How to solve small sample size problems in time-series soil organic carbon mapping: New insights from the Third Law of Geography

Accurate and up-to-date mapping of soil organic carbon density (SOCD) spatial distribution and temporal dynamics is essential for understanding terrestrial ecosystem carbon fluxes and monitoring global climate change. However, the available historical soil sample data remained insufficient to meet the high-precision spatiotemporal mapping requirements of SOCD across large regions. Therefore, we attempted to apply the Third Law of Geography (also known as the Law of Geographic Similarity) to address the issue of small sample size in modelling. In this study, we proposed a weighted multivariate similarity index and a similarity threshold index, along with the identification of optimal thresholds for measuring geographic similarity, to effectively increase the soil sample size. Based on the different input samples, we designed various modeling schemes for SOCD mapping. Our results suggest that the geographic similarity threshold-driven framework successfully reconciles the trade-off between sample quantity and quality, increasing sample sizes by up to three times while enhancing spatial representativeness and reducing prediction uncertainty. Accuracy evaluation and uncertainty analysis consistently demonstrated that models incorporating similarity-based input samples outperformed those relying solely on limited local samples. In comparison to the model utilizing only a limited data sample, the S1-1980 s model, achieved a coefficient of determination ($R^2$) of 0.04 and a root mean square error ($RMSE$) of 2.47 Kg C m⁻². Conversely, the S3-1980 s model, based on similarity-expanded samples, demonstrated a significant improvement, achieving an $R^2$ of 0.64 and a $RMSE$ of 1.36 Kg C m⁻². Consequently, the prediction using the improved model achieved accurate detection of regional spatiotemporal patterns of SOCD. This study provides a reference for addressing small sample size issues in time-series soil organic carbon mapping.