Quantitative prediction of soil heavy metals Cu and Cr based on visible-near infrared spectra with fractional-order derivatives

A long-established gold mine is located in Mojiang Hani Autonomous County, Yunnan Province, and long-term mining activities may have caused heavy metal contamination in the surrounding soils. However, studies on the prediction of soil heavy metal concentrations in this region remain limited. Rapid monitoring of copper (Cu) and chromium (Cr) concentrations in farmland soils is therefore of considerable practical importance, and visible–near-infrared (Vis–NIR) spectroscopy has substantial potential for this purpose. In this study, 60 soil samples were collected from farmland surrounding the gold mine. After the dataset was partitioned using the Kennard–Stone algorithm, fractional-order derivatives (FOD) preprocessing was applied to enhance informative spectral features and reduce the influence of noise. A total of 41 derivative orders ranging from 0 to 2.0 were evaluated at intervals of 0.05, with the 0-order spectra representing the raw spectra. Stepwise multiple linear regression (SMLR) models were developed using full-spectrum data, whereas uninformative variable elimination (UVE) was used to select characteristic bands before partial least squares regression (PLSR) modeling. The performance of the FOD-SMLR and FOD-UVE-PLSR models was compared across different FOD orders. For Cu, the optimal SMLR and UVE-PLSR models were obtained with 0.1- and 0.35-order FOD preprocessing, respectively, yielding ratio of performance to deviation (RPD) values of 1.575 and 2.023. For Cr, the optimal SMLR and UVE-PLSR models were obtained with 0.75- and 0.15-order FOD preprocessing, respectively, yielding RPD values of 4.488 and 3.064. The main findings were as follows: (1) compared with integer-order derivatives (IOD) preprocessing, lower-order FOD preprocessing more effectively enhanced spectral information associated with Cu and Cr concentrations, mitigated the tendency toward model overfitting, and improved prediction performance; and (2) compared with SMLR, UVE-PLSR exhibited greater compatibility with different FOD orders and more stable predictive performance, while requiring substantially less computational time. Overall, UVE-PLSR provided greater practical advantages when prediction performance, computational cost, and compatibility with different FOD orders were considered together.