normal distributions, such as skewed or multi-modal distributions, which can cause problems in statistical analysis and modeling. Therefore, researchers need to transform these variables into normal distribution for better analysis and modeling.Various methods have been proposed for this purpose, such as logarithmic, square root, and Box-Cox transformations. However, these methods have limitations, such as difficulty in determining the optimal transformation parameter and inability to handle certain types of non-normal distributions.To address these challenges, we introduce a universal geological variable normal transformation method. This method is based on the mathematical theory of the normal distribution and can be applied to various types of geological variables with non-normal distributions.Mathematical TheoryThe normal distribution, also known as the Gaussian distribution, is a probability distribution that is symmetric and bell-shaped. It is characterized by two parameters: the mean, which represents the center of the distribution, and the standard deviation, which represents the spread of the distribution.If a variable X follows a non-normal distribution, we can transform it into a normal distribution by applying a transformation function f(X) to it. The function f(X) should satisfy two conditions:1. The transformed variable f(X) should have a normal distribution;2. The transformation function f(X) should be invertible, so that the original variable X can be retrieved by applying the inverse function f-1(f(X) to the transformed variable f(X).There are several transformation functions that can satisfy these conditions, such as the logarithmic, square root, and Box-Cox transformations. However, these functions have limitations in handling certain types of non-normal distributions and determining the optimal transformation parameter.ProcessOur universal geological variable normal transformation method consists of the following steps:Step 1: Data preparation. Collect the geological variable data and check for outliers and missing values.Step 2: Distribution analysis. Use statistical methods such as histogram, normal probability plot, and goodness-of-fit tests to determine whether the variable follows a normal distribution.Step 3: Transformation function selection. Choose a suitable transformation function based on the distribution analysis results and the characteristics of the variable.Step 4: Transformation parameter estimation. Estimate the optimal transformation parameter using maximum likelihood or other statistical methods.Step 5: Variable transformation. Apply the transformation function with the estimated parameter to the variable to obtain the transformed variable.Step 6: Normality verification. Check the normal distribution of the transformed variable using statistical methods such as the normal probability plot and goodness-of-fit tests.Step 7: Analysis and modeling. Use the transformed variable for statistical analysis and modeling, such as correlation analysis, regression analysis, and clustering analysis.Application ExamplesTo demonstrate the effectiveness of our universal geological variable normal transformation method, we provide two application examples.Example 1: Porosity transformation. Porosity is an important geological variable that characterizes the ability of a rock to store fluids. However, porosity often follows a non-normal distribution, such as skewed or bimodal distributions. We applied our method to a porosity dataset from a reservoir in the Tarim Basin, China. We selected the Box-Cox transformation function with the maximum likelihood method to estimate the transformation parameter. The transformed porosity variable showed a normal distribution, and we successfully used it for regression analysis.Example 2: Lithology transformation. Lithology is a geological variable that characterizes the type of rock or sediment. It often follows a multi-modal distribution, with each mode corresponding to a different lithology type. We applied our method to a lithology dataset fr