A modified spherical variogram model with constrained optimization for spatial volume estimation

In this study, we proposed a modified spherical variogram model aimed at improving the accuracy of spatial modeling in volume estimation. The model enhances the flexibility of the traditional spherical variogram structure by incorporating additional polynomial terms to better capture spatial variability in structured plantation datasets. Parameters such as nugget, sill, range, and the coefficients of the polynomial terms were estimated using the L-BFGS-B optimization algorithm under box constraints, ensuring numerical stability and physically meaningful values. The performance of the modified model was evaluated using real-world volume data from Tectona grandis Linn. f. (teak) trees planted in a multiclonal block in Brumas Camp, Tawau, Sabah, Malaysia. To assess model accuracy and generalizability, predicted volumes derived from the fitted variogram model were compared to measured values using three validation strategies: Full dataset fitting, Leave-One-Out Cross-Validation (LOOCV), and K-Fold Cross-Validation. The modified spherical variogram model demonstrated superior performance over the classical version in terms of weighted root mean squared error (RMSE) and coefficient of determination (R²). These findings highlighted the value of refining variogram structures to improve estimation precision in geostatistical applications, particularly when modeling spatially complex forest data.