## 1 Introduction  
Chromium (Cr) is a significant focus of research endeavors concerning soil environmental quality assessment and mineral resource exploration [1, 2, 3, 4]. Establishing the geochemical background value of Cr is indispensable for assessing environmental quality and tracing pollution sources, as well as identifying and assessing anomalies in the context of mineral exploration [5, 6, 7, 8]. Traditional geochemical background values are defined as the natural concentrations or distributions of chemical elements or compounds in a specific geological setting [9, 10]. These values are typically influenced by the regional geological and geographical factors [11, 12, 13]. Determining these geochemical background valuesof elements has relied on statistical parameters derived from data distributions [14,15]. When data closely adhere to a normal or log-normal distribution, the mean or geometric mean, calculated after the removal of outliers, is often used to represent background values [16, 17, 18, 19]. In contrast, for scenarios exhibiting multimodal distributions, the median is commonly utilized to define the geochemical background value [5,12,19,20]. However, these traditional approaches depend on large datasets and specific distribution patterns, posing significant challenges to determine the background for a single sample. Recently, a new approach has been proposed to estimate trace element background values based on major element abundances. This method originated from the work of [PERSON] et al. [21], who developed an index for the weathering intensity of Linglong granite in Jiaodong, China. The Weathering Index of Granite (WIG) was subsequently confirmed to also represent other rocks and their weathering products [22,23]. The formula for WIG is as follows:  
\[\text{WIG}=100[\text{Na}_{2}\text{O}+\text{K}_{2}\text{O}+(\text{CaO}-10/3 \text{P}_{2}\text{O}_{5})]/(\text{Al}_{2}\text{O}_{3}+\text{T}\text{Fe}_{2} \text{O}_{3}+\text{TiO}_{2}) \tag{1}\]  
where the oxides represent their molar quantities per 100 g of sample, and any negative values of (CaO \(-\) 10/3P\({}_{2}\)O\({}_{5}\)) are set to zero. To establish the geochemical background values for Cr in samples with varying lithologies and degrees of weathering, [PERSON] et al. [24] introduced an empirical formula based on WIG, Al\({}_{2}\)O\({}_{3}\)/Ti, and K\({}_{2}\)O/SiO\({}_{2}\):  
\[\text{lgCr}=1.135\times(1.2-\text{WIG}/100)-1.304\times\text{lg}(\text{Al}_{2 }\text{O}_{3}/\text{Ti})+0.243\times\text{lg}(\text{K}_{2}\text{O}/\text{SiO}_{2})-1.88 \tag{2}\]  
where the oxide contents are given in weight percent (%), and Cr and Ti are in units of \(\text{\mu g}\)/g. Unlike traditional methods, this approach directly calculates the Cr background values based on their major component contents, irrespective of data distribution characteristics, and is applicable even for single samples. However, the equation does not apply to samples with WIG \(\geq\) 120. In a dataset of 791 abundance records for rocks, soils, and sediments, treated as virtual background samples, application of Equation (2) revealed that 63 records (those with WIG \(>\) 120) could not be calculated. Among the remaining 728 records, 425 displayed logarithmic residuals exceeding \(\pm\)0.2 between calculated and observed values, accounting for 58.4% of the total. Consequently, Equation (2) requires refinement to improve its representativeness of Cr geochemical background values in geological samples. In this paper, we propose an innovative equation to calculate the geochemical background of Cr, leveraging a comprehensive dataset of 791 abundance records sourced from rocks, soils, and sediments, all treated as background samples. Following its development, we rigorously tested this equation on a diverse range of background samples from various regions with varying degrees of weathering.