3.1. Nemerow Pollution Index with MCS

The Nemerow index has been proven to be effective in quantifying the overall pollution level of soil heavy metal(loid)s [11]. It assesses the average pollution level of various soil contaminants and highlights the impact and significance of pollutants with the highest pollution index on environmental quality [19]. The calculations for the single-factor pollution index and the Nemerow pollution index are shown in Formula (1) and (2), respectively.

where Pi is the pollution index of a single heavy metal(loid) i; Ci is the concentration of heavy metal(loid)s i (mg/kg); and Si is the evaluation standard of heavy metal(loid)s i (mg/kg) in soil. In this study, GB 15618-2018 [26] was adopted as the evaluation standard, and specified risk screening values are outlined in Table S3.

where Pn is the Nemerow pollution index of heavy metal(loid)s in soil; Pave is the average value of the single factor index; Pmax is the maximum value of the single factor pollution index. The classification criteria for the assessment of soil heavy metal(loid) pollution are presented in Table S4.

MCS represents one of the most prevalent and effective approaches for characterizing uncertainty in numerous risk-related problems [29]. MCS involves several steps: (1) defining random variables of the assessment model, (2) setting distribution models for these random variables, (3) configuring simulation parameters and executing the model, and (4) analyzing the simulation outcomes. In this study, the PyMC package was employed for MCS (https://www.pymc.io/, accessed on 30 September 2023), with 5000 iterations conducted, and the results were derived from the last 3000 iterations.

The uncertainty assessment for heavy metal(loid) pollution evaluation in Hunan Province is detailed as follows: (1) MCS was utilized to model random variables, including X and Y coordinates of sampling points and the concentrations of five heavy metal(loid)s, with the variable distribution specified in Table 1. (2) Based on multiple simulation outcomes, inverse distance weighted (IDW) interpolation was employed to generate distribution maps of five heavy metal(loid)s with a 1 × 1 km grid size in Hunan Province. (3) The Nemerow pollution indices were repeatedly computed according to Equations (1) and (2), and their corresponding distribution diagrams were graded into different pollution degrees. (4) The uncertainty of pollution assessment results was quantitatively represented by information entropy.

The probabilistic distribution of sampling point attributes and location.

Information entropy, originally defined by Shannon [30] to quantify uncertainty in information, involves partitioning the entire model space into regular grids with uniform pixel size. For each grid cell, if there is only one possible outcome with Pi = 1, the entropy value is 0, indicating no uncertainty. However, the more possible M outcomes, the greater the entropy and the greater the uncertainty. The general expression for information entropy is calculated by Formula (3).