Direction-Dependent Applicability of Grey Forecasting Models for Spatiotemporal Chloride Concentration Prediction in Concrete

Accurately predicting chloride concentration is crucial for the durability assessment of reinforced concrete structures. This study evaluates the direction-dependent applicability of four grey forecasting models, including conventional GM (1,1), Markov residual-corrected GM (1,1), GM (1,1) with full order time power terms (FOTP-GM (1,1)), and improved metabolic GM (1,1), for temporal and spatial chloride concentration prediction. Temporal sequences were constructed at fixed depths of 10 and 14 mm, while spatial sequences were obtained at exposure times of 280 and 320 d. Holdout, rolling one-step, and rolling two-step forecasting were employed, and model performance was assessed using statistical accuracy, numerical stability, and physical admissibility. The results showed that the temporal sequences exhibited quasi-linear growth, whereas the spatial profiles showed approximately exponential-like attenuation. Among the four grey models, FOTP-GM showed the most consistent temporal forecasting performance, with holdout MAPEs of only 2.1–2.3% and a pooled rolling one-step WAPE of 7.49%. Comparison with reference benchmarks further showed that simple linear and Fick-based models remained competitive for temporal forecasting, while the Fick-based model provided the highest accuracy for spatial extrapolation. In contrast, no single model was consistently superior for spatial prediction; although Markov-GM and FOTP-GM reduced overall statistical errors, both produced nonphysical negative concentrations in the deep low-concentration region. The optimal FOTP order was also generally higher for temporal sequences than for spatial sequences, further demonstrating the dependence of model complexity on forecasting direction. These findings reveal that under the investigated 28-d curing and exposure conditions, the grey forecasting models exhibited clear direction-dependent applicability associated with the distinct characteristics of temporal and spatial chloride sequences, owing to the distinct sequence characteristics of temporal accumulation and spatial attenuation. The proposed framework provides a rational basis for grey-model selection in spatiotemporal chloride concentration prediction and offers a practical small-sample approach for concrete durability assessment.