Spatial Correlation Between Soil Phosphorus Distribution and Its Uptake in Bean Roots Using Lagrange, Hermite and IDW Interpolation Methods

Phosphorus (P) is a key macronutrient required for plant growth, energy transfer, and root development, yet its mobility and bioavailability in soil are often limited due to fixation in insoluble forms. Understanding the spatial variability of soil phosphorus is therefore essential for optimizing fertilization strategies, particularly in legume crops such as common bean (Phaseolus vulgaris), where P uptake is strongly influenced by soil properties, microbial activity, and root physiology. In this study, twelve soil samples were collected from an agricultural field in northern Iran to quantify available soil phosphorus and the corresponding phosphorus absorbed in bean roots. Three numerical interpolation approaches-Lagrange polynomial interpolation, bicubic Hermite interpolation, and inverse distance weighting (IDW)-were applied to model and map the spatial distribution of soil P. Pearson's correlation analysis was conducted to examine the relationship between soil P concentration and P uptake in bean roots. The interpolation results revealed clear differences among methods. Hermite produced the strongest soil-plant agreement (r = 0.926), closely followed by Lagrange (r = 0.922), while IDW showed a slightly weaker correlation (r = 0.854). Despite these strong statistical correlations, the spatial correspondence between soil P and plant uptake remained only moderate at the field scale, indicating that soil P distribution alone does not fully determine plant phosphorus acquisition. Error metrics (RMSE and MAE) showed comparable performance among methods, with IDW producing slightly lower prediction errors but weaker biological alignment. Overall, Hermite provided the most reliable representation of phosphorus gradients relevant to plant uptake, highlighting the importance of gradient-preserving interpolation for site-specific phosphorus management in calcareous soils.